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Research ArticleDetermining the Optimal Order Quantity withCompound Erlang Demand under (TQ) Policy
Aiping Jiang 1 Kwok Leung Tam2 Yingzi Bao1 and Jialing Lu 1
1SHU-UTS SILC Business School Shanghai University 20 Chengzhong Road Jiading District Shanghai 201899 China2School of Mathematics and Statistics UNSW Sydney NSW 2052 Australia
Correspondence should be addressed to Aiping Jiang ap724shueducn
Received 10 March 2018 Revised 25 June 2018 Accepted 12 July 2018 Published 19 August 2018
Academic Editor Neale R Smith
Copyright copy 2018 Aiping Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Management of electric equipment has a direct impact on companiesrsquo performance and profitability Considering the critical rolethat electric powermaterials play in supportingmaintenance operations andpreventing equipment failure it is essential tomaintainan inventory to a reasonable level However a majority of these electric power materials exhibit an intermittent demand patterncharacterized by random arrivals interspersed with time periods with no demand at all These characteristics cause additionaldifficulty for companies in managing these electric power material inventories In response to the above problem this paper basedon the electric powermaterial demand data of Shanghai Electric Power Company develops a newmethod to determine the optimalorder quantity 119876lowast in a cost-oriented periodic review (119879119876) system whereby unsatisfied demands are backordered and demandfollows a compound Erlang distribution119876lowastcorresponds to the value of119876 that gives theminimum expected total inventory holdingand backordering cost Subsequently an empirical investigation is conducted to compare this methodwith the Newsvendor modelResults verify its superiority in cost savings Ultimately considering the complicated calculation and low efficiency of that algorithmthis paper proposes an approximation and a heuristic algorithm which have a higher level of utility in a real industrial contextTheapproximation algorithm simplifies the calculation process by reducing iterative times while the heuristic algorithm achieves it bygeneralizing the relationship between the optimal order quantity 119876lowast and mean demand interarrival rate 120582
1 Introduction
Whenmanaging electric power material inventories compa-nies need to decide when to order and how much to order soas to maintain the appropriate inventory level Consideringthat amajority of electric powermaterials exhibit an intermit-tent demand pattern effective management of these electricpower materials is essential to electric power companiesIntermittent demand appears at random with time intervalsduring which no demand occurs These characteristics makeinventory control a rather challenging task These electricpower materials are usually characterized by low usage fre-quency long intervals and irregular demand Furthermoredemand arrivals and sizes vary with usage and maintenancepatterns further adding to the degree of intermittenceMoreover the historical data of electric power materials arelimited coupled with constraints on the forecasting methoddemand forecast for these electric powermaterials is difficult
However such electric power materials are usually essentialcomponents of expensive electric equipment In the caseof equipment failure when electric power materials areunavailable and need to be ordered immediately the cost ofthese urgent orders is higher than regular orders significantlyincreasing the maintenance cost In addition to this thefailure of a component in the electrical equipment is likelyto lead to equipment failure generating a high backorderingcost and economic loss On the other hand excessive stockof electric power materials leads to backlogs and erosionresulting in a significant waste of capital as well as thecorresponding corporate cost Some electric power materialsstored in warehouses for a long time erode before they areput into use while some are permanently idle due to thecontinuous changes and updates of electrical equipmentSince traditional inventory control is generally designed forregular demand its application to intermittent demand islikely to lead to frequent stock-out events or excessive stock
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6085342 23 pageshttpsdoiorg10115520186085342
2 Mathematical Problems in Engineering
making the inventory control of electric power materialsa more challenging task Thus it is essential for electricpower companies to control their intermittent electric powermaterial inventories
2 Research Background
In this section a brief review of the literature related tointermittent material inventory control is undertaken
Traditional inventory control systems generally focus onregular demand and are not suitable for intermittent demand(Adelson [1] Friend [2]) As a representative of traditionalinventory control models the Newsvendor model had beenextensively used since it was introduced in 1962 (Hanssman[3])Thismodel states that if inventory is larger than demandthe remaining inventory will be sold at a discounted priceor disposed at the end of period however if inventory issmaller than demand some profit will be lost Thus thismodel focuses on optimizing cost or profit and developsrelevant inventory policy Subsequently a great number ofscholars make extensions to this traditional model which canbe divided into three categories Firstly traditional modelsusually assume a fixed purchase price which departs fromthe reality Thus price-sensitive demand is introduced Forinstance Xiao et al [4] study the price-dependent demandin a multi-product Newsvendor model and propose corre-sponding methods of obtaining the optimal order quantityand optimal sales price Kebe et al [5] consider a two-echeloninventory lot-sizing problem with price-dependent demandand develop a mixed-integer programming formulation aswell as a Lagrangian relaxation solution procedure for solvingthis problem Secondly considering the profound influenceof demand variation on expected profits or costs plenty ofscholars focus on the modelling of demand when studyingNewsvendor model Uduman [6] employs demand distri-butions satisfying the SCBZ (clock back to zero) propertyto model the single demand for a single product namelynewspaper and subsequently obtain the optimal order quan-tity Fathima et al [7] study the generalization of Udumanrsquosmodel with several individual demands for a single productfollowed by numerical illustrations Furthermore Faithma ampUduman [8] propose an approximation closed-form optimalsolution for three cases of single period inventory modelin which the demand varies with the SCBZ property Inaddition to SCBZ property Kamburowski [9] developeda theoretical framework for analyzing the distribution-freeNewsboy problems due to incomplete information SimilarlyChen [10] considers unknown demand and demonstratesthe appropriateness of applying the bootstrap method toNewsvendor model Hnaien et al [11] focus on discretedistributions of probabilities of demand Besides Priyan ampUthayakumar [12] consider a two-echelon multi-productmulti-constraint inventory model with product returns andrecovery Lastly two-level Newsvendor model includinginventories on raw materials and finished products hasrecently received increasingly attention In suchmodel Kera-matpour et al [13] develop a meta-heuristic algorithm toobtain the optimal solution which optimize profit and servicelevel simultaneously
Axsater [14] reveals that a majority of inventory controlmodels including Newsvendor model are built on someassumptions for instance that lead-time demand follows thenormal distribution However they point out that althoughthis assumption is reasonable in most cases it is inconsistentwith the reality in certain circumstances and leads to a wasteof capital or low service levels Thus they propose the (RQ) policy with the compound Poisson demand Consider-ing that the compound distribution model can determinearrival and demand sizes respectively and its structure issimilar to the intermittent demand process a vast numberof scholars propose that compound distributions compoundPoisson distribution in particular can offer a good fit forintermittent material inventory (Adelson [1] Friend [2])Similarly Matheus and Gelder [15] also employ compoundPoisson distribution to model demand and determine theorder quantity and reorder points Babai et al [16] havedeveloped a method to determine the optimal order-up-tolevel subject to a compound Poisson distribution
Dunsmuir and Snyder [17] present a method for deter-mining an optimal reorder point for the continuous review(S Q) inventory policy subject to a compound Bernoullidemand However they fail to take the impact of lead-timedemand on the service level into account To fill the gapJanssen et al [18] make some modifications and adopt the(R s Q) inventory model to calculate the reorder point witha service-level restriction Teunter et al [19] restrict theirstudy to the calculation of the mean and variance of lead timedemand and derive closed-form expressions for the servicelevel and the expected total cost under the (T S) policy with acompound binomial demand thereby avoiding complicatedcalculation of the mean and variance
Of all the literature with compound demand distributionassumptions the compound Poisson is the most widely uti-lized Its popularity can be mainly attributed to the simplicityarising from the memoryless propensity of the exponentialdistribution of the demand interarrivals Gupta et al [20]introduce the Palmrsquos theorem to justify such demand mod-elling for whole complicated systems However in the caseof one single material this modelling is reported to lead toexcess stock and imposes an additional financial burden oncompanies (Smith andDekker [21])Thememoryless propen-sity implies a constant failure rate which is inconsistentwith the reality where the likelihood of the failure of anitem and the demand for it grows with time Smith andDekker [21] initially propose amore realistic renewal demandarrival process which has an Increasing Failure Rate (IFR)but they do not propose specific demand distribution tomodel the renewal demand arrival process SubsequentlyLarsen and Kiesmuller [22] provide the first insight intothe employment of compound Erlang distribution-k in theinventory control system and they derive a closed-form costexpression of the optimal reorder level for an (R s nQ)policy Based on the same demand distribution Larsen andThorstenson [23] consider a continuous review base stockpolicy and demonstrate the use of a computer program toobtain the base stock level needed to achieve the specifiedorder fill rate (OFR) and volume fill rate (VFR) respectivelyMoreover the respective performances of the OFR and VFR
Mathematical Problems in Engineering 3
for different inventory control systems are compared (Larsenand Thorstenson [24]) With demand arrivals following theErlang distribution in conjunction with a Gamma distribu-tion for demand size Saidane et al [25 26] propose an algo-rithm to determine the optimal base stock level under a cost-oriented continuous review inventory system and conductnumerical investigations to compare the performance of thismodel in cost reduction with that of the compound Poissondistribution and the classic Newsvendor model respectivelySyntetos et al [27] not only demonstrate the superiority ofdeviating from the memoryless demand model but also takethe degree of intermittence into consideration and exploreits connection with forecast accuracy through a numericalinvestigation In sum the above literature under a compoundErlang process all assume that the lead time is constant
In spite of the many contributions to the material inven-tory management field there are some shortcomings thatneed to be addressed Firstly the majority of research focuseson the service-oriented rather than cost-oriented inventorysystems For instanceMatheus andGelder [15] demonstrate asimpler method to calculate the optimal reorder point subjectto a compound Poisson demand due to the assumption of atarget service level Fung et al [28] build a model of period-review with an order-up-to-level (T S) policy for single-iteminventory systems under compound Poisson demands andservice-level constraints Zhao [29] evaluates an Assemble-to-Order (ATO) system under compound Poisson demandwith a batch ordering policy bymeans of service level includ-ing the delivery lead-times and the order-based fill ratesTopan and Bayindir [30] explore the multi-item two-echelonmaterial inventory system with compound Poisson demandand constraints on the aggregate mean response time whichis an indicator of a target service level In addition theproposed algorithm is not able to be implemented since it iscomplicated and does not fit bulk operation
In response to these shortcomings this paper is based onthe electric power material demand data of Shanghai ElectricPower Company and develops a method to determine theoptimal order quantity with the (119879119876) policy subject to thecompound Erlang demand Empirical analysis is conductedto verify the superiority of the proposed theoretical modelin cost reductions as compared to the Newsvendor modelThis paper also proposes an approximation and a heuristicalgorithm to overcome the inconvenience and low efficiencyof existing algorithms in the real world
3 The Optimal Order Quantity Model withCompound Erlang Demand and (TQ) Policy
31 Problem Description and Model Assumptions
311 Problem Description This paper explores the inventorycontrol of electric powermaterials with intermittent demandwhich is a rather challenging task with tremendous costimplications for companies holding these inventories Therelevant costmainly consists of holding cost incurred for eachunit kept in stock and backordering cost arising from excessdemand Intermittent demand is modelled as a more realisticcompound Erlang-k distribution which has an IFR We
consider a cost-oriented single echelon single-item inventorysystem where unfilled demand is backordered and lead timeis deterministic The stock level is controlled according toa periodic review (119879 119876) policy which means that an orderwith fixed-amount quantity 119876 is triggered every inspectioninterval T The stock level is recorded as holding cost if 119876exceeds the total demand sizes within the lead time and asbackordering cost if not
312 Model Assumptions This section shows relevant nota-tions and assumptions of the inventory control model pro-posed in this paper
We introduce the following notations
119871 lead time (constant)119876 order quantity119883 demand size (random variable)0X probability distribution function of demand sizes120573 mean of nonzero demand sizes120582 mean demand arrival rate119896 the number of Erlang stages119879 inspection intervalℎ inventory holding cost per SKU (Stock KeepingUnit) per unit of time119887 inventory backordering cost per SKU per unit oftime119894 the complex number
For the proposed model the following assumptions aremade
(1) A single echelon single-item inventory system whichmeans that only a particular item in a particular echelon ofthe supply chain is considered in the inventory system
(2)Aperiodic review (119879 119876)policy where periodic inspec-tions are performed every cycle time of T and a fixed-amount119876order quantity is subsequently generatedThis policymeansthat organizations do nothing to control inventory within acycle
(3) The compound Erlang demand process wheredemand arrivals are governed by Erlang distribution anddemand sizes are governed by an arbitrary nonnegativeprobability distribution
(3) Constant lead time(4) The total inventory cost consisting of inventory
holding cost and backordering cost Inventory holding cost isincurred for each unit kept in stock while backordering costis due to the excess demand
32 Model Analysis The model proposed in this paper pre-sents demand as a compound Erlang process where demandarrivals are governed by an Erlang-k distribution and demandsizes are governed by an arbitrary nonnegative probabilitydistribution According to the above assumptions the totalexpected inventory cost consists of the holding cost ℎ andbackordering cost 119887 Assuming that the probability of no-demand arrivals during the lead time is 1198750 the expected
4 Mathematical Problems in Engineering
cost equals ℎ1198761198750 When there are 119898 demand arrivals witha total size of119883119898 during the lead time the inventory holdingcost constitutes the total cost when Q minus 119883119898 gt 0 while thebackordering cost constitutes the total cost whenQminus119883119898 lt 0Thus the expected total cost is given as follows
E [119862 (119876)] = ℎ1198761198750+ +infinsum119898=1
[ℎint1198760
(119876 minus 119883119898) 1198751198981198890(119898)119883119898
(119883119898)
+ 119887int+infin119876
(119883119898 minus 119876)1198751198981198890(119898)119883119898
(119883119898)](1)
Note that the probability distribution of 119883119898demand size0(119898)119883119898
is the m-fold iterated convolution of 0119883 Besides theprobability of 119898 demand arrivals within the lead time 119875119898 informula (1) can be derived based on the queuing and renewaltheories (Saidane et al [26])
119875119898 = 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871 (119898 ge 0) (2)
The objective of this paper is to derive the optimalorder quantity 119876lowast minimizing the expected total cost SinceE[119862(119876)] is a convex function it can be obtained by setting thederivative of the expected total cost with respect to 119876 equalto zero
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (3)
Proposition 1 The optimal order quantity 119876lowast is the solutionof
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(4)
Proof The optimal order quantity 119876lowastis the solution of thederivative of the expected total cost with respect to Q whichequals zero
dE [119862 (119876)]dQ
= 0 (5)
which is equivalent to
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (6)
Thus+infinsum119898=1
0(119898)119883119898
(119876) 119875119898 = 119887ℎ + 119887+infinsum119898=1
119875119898 minus ℎℎ + 1198871198750 (7)
Considering that sum+infin119898=1 119875119898 = 1 minus 1198750 the optimal orderquantity 119876lowastcan be derived from
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(8)
Since formula (8) is the infinite sum this paper uses anupper bound 119876119880 and a lower bound 119876119871 to approximate theoptimal order quantity 119876lowast In order to derive 119876119880 and 119876119871 wedenote that
119866 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ (1 minus 119899sum
119898=0
119875119898)
119867 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
(9)
Property 2 119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899) for all 119883119898 119899 ge0Proof
119874 (119883119898) = +infinsum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ 119899+infinsum119898=119899
0(119898)119883119898
(119876) 119875119898= 119867(119883119898 119899) + +infinsum
119898=119899
0(119898)119883119898
(119876) 119875119898
(10)
Since 0 le 0(119898)119883119898
(119876) le 1 for any 119883119898 119898 ge 00 le +infinsum119898=119899
0(119898)119883119898
(119876) 119875119898 le +infinsum119898=119899
119875119898 for all 119883119898 119899 ge 0 (11)
Thus119867(119883119898 119899) le 119874(119883119898 119899)119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
Mathematical Problems in Engineering 5
+ (1 minus 119899sum119898=0
119875119898) + 119899+infinsum119898=119899+1
0(119898)119883119898
(119876) 119875119898minus (1 minus 119899sum
119898=0
119875119898)
= 119866 (119883119898 119899) minus +infinsum119898=119899+1
[1 minus 0(119898)119883119898
(119876)] 119875119898(12)
Since 0 le 1 minus 0(119898)119883119898
(119876) le 1sum+infin119898=119899+1[1 minus 0(119898)119883119898
(119876)]119875119898 ge 0Thus 119874(119883119898) le 119866(119883119898 119899)To sum up119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899)
Property 3 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are strictlyincreasing functions of 119883119898Proof Since 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are sums ofvaried cumulative probability distribution functions whichare strictly increasing these three functions are strictlyincreasing
Property 4
lim119899997888rarr+infin
119866 (119883119898 119899) 997888rarr 119874 (119883119898) lim119899997888rarr+infin
119867(119883119898 119899) 997888rarr 119874 (119883119898) (13)
Proof When n tends to infinity sum+infin119898=119899 0(119898)119883119898
(119876)119875119898 andsum+infin119898=119899+1[1 minus 0(119898)119883
119898
(119876)]119875119898 tend to zeroAssume that 119876119871 119876119880 and 119876lowast are the solutions of119866(119883119898 119899) = 0 119867(119883119898 119899) = 0 and 119874(119883119898) = 0 respec-
tively Based on the above three properties the followingProposition 5 is correct when n tends to infinity
Proposition 5 119876lowastcan be approximated by an upper bound119876119880 and a lower bound 119876119871 based on 119876119871 le 119876lowast le 119876119880 where119876119880 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(14)
and 119876119871 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871
minus (1 minus 119899sum119898=0
(119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871))
(15)
For a given arbitrarily small value
Substitute n into formula (15)
No
Yes
( gt 0) initialize n =1
1 minusnsum
m=0
Pm = 1 minusnsum
m=0
(kminus1sumi=0
(kL)(km+i)
(km + i)eminuskL)
n = n + 1
Obtain Qlowast
1 minus sumnm=0 Pm le
Figure 1 The flow chart of the basic algorithm
33 Basic Algorithm Based the above analysis the algorithmto compute the optimal order quantity 119876lowast is comprised thefollowing steps
Step 0 For a given arbitrarily small value 120575 (120575 gt 0) initialize119899 = 1Step 1 Compute 1 minus sum119899119898=0 119875119898 = 1 minus sum119899119898=0(sum119896minus1119894=0 ((119896120582119871)(119896119898+119894)(119896119898 + 119894))119890minus119896120582119871)Step 2
(i) If 1 minus sum119899119898=0 119875119898 le 120575 go to Step 3
(ii) Otherwise do 119899 = 119899 + 1 and go to Step 1
Step 3 Utilize the derived n to solve (15) so as to obtain 119876lowastStep 4 Stop
In order to help understand the basic algorithm all aboveprocedures are illustrated in the flow chart ie Figure 1
In this flow chart the optimal order quantity 119876lowast isapproximated by an upper bound 119876119880 and a lower bound 119876119871both of which are calculated with finite varying 119899 As 119899 getslarger the lower and upper bounds get tighter and the gapbetween them gets smaller The algorithm cannot be stoppeduntil the gap is smaller than the given stopping criterion120575 In this circumstance 119876119871is obtained by substituting thecorresponding 119899 into formula (15) which is considered as analternative to the optimal order quantity 119876lowast In addition itshould be noted that 119876lowastcan be approximately as closely asdesired by setting varied stopping criterion 120575
6 Mathematical Problems in Engineering
500000000500010000500020000500030000500040000500050000500060000500070000500080000
Stat
e Gri
d C
ode
010 020 030 040 050 060000Mean of demand arrival rate
Figure 2The scatter diagram of demand interarrival rate of electricpower materials
4 Numerical Investigation
41 Data Processing There are 3839 types of electric powermaterials in the StateGrid Shanghai Electric PowerCompanyBecause most of the 3839 electrical power materials arerelatively less important (eg circuits) we do not needto pay much attention to the inventory management forthese electrical power materials with the model proposedin our paper Moreover for those electrical power materialswhich are also suitable for our model some of them arenot standardized and therefore eliminated (or are beingeliminated) For standardized electrical power materials dueto similarity only representative electrical power materialsare selectedThereby we gather the historical data of demandsof 53 types of electric power materials from 2015 to 2017the demand arrivals of which follow an Erlang distributionin conjunction with a Gamma distribution of demand sizesSince the Gamma distribution covers a large spectrum ofprobability distributions suchmodelling ismore appropriateBasic information of the 53 types of electric power materialsis presented in Table 1
As shown in Table 1 the mean nonzero demand sizes 120573of the 53 types of electric power materials vary sharply forexample the minimum mean nonzero demand size120573 is 319while the maximummean nonzero demand size 120573 is 1043716Moreover the variance of the mean nonzero demand size120573 is 3041691 In a word there exists a high volatility ofmean nonzero demand size And demand arrivals of the53 types of electric power materials mostly follow a two-stage Erlang distribution There are only four electric powermaterials consisting of 500014661500017325 500020638 and500064132 whose demand interarrivals follow three stageErlang distribution
Figure 2 shows the scatter diagram of demand arrival rateof the 53 types of electric power materials
As shown in Figure 2 the mean demand interarrival rate120582 of these 53 types of electric power materials within 3 yearsis between 001 and 052 denoted by 120582 isin [001 052] Thevariance of the mean demand interarrival rate 120582 is 0019reflecting its low volatility According to the classificationmethod proposed by Eaves [31] these electric power mate-rials exhibit a high level of intermittence
Subsequently we adopt the above model to model theinventory cost of these materials compute the optimal order
State grid code of spare parts
New modelNewsvendor model
minus100000100200300400500600700
Logc
Figure 3 Average inventory costs calculated with the two models
quantity of every electric power material and the corre-sponding average inventory cost and then compare themwith results of the classic Newsvendor model (Trabka andMarchand [32]) to examine the validity of this method
42 Comparison with the Newsvendor Model The aim of thissubsection is to examine the performance of the proposedtheoretical model in cost reduction We initially employedthis new model to obtain the optimal order quantity ofeach electric power material and the corresponding averageinventory cost and followed this with a comparison to theclassic Newsvendor model under a Gamma demand processConsidering that the backordering cost is higher than theinventory holding cost this paper assumes that ℎ = 1 119887 = 10Lead-time L is assumed to be 1 Results are as in Table 2
Table 2 illustrates the performances of the two modelswith respect to the optimal order quantity and inventorycost Results show that the proposed theoretical model ienew model suggests a higher optimal order quantity andlower average inventory cost compared with the Newsvendormodel Take the electric power material 500065644 forexample the optimal order quantity 119876 obtained with the newmodel is 32 which is approximately 45 times of that obtainedwith the Newsvendor model which equals 7 However interms of average inventory cost 119862 the new model suggests acost of 3063 which is significantly less than that suggested bythe Newsvendor model which equals 2162199 According toTable 2 other electric power materials show similar patternas the above material 500065644
In order to more vividly demonstrate the significantdifference of average inventory cost 119862 between these twomodels we make the line chart ie Figure 3
Since that there is huge gap between the average inventorycost 119862 calculated with the above two models we use thenatural logarithm of average inventory cost 119862 as the ordinateso as to narrow the gap As shown in the Figure 3 the averageinventory cost of new model is significantly less than thatof Newsvendor model which is consistent with the findingrevealed by the Table 2 In a word larger optimal orderquantity and lower average inventory cost are advised by thenewmodelThe conflicting change in these two variables canbe explained by the fact that although higher order quantityincreases the amount of electric powermaterials kept in stockand thus holding cost the backordering cost decreases due
Mathematical Problems in Engineering 7
Table 1 Basic information of 53 types of electric power materials
State grid code of spare parts Class Division Section k 120573 120582
500012723 Hardware Mechanical hardware Bolt 2 213777 047500012771 Hardware Mechanical hardware Bolt 2 84070 039500012797 Hardware Mechanical hardware Bolt 2 35231 006500012801 Hardware Mechanical hardware Bolt 2 11876 013500014661 Installation Wire Aerial insulation wire 3 947 051500014662 Installation Wire Aerial insulation wire 2 1437 023500014670 Installation Wire Aerial insulation wire 2 3759 046500014711 Installation Wire Aerial insulation wire 2 518 041500014713 Installation Wire Aerial insulation wire 2 453 035500014805 Installation Cable Cloth wire 2 1043716 005500016589 Installation Cable Control cable 2 116167 020500016592 Installation Cable Control cable 2 108033 016500017293 Installation Insulator Post porcelain insulator 2 25476 033500017320 Installation Insulator Strain insulator 2 27917 019500017324 Installation Fitting Shackle insulator 2 647462 042500017325 Installation Fitting Shackle insulator 3 184946 052500020609 Installation Cable accessories Splicing fittingmdashGT-G 2 2705 026500020612 Installation Cable accessories Splicing fittingmdashGT-G 2 309122 007500020638 Installation Cable accessories Splicing fittingmdashGTL 3 2204 001500021412 Installation Cable accessories 35kV cable connector 2 664 019500021803 Installation Cable accessories Cable terminals 2 7997 024500021830 Installation Low voltage apparatus Cable terminals 2 28081 033500021862 Installation Low voltage apparatus Cable terminals 2 136348 030500021869 Installation Surge arrester Cable terminals 2 30576 033500022269 Low voltage apparatus Traverse earth wire Low-tension fuse 2 21886 019500022271 Low voltage apparatus Fitting Low-tension fuse 2 229308 018500027151 Primary equipment High-pressure fuse fittings AC surge arrester 2 21094 043500027464 Installation High-tension fuse accessories Aerial insulation traverse 2 505 026500029690 Installation High-tension fuse accessories Guy wire fittingmdashwedge clamps 2 28030 047500032681 Spare part High-tension fuse accessories Fuse 2 8564 031500032682 Spare part Iron fittings Fuse 2 2735 003500032684 Spare part Mechanical hardware Fuse 2 15188 047500032687 Spare part Cable accessories Fuse 2 6044 036500035137 Installation Mechanical hardware Anchor rod 2 25067 044500051829 Hardware Low voltage apparatus Bolt 2 34951 032500053694 Installation Low voltage apparatus 35kV cable termination 2 826 016500058470 Hardware Security burglary fire Bolt 2 154798 030500061526 Low voltage apparatus Security burglary fire Low-tension fuse accessories 2 13157 022500061564 Low voltage apparatus Fitting Low-tension fuse accessories 2 11372 031500061758 Supplementary equipment Fitting Anti-lighting device 2 21126 025500062282 Supplementary equipment Traverse earth wire Anti-lighting device 2 16365 018500063419 Installation Traverse earth wire Strain clampsmdashpre-twisted type 2 16665 033500064132 Installation Iron fittings Guy wire fittingsmdashguy clips 3 146533 051500065644 Installation Mechanical hardware Aerial insulation traverse 2 319 048500065658 Installation Mechanical hardware Aerial insulation traverse 2 3122 044500066495 Installation Fittings Iron connection 2 282708 031500067251 Hardware Iron fittings Bolt 2 35250 019500067292 Hardware Iron fittings Bolt 2 46955 043500067551 Installation Iron fittings Guy wire fittingsmdashguy clips 2 58101 027
8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
100
Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
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n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
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074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
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904213938
0322
1221
3039
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6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
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543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
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6877
8593
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450514
577639
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17662240
26883116
35273924
43084682
1074017299
0412
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5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
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30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
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30443543
40254493
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1287621713
0592
1324
3444
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92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
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92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
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92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
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1324
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93102
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350426
500500
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7821426
20172574
31063618
41134594
50625519
1326922394
0633
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5564
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351427
502502
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7851431
20252584
31193633
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5565
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93102
193273
352429
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20322594
31313648
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0653
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5565
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193275
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654725
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94103
194276
357434
510510
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7991459
20672641
31893718
42304727
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1377423411
073
1324
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5665
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94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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2 Mathematical Problems in Engineering
making the inventory control of electric power materialsa more challenging task Thus it is essential for electricpower companies to control their intermittent electric powermaterial inventories
2 Research Background
In this section a brief review of the literature related tointermittent material inventory control is undertaken
Traditional inventory control systems generally focus onregular demand and are not suitable for intermittent demand(Adelson [1] Friend [2]) As a representative of traditionalinventory control models the Newsvendor model had beenextensively used since it was introduced in 1962 (Hanssman[3])Thismodel states that if inventory is larger than demandthe remaining inventory will be sold at a discounted priceor disposed at the end of period however if inventory issmaller than demand some profit will be lost Thus thismodel focuses on optimizing cost or profit and developsrelevant inventory policy Subsequently a great number ofscholars make extensions to this traditional model which canbe divided into three categories Firstly traditional modelsusually assume a fixed purchase price which departs fromthe reality Thus price-sensitive demand is introduced Forinstance Xiao et al [4] study the price-dependent demandin a multi-product Newsvendor model and propose corre-sponding methods of obtaining the optimal order quantityand optimal sales price Kebe et al [5] consider a two-echeloninventory lot-sizing problem with price-dependent demandand develop a mixed-integer programming formulation aswell as a Lagrangian relaxation solution procedure for solvingthis problem Secondly considering the profound influenceof demand variation on expected profits or costs plenty ofscholars focus on the modelling of demand when studyingNewsvendor model Uduman [6] employs demand distri-butions satisfying the SCBZ (clock back to zero) propertyto model the single demand for a single product namelynewspaper and subsequently obtain the optimal order quan-tity Fathima et al [7] study the generalization of Udumanrsquosmodel with several individual demands for a single productfollowed by numerical illustrations Furthermore Faithma ampUduman [8] propose an approximation closed-form optimalsolution for three cases of single period inventory modelin which the demand varies with the SCBZ property Inaddition to SCBZ property Kamburowski [9] developeda theoretical framework for analyzing the distribution-freeNewsboy problems due to incomplete information SimilarlyChen [10] considers unknown demand and demonstratesthe appropriateness of applying the bootstrap method toNewsvendor model Hnaien et al [11] focus on discretedistributions of probabilities of demand Besides Priyan ampUthayakumar [12] consider a two-echelon multi-productmulti-constraint inventory model with product returns andrecovery Lastly two-level Newsvendor model includinginventories on raw materials and finished products hasrecently received increasingly attention In suchmodel Kera-matpour et al [13] develop a meta-heuristic algorithm toobtain the optimal solution which optimize profit and servicelevel simultaneously
Axsater [14] reveals that a majority of inventory controlmodels including Newsvendor model are built on someassumptions for instance that lead-time demand follows thenormal distribution However they point out that althoughthis assumption is reasonable in most cases it is inconsistentwith the reality in certain circumstances and leads to a wasteof capital or low service levels Thus they propose the (RQ) policy with the compound Poisson demand Consider-ing that the compound distribution model can determinearrival and demand sizes respectively and its structure issimilar to the intermittent demand process a vast numberof scholars propose that compound distributions compoundPoisson distribution in particular can offer a good fit forintermittent material inventory (Adelson [1] Friend [2])Similarly Matheus and Gelder [15] also employ compoundPoisson distribution to model demand and determine theorder quantity and reorder points Babai et al [16] havedeveloped a method to determine the optimal order-up-tolevel subject to a compound Poisson distribution
Dunsmuir and Snyder [17] present a method for deter-mining an optimal reorder point for the continuous review(S Q) inventory policy subject to a compound Bernoullidemand However they fail to take the impact of lead-timedemand on the service level into account To fill the gapJanssen et al [18] make some modifications and adopt the(R s Q) inventory model to calculate the reorder point witha service-level restriction Teunter et al [19] restrict theirstudy to the calculation of the mean and variance of lead timedemand and derive closed-form expressions for the servicelevel and the expected total cost under the (T S) policy with acompound binomial demand thereby avoiding complicatedcalculation of the mean and variance
Of all the literature with compound demand distributionassumptions the compound Poisson is the most widely uti-lized Its popularity can be mainly attributed to the simplicityarising from the memoryless propensity of the exponentialdistribution of the demand interarrivals Gupta et al [20]introduce the Palmrsquos theorem to justify such demand mod-elling for whole complicated systems However in the caseof one single material this modelling is reported to lead toexcess stock and imposes an additional financial burden oncompanies (Smith andDekker [21])Thememoryless propen-sity implies a constant failure rate which is inconsistentwith the reality where the likelihood of the failure of anitem and the demand for it grows with time Smith andDekker [21] initially propose amore realistic renewal demandarrival process which has an Increasing Failure Rate (IFR)but they do not propose specific demand distribution tomodel the renewal demand arrival process SubsequentlyLarsen and Kiesmuller [22] provide the first insight intothe employment of compound Erlang distribution-k in theinventory control system and they derive a closed-form costexpression of the optimal reorder level for an (R s nQ)policy Based on the same demand distribution Larsen andThorstenson [23] consider a continuous review base stockpolicy and demonstrate the use of a computer program toobtain the base stock level needed to achieve the specifiedorder fill rate (OFR) and volume fill rate (VFR) respectivelyMoreover the respective performances of the OFR and VFR
Mathematical Problems in Engineering 3
for different inventory control systems are compared (Larsenand Thorstenson [24]) With demand arrivals following theErlang distribution in conjunction with a Gamma distribu-tion for demand size Saidane et al [25 26] propose an algo-rithm to determine the optimal base stock level under a cost-oriented continuous review inventory system and conductnumerical investigations to compare the performance of thismodel in cost reduction with that of the compound Poissondistribution and the classic Newsvendor model respectivelySyntetos et al [27] not only demonstrate the superiority ofdeviating from the memoryless demand model but also takethe degree of intermittence into consideration and exploreits connection with forecast accuracy through a numericalinvestigation In sum the above literature under a compoundErlang process all assume that the lead time is constant
In spite of the many contributions to the material inven-tory management field there are some shortcomings thatneed to be addressed Firstly the majority of research focuseson the service-oriented rather than cost-oriented inventorysystems For instanceMatheus andGelder [15] demonstrate asimpler method to calculate the optimal reorder point subjectto a compound Poisson demand due to the assumption of atarget service level Fung et al [28] build a model of period-review with an order-up-to-level (T S) policy for single-iteminventory systems under compound Poisson demands andservice-level constraints Zhao [29] evaluates an Assemble-to-Order (ATO) system under compound Poisson demandwith a batch ordering policy bymeans of service level includ-ing the delivery lead-times and the order-based fill ratesTopan and Bayindir [30] explore the multi-item two-echelonmaterial inventory system with compound Poisson demandand constraints on the aggregate mean response time whichis an indicator of a target service level In addition theproposed algorithm is not able to be implemented since it iscomplicated and does not fit bulk operation
In response to these shortcomings this paper is based onthe electric power material demand data of Shanghai ElectricPower Company and develops a method to determine theoptimal order quantity with the (119879119876) policy subject to thecompound Erlang demand Empirical analysis is conductedto verify the superiority of the proposed theoretical modelin cost reductions as compared to the Newsvendor modelThis paper also proposes an approximation and a heuristicalgorithm to overcome the inconvenience and low efficiencyof existing algorithms in the real world
3 The Optimal Order Quantity Model withCompound Erlang Demand and (TQ) Policy
31 Problem Description and Model Assumptions
311 Problem Description This paper explores the inventorycontrol of electric powermaterials with intermittent demandwhich is a rather challenging task with tremendous costimplications for companies holding these inventories Therelevant costmainly consists of holding cost incurred for eachunit kept in stock and backordering cost arising from excessdemand Intermittent demand is modelled as a more realisticcompound Erlang-k distribution which has an IFR We
consider a cost-oriented single echelon single-item inventorysystem where unfilled demand is backordered and lead timeis deterministic The stock level is controlled according toa periodic review (119879 119876) policy which means that an orderwith fixed-amount quantity 119876 is triggered every inspectioninterval T The stock level is recorded as holding cost if 119876exceeds the total demand sizes within the lead time and asbackordering cost if not
312 Model Assumptions This section shows relevant nota-tions and assumptions of the inventory control model pro-posed in this paper
We introduce the following notations
119871 lead time (constant)119876 order quantity119883 demand size (random variable)0X probability distribution function of demand sizes120573 mean of nonzero demand sizes120582 mean demand arrival rate119896 the number of Erlang stages119879 inspection intervalℎ inventory holding cost per SKU (Stock KeepingUnit) per unit of time119887 inventory backordering cost per SKU per unit oftime119894 the complex number
For the proposed model the following assumptions aremade
(1) A single echelon single-item inventory system whichmeans that only a particular item in a particular echelon ofthe supply chain is considered in the inventory system
(2)Aperiodic review (119879 119876)policy where periodic inspec-tions are performed every cycle time of T and a fixed-amount119876order quantity is subsequently generatedThis policymeansthat organizations do nothing to control inventory within acycle
(3) The compound Erlang demand process wheredemand arrivals are governed by Erlang distribution anddemand sizes are governed by an arbitrary nonnegativeprobability distribution
(3) Constant lead time(4) The total inventory cost consisting of inventory
holding cost and backordering cost Inventory holding cost isincurred for each unit kept in stock while backordering costis due to the excess demand
32 Model Analysis The model proposed in this paper pre-sents demand as a compound Erlang process where demandarrivals are governed by an Erlang-k distribution and demandsizes are governed by an arbitrary nonnegative probabilitydistribution According to the above assumptions the totalexpected inventory cost consists of the holding cost ℎ andbackordering cost 119887 Assuming that the probability of no-demand arrivals during the lead time is 1198750 the expected
4 Mathematical Problems in Engineering
cost equals ℎ1198761198750 When there are 119898 demand arrivals witha total size of119883119898 during the lead time the inventory holdingcost constitutes the total cost when Q minus 119883119898 gt 0 while thebackordering cost constitutes the total cost whenQminus119883119898 lt 0Thus the expected total cost is given as follows
E [119862 (119876)] = ℎ1198761198750+ +infinsum119898=1
[ℎint1198760
(119876 minus 119883119898) 1198751198981198890(119898)119883119898
(119883119898)
+ 119887int+infin119876
(119883119898 minus 119876)1198751198981198890(119898)119883119898
(119883119898)](1)
Note that the probability distribution of 119883119898demand size0(119898)119883119898
is the m-fold iterated convolution of 0119883 Besides theprobability of 119898 demand arrivals within the lead time 119875119898 informula (1) can be derived based on the queuing and renewaltheories (Saidane et al [26])
119875119898 = 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871 (119898 ge 0) (2)
The objective of this paper is to derive the optimalorder quantity 119876lowast minimizing the expected total cost SinceE[119862(119876)] is a convex function it can be obtained by setting thederivative of the expected total cost with respect to 119876 equalto zero
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (3)
Proposition 1 The optimal order quantity 119876lowast is the solutionof
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(4)
Proof The optimal order quantity 119876lowastis the solution of thederivative of the expected total cost with respect to Q whichequals zero
dE [119862 (119876)]dQ
= 0 (5)
which is equivalent to
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (6)
Thus+infinsum119898=1
0(119898)119883119898
(119876) 119875119898 = 119887ℎ + 119887+infinsum119898=1
119875119898 minus ℎℎ + 1198871198750 (7)
Considering that sum+infin119898=1 119875119898 = 1 minus 1198750 the optimal orderquantity 119876lowastcan be derived from
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(8)
Since formula (8) is the infinite sum this paper uses anupper bound 119876119880 and a lower bound 119876119871 to approximate theoptimal order quantity 119876lowast In order to derive 119876119880 and 119876119871 wedenote that
119866 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ (1 minus 119899sum
119898=0
119875119898)
119867 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
(9)
Property 2 119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899) for all 119883119898 119899 ge0Proof
119874 (119883119898) = +infinsum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ 119899+infinsum119898=119899
0(119898)119883119898
(119876) 119875119898= 119867(119883119898 119899) + +infinsum
119898=119899
0(119898)119883119898
(119876) 119875119898
(10)
Since 0 le 0(119898)119883119898
(119876) le 1 for any 119883119898 119898 ge 00 le +infinsum119898=119899
0(119898)119883119898
(119876) 119875119898 le +infinsum119898=119899
119875119898 for all 119883119898 119899 ge 0 (11)
Thus119867(119883119898 119899) le 119874(119883119898 119899)119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
Mathematical Problems in Engineering 5
+ (1 minus 119899sum119898=0
119875119898) + 119899+infinsum119898=119899+1
0(119898)119883119898
(119876) 119875119898minus (1 minus 119899sum
119898=0
119875119898)
= 119866 (119883119898 119899) minus +infinsum119898=119899+1
[1 minus 0(119898)119883119898
(119876)] 119875119898(12)
Since 0 le 1 minus 0(119898)119883119898
(119876) le 1sum+infin119898=119899+1[1 minus 0(119898)119883119898
(119876)]119875119898 ge 0Thus 119874(119883119898) le 119866(119883119898 119899)To sum up119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899)
Property 3 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are strictlyincreasing functions of 119883119898Proof Since 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are sums ofvaried cumulative probability distribution functions whichare strictly increasing these three functions are strictlyincreasing
Property 4
lim119899997888rarr+infin
119866 (119883119898 119899) 997888rarr 119874 (119883119898) lim119899997888rarr+infin
119867(119883119898 119899) 997888rarr 119874 (119883119898) (13)
Proof When n tends to infinity sum+infin119898=119899 0(119898)119883119898
(119876)119875119898 andsum+infin119898=119899+1[1 minus 0(119898)119883
119898
(119876)]119875119898 tend to zeroAssume that 119876119871 119876119880 and 119876lowast are the solutions of119866(119883119898 119899) = 0 119867(119883119898 119899) = 0 and 119874(119883119898) = 0 respec-
tively Based on the above three properties the followingProposition 5 is correct when n tends to infinity
Proposition 5 119876lowastcan be approximated by an upper bound119876119880 and a lower bound 119876119871 based on 119876119871 le 119876lowast le 119876119880 where119876119880 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(14)
and 119876119871 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871
minus (1 minus 119899sum119898=0
(119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871))
(15)
For a given arbitrarily small value
Substitute n into formula (15)
No
Yes
( gt 0) initialize n =1
1 minusnsum
m=0
Pm = 1 minusnsum
m=0
(kminus1sumi=0
(kL)(km+i)
(km + i)eminuskL)
n = n + 1
Obtain Qlowast
1 minus sumnm=0 Pm le
Figure 1 The flow chart of the basic algorithm
33 Basic Algorithm Based the above analysis the algorithmto compute the optimal order quantity 119876lowast is comprised thefollowing steps
Step 0 For a given arbitrarily small value 120575 (120575 gt 0) initialize119899 = 1Step 1 Compute 1 minus sum119899119898=0 119875119898 = 1 minus sum119899119898=0(sum119896minus1119894=0 ((119896120582119871)(119896119898+119894)(119896119898 + 119894))119890minus119896120582119871)Step 2
(i) If 1 minus sum119899119898=0 119875119898 le 120575 go to Step 3
(ii) Otherwise do 119899 = 119899 + 1 and go to Step 1
Step 3 Utilize the derived n to solve (15) so as to obtain 119876lowastStep 4 Stop
In order to help understand the basic algorithm all aboveprocedures are illustrated in the flow chart ie Figure 1
In this flow chart the optimal order quantity 119876lowast isapproximated by an upper bound 119876119880 and a lower bound 119876119871both of which are calculated with finite varying 119899 As 119899 getslarger the lower and upper bounds get tighter and the gapbetween them gets smaller The algorithm cannot be stoppeduntil the gap is smaller than the given stopping criterion120575 In this circumstance 119876119871is obtained by substituting thecorresponding 119899 into formula (15) which is considered as analternative to the optimal order quantity 119876lowast In addition itshould be noted that 119876lowastcan be approximately as closely asdesired by setting varied stopping criterion 120575
6 Mathematical Problems in Engineering
500000000500010000500020000500030000500040000500050000500060000500070000500080000
Stat
e Gri
d C
ode
010 020 030 040 050 060000Mean of demand arrival rate
Figure 2The scatter diagram of demand interarrival rate of electricpower materials
4 Numerical Investigation
41 Data Processing There are 3839 types of electric powermaterials in the StateGrid Shanghai Electric PowerCompanyBecause most of the 3839 electrical power materials arerelatively less important (eg circuits) we do not needto pay much attention to the inventory management forthese electrical power materials with the model proposedin our paper Moreover for those electrical power materialswhich are also suitable for our model some of them arenot standardized and therefore eliminated (or are beingeliminated) For standardized electrical power materials dueto similarity only representative electrical power materialsare selectedThereby we gather the historical data of demandsof 53 types of electric power materials from 2015 to 2017the demand arrivals of which follow an Erlang distributionin conjunction with a Gamma distribution of demand sizesSince the Gamma distribution covers a large spectrum ofprobability distributions suchmodelling ismore appropriateBasic information of the 53 types of electric power materialsis presented in Table 1
As shown in Table 1 the mean nonzero demand sizes 120573of the 53 types of electric power materials vary sharply forexample the minimum mean nonzero demand size120573 is 319while the maximummean nonzero demand size 120573 is 1043716Moreover the variance of the mean nonzero demand size120573 is 3041691 In a word there exists a high volatility ofmean nonzero demand size And demand arrivals of the53 types of electric power materials mostly follow a two-stage Erlang distribution There are only four electric powermaterials consisting of 500014661500017325 500020638 and500064132 whose demand interarrivals follow three stageErlang distribution
Figure 2 shows the scatter diagram of demand arrival rateof the 53 types of electric power materials
As shown in Figure 2 the mean demand interarrival rate120582 of these 53 types of electric power materials within 3 yearsis between 001 and 052 denoted by 120582 isin [001 052] Thevariance of the mean demand interarrival rate 120582 is 0019reflecting its low volatility According to the classificationmethod proposed by Eaves [31] these electric power mate-rials exhibit a high level of intermittence
Subsequently we adopt the above model to model theinventory cost of these materials compute the optimal order
State grid code of spare parts
New modelNewsvendor model
minus100000100200300400500600700
Logc
Figure 3 Average inventory costs calculated with the two models
quantity of every electric power material and the corre-sponding average inventory cost and then compare themwith results of the classic Newsvendor model (Trabka andMarchand [32]) to examine the validity of this method
42 Comparison with the Newsvendor Model The aim of thissubsection is to examine the performance of the proposedtheoretical model in cost reduction We initially employedthis new model to obtain the optimal order quantity ofeach electric power material and the corresponding averageinventory cost and followed this with a comparison to theclassic Newsvendor model under a Gamma demand processConsidering that the backordering cost is higher than theinventory holding cost this paper assumes that ℎ = 1 119887 = 10Lead-time L is assumed to be 1 Results are as in Table 2
Table 2 illustrates the performances of the two modelswith respect to the optimal order quantity and inventorycost Results show that the proposed theoretical model ienew model suggests a higher optimal order quantity andlower average inventory cost compared with the Newsvendormodel Take the electric power material 500065644 forexample the optimal order quantity 119876 obtained with the newmodel is 32 which is approximately 45 times of that obtainedwith the Newsvendor model which equals 7 However interms of average inventory cost 119862 the new model suggests acost of 3063 which is significantly less than that suggested bythe Newsvendor model which equals 2162199 According toTable 2 other electric power materials show similar patternas the above material 500065644
In order to more vividly demonstrate the significantdifference of average inventory cost 119862 between these twomodels we make the line chart ie Figure 3
Since that there is huge gap between the average inventorycost 119862 calculated with the above two models we use thenatural logarithm of average inventory cost 119862 as the ordinateso as to narrow the gap As shown in the Figure 3 the averageinventory cost of new model is significantly less than thatof Newsvendor model which is consistent with the findingrevealed by the Table 2 In a word larger optimal orderquantity and lower average inventory cost are advised by thenewmodelThe conflicting change in these two variables canbe explained by the fact that although higher order quantityincreases the amount of electric powermaterials kept in stockand thus holding cost the backordering cost decreases due
Mathematical Problems in Engineering 7
Table 1 Basic information of 53 types of electric power materials
State grid code of spare parts Class Division Section k 120573 120582
500012723 Hardware Mechanical hardware Bolt 2 213777 047500012771 Hardware Mechanical hardware Bolt 2 84070 039500012797 Hardware Mechanical hardware Bolt 2 35231 006500012801 Hardware Mechanical hardware Bolt 2 11876 013500014661 Installation Wire Aerial insulation wire 3 947 051500014662 Installation Wire Aerial insulation wire 2 1437 023500014670 Installation Wire Aerial insulation wire 2 3759 046500014711 Installation Wire Aerial insulation wire 2 518 041500014713 Installation Wire Aerial insulation wire 2 453 035500014805 Installation Cable Cloth wire 2 1043716 005500016589 Installation Cable Control cable 2 116167 020500016592 Installation Cable Control cable 2 108033 016500017293 Installation Insulator Post porcelain insulator 2 25476 033500017320 Installation Insulator Strain insulator 2 27917 019500017324 Installation Fitting Shackle insulator 2 647462 042500017325 Installation Fitting Shackle insulator 3 184946 052500020609 Installation Cable accessories Splicing fittingmdashGT-G 2 2705 026500020612 Installation Cable accessories Splicing fittingmdashGT-G 2 309122 007500020638 Installation Cable accessories Splicing fittingmdashGTL 3 2204 001500021412 Installation Cable accessories 35kV cable connector 2 664 019500021803 Installation Cable accessories Cable terminals 2 7997 024500021830 Installation Low voltage apparatus Cable terminals 2 28081 033500021862 Installation Low voltage apparatus Cable terminals 2 136348 030500021869 Installation Surge arrester Cable terminals 2 30576 033500022269 Low voltage apparatus Traverse earth wire Low-tension fuse 2 21886 019500022271 Low voltage apparatus Fitting Low-tension fuse 2 229308 018500027151 Primary equipment High-pressure fuse fittings AC surge arrester 2 21094 043500027464 Installation High-tension fuse accessories Aerial insulation traverse 2 505 026500029690 Installation High-tension fuse accessories Guy wire fittingmdashwedge clamps 2 28030 047500032681 Spare part High-tension fuse accessories Fuse 2 8564 031500032682 Spare part Iron fittings Fuse 2 2735 003500032684 Spare part Mechanical hardware Fuse 2 15188 047500032687 Spare part Cable accessories Fuse 2 6044 036500035137 Installation Mechanical hardware Anchor rod 2 25067 044500051829 Hardware Low voltage apparatus Bolt 2 34951 032500053694 Installation Low voltage apparatus 35kV cable termination 2 826 016500058470 Hardware Security burglary fire Bolt 2 154798 030500061526 Low voltage apparatus Security burglary fire Low-tension fuse accessories 2 13157 022500061564 Low voltage apparatus Fitting Low-tension fuse accessories 2 11372 031500061758 Supplementary equipment Fitting Anti-lighting device 2 21126 025500062282 Supplementary equipment Traverse earth wire Anti-lighting device 2 16365 018500063419 Installation Traverse earth wire Strain clampsmdashpre-twisted type 2 16665 033500064132 Installation Iron fittings Guy wire fittingsmdashguy clips 3 146533 051500065644 Installation Mechanical hardware Aerial insulation traverse 2 319 048500065658 Installation Mechanical hardware Aerial insulation traverse 2 3122 044500066495 Installation Fittings Iron connection 2 282708 031500067251 Hardware Iron fittings Bolt 2 35250 019500067292 Hardware Iron fittings Bolt 2 46955 043500067551 Installation Iron fittings Guy wire fittingsmdashguy clips 2 58101 027
8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
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30
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05 1 150Lamda
5
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Q
05 1 150Lamda
20
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Q
0
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Q
20
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Q
0
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Q
0
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600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
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600
Q
0
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800
Q
0
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2000Q
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Q
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Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
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2431
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5157
6368
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217259
300340
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10171241
14391617
17791926
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32462311
0091
1018
2532
3946
5258
6470
126177
225270
312354
393432
469799
10771321
15391738
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22412385
38503518
011
1018
2633
4047
5360
6672
130183
232278
323366
407448
487835
11301391
16281843
20432277
24002561
43784574
0111
1018
2633
4147
5461
6674
133187
238286
332377
420462
502866
11771454
17051937
21522352
25402717
48455509
0122
1119
2735
4350
5764
7178
141199
253305
355404
451496
541943
12921608
18992168
24212660
30013101
59997817
0132
1119
2836
4351
5865
7279
144203
259313
364414
462510
556972
13361666
19712255
25232777
30183247
64388861
0142
1120
2836
4351
5966
7380
146207
265319
372423
473522
569999
13761720
20382336
26172884
31383382
68419500
0152
1120
2837
4552
6067
7482
149211
270326
380432
483533
5811024
14131769
21002410
27042893
32503506
721310244
0162
1120
2937
4553
6168
7583
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274331
387440
492543
5931047
14481815
21572479
27853075
33543621
755811577
0172
1120
2937
4654
6169
7684
153218
278337
393448
501553
6041068
14801858
22112544
28603161
34503728
787912178
0182
1121
2938
4654
6270
7785
155221
282342
399455
509562
6141088
15101898
22612604
29303241
35403828
818012742
0192
1121
3038
4755
6371
7886
157223
286346
405461
516570
6231107
15381936
23082660
29853316
36253922
846212742
022
1221
3039
4755
6371
7987
159226
290351
410467
523578
6321125
15651971
23522713
30573387
37044010
872713272
0212
1221
3039
4756
6472
8087
161229
293355
415473
530586
6401141
15902005
23942763
31163454
37794094
897713772
0222
1221
3039
4856
6473
8088
162231
296359
420479
536593
6481157
16132036
24332810
31713517
38504172
921314244
0232
1221
3140
4857
6573
8189
164233
299363
424484
542599
6551172
16362066
24712855
32233576
39174247
943614691
0242
1222
3140
4957
6574
8290
165235
302366
429489
548606
6631186
16572094
25062897
32723633
39814317
964815115
0252
1222
3140
4958
6674
8290
166237
305370
432494
553612
6691199
16772121
25392938
33193686
40414385
984915517
0262
1222
3140
4958
6675
8391
168239
307373
436498
558618
6761212
16962146
25712976
33643738
40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
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0272
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506558
6101080
14711883
22422580
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0282
1121
3038
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6270
7885
156222
284344
402458
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22852633
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6371
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169241
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176252
324394
462528
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45274924
1152118830
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330401
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28523313
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7180
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331403
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7361333
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28733337
37864219
46415051
1184819602
0492
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5262
7180
8897
181258
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610675
7401341
18892403
28923361
38134251
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1196719844
052
1223
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7180
8998
181260
334407
477546
613679
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19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
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38664311
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1219220525
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3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
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38914340
47765201
1230020305
0532
1323
3444
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7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
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1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
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0552
1324
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7282
91100
185265
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7611383
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063
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640709
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50135464
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642711
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0633
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31193633
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5565
7484
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193273
352429
503503
648718
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20322594
31313648
41494635
51085570
1342422706
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5565
7484
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353430
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7901441
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7584
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193274
354431
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652723
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31563677
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193275
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508508
654725
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31673691
41824692
51725642
1364023142
0683
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5565
7584
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655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
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7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
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661732
8031467
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42594761
52505728
1390023666
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359437
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8051471
20852665
32203755
42734777
52685748
1396023788
0733
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360438
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665737
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7686
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32483778
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1413124133
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0793
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7686
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8181496
21222715
32823819
43594876
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083
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196282
365444
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673747
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21272721
32903829
43704888
53935888
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0813
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5667
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197283
365445
523523
674748
8211502
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54075903
1443024738
0823
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5767
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96106
197283
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524524
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21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
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197283
366447
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677751
8231508
21402739
33123856
44014923
54335934
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0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
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97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
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200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
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5768
7888
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200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 3
for different inventory control systems are compared (Larsenand Thorstenson [24]) With demand arrivals following theErlang distribution in conjunction with a Gamma distribu-tion for demand size Saidane et al [25 26] propose an algo-rithm to determine the optimal base stock level under a cost-oriented continuous review inventory system and conductnumerical investigations to compare the performance of thismodel in cost reduction with that of the compound Poissondistribution and the classic Newsvendor model respectivelySyntetos et al [27] not only demonstrate the superiority ofdeviating from the memoryless demand model but also takethe degree of intermittence into consideration and exploreits connection with forecast accuracy through a numericalinvestigation In sum the above literature under a compoundErlang process all assume that the lead time is constant
In spite of the many contributions to the material inven-tory management field there are some shortcomings thatneed to be addressed Firstly the majority of research focuseson the service-oriented rather than cost-oriented inventorysystems For instanceMatheus andGelder [15] demonstrate asimpler method to calculate the optimal reorder point subjectto a compound Poisson demand due to the assumption of atarget service level Fung et al [28] build a model of period-review with an order-up-to-level (T S) policy for single-iteminventory systems under compound Poisson demands andservice-level constraints Zhao [29] evaluates an Assemble-to-Order (ATO) system under compound Poisson demandwith a batch ordering policy bymeans of service level includ-ing the delivery lead-times and the order-based fill ratesTopan and Bayindir [30] explore the multi-item two-echelonmaterial inventory system with compound Poisson demandand constraints on the aggregate mean response time whichis an indicator of a target service level In addition theproposed algorithm is not able to be implemented since it iscomplicated and does not fit bulk operation
In response to these shortcomings this paper is based onthe electric power material demand data of Shanghai ElectricPower Company and develops a method to determine theoptimal order quantity with the (119879119876) policy subject to thecompound Erlang demand Empirical analysis is conductedto verify the superiority of the proposed theoretical modelin cost reductions as compared to the Newsvendor modelThis paper also proposes an approximation and a heuristicalgorithm to overcome the inconvenience and low efficiencyof existing algorithms in the real world
3 The Optimal Order Quantity Model withCompound Erlang Demand and (TQ) Policy
31 Problem Description and Model Assumptions
311 Problem Description This paper explores the inventorycontrol of electric powermaterials with intermittent demandwhich is a rather challenging task with tremendous costimplications for companies holding these inventories Therelevant costmainly consists of holding cost incurred for eachunit kept in stock and backordering cost arising from excessdemand Intermittent demand is modelled as a more realisticcompound Erlang-k distribution which has an IFR We
consider a cost-oriented single echelon single-item inventorysystem where unfilled demand is backordered and lead timeis deterministic The stock level is controlled according toa periodic review (119879 119876) policy which means that an orderwith fixed-amount quantity 119876 is triggered every inspectioninterval T The stock level is recorded as holding cost if 119876exceeds the total demand sizes within the lead time and asbackordering cost if not
312 Model Assumptions This section shows relevant nota-tions and assumptions of the inventory control model pro-posed in this paper
We introduce the following notations
119871 lead time (constant)119876 order quantity119883 demand size (random variable)0X probability distribution function of demand sizes120573 mean of nonzero demand sizes120582 mean demand arrival rate119896 the number of Erlang stages119879 inspection intervalℎ inventory holding cost per SKU (Stock KeepingUnit) per unit of time119887 inventory backordering cost per SKU per unit oftime119894 the complex number
For the proposed model the following assumptions aremade
(1) A single echelon single-item inventory system whichmeans that only a particular item in a particular echelon ofthe supply chain is considered in the inventory system
(2)Aperiodic review (119879 119876)policy where periodic inspec-tions are performed every cycle time of T and a fixed-amount119876order quantity is subsequently generatedThis policymeansthat organizations do nothing to control inventory within acycle
(3) The compound Erlang demand process wheredemand arrivals are governed by Erlang distribution anddemand sizes are governed by an arbitrary nonnegativeprobability distribution
(3) Constant lead time(4) The total inventory cost consisting of inventory
holding cost and backordering cost Inventory holding cost isincurred for each unit kept in stock while backordering costis due to the excess demand
32 Model Analysis The model proposed in this paper pre-sents demand as a compound Erlang process where demandarrivals are governed by an Erlang-k distribution and demandsizes are governed by an arbitrary nonnegative probabilitydistribution According to the above assumptions the totalexpected inventory cost consists of the holding cost ℎ andbackordering cost 119887 Assuming that the probability of no-demand arrivals during the lead time is 1198750 the expected
4 Mathematical Problems in Engineering
cost equals ℎ1198761198750 When there are 119898 demand arrivals witha total size of119883119898 during the lead time the inventory holdingcost constitutes the total cost when Q minus 119883119898 gt 0 while thebackordering cost constitutes the total cost whenQminus119883119898 lt 0Thus the expected total cost is given as follows
E [119862 (119876)] = ℎ1198761198750+ +infinsum119898=1
[ℎint1198760
(119876 minus 119883119898) 1198751198981198890(119898)119883119898
(119883119898)
+ 119887int+infin119876
(119883119898 minus 119876)1198751198981198890(119898)119883119898
(119883119898)](1)
Note that the probability distribution of 119883119898demand size0(119898)119883119898
is the m-fold iterated convolution of 0119883 Besides theprobability of 119898 demand arrivals within the lead time 119875119898 informula (1) can be derived based on the queuing and renewaltheories (Saidane et al [26])
119875119898 = 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871 (119898 ge 0) (2)
The objective of this paper is to derive the optimalorder quantity 119876lowast minimizing the expected total cost SinceE[119862(119876)] is a convex function it can be obtained by setting thederivative of the expected total cost with respect to 119876 equalto zero
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (3)
Proposition 1 The optimal order quantity 119876lowast is the solutionof
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(4)
Proof The optimal order quantity 119876lowastis the solution of thederivative of the expected total cost with respect to Q whichequals zero
dE [119862 (119876)]dQ
= 0 (5)
which is equivalent to
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (6)
Thus+infinsum119898=1
0(119898)119883119898
(119876) 119875119898 = 119887ℎ + 119887+infinsum119898=1
119875119898 minus ℎℎ + 1198871198750 (7)
Considering that sum+infin119898=1 119875119898 = 1 minus 1198750 the optimal orderquantity 119876lowastcan be derived from
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(8)
Since formula (8) is the infinite sum this paper uses anupper bound 119876119880 and a lower bound 119876119871 to approximate theoptimal order quantity 119876lowast In order to derive 119876119880 and 119876119871 wedenote that
119866 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ (1 minus 119899sum
119898=0
119875119898)
119867 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
(9)
Property 2 119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899) for all 119883119898 119899 ge0Proof
119874 (119883119898) = +infinsum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ 119899+infinsum119898=119899
0(119898)119883119898
(119876) 119875119898= 119867(119883119898 119899) + +infinsum
119898=119899
0(119898)119883119898
(119876) 119875119898
(10)
Since 0 le 0(119898)119883119898
(119876) le 1 for any 119883119898 119898 ge 00 le +infinsum119898=119899
0(119898)119883119898
(119876) 119875119898 le +infinsum119898=119899
119875119898 for all 119883119898 119899 ge 0 (11)
Thus119867(119883119898 119899) le 119874(119883119898 119899)119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
Mathematical Problems in Engineering 5
+ (1 minus 119899sum119898=0
119875119898) + 119899+infinsum119898=119899+1
0(119898)119883119898
(119876) 119875119898minus (1 minus 119899sum
119898=0
119875119898)
= 119866 (119883119898 119899) minus +infinsum119898=119899+1
[1 minus 0(119898)119883119898
(119876)] 119875119898(12)
Since 0 le 1 minus 0(119898)119883119898
(119876) le 1sum+infin119898=119899+1[1 minus 0(119898)119883119898
(119876)]119875119898 ge 0Thus 119874(119883119898) le 119866(119883119898 119899)To sum up119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899)
Property 3 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are strictlyincreasing functions of 119883119898Proof Since 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are sums ofvaried cumulative probability distribution functions whichare strictly increasing these three functions are strictlyincreasing
Property 4
lim119899997888rarr+infin
119866 (119883119898 119899) 997888rarr 119874 (119883119898) lim119899997888rarr+infin
119867(119883119898 119899) 997888rarr 119874 (119883119898) (13)
Proof When n tends to infinity sum+infin119898=119899 0(119898)119883119898
(119876)119875119898 andsum+infin119898=119899+1[1 minus 0(119898)119883
119898
(119876)]119875119898 tend to zeroAssume that 119876119871 119876119880 and 119876lowast are the solutions of119866(119883119898 119899) = 0 119867(119883119898 119899) = 0 and 119874(119883119898) = 0 respec-
tively Based on the above three properties the followingProposition 5 is correct when n tends to infinity
Proposition 5 119876lowastcan be approximated by an upper bound119876119880 and a lower bound 119876119871 based on 119876119871 le 119876lowast le 119876119880 where119876119880 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(14)
and 119876119871 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871
minus (1 minus 119899sum119898=0
(119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871))
(15)
For a given arbitrarily small value
Substitute n into formula (15)
No
Yes
( gt 0) initialize n =1
1 minusnsum
m=0
Pm = 1 minusnsum
m=0
(kminus1sumi=0
(kL)(km+i)
(km + i)eminuskL)
n = n + 1
Obtain Qlowast
1 minus sumnm=0 Pm le
Figure 1 The flow chart of the basic algorithm
33 Basic Algorithm Based the above analysis the algorithmto compute the optimal order quantity 119876lowast is comprised thefollowing steps
Step 0 For a given arbitrarily small value 120575 (120575 gt 0) initialize119899 = 1Step 1 Compute 1 minus sum119899119898=0 119875119898 = 1 minus sum119899119898=0(sum119896minus1119894=0 ((119896120582119871)(119896119898+119894)(119896119898 + 119894))119890minus119896120582119871)Step 2
(i) If 1 minus sum119899119898=0 119875119898 le 120575 go to Step 3
(ii) Otherwise do 119899 = 119899 + 1 and go to Step 1
Step 3 Utilize the derived n to solve (15) so as to obtain 119876lowastStep 4 Stop
In order to help understand the basic algorithm all aboveprocedures are illustrated in the flow chart ie Figure 1
In this flow chart the optimal order quantity 119876lowast isapproximated by an upper bound 119876119880 and a lower bound 119876119871both of which are calculated with finite varying 119899 As 119899 getslarger the lower and upper bounds get tighter and the gapbetween them gets smaller The algorithm cannot be stoppeduntil the gap is smaller than the given stopping criterion120575 In this circumstance 119876119871is obtained by substituting thecorresponding 119899 into formula (15) which is considered as analternative to the optimal order quantity 119876lowast In addition itshould be noted that 119876lowastcan be approximately as closely asdesired by setting varied stopping criterion 120575
6 Mathematical Problems in Engineering
500000000500010000500020000500030000500040000500050000500060000500070000500080000
Stat
e Gri
d C
ode
010 020 030 040 050 060000Mean of demand arrival rate
Figure 2The scatter diagram of demand interarrival rate of electricpower materials
4 Numerical Investigation
41 Data Processing There are 3839 types of electric powermaterials in the StateGrid Shanghai Electric PowerCompanyBecause most of the 3839 electrical power materials arerelatively less important (eg circuits) we do not needto pay much attention to the inventory management forthese electrical power materials with the model proposedin our paper Moreover for those electrical power materialswhich are also suitable for our model some of them arenot standardized and therefore eliminated (or are beingeliminated) For standardized electrical power materials dueto similarity only representative electrical power materialsare selectedThereby we gather the historical data of demandsof 53 types of electric power materials from 2015 to 2017the demand arrivals of which follow an Erlang distributionin conjunction with a Gamma distribution of demand sizesSince the Gamma distribution covers a large spectrum ofprobability distributions suchmodelling ismore appropriateBasic information of the 53 types of electric power materialsis presented in Table 1
As shown in Table 1 the mean nonzero demand sizes 120573of the 53 types of electric power materials vary sharply forexample the minimum mean nonzero demand size120573 is 319while the maximummean nonzero demand size 120573 is 1043716Moreover the variance of the mean nonzero demand size120573 is 3041691 In a word there exists a high volatility ofmean nonzero demand size And demand arrivals of the53 types of electric power materials mostly follow a two-stage Erlang distribution There are only four electric powermaterials consisting of 500014661500017325 500020638 and500064132 whose demand interarrivals follow three stageErlang distribution
Figure 2 shows the scatter diagram of demand arrival rateof the 53 types of electric power materials
As shown in Figure 2 the mean demand interarrival rate120582 of these 53 types of electric power materials within 3 yearsis between 001 and 052 denoted by 120582 isin [001 052] Thevariance of the mean demand interarrival rate 120582 is 0019reflecting its low volatility According to the classificationmethod proposed by Eaves [31] these electric power mate-rials exhibit a high level of intermittence
Subsequently we adopt the above model to model theinventory cost of these materials compute the optimal order
State grid code of spare parts
New modelNewsvendor model
minus100000100200300400500600700
Logc
Figure 3 Average inventory costs calculated with the two models
quantity of every electric power material and the corre-sponding average inventory cost and then compare themwith results of the classic Newsvendor model (Trabka andMarchand [32]) to examine the validity of this method
42 Comparison with the Newsvendor Model The aim of thissubsection is to examine the performance of the proposedtheoretical model in cost reduction We initially employedthis new model to obtain the optimal order quantity ofeach electric power material and the corresponding averageinventory cost and followed this with a comparison to theclassic Newsvendor model under a Gamma demand processConsidering that the backordering cost is higher than theinventory holding cost this paper assumes that ℎ = 1 119887 = 10Lead-time L is assumed to be 1 Results are as in Table 2
Table 2 illustrates the performances of the two modelswith respect to the optimal order quantity and inventorycost Results show that the proposed theoretical model ienew model suggests a higher optimal order quantity andlower average inventory cost compared with the Newsvendormodel Take the electric power material 500065644 forexample the optimal order quantity 119876 obtained with the newmodel is 32 which is approximately 45 times of that obtainedwith the Newsvendor model which equals 7 However interms of average inventory cost 119862 the new model suggests acost of 3063 which is significantly less than that suggested bythe Newsvendor model which equals 2162199 According toTable 2 other electric power materials show similar patternas the above material 500065644
In order to more vividly demonstrate the significantdifference of average inventory cost 119862 between these twomodels we make the line chart ie Figure 3
Since that there is huge gap between the average inventorycost 119862 calculated with the above two models we use thenatural logarithm of average inventory cost 119862 as the ordinateso as to narrow the gap As shown in the Figure 3 the averageinventory cost of new model is significantly less than thatof Newsvendor model which is consistent with the findingrevealed by the Table 2 In a word larger optimal orderquantity and lower average inventory cost are advised by thenewmodelThe conflicting change in these two variables canbe explained by the fact that although higher order quantityincreases the amount of electric powermaterials kept in stockand thus holding cost the backordering cost decreases due
Mathematical Problems in Engineering 7
Table 1 Basic information of 53 types of electric power materials
State grid code of spare parts Class Division Section k 120573 120582
500012723 Hardware Mechanical hardware Bolt 2 213777 047500012771 Hardware Mechanical hardware Bolt 2 84070 039500012797 Hardware Mechanical hardware Bolt 2 35231 006500012801 Hardware Mechanical hardware Bolt 2 11876 013500014661 Installation Wire Aerial insulation wire 3 947 051500014662 Installation Wire Aerial insulation wire 2 1437 023500014670 Installation Wire Aerial insulation wire 2 3759 046500014711 Installation Wire Aerial insulation wire 2 518 041500014713 Installation Wire Aerial insulation wire 2 453 035500014805 Installation Cable Cloth wire 2 1043716 005500016589 Installation Cable Control cable 2 116167 020500016592 Installation Cable Control cable 2 108033 016500017293 Installation Insulator Post porcelain insulator 2 25476 033500017320 Installation Insulator Strain insulator 2 27917 019500017324 Installation Fitting Shackle insulator 2 647462 042500017325 Installation Fitting Shackle insulator 3 184946 052500020609 Installation Cable accessories Splicing fittingmdashGT-G 2 2705 026500020612 Installation Cable accessories Splicing fittingmdashGT-G 2 309122 007500020638 Installation Cable accessories Splicing fittingmdashGTL 3 2204 001500021412 Installation Cable accessories 35kV cable connector 2 664 019500021803 Installation Cable accessories Cable terminals 2 7997 024500021830 Installation Low voltage apparatus Cable terminals 2 28081 033500021862 Installation Low voltage apparatus Cable terminals 2 136348 030500021869 Installation Surge arrester Cable terminals 2 30576 033500022269 Low voltage apparatus Traverse earth wire Low-tension fuse 2 21886 019500022271 Low voltage apparatus Fitting Low-tension fuse 2 229308 018500027151 Primary equipment High-pressure fuse fittings AC surge arrester 2 21094 043500027464 Installation High-tension fuse accessories Aerial insulation traverse 2 505 026500029690 Installation High-tension fuse accessories Guy wire fittingmdashwedge clamps 2 28030 047500032681 Spare part High-tension fuse accessories Fuse 2 8564 031500032682 Spare part Iron fittings Fuse 2 2735 003500032684 Spare part Mechanical hardware Fuse 2 15188 047500032687 Spare part Cable accessories Fuse 2 6044 036500035137 Installation Mechanical hardware Anchor rod 2 25067 044500051829 Hardware Low voltage apparatus Bolt 2 34951 032500053694 Installation Low voltage apparatus 35kV cable termination 2 826 016500058470 Hardware Security burglary fire Bolt 2 154798 030500061526 Low voltage apparatus Security burglary fire Low-tension fuse accessories 2 13157 022500061564 Low voltage apparatus Fitting Low-tension fuse accessories 2 11372 031500061758 Supplementary equipment Fitting Anti-lighting device 2 21126 025500062282 Supplementary equipment Traverse earth wire Anti-lighting device 2 16365 018500063419 Installation Traverse earth wire Strain clampsmdashpre-twisted type 2 16665 033500064132 Installation Iron fittings Guy wire fittingsmdashguy clips 3 146533 051500065644 Installation Mechanical hardware Aerial insulation traverse 2 319 048500065658 Installation Mechanical hardware Aerial insulation traverse 2 3122 044500066495 Installation Fittings Iron connection 2 282708 031500067251 Hardware Iron fittings Bolt 2 35250 019500067292 Hardware Iron fittings Bolt 2 46955 043500067551 Installation Iron fittings Guy wire fittingsmdashguy clips 2 58101 027
8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
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60
Q
20
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80
Q
20
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100
Q
0
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Q
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Q
0
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300
Q
0
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600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
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Q
0
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Q
0
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2000Q
0
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Q
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Q
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Q
0
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8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
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Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
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0011
69
1316
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2325
2729
4454
6063
6565
6361
570
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712
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814
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543542
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250281
310338
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0061
916
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4753
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64388861
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1120
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483533
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14131769
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27042893
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0162
1120
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4553
6168
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492543
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21572479
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1120
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6169
7684
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6041068
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1121
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6270
7785
155221
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399455
509562
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22612604
29303241
35403828
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286346
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1221
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290351
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6321125
15651971
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0212
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921314244
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92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
5059
6877
8593
172246
317384
450514
577639
6991259
17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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4 Mathematical Problems in Engineering
cost equals ℎ1198761198750 When there are 119898 demand arrivals witha total size of119883119898 during the lead time the inventory holdingcost constitutes the total cost when Q minus 119883119898 gt 0 while thebackordering cost constitutes the total cost whenQminus119883119898 lt 0Thus the expected total cost is given as follows
E [119862 (119876)] = ℎ1198761198750+ +infinsum119898=1
[ℎint1198760
(119876 minus 119883119898) 1198751198981198890(119898)119883119898
(119883119898)
+ 119887int+infin119876
(119883119898 minus 119876)1198751198981198890(119898)119883119898
(119883119898)](1)
Note that the probability distribution of 119883119898demand size0(119898)119883119898
is the m-fold iterated convolution of 0119883 Besides theprobability of 119898 demand arrivals within the lead time 119875119898 informula (1) can be derived based on the queuing and renewaltheories (Saidane et al [26])
119875119898 = 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871 (119898 ge 0) (2)
The objective of this paper is to derive the optimalorder quantity 119876lowast minimizing the expected total cost SinceE[119862(119876)] is a convex function it can be obtained by setting thederivative of the expected total cost with respect to 119876 equalto zero
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (3)
Proposition 1 The optimal order quantity 119876lowast is the solutionof
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(4)
Proof The optimal order quantity 119876lowastis the solution of thederivative of the expected total cost with respect to Q whichequals zero
dE [119862 (119876)]dQ
= 0 (5)
which is equivalent to
dE [119862 (119876)]dQ
= ℎ1198750 + +infinsum119898=1
[(ℎ + 119887) 0(119898)119883119898
(119876) minus 119887] 119875119898 = 0 (6)
Thus+infinsum119898=1
0(119898)119883119898
(119876) 119875119898 = 119887ℎ + 119887+infinsum119898=1
119875119898 minus ℎℎ + 1198871198750 (7)
Considering that sum+infin119898=1 119875119898 = 1 minus 1198750 the optimal orderquantity 119876lowastcan be derived from
+infinsum119898=1
0(119898)119883119898
(119876lowast) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(8)
Since formula (8) is the infinite sum this paper uses anupper bound 119876119880 and a lower bound 119876119871 to approximate theoptimal order quantity 119876lowast In order to derive 119876119880 and 119876119871 wedenote that
119866 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ (1 minus 119899sum
119898=0
119875119898)
119867 (119883119898 119899) = 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
(9)
Property 2 119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899) for all 119883119898 119899 ge0Proof
119874 (119883119898) = +infinsum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750+ 119899+infinsum119898=119899
0(119898)119883119898
(119876) 119875119898= 119867(119883119898 119899) + +infinsum
119898=119899
0(119898)119883119898
(119876) 119875119898
(10)
Since 0 le 0(119898)119883119898
(119876) le 1 for any 119883119898 119898 ge 00 le +infinsum119898=119899
0(119898)119883119898
(119876) 119875119898 le +infinsum119898=119899
119875119898 for all 119883119898 119899 ge 0 (11)
Thus119867(119883119898 119899) le 119874(119883119898 119899)119874 (119883119898) = +infinsum
119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750= 119899sum119898=1
0(119898)119883119898
(119876) 119875119898 minus 119887ℎ + 119887 + 1198750
Mathematical Problems in Engineering 5
+ (1 minus 119899sum119898=0
119875119898) + 119899+infinsum119898=119899+1
0(119898)119883119898
(119876) 119875119898minus (1 minus 119899sum
119898=0
119875119898)
= 119866 (119883119898 119899) minus +infinsum119898=119899+1
[1 minus 0(119898)119883119898
(119876)] 119875119898(12)
Since 0 le 1 minus 0(119898)119883119898
(119876) le 1sum+infin119898=119899+1[1 minus 0(119898)119883119898
(119876)]119875119898 ge 0Thus 119874(119883119898) le 119866(119883119898 119899)To sum up119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899)
Property 3 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are strictlyincreasing functions of 119883119898Proof Since 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are sums ofvaried cumulative probability distribution functions whichare strictly increasing these three functions are strictlyincreasing
Property 4
lim119899997888rarr+infin
119866 (119883119898 119899) 997888rarr 119874 (119883119898) lim119899997888rarr+infin
119867(119883119898 119899) 997888rarr 119874 (119883119898) (13)
Proof When n tends to infinity sum+infin119898=119899 0(119898)119883119898
(119876)119875119898 andsum+infin119898=119899+1[1 minus 0(119898)119883
119898
(119876)]119875119898 tend to zeroAssume that 119876119871 119876119880 and 119876lowast are the solutions of119866(119883119898 119899) = 0 119867(119883119898 119899) = 0 and 119874(119883119898) = 0 respec-
tively Based on the above three properties the followingProposition 5 is correct when n tends to infinity
Proposition 5 119876lowastcan be approximated by an upper bound119876119880 and a lower bound 119876119871 based on 119876119871 le 119876lowast le 119876119880 where119876119880 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(14)
and 119876119871 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871
minus (1 minus 119899sum119898=0
(119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871))
(15)
For a given arbitrarily small value
Substitute n into formula (15)
No
Yes
( gt 0) initialize n =1
1 minusnsum
m=0
Pm = 1 minusnsum
m=0
(kminus1sumi=0
(kL)(km+i)
(km + i)eminuskL)
n = n + 1
Obtain Qlowast
1 minus sumnm=0 Pm le
Figure 1 The flow chart of the basic algorithm
33 Basic Algorithm Based the above analysis the algorithmto compute the optimal order quantity 119876lowast is comprised thefollowing steps
Step 0 For a given arbitrarily small value 120575 (120575 gt 0) initialize119899 = 1Step 1 Compute 1 minus sum119899119898=0 119875119898 = 1 minus sum119899119898=0(sum119896minus1119894=0 ((119896120582119871)(119896119898+119894)(119896119898 + 119894))119890minus119896120582119871)Step 2
(i) If 1 minus sum119899119898=0 119875119898 le 120575 go to Step 3
(ii) Otherwise do 119899 = 119899 + 1 and go to Step 1
Step 3 Utilize the derived n to solve (15) so as to obtain 119876lowastStep 4 Stop
In order to help understand the basic algorithm all aboveprocedures are illustrated in the flow chart ie Figure 1
In this flow chart the optimal order quantity 119876lowast isapproximated by an upper bound 119876119880 and a lower bound 119876119871both of which are calculated with finite varying 119899 As 119899 getslarger the lower and upper bounds get tighter and the gapbetween them gets smaller The algorithm cannot be stoppeduntil the gap is smaller than the given stopping criterion120575 In this circumstance 119876119871is obtained by substituting thecorresponding 119899 into formula (15) which is considered as analternative to the optimal order quantity 119876lowast In addition itshould be noted that 119876lowastcan be approximately as closely asdesired by setting varied stopping criterion 120575
6 Mathematical Problems in Engineering
500000000500010000500020000500030000500040000500050000500060000500070000500080000
Stat
e Gri
d C
ode
010 020 030 040 050 060000Mean of demand arrival rate
Figure 2The scatter diagram of demand interarrival rate of electricpower materials
4 Numerical Investigation
41 Data Processing There are 3839 types of electric powermaterials in the StateGrid Shanghai Electric PowerCompanyBecause most of the 3839 electrical power materials arerelatively less important (eg circuits) we do not needto pay much attention to the inventory management forthese electrical power materials with the model proposedin our paper Moreover for those electrical power materialswhich are also suitable for our model some of them arenot standardized and therefore eliminated (or are beingeliminated) For standardized electrical power materials dueto similarity only representative electrical power materialsare selectedThereby we gather the historical data of demandsof 53 types of electric power materials from 2015 to 2017the demand arrivals of which follow an Erlang distributionin conjunction with a Gamma distribution of demand sizesSince the Gamma distribution covers a large spectrum ofprobability distributions suchmodelling ismore appropriateBasic information of the 53 types of electric power materialsis presented in Table 1
As shown in Table 1 the mean nonzero demand sizes 120573of the 53 types of electric power materials vary sharply forexample the minimum mean nonzero demand size120573 is 319while the maximummean nonzero demand size 120573 is 1043716Moreover the variance of the mean nonzero demand size120573 is 3041691 In a word there exists a high volatility ofmean nonzero demand size And demand arrivals of the53 types of electric power materials mostly follow a two-stage Erlang distribution There are only four electric powermaterials consisting of 500014661500017325 500020638 and500064132 whose demand interarrivals follow three stageErlang distribution
Figure 2 shows the scatter diagram of demand arrival rateof the 53 types of electric power materials
As shown in Figure 2 the mean demand interarrival rate120582 of these 53 types of electric power materials within 3 yearsis between 001 and 052 denoted by 120582 isin [001 052] Thevariance of the mean demand interarrival rate 120582 is 0019reflecting its low volatility According to the classificationmethod proposed by Eaves [31] these electric power mate-rials exhibit a high level of intermittence
Subsequently we adopt the above model to model theinventory cost of these materials compute the optimal order
State grid code of spare parts
New modelNewsvendor model
minus100000100200300400500600700
Logc
Figure 3 Average inventory costs calculated with the two models
quantity of every electric power material and the corre-sponding average inventory cost and then compare themwith results of the classic Newsvendor model (Trabka andMarchand [32]) to examine the validity of this method
42 Comparison with the Newsvendor Model The aim of thissubsection is to examine the performance of the proposedtheoretical model in cost reduction We initially employedthis new model to obtain the optimal order quantity ofeach electric power material and the corresponding averageinventory cost and followed this with a comparison to theclassic Newsvendor model under a Gamma demand processConsidering that the backordering cost is higher than theinventory holding cost this paper assumes that ℎ = 1 119887 = 10Lead-time L is assumed to be 1 Results are as in Table 2
Table 2 illustrates the performances of the two modelswith respect to the optimal order quantity and inventorycost Results show that the proposed theoretical model ienew model suggests a higher optimal order quantity andlower average inventory cost compared with the Newsvendormodel Take the electric power material 500065644 forexample the optimal order quantity 119876 obtained with the newmodel is 32 which is approximately 45 times of that obtainedwith the Newsvendor model which equals 7 However interms of average inventory cost 119862 the new model suggests acost of 3063 which is significantly less than that suggested bythe Newsvendor model which equals 2162199 According toTable 2 other electric power materials show similar patternas the above material 500065644
In order to more vividly demonstrate the significantdifference of average inventory cost 119862 between these twomodels we make the line chart ie Figure 3
Since that there is huge gap between the average inventorycost 119862 calculated with the above two models we use thenatural logarithm of average inventory cost 119862 as the ordinateso as to narrow the gap As shown in the Figure 3 the averageinventory cost of new model is significantly less than thatof Newsvendor model which is consistent with the findingrevealed by the Table 2 In a word larger optimal orderquantity and lower average inventory cost are advised by thenewmodelThe conflicting change in these two variables canbe explained by the fact that although higher order quantityincreases the amount of electric powermaterials kept in stockand thus holding cost the backordering cost decreases due
Mathematical Problems in Engineering 7
Table 1 Basic information of 53 types of electric power materials
State grid code of spare parts Class Division Section k 120573 120582
500012723 Hardware Mechanical hardware Bolt 2 213777 047500012771 Hardware Mechanical hardware Bolt 2 84070 039500012797 Hardware Mechanical hardware Bolt 2 35231 006500012801 Hardware Mechanical hardware Bolt 2 11876 013500014661 Installation Wire Aerial insulation wire 3 947 051500014662 Installation Wire Aerial insulation wire 2 1437 023500014670 Installation Wire Aerial insulation wire 2 3759 046500014711 Installation Wire Aerial insulation wire 2 518 041500014713 Installation Wire Aerial insulation wire 2 453 035500014805 Installation Cable Cloth wire 2 1043716 005500016589 Installation Cable Control cable 2 116167 020500016592 Installation Cable Control cable 2 108033 016500017293 Installation Insulator Post porcelain insulator 2 25476 033500017320 Installation Insulator Strain insulator 2 27917 019500017324 Installation Fitting Shackle insulator 2 647462 042500017325 Installation Fitting Shackle insulator 3 184946 052500020609 Installation Cable accessories Splicing fittingmdashGT-G 2 2705 026500020612 Installation Cable accessories Splicing fittingmdashGT-G 2 309122 007500020638 Installation Cable accessories Splicing fittingmdashGTL 3 2204 001500021412 Installation Cable accessories 35kV cable connector 2 664 019500021803 Installation Cable accessories Cable terminals 2 7997 024500021830 Installation Low voltage apparatus Cable terminals 2 28081 033500021862 Installation Low voltage apparatus Cable terminals 2 136348 030500021869 Installation Surge arrester Cable terminals 2 30576 033500022269 Low voltage apparatus Traverse earth wire Low-tension fuse 2 21886 019500022271 Low voltage apparatus Fitting Low-tension fuse 2 229308 018500027151 Primary equipment High-pressure fuse fittings AC surge arrester 2 21094 043500027464 Installation High-tension fuse accessories Aerial insulation traverse 2 505 026500029690 Installation High-tension fuse accessories Guy wire fittingmdashwedge clamps 2 28030 047500032681 Spare part High-tension fuse accessories Fuse 2 8564 031500032682 Spare part Iron fittings Fuse 2 2735 003500032684 Spare part Mechanical hardware Fuse 2 15188 047500032687 Spare part Cable accessories Fuse 2 6044 036500035137 Installation Mechanical hardware Anchor rod 2 25067 044500051829 Hardware Low voltage apparatus Bolt 2 34951 032500053694 Installation Low voltage apparatus 35kV cable termination 2 826 016500058470 Hardware Security burglary fire Bolt 2 154798 030500061526 Low voltage apparatus Security burglary fire Low-tension fuse accessories 2 13157 022500061564 Low voltage apparatus Fitting Low-tension fuse accessories 2 11372 031500061758 Supplementary equipment Fitting Anti-lighting device 2 21126 025500062282 Supplementary equipment Traverse earth wire Anti-lighting device 2 16365 018500063419 Installation Traverse earth wire Strain clampsmdashpre-twisted type 2 16665 033500064132 Installation Iron fittings Guy wire fittingsmdashguy clips 3 146533 051500065644 Installation Mechanical hardware Aerial insulation traverse 2 319 048500065658 Installation Mechanical hardware Aerial insulation traverse 2 3122 044500066495 Installation Fittings Iron connection 2 282708 031500067251 Hardware Iron fittings Bolt 2 35250 019500067292 Hardware Iron fittings Bolt 2 46955 043500067551 Installation Iron fittings Guy wire fittingsmdashguy clips 2 58101 027
8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
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Q
10
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05 1 150Lamda
5
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05 1 150Lamda
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0
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Q
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Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
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800
Q
0
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Q
0
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0
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2000Q
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Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
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40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
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857412997
032
1121
3039
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6471
7987
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0312
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0342
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6675
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0372
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6775
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0382
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5059
6776
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0392
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6876
8593
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573634
6941248
17502218
26613084
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1057916976
042
1222
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6877
8593
172246
317384
450514
577639
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26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 5
+ (1 minus 119899sum119898=0
119875119898) + 119899+infinsum119898=119899+1
0(119898)119883119898
(119876) 119875119898minus (1 minus 119899sum
119898=0
119875119898)
= 119866 (119883119898 119899) minus +infinsum119898=119899+1
[1 minus 0(119898)119883119898
(119876)] 119875119898(12)
Since 0 le 1 minus 0(119898)119883119898
(119876) le 1sum+infin119898=119899+1[1 minus 0(119898)119883119898
(119876)]119875119898 ge 0Thus 119874(119883119898) le 119866(119883119898 119899)To sum up119867(119883119898 119899) le 119874(119883119898) le 119866(119883119898 119899)
Property 3 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are strictlyincreasing functions of 119883119898Proof Since 119867(119883119898 119899) 119874(119883119898) and 119866(119883119898 119899) are sums ofvaried cumulative probability distribution functions whichare strictly increasing these three functions are strictlyincreasing
Property 4
lim119899997888rarr+infin
119866 (119883119898 119899) 997888rarr 119874 (119883119898) lim119899997888rarr+infin
119867(119883119898 119899) 997888rarr 119874 (119883119898) (13)
Proof When n tends to infinity sum+infin119898=119899 0(119898)119883119898
(119876)119875119898 andsum+infin119898=119899+1[1 minus 0(119898)119883
119898
(119876)]119875119898 tend to zeroAssume that 119876119871 119876119880 and 119876lowast are the solutions of119866(119883119898 119899) = 0 119867(119883119898 119899) = 0 and 119874(119883119898) = 0 respec-
tively Based on the above three properties the followingProposition 5 is correct when n tends to infinity
Proposition 5 119876lowastcan be approximated by an upper bound119876119880 and a lower bound 119876119871 based on 119876119871 le 119876lowast le 119876119880 where119876119880 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871(14)
and 119876119871 is the solution of
119899sum119898=1
0(119898)119883119898
(119876) 119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871
= 119887ℎ + 119887 minus119896minus1sum119894=0
(119896120582119871)(119894)119894 119890minus119896120582119871
minus (1 minus 119899sum119898=0
(119896minus1sum119894=0
(119896120582119871)(119896119898+119894)(119896119898 + 119894) 119890minus119896120582119871))
(15)
For a given arbitrarily small value
Substitute n into formula (15)
No
Yes
( gt 0) initialize n =1
1 minusnsum
m=0
Pm = 1 minusnsum
m=0
(kminus1sumi=0
(kL)(km+i)
(km + i)eminuskL)
n = n + 1
Obtain Qlowast
1 minus sumnm=0 Pm le
Figure 1 The flow chart of the basic algorithm
33 Basic Algorithm Based the above analysis the algorithmto compute the optimal order quantity 119876lowast is comprised thefollowing steps
Step 0 For a given arbitrarily small value 120575 (120575 gt 0) initialize119899 = 1Step 1 Compute 1 minus sum119899119898=0 119875119898 = 1 minus sum119899119898=0(sum119896minus1119894=0 ((119896120582119871)(119896119898+119894)(119896119898 + 119894))119890minus119896120582119871)Step 2
(i) If 1 minus sum119899119898=0 119875119898 le 120575 go to Step 3
(ii) Otherwise do 119899 = 119899 + 1 and go to Step 1
Step 3 Utilize the derived n to solve (15) so as to obtain 119876lowastStep 4 Stop
In order to help understand the basic algorithm all aboveprocedures are illustrated in the flow chart ie Figure 1
In this flow chart the optimal order quantity 119876lowast isapproximated by an upper bound 119876119880 and a lower bound 119876119871both of which are calculated with finite varying 119899 As 119899 getslarger the lower and upper bounds get tighter and the gapbetween them gets smaller The algorithm cannot be stoppeduntil the gap is smaller than the given stopping criterion120575 In this circumstance 119876119871is obtained by substituting thecorresponding 119899 into formula (15) which is considered as analternative to the optimal order quantity 119876lowast In addition itshould be noted that 119876lowastcan be approximately as closely asdesired by setting varied stopping criterion 120575
6 Mathematical Problems in Engineering
500000000500010000500020000500030000500040000500050000500060000500070000500080000
Stat
e Gri
d C
ode
010 020 030 040 050 060000Mean of demand arrival rate
Figure 2The scatter diagram of demand interarrival rate of electricpower materials
4 Numerical Investigation
41 Data Processing There are 3839 types of electric powermaterials in the StateGrid Shanghai Electric PowerCompanyBecause most of the 3839 electrical power materials arerelatively less important (eg circuits) we do not needto pay much attention to the inventory management forthese electrical power materials with the model proposedin our paper Moreover for those electrical power materialswhich are also suitable for our model some of them arenot standardized and therefore eliminated (or are beingeliminated) For standardized electrical power materials dueto similarity only representative electrical power materialsare selectedThereby we gather the historical data of demandsof 53 types of electric power materials from 2015 to 2017the demand arrivals of which follow an Erlang distributionin conjunction with a Gamma distribution of demand sizesSince the Gamma distribution covers a large spectrum ofprobability distributions suchmodelling ismore appropriateBasic information of the 53 types of electric power materialsis presented in Table 1
As shown in Table 1 the mean nonzero demand sizes 120573of the 53 types of electric power materials vary sharply forexample the minimum mean nonzero demand size120573 is 319while the maximummean nonzero demand size 120573 is 1043716Moreover the variance of the mean nonzero demand size120573 is 3041691 In a word there exists a high volatility ofmean nonzero demand size And demand arrivals of the53 types of electric power materials mostly follow a two-stage Erlang distribution There are only four electric powermaterials consisting of 500014661500017325 500020638 and500064132 whose demand interarrivals follow three stageErlang distribution
Figure 2 shows the scatter diagram of demand arrival rateof the 53 types of electric power materials
As shown in Figure 2 the mean demand interarrival rate120582 of these 53 types of electric power materials within 3 yearsis between 001 and 052 denoted by 120582 isin [001 052] Thevariance of the mean demand interarrival rate 120582 is 0019reflecting its low volatility According to the classificationmethod proposed by Eaves [31] these electric power mate-rials exhibit a high level of intermittence
Subsequently we adopt the above model to model theinventory cost of these materials compute the optimal order
State grid code of spare parts
New modelNewsvendor model
minus100000100200300400500600700
Logc
Figure 3 Average inventory costs calculated with the two models
quantity of every electric power material and the corre-sponding average inventory cost and then compare themwith results of the classic Newsvendor model (Trabka andMarchand [32]) to examine the validity of this method
42 Comparison with the Newsvendor Model The aim of thissubsection is to examine the performance of the proposedtheoretical model in cost reduction We initially employedthis new model to obtain the optimal order quantity ofeach electric power material and the corresponding averageinventory cost and followed this with a comparison to theclassic Newsvendor model under a Gamma demand processConsidering that the backordering cost is higher than theinventory holding cost this paper assumes that ℎ = 1 119887 = 10Lead-time L is assumed to be 1 Results are as in Table 2
Table 2 illustrates the performances of the two modelswith respect to the optimal order quantity and inventorycost Results show that the proposed theoretical model ienew model suggests a higher optimal order quantity andlower average inventory cost compared with the Newsvendormodel Take the electric power material 500065644 forexample the optimal order quantity 119876 obtained with the newmodel is 32 which is approximately 45 times of that obtainedwith the Newsvendor model which equals 7 However interms of average inventory cost 119862 the new model suggests acost of 3063 which is significantly less than that suggested bythe Newsvendor model which equals 2162199 According toTable 2 other electric power materials show similar patternas the above material 500065644
In order to more vividly demonstrate the significantdifference of average inventory cost 119862 between these twomodels we make the line chart ie Figure 3
Since that there is huge gap between the average inventorycost 119862 calculated with the above two models we use thenatural logarithm of average inventory cost 119862 as the ordinateso as to narrow the gap As shown in the Figure 3 the averageinventory cost of new model is significantly less than thatof Newsvendor model which is consistent with the findingrevealed by the Table 2 In a word larger optimal orderquantity and lower average inventory cost are advised by thenewmodelThe conflicting change in these two variables canbe explained by the fact that although higher order quantityincreases the amount of electric powermaterials kept in stockand thus holding cost the backordering cost decreases due
Mathematical Problems in Engineering 7
Table 1 Basic information of 53 types of electric power materials
State grid code of spare parts Class Division Section k 120573 120582
500012723 Hardware Mechanical hardware Bolt 2 213777 047500012771 Hardware Mechanical hardware Bolt 2 84070 039500012797 Hardware Mechanical hardware Bolt 2 35231 006500012801 Hardware Mechanical hardware Bolt 2 11876 013500014661 Installation Wire Aerial insulation wire 3 947 051500014662 Installation Wire Aerial insulation wire 2 1437 023500014670 Installation Wire Aerial insulation wire 2 3759 046500014711 Installation Wire Aerial insulation wire 2 518 041500014713 Installation Wire Aerial insulation wire 2 453 035500014805 Installation Cable Cloth wire 2 1043716 005500016589 Installation Cable Control cable 2 116167 020500016592 Installation Cable Control cable 2 108033 016500017293 Installation Insulator Post porcelain insulator 2 25476 033500017320 Installation Insulator Strain insulator 2 27917 019500017324 Installation Fitting Shackle insulator 2 647462 042500017325 Installation Fitting Shackle insulator 3 184946 052500020609 Installation Cable accessories Splicing fittingmdashGT-G 2 2705 026500020612 Installation Cable accessories Splicing fittingmdashGT-G 2 309122 007500020638 Installation Cable accessories Splicing fittingmdashGTL 3 2204 001500021412 Installation Cable accessories 35kV cable connector 2 664 019500021803 Installation Cable accessories Cable terminals 2 7997 024500021830 Installation Low voltage apparatus Cable terminals 2 28081 033500021862 Installation Low voltage apparatus Cable terminals 2 136348 030500021869 Installation Surge arrester Cable terminals 2 30576 033500022269 Low voltage apparatus Traverse earth wire Low-tension fuse 2 21886 019500022271 Low voltage apparatus Fitting Low-tension fuse 2 229308 018500027151 Primary equipment High-pressure fuse fittings AC surge arrester 2 21094 043500027464 Installation High-tension fuse accessories Aerial insulation traverse 2 505 026500029690 Installation High-tension fuse accessories Guy wire fittingmdashwedge clamps 2 28030 047500032681 Spare part High-tension fuse accessories Fuse 2 8564 031500032682 Spare part Iron fittings Fuse 2 2735 003500032684 Spare part Mechanical hardware Fuse 2 15188 047500032687 Spare part Cable accessories Fuse 2 6044 036500035137 Installation Mechanical hardware Anchor rod 2 25067 044500051829 Hardware Low voltage apparatus Bolt 2 34951 032500053694 Installation Low voltage apparatus 35kV cable termination 2 826 016500058470 Hardware Security burglary fire Bolt 2 154798 030500061526 Low voltage apparatus Security burglary fire Low-tension fuse accessories 2 13157 022500061564 Low voltage apparatus Fitting Low-tension fuse accessories 2 11372 031500061758 Supplementary equipment Fitting Anti-lighting device 2 21126 025500062282 Supplementary equipment Traverse earth wire Anti-lighting device 2 16365 018500063419 Installation Traverse earth wire Strain clampsmdashpre-twisted type 2 16665 033500064132 Installation Iron fittings Guy wire fittingsmdashguy clips 3 146533 051500065644 Installation Mechanical hardware Aerial insulation traverse 2 319 048500065658 Installation Mechanical hardware Aerial insulation traverse 2 3122 044500066495 Installation Fittings Iron connection 2 282708 031500067251 Hardware Iron fittings Bolt 2 35250 019500067292 Hardware Iron fittings Bolt 2 46955 043500067551 Installation Iron fittings Guy wire fittingsmdashguy clips 2 58101 027
8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
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Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
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5461
6674
133187
238286
332377
420462
502866
11771454
17051937
21522352
25402717
48455509
0122
1119
2735
4350
5764
7178
141199
253305
355404
451496
541943
12921608
18992168
24212660
30013101
59997817
0132
1119
2836
4351
5865
7279
144203
259313
364414
462510
556972
13361666
19712255
25232777
30183247
64388861
0142
1120
2836
4351
5966
7380
146207
265319
372423
473522
569999
13761720
20382336
26172884
31383382
68419500
0152
1120
2837
4552
6067
7482
149211
270326
380432
483533
5811024
14131769
21002410
27042893
32503506
721310244
0162
1120
2937
4553
6168
7583
151214
274331
387440
492543
5931047
14481815
21572479
27853075
33543621
755811577
0172
1120
2937
4654
6169
7684
153218
278337
393448
501553
6041068
14801858
22112544
28603161
34503728
787912178
0182
1121
2938
4654
6270
7785
155221
282342
399455
509562
6141088
15101898
22612604
29303241
35403828
818012742
0192
1121
3038
4755
6371
7886
157223
286346
405461
516570
6231107
15381936
23082660
29853316
36253922
846212742
022
1221
3039
4755
6371
7987
159226
290351
410467
523578
6321125
15651971
23522713
30573387
37044010
872713272
0212
1221
3039
4756
6472
8087
161229
293355
415473
530586
6401141
15902005
23942763
31163454
37794094
897713772
0222
1221
3039
4856
6473
8088
162231
296359
420479
536593
6481157
16132036
24332810
31713517
38504172
921314244
0232
1221
3140
4857
6573
8189
164233
299363
424484
542599
6551172
16362066
24712855
32233576
39174247
943614691
0242
1222
3140
4957
6574
8290
165235
302366
429489
548606
6631186
16572094
25062897
32723633
39814317
964815115
0252
1222
3140
4958
6674
8290
166237
305370
432494
553612
6691199
16772121
25392938
33193686
40414385
984915517
0262
1222
3140
4958
6675
8391
168239
307373
436498
558618
6761212
16962146
25712976
33643738
40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
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624692
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19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
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910
2030
4050
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13
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7888
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373456
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21942810
34023973
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1507326360
114
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376459
539539
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45715118
56526175
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124
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203291
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698761
8491558
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56696199
1538726624
134
1425
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5868
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203292
378460
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34504024
45915143
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1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
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Beta=4
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5
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Q5
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Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
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Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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6 Mathematical Problems in Engineering
500000000500010000500020000500030000500040000500050000500060000500070000500080000
Stat
e Gri
d C
ode
010 020 030 040 050 060000Mean of demand arrival rate
Figure 2The scatter diagram of demand interarrival rate of electricpower materials
4 Numerical Investigation
41 Data Processing There are 3839 types of electric powermaterials in the StateGrid Shanghai Electric PowerCompanyBecause most of the 3839 electrical power materials arerelatively less important (eg circuits) we do not needto pay much attention to the inventory management forthese electrical power materials with the model proposedin our paper Moreover for those electrical power materialswhich are also suitable for our model some of them arenot standardized and therefore eliminated (or are beingeliminated) For standardized electrical power materials dueto similarity only representative electrical power materialsare selectedThereby we gather the historical data of demandsof 53 types of electric power materials from 2015 to 2017the demand arrivals of which follow an Erlang distributionin conjunction with a Gamma distribution of demand sizesSince the Gamma distribution covers a large spectrum ofprobability distributions suchmodelling ismore appropriateBasic information of the 53 types of electric power materialsis presented in Table 1
As shown in Table 1 the mean nonzero demand sizes 120573of the 53 types of electric power materials vary sharply forexample the minimum mean nonzero demand size120573 is 319while the maximummean nonzero demand size 120573 is 1043716Moreover the variance of the mean nonzero demand size120573 is 3041691 In a word there exists a high volatility ofmean nonzero demand size And demand arrivals of the53 types of electric power materials mostly follow a two-stage Erlang distribution There are only four electric powermaterials consisting of 500014661500017325 500020638 and500064132 whose demand interarrivals follow three stageErlang distribution
Figure 2 shows the scatter diagram of demand arrival rateof the 53 types of electric power materials
As shown in Figure 2 the mean demand interarrival rate120582 of these 53 types of electric power materials within 3 yearsis between 001 and 052 denoted by 120582 isin [001 052] Thevariance of the mean demand interarrival rate 120582 is 0019reflecting its low volatility According to the classificationmethod proposed by Eaves [31] these electric power mate-rials exhibit a high level of intermittence
Subsequently we adopt the above model to model theinventory cost of these materials compute the optimal order
State grid code of spare parts
New modelNewsvendor model
minus100000100200300400500600700
Logc
Figure 3 Average inventory costs calculated with the two models
quantity of every electric power material and the corre-sponding average inventory cost and then compare themwith results of the classic Newsvendor model (Trabka andMarchand [32]) to examine the validity of this method
42 Comparison with the Newsvendor Model The aim of thissubsection is to examine the performance of the proposedtheoretical model in cost reduction We initially employedthis new model to obtain the optimal order quantity ofeach electric power material and the corresponding averageinventory cost and followed this with a comparison to theclassic Newsvendor model under a Gamma demand processConsidering that the backordering cost is higher than theinventory holding cost this paper assumes that ℎ = 1 119887 = 10Lead-time L is assumed to be 1 Results are as in Table 2
Table 2 illustrates the performances of the two modelswith respect to the optimal order quantity and inventorycost Results show that the proposed theoretical model ienew model suggests a higher optimal order quantity andlower average inventory cost compared with the Newsvendormodel Take the electric power material 500065644 forexample the optimal order quantity 119876 obtained with the newmodel is 32 which is approximately 45 times of that obtainedwith the Newsvendor model which equals 7 However interms of average inventory cost 119862 the new model suggests acost of 3063 which is significantly less than that suggested bythe Newsvendor model which equals 2162199 According toTable 2 other electric power materials show similar patternas the above material 500065644
In order to more vividly demonstrate the significantdifference of average inventory cost 119862 between these twomodels we make the line chart ie Figure 3
Since that there is huge gap between the average inventorycost 119862 calculated with the above two models we use thenatural logarithm of average inventory cost 119862 as the ordinateso as to narrow the gap As shown in the Figure 3 the averageinventory cost of new model is significantly less than thatof Newsvendor model which is consistent with the findingrevealed by the Table 2 In a word larger optimal orderquantity and lower average inventory cost are advised by thenewmodelThe conflicting change in these two variables canbe explained by the fact that although higher order quantityincreases the amount of electric powermaterials kept in stockand thus holding cost the backordering cost decreases due
Mathematical Problems in Engineering 7
Table 1 Basic information of 53 types of electric power materials
State grid code of spare parts Class Division Section k 120573 120582
500012723 Hardware Mechanical hardware Bolt 2 213777 047500012771 Hardware Mechanical hardware Bolt 2 84070 039500012797 Hardware Mechanical hardware Bolt 2 35231 006500012801 Hardware Mechanical hardware Bolt 2 11876 013500014661 Installation Wire Aerial insulation wire 3 947 051500014662 Installation Wire Aerial insulation wire 2 1437 023500014670 Installation Wire Aerial insulation wire 2 3759 046500014711 Installation Wire Aerial insulation wire 2 518 041500014713 Installation Wire Aerial insulation wire 2 453 035500014805 Installation Cable Cloth wire 2 1043716 005500016589 Installation Cable Control cable 2 116167 020500016592 Installation Cable Control cable 2 108033 016500017293 Installation Insulator Post porcelain insulator 2 25476 033500017320 Installation Insulator Strain insulator 2 27917 019500017324 Installation Fitting Shackle insulator 2 647462 042500017325 Installation Fitting Shackle insulator 3 184946 052500020609 Installation Cable accessories Splicing fittingmdashGT-G 2 2705 026500020612 Installation Cable accessories Splicing fittingmdashGT-G 2 309122 007500020638 Installation Cable accessories Splicing fittingmdashGTL 3 2204 001500021412 Installation Cable accessories 35kV cable connector 2 664 019500021803 Installation Cable accessories Cable terminals 2 7997 024500021830 Installation Low voltage apparatus Cable terminals 2 28081 033500021862 Installation Low voltage apparatus Cable terminals 2 136348 030500021869 Installation Surge arrester Cable terminals 2 30576 033500022269 Low voltage apparatus Traverse earth wire Low-tension fuse 2 21886 019500022271 Low voltage apparatus Fitting Low-tension fuse 2 229308 018500027151 Primary equipment High-pressure fuse fittings AC surge arrester 2 21094 043500027464 Installation High-tension fuse accessories Aerial insulation traverse 2 505 026500029690 Installation High-tension fuse accessories Guy wire fittingmdashwedge clamps 2 28030 047500032681 Spare part High-tension fuse accessories Fuse 2 8564 031500032682 Spare part Iron fittings Fuse 2 2735 003500032684 Spare part Mechanical hardware Fuse 2 15188 047500032687 Spare part Cable accessories Fuse 2 6044 036500035137 Installation Mechanical hardware Anchor rod 2 25067 044500051829 Hardware Low voltage apparatus Bolt 2 34951 032500053694 Installation Low voltage apparatus 35kV cable termination 2 826 016500058470 Hardware Security burglary fire Bolt 2 154798 030500061526 Low voltage apparatus Security burglary fire Low-tension fuse accessories 2 13157 022500061564 Low voltage apparatus Fitting Low-tension fuse accessories 2 11372 031500061758 Supplementary equipment Fitting Anti-lighting device 2 21126 025500062282 Supplementary equipment Traverse earth wire Anti-lighting device 2 16365 018500063419 Installation Traverse earth wire Strain clampsmdashpre-twisted type 2 16665 033500064132 Installation Iron fittings Guy wire fittingsmdashguy clips 3 146533 051500065644 Installation Mechanical hardware Aerial insulation traverse 2 319 048500065658 Installation Mechanical hardware Aerial insulation traverse 2 3122 044500066495 Installation Fittings Iron connection 2 282708 031500067251 Hardware Iron fittings Bolt 2 35250 019500067292 Hardware Iron fittings Bolt 2 46955 043500067551 Installation Iron fittings Guy wire fittingsmdashguy clips 2 58101 027
8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
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100
Q
0
50
100
150
Q
20
40
60
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Q
0
100
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300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
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6000
Q
0
2000
4000
6000
Q
0
2000
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6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
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4957
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8290
165235
302366
429489
548606
6631186
16572094
25062897
32723633
39814317
964815115
0252
1222
3140
4958
6674
8290
166237
305370
432494
553612
6691199
16772121
25392938
33193686
40414385
984915517
0262
1222
3140
4958
6675
8391
168239
307373
436498
558618
6761212
16962146
25712976
33643738
40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
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6101080
14711883
22422580
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806311971
0282
1121
3038
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6270
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156222
284344
402458
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22852633
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0292
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6371
7986
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288348
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23272683
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032
1121
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0312
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0342
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0352
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3140
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6675
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307373
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0372
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6775
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169241
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564624
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1023816294
0382
1222
3241
5059
6776
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170243
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444507
568629
6881237
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26333050
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1041116641
0392
1222
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6876
8593
171245
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573634
6941248
17502218
26613084
34893881
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1057916976
042
1222
3241
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6877
8593
172246
317384
450514
577639
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17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 7
Table 1 Basic information of 53 types of electric power materials
State grid code of spare parts Class Division Section k 120573 120582
500012723 Hardware Mechanical hardware Bolt 2 213777 047500012771 Hardware Mechanical hardware Bolt 2 84070 039500012797 Hardware Mechanical hardware Bolt 2 35231 006500012801 Hardware Mechanical hardware Bolt 2 11876 013500014661 Installation Wire Aerial insulation wire 3 947 051500014662 Installation Wire Aerial insulation wire 2 1437 023500014670 Installation Wire Aerial insulation wire 2 3759 046500014711 Installation Wire Aerial insulation wire 2 518 041500014713 Installation Wire Aerial insulation wire 2 453 035500014805 Installation Cable Cloth wire 2 1043716 005500016589 Installation Cable Control cable 2 116167 020500016592 Installation Cable Control cable 2 108033 016500017293 Installation Insulator Post porcelain insulator 2 25476 033500017320 Installation Insulator Strain insulator 2 27917 019500017324 Installation Fitting Shackle insulator 2 647462 042500017325 Installation Fitting Shackle insulator 3 184946 052500020609 Installation Cable accessories Splicing fittingmdashGT-G 2 2705 026500020612 Installation Cable accessories Splicing fittingmdashGT-G 2 309122 007500020638 Installation Cable accessories Splicing fittingmdashGTL 3 2204 001500021412 Installation Cable accessories 35kV cable connector 2 664 019500021803 Installation Cable accessories Cable terminals 2 7997 024500021830 Installation Low voltage apparatus Cable terminals 2 28081 033500021862 Installation Low voltage apparatus Cable terminals 2 136348 030500021869 Installation Surge arrester Cable terminals 2 30576 033500022269 Low voltage apparatus Traverse earth wire Low-tension fuse 2 21886 019500022271 Low voltage apparatus Fitting Low-tension fuse 2 229308 018500027151 Primary equipment High-pressure fuse fittings AC surge arrester 2 21094 043500027464 Installation High-tension fuse accessories Aerial insulation traverse 2 505 026500029690 Installation High-tension fuse accessories Guy wire fittingmdashwedge clamps 2 28030 047500032681 Spare part High-tension fuse accessories Fuse 2 8564 031500032682 Spare part Iron fittings Fuse 2 2735 003500032684 Spare part Mechanical hardware Fuse 2 15188 047500032687 Spare part Cable accessories Fuse 2 6044 036500035137 Installation Mechanical hardware Anchor rod 2 25067 044500051829 Hardware Low voltage apparatus Bolt 2 34951 032500053694 Installation Low voltage apparatus 35kV cable termination 2 826 016500058470 Hardware Security burglary fire Bolt 2 154798 030500061526 Low voltage apparatus Security burglary fire Low-tension fuse accessories 2 13157 022500061564 Low voltage apparatus Fitting Low-tension fuse accessories 2 11372 031500061758 Supplementary equipment Fitting Anti-lighting device 2 21126 025500062282 Supplementary equipment Traverse earth wire Anti-lighting device 2 16365 018500063419 Installation Traverse earth wire Strain clampsmdashpre-twisted type 2 16665 033500064132 Installation Iron fittings Guy wire fittingsmdashguy clips 3 146533 051500065644 Installation Mechanical hardware Aerial insulation traverse 2 319 048500065658 Installation Mechanical hardware Aerial insulation traverse 2 3122 044500066495 Installation Fittings Iron connection 2 282708 031500067251 Hardware Iron fittings Bolt 2 35250 019500067292 Hardware Iron fittings Bolt 2 46955 043500067551 Installation Iron fittings Guy wire fittingsmdashguy clips 2 58101 027
8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
100
Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
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5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
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5059
6776
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0392
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042
1222
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6877
8593
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0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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8 Mathematical Problems in Engineering
Table 1 Continued
State grid code of spare parts Class Division Section k 120573 120582
500067567 Installation Description of the class Cable hold hoop 2 5625 013500067652 Installation Mechanical hardware Semicircular hold hoop 2 9210 028500067654 Installation Mechanical hardware Semicircular hold hoop 2 20246 029500016521 Installation Mechanical hardware Control cable 2 57627 015
L=1
L=2
L=3
L=4
L=5
L=6
L=7
L=8
L=9
L=10
L=11
L=12
000100020003000400050006000700080009000
10000
Figure 4 The average inventory cost 119862 with varying lead time 119871for the electric power materials with compound Erlang-distributeddemand
to the less unfilled demand leading to a reduction in totalinventory cost Thus the new model outperforms the classicNewsboymodel in cost reductions which ismore suitable forcost-oriented inventory system
43 Sensitivity Analysis of Lead Time In this subsection wevary the value of lead time L in order to analyze its impactson the average inventory costCThuswe consider the variousvalues of lead time L in the interval [1 10] and computethe corresponding average inventory cost C respectively Theoutcomes are shown in Table 3 and Figure 4
As shown in the Table 3 an overwhelming percent ofelectric power materials with compound Erlang-distributeddemand demonstrates a nonliner relationship between leadtime 119871 and average inventory cost 119862 Specifically speaking aslead time 119871 increases the average inventory cost 119862 of mostelectric power material rises first and then falls For examplefor electric power material 500012723 the correspondingaverage inventory cost 119862 experiences an increase from 078to 188 as the lead time 119871 increases from 1 to 3 After reachingits peak of 188 the average inventory cost begins to decreaseand reaches 029 when lead time is 10
In order to more vividly depict the effect of lead time onthe average inventory cost wemake the line chart in Figure 4based on Table 3
Figure 4 illustrates that for most electric power materialwith compound Erlang-distributed demand average inven-tory cost initially rises in the lead time and subsequentlythis trend reverses However there are three types of electricpowermaterial such as 500012797 500020612 and 500032682which do not show the above relationship Instead theaverage inventory cost rises with the lead time Maybe it isbecause that the lead time is not large enough If lead time islarger the corresponding inventory cost maybe will provide
the same pattern as the other most electric power materialwith compound Erlang-distributed demand
5 The Approximation andHeuristic Algorithm
Given the complexity of the basic algorithm this paperproposes easily implementable approximation and heuristicalgorithms for determining the optimal order quantity underthe (119879119876) policy These two methods simplify complicatedmathematical calculation and solve the lot-sizing problemwhich are of great utility in the real industrial context
51 The Approximation Algorithm This paper proposes anapproximation algorithm to determine the optimal orderquantity by seeking the minimum iterative numbers withinthe desired accuracy under the varied average demandinterarrival rate 120582 and average nonzero demand size 120573 Thusthe calculation process is simplified due to reduced iterativenumbers Electric power materials vary with the iterativenumber corresponding to the optimal order quantity in thisproposed model If practitioners control material inventoryand determine the optimal order quantity based on the aboveiterative numbers the efficiency of inventory managementwill be rather low due to the wide range of electric powermaterials Therefore if the iterative number can be reducedwhile guaranteeing accuracy the simplification can signifi-cantly increase companiesrsquo efficiency
Calculation is as follows
Step 1 Adopt the basic algorithm to calculate the optimaliterative numbers of each electric power material with dif-ferent 120582 120573 119896When the value of 1 minus sum119899119898=0 119875119898 in formula (15)converges to zero the right side of formula (14) and (15)is almost equal The corresponding iterative number is theoptimal denoted by 119899lowast We assumed that the optimal iterativenumber 119899lowast is derived when 120575 equals 10minus4 or equivalently1 minus sum119899119898=0 119875119898 le 10minus4Step 2 Calculate the corresponding optimal order quantity119876lowastStep 3 Calculate the effective iterative number within theacceptable error range denoted by 119899 We assume that theacceptable error range is 10
The effective iterative number 119899 is derived as in Figure 5In this flow chart 119876lowast and 119899 are derived by comparing the
corresponding order quantity 119876119899with optimal order quantity119876lowast This algorithm can be stopped when the differencebetween 119876119899 and 119876lowast is small enough and conforms to the
Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
100
Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
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30006000
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1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
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4050
6070
8090
100200
300400
500600
700800
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20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
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6877
8593
172246
317384
450514
577639
6991259
17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
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30443543
40254493
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1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
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7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
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5564
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93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
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0643
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3545
5565
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93102
193273
352429
503503
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20322594
31313648
41494635
51085570
1342422706
0653
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5565
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31443663
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1357023001
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193275
355432
508508
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31673691
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1364023278
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7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 9
Table 2 The optimal order quantity and corresponding average inventory cost of the two methods
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500012723 7475 4697 078 2500012771 3669 1847 091 22500012797 935 774 132 154500012801 613 261 159 1490500014661 77 21 2403 244424500014662 116 32 523 106020500014670 283 83 391 15360500014711 50 11 1451 819737500014713 44 10 1465 1069736500016589 3987 2552 099 10500016592 3506 2374 102 12500017293 1392 560 122 307500017320 1332 613 122 253500017325 5282 4064 089 3500020609 203 59 363 29781500020612 2557 6792 108 1500021412 57 15 911 498517500021803 505 176 189 3338500021830 1507 617 119 250500021862 5069 2996 091 7500021869 1615 672 116 209500022269 1111 481 130 422500022271 6100 5038 096 1500027151 1219 463 135 455500027464 47 11 1133 863026500029690 1541 616 122 251500032681 559 188 191 2904500032682 110 60 402 29123500032684 930 334 164 899500032687 421 133 245 5889500035137 1408 551 126 317500051829 1794 768 112 157500053694 67 18 743 321886500058470 5567 3401 091 5500061526 753 289 152 1208500061564 707 250 167 1629500061758 1145 464 130 454500062282 862 360 141 771500063419 980 366 143 743500064132 4528 3220 094 5500065644 32 7 3063 2162199500065658 240 69 439 22318500066495 8558 6212 088 1500067251 1622 775 116 154500067292 2340 1032 101 83
10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
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Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
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5461
6674
133187
238286
332377
420462
502866
11771454
17051937
21522352
25402717
48455509
0122
1119
2735
4350
5764
7178
141199
253305
355404
451496
541943
12921608
18992168
24212660
30013101
59997817
0132
1119
2836
4351
5865
7279
144203
259313
364414
462510
556972
13361666
19712255
25232777
30183247
64388861
0142
1120
2836
4351
5966
7380
146207
265319
372423
473522
569999
13761720
20382336
26172884
31383382
68419500
0152
1120
2837
4552
6067
7482
149211
270326
380432
483533
5811024
14131769
21002410
27042893
32503506
721310244
0162
1120
2937
4553
6168
7583
151214
274331
387440
492543
5931047
14481815
21572479
27853075
33543621
755811577
0172
1120
2937
4654
6169
7684
153218
278337
393448
501553
6041068
14801858
22112544
28603161
34503728
787912178
0182
1121
2938
4654
6270
7785
155221
282342
399455
509562
6141088
15101898
22612604
29303241
35403828
818012742
0192
1121
3038
4755
6371
7886
157223
286346
405461
516570
6231107
15381936
23082660
29853316
36253922
846212742
022
1221
3039
4755
6371
7987
159226
290351
410467
523578
6321125
15651971
23522713
30573387
37044010
872713272
0212
1221
3039
4756
6472
8087
161229
293355
415473
530586
6401141
15902005
23942763
31163454
37794094
897713772
0222
1221
3039
4856
6473
8088
162231
296359
420479
536593
6481157
16132036
24332810
31713517
38504172
921314244
0232
1221
3140
4857
6573
8189
164233
299363
424484
542599
6551172
16362066
24712855
32233576
39174247
943614691
0242
1222
3140
4957
6574
8290
165235
302366
429489
548606
6631186
16572094
25062897
32723633
39814317
964815115
0252
1222
3140
4958
6674
8290
166237
305370
432494
553612
6691199
16772121
25392938
33193686
40414385
984915517
0262
1222
3140
4958
6675
8391
168239
307373
436498
558618
6761212
16962146
25712976
33643738
40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
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624692
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19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
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910
2030
4050
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13
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7888
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373456
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21942810
34023973
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1507326360
114
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376459
539539
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45715118
56526175
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124
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203291
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698761
8491558
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56696199
1538726624
134
1425
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5868
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203292
378460
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34504024
45915143
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1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
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Beta=4
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5
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Q5
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Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
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Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
Hindawiwwwhindawicom Volume 2018
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10 Mathematical Problems in Engineering
Table 2 Continued
State grid code ofmaterials
Optimal order quantity Q Average inventory cost CNew model Newsvendor model New model Newsvendor model
500067551 2601 1277 102 51500067567 329 124 218 6809500067652 585 202 181 2505500067654 1135 445 132 496
No
Yes
Substitute n into formula (15)
Input and to calculate Qlowast
n = 1
Obtain 1n
Obtain n
n = n + 11n isin [09Qlowast 11Qlowast]
Figure 5Theflow chart of calculation of 119899 under the approximationapproach
predefined criterion The effective iterative number 119899 is thusobtained
Based on the above approximation algorithm 119876lowast and 119899under varied 120582 120573 k are calculated and the results are shownin Appendices A and B (Figures 7 and 8 correspond to theresult of 119899 and Figures 9 and 10 correspond to the result of119876lowast) It reveals that there is an overwhelming percentage ofelectric power materials whose effective iterative numbers 119899equal 2 while the effective iterative number for the remainingelectric power materials is 1 This means for most electricpowermaterial the approximately optimal order quantity 119876lowastcan be derived after only two iterations under the acceptableerror range of 10 percent Given the positive relationshipsbetween the accuracy and the number of iterations theuniform iterative number for all electric power materials ofelectric power companieswith a compoundErlang-k demanddistribution is set to be 2Thus practitioners can calculate theorder quantity through two iterations for all electric powermaterial with compound Erlang-distributed demand
52 The Heuristic Algorithm Given the complexity of thebasic algorithm and the shortage of relevant experts in enter-prises this paper proposes a more implementable heuristic
algorithm This algorithm attempts to generalize the rela-tionship between average demand interarrival rate 120582 and theoptimal order quantity 119876lowast for varied 120573 and k In this paperwe utilize the scatter diagrams to illustrate their relationshipsin the cases of k is equal to 2 and 3
When k = 2 the relationships between the independentvariable 120582 and the dependent variable 119876lowast under varied 120573 areas in Figure 6
Because of space constraints the relationships between 120582and 119876lowast with respect to varied 120573 when k = 3 are shown in theFigure 11 which is put in Appendix C
As shown in Figures 6 and 11 the optimal quantity119876lowast also increases when the average demand arrival rate 120582increases As 120582 increases ie the intermittence get weakerthe optimal order quantity 119876lowast increases Such relationshipimplies that for electric power materials whose demandexhibits a higher intermittence it is more economic fororganizations to experience stock-outs occasionally than tokeep high quantities in stockMoreover as120582 rises the optimalorder quantity tends to be deterministic In addition keeping120582 constant the optimal order quantity 119876lowastgrows with themean nonnegative demand sizes 120573 Moreover the optimalorder quantity 119876lowast also increases with k As 119896 increasesie the degree of certainty on demand arrival increases theorder quantity increases For instance when 120573 is 500 and 120582is 05 the optimal order quantity 119876lowast is from 750 to 3000respectively when 119896 shifts from 2 to 3
Based on the above analysis the relationships between 120582and 119876lowast can be generalized as
119876lowast = 119888 minus eminusa120582 (119886 ge 0) (16)
Subsequently we employ the calculated120582 and119876lowastas inputsto determine parameters 119888 and 119886 followed by an analysis ofaccuracy Results are as in Table 4
As the average demand sizes 120573 increases the parameter cincreases obviously while the parameter a remains relativelystable When 120582 is very small ie demand interval T is largethe optimal order quantity 119876lowast is approximately equal toparameter c In this case when average demand size is largecoupled with long demand interval organizations will ordermore quantity to satisfy the large demand leading to therise in parameter c This is the reason why c grows with 120573Moreover parameters c and a decrease when 119896 increasesie the certainty degree on demand arrivals increases Inaddition MAPE is computed to examine the goodness offit of these relationships to real data The value of MAPE ismostly around 15 indicating that this algorithm is effectivein the practical application Moreover MAPE gets larger with
Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
100
Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
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Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
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0011
69
1316
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2325
2729
4454
6063
6565
6361
570
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712
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814
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543542
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250281
310338
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0061
916
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4753
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64388861
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1120
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483533
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14131769
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27042893
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0162
1120
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4553
6168
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492543
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21572479
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1120
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6169
7684
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6041068
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1121
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6270
7785
155221
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399455
509562
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22612604
29303241
35403828
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286346
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1221
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290351
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6321125
15651971
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0212
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921314244
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92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
5059
6877
8593
172246
317384
450514
577639
6991259
17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 11
Table 3 Average inventory cost C of varying lead time L for the electric power materials with compound Erlang distributed demand
State grid code of materials Average inventory cost CL=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10
500012723 078 161 188 175 146 113 071 060 042 029500012771 091 208 265 264 230 185 121 103 073 051500012797 132 489 1019 1679 2431 3246 4531 4966 5834 6688500012801 159 528 981 1435 1838 2165 2490 2556 2629 2633500014661 2403 6203 6513 4620 2611 1270 357 225 086 031500014662 523 1490 2358 2915 3140 3093 2704 2529 2155 1784500014670 391 774 831 688 492 320 149 113 063 034500014711 1451 3123 3663 3320 2600 1852 988 782 477 282500014713 1465 3444 4439 4435 3838 3025 1882 1569 1062 697500016589 099 302 517 700 833 914 954 949 921 873500016592 102 326 587 836 1048 1211 1363 1392 1420 1416500017293 122 302 411 436 402 340 237 206 151 108500017320 122 376 647 875 1035 1125 1148 1132 1074 993500017325 089 246 308 294 248 198 131 113 082 058500020609 363 992 1500 1768 1815 1702 1381 1259 1020 803500020612 108 393 810 1323 1903 2528 3515 3851 4525 5198500021412 911 2771 4694 6228 7213 7653 7493 7280 6698 5992500021803 189 537 848 1047 1128 1112 974 912 779 648500021830 119 293 399 425 393 333 233 203 149 107500021862 091 241 356 414 424 401 336 311 261 215500021869 116 286 390 415 385 327 230 200 148 106500022269 130 393 666 885 1030 1099 1089 1063 990 897500022271 096 302 536 753 935 1074 1207 1233 1262 1266500027151 135 284 326 292 226 160 085 068 042 025500027464 1133 3110 4726 5603 5778 5446 4448 4065 3307 2612500029690 122 242 261 219 159 106 052 040 023 013500032681 191 480 665 717 672 575 406 354 261 188500032682 402 1543 3335 5692 8537 11799 17330 19313 23449 27768500032684 164 318 336 274 194 125 058 044 024 013500032687 245 569 725 717 614 480 296 246 165 108500035137 126 259 290 253 191 132 068 053 032 019500051829 112 279 386 417 392 338 243 213 161 118500053694 743 2373 4231 5923 7248 8135 8683 8686 8477 8047500058470 091 239 353 412 424 403 341 316 268 222500061526 152 445 726 927 1033 1056 977 932 828 715500061564 167 424 594 649 616 535 387 340 255 186500061758 130 360 555 670 705 680 578 535 449 367500062282 141 438 758 1030 1225 1336 1368 1351 1284 1189500063419 143 352 480 510 471 397 276 239 175 124500064132 094 263 329 307 252 193 121 102 072 049500065644 3063 5831 6009 4763 3254 2018 872 643 340 175500065658 439 900 1003 863 641 434 214 165 096 054500066495 088 232 347 412 433 424 377 357 315 273500067251 116 352 598 798 933 1001 1003 983 923 844500067292 101 214 250 227 180 130 073 059 037 023500067551 102 279 422 501 520 497 418 387 324 264500067567 218 727 1362 2006 2589 3069 3564 3670 3797 3824500067652 181 480 703 803 799 727 564 506 399 305500067654 132 343 493 554 543 486 369 330 257 194
12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
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Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
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5461
6674
133187
238286
332377
420462
502866
11771454
17051937
21522352
25402717
48455509
0122
1119
2735
4350
5764
7178
141199
253305
355404
451496
541943
12921608
18992168
24212660
30013101
59997817
0132
1119
2836
4351
5865
7279
144203
259313
364414
462510
556972
13361666
19712255
25232777
30183247
64388861
0142
1120
2836
4351
5966
7380
146207
265319
372423
473522
569999
13761720
20382336
26172884
31383382
68419500
0152
1120
2837
4552
6067
7482
149211
270326
380432
483533
5811024
14131769
21002410
27042893
32503506
721310244
0162
1120
2937
4553
6168
7583
151214
274331
387440
492543
5931047
14481815
21572479
27853075
33543621
755811577
0172
1120
2937
4654
6169
7684
153218
278337
393448
501553
6041068
14801858
22112544
28603161
34503728
787912178
0182
1121
2938
4654
6270
7785
155221
282342
399455
509562
6141088
15101898
22612604
29303241
35403828
818012742
0192
1121
3038
4755
6371
7886
157223
286346
405461
516570
6231107
15381936
23082660
29853316
36253922
846212742
022
1221
3039
4755
6371
7987
159226
290351
410467
523578
6321125
15651971
23522713
30573387
37044010
872713272
0212
1221
3039
4756
6472
8087
161229
293355
415473
530586
6401141
15902005
23942763
31163454
37794094
897713772
0222
1221
3039
4856
6473
8088
162231
296359
420479
536593
6481157
16132036
24332810
31713517
38504172
921314244
0232
1221
3140
4857
6573
8189
164233
299363
424484
542599
6551172
16362066
24712855
32233576
39174247
943614691
0242
1222
3140
4957
6574
8290
165235
302366
429489
548606
6631186
16572094
25062897
32723633
39814317
964815115
0252
1222
3140
4958
6674
8290
166237
305370
432494
553612
6691199
16772121
25392938
33193686
40414385
984915517
0262
1222
3140
4958
6675
8391
168239
307373
436498
558618
6761212
16962146
25712976
33643738
40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
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624692
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19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
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910
2030
4050
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13
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7888
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373456
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21942810
34023973
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1507326360
114
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376459
539539
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45715118
56526175
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124
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203291
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698761
8491558
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56696199
1538726624
134
1425
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5868
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203292
378460
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34504024
45915143
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1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
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Beta=4
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5
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Q5
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Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
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Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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12 Mathematical Problems in Engineering
Beta=1 Beta=2
Beta=3 Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
6
8
10
12
14
Q05 1 150
Lamda
05 1 150Lamda
10
20
30
40
Q
10
20
30
40
50
Q
05 1 150Lamda
5
10
15
20
25
Q
05 1 150Lamda
20
40
60
80
Q
0
20
40
60
Q
20
40
60
80
Q
20
40
60
80
100
Q
0
50
100
150
Q
20
40
60
80
100
Q
0
100
200
300
Q
0
100
200
300
Q
0
200
400
600
Q
0
100
200
300
400
Q
Figure 6 Continued
Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
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n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
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074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
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904213938
0322
1221
3039
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6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
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543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
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6877
8593
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450514
577639
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17662240
26883116
35273924
43084682
1074017299
0412
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5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
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30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
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30443543
40254493
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1287621713
0592
1324
3444
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92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
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92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
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92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
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1324
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93102
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350426
500500
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7821426
20172574
31063618
41134594
50625519
1326922394
0633
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5564
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351427
502502
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7851431
20252584
31193633
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5565
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93102
193273
352429
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20322594
31313648
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0653
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5565
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193275
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654725
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94103
194276
357434
510510
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7991459
20672641
31893718
42304727
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1377423411
073
1324
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5665
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94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 13
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=700
0 05 1 15Lamda
Beta=800
0 05 1 15Lamda
Beta=900
0 05 1 15Lamda
Beta=1000
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
500
1000
Q
0
500
1000
1500
2000Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
Q
0
2000
4000
6000
8000
Q
0 05 1 15Lamda
0 05 1 15Lamda
x 104 Beta=3000
0
05
1
15
2
Q
x 104 Beta=60000
0
1
3
2
Q
Figure 6 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 2
14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
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38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
5059
6877
8593
172246
317384
450514
577639
6991259
17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
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14 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 7 119899 under varied 120582 120573 when 119896 = 2
Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
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6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
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042
1222
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0412
1222
3242
5160
6877
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173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 15
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 3000 6000Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast Qlowast
0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
n
Figure 8 119899 under varied 120582 120573 when 119896 = 3
the increases of the mean demand sizes 120573 and the numberof stages k implying that this algorithm is more suitable forthose electric powermaterials whose demand size is not largeand demand arrivals follow two-stage Erlang distribution
6 Conclusion and Future Research
Currently the compound Poisson distribution is the mostoften utilized model for intermittent demand to control
16 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0011
69
1316
1821
2325
2729
4454
6063
6565
6361
570
00
00
00
00
00
0021
712
1721
2529
3336
3943
7194
114131
146160
172183
192245
0213
15475
00
00
00
0031
814
1924
2934
3842
4650
87117
145170
193214
234253
270401
0524
543542
524492
447391
00
0041
815
2126
3237
4247
5156
97134
167197
225252
277301
324509
0740
814867
903925
935933
00
0051
916
2228
3439
4550
5560
106146
183218
250281
310338
365592
766905
10201114
11921255
13061346
7330
0061
916
2329
3541
4753
5863
112156
196234
270304
337368
398658
8651038
11851313
14241520
16041676
17250
0071
916
2430
3743
4955
6066
118164
207248
270323
359393
426713
9471147
13221477
16161739
18511951
2548914
0081
1017
2431
3844
5157
6368
112171
217259
300340
377414
449759
10171241
14391617
17791926
20602184
32462311
0091
1018
2532
3946
5258
6470
126177
225270
312354
393432
469799
10771321
15391738
19192087
22412385
38503518
011
1018
2633
4047
5360
6672
130183
232278
323366
407448
487835
11301391
16281843
20432277
24002561
43784574
0111
1018
2633
4147
5461
6674
133187
238286
332377
420462
502866
11771454
17051937
21522352
25402717
48455509
0122
1119
2735
4350
5764
7178
141199
253305
355404
451496
541943
12921608
18992168
24212660
30013101
59997817
0132
1119
2836
4351
5865
7279
144203
259313
364414
462510
556972
13361666
19712255
25232777
30183247
64388861
0142
1120
2836
4351
5966
7380
146207
265319
372423
473522
569999
13761720
20382336
26172884
31383382
68419500
0152
1120
2837
4552
6067
7482
149211
270326
380432
483533
5811024
14131769
21002410
27042893
32503506
721310244
0162
1120
2937
4553
6168
7583
151214
274331
387440
492543
5931047
14481815
21572479
27853075
33543621
755811577
0172
1120
2937
4654
6169
7684
153218
278337
393448
501553
6041068
14801858
22112544
28603161
34503728
787912178
0182
1121
2938
4654
6270
7785
155221
282342
399455
509562
6141088
15101898
22612604
29303241
35403828
818012742
0192
1121
3038
4755
6371
7886
157223
286346
405461
516570
6231107
15381936
23082660
29853316
36253922
846212742
022
1221
3039
4755
6371
7987
159226
290351
410467
523578
6321125
15651971
23522713
30573387
37044010
872713272
0212
1221
3039
4756
6472
8087
161229
293355
415473
530586
6401141
15902005
23942763
31163454
37794094
897713772
0222
1221
3039
4856
6473
8088
162231
296359
420479
536593
6481157
16132036
24332810
31713517
38504172
921314244
0232
1221
3140
4857
6573
8189
164233
299363
424484
542599
6551172
16362066
24712855
32233576
39174247
943614691
0242
1222
3140
4957
6574
8290
165235
302366
429489
548606
6631186
16572094
25062897
32723633
39814317
964815115
0252
1222
3140
4958
6674
8290
166237
305370
432494
553612
6691199
16772121
25392938
33193686
40414385
984915517
0262
1222
3140
4958
6675
8391
168239
307373
436498
558618
6761212
16962146
25712976
33643738
40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
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474542
608674
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18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
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47295149
1216920213
0423
1323
3343
5362
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0443
1323
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1324
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5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
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0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
5059
6877
8593
172246
317384
450514
577639
6991259
17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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16 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0011
69
1316
1821
2325
2729
4454
6063
6565
6361
570
00
00
00
00
00
0021
712
1721
2529
3336
3943
7194
114131
146160
172183
192245
0213
15475
00
00
00
0031
814
1924
2934
3842
4650
87117
145170
193214
234253
270401
0524
543542
524492
447391
00
0041
815
2126
3237
4247
5156
97134
167197
225252
277301
324509
0740
814867
903925
935933
00
0051
916
2228
3439
4550
5560
106146
183218
250281
310338
365592
766905
10201114
11921255
13061346
7330
0061
916
2329
3541
4753
5863
112156
196234
270304
337368
398658
8651038
11851313
14241520
16041676
17250
0071
916
2430
3743
4955
6066
118164
207248
270323
359393
426713
9471147
13221477
16161739
18511951
2548914
0081
1017
2431
3844
5157
6368
112171
217259
300340
377414
449759
10171241
14391617
17791926
20602184
32462311
0091
1018
2532
3946
5258
6470
126177
225270
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393432
469799
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15391738
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22412385
38503518
011
1018
2633
4047
5360
6672
130183
232278
323366
407448
487835
11301391
16281843
20432277
24002561
43784574
0111
1018
2633
4147
5461
6674
133187
238286
332377
420462
502866
11771454
17051937
21522352
25402717
48455509
0122
1119
2735
4350
5764
7178
141199
253305
355404
451496
541943
12921608
18992168
24212660
30013101
59997817
0132
1119
2836
4351
5865
7279
144203
259313
364414
462510
556972
13361666
19712255
25232777
30183247
64388861
0142
1120
2836
4351
5966
7380
146207
265319
372423
473522
569999
13761720
20382336
26172884
31383382
68419500
0152
1120
2837
4552
6067
7482
149211
270326
380432
483533
5811024
14131769
21002410
27042893
32503506
721310244
0162
1120
2937
4553
6168
7583
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274331
387440
492543
5931047
14481815
21572479
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33543621
755811577
0172
1120
2937
4654
6169
7684
153218
278337
393448
501553
6041068
14801858
22112544
28603161
34503728
787912178
0182
1121
2938
4654
6270
7785
155221
282342
399455
509562
6141088
15101898
22612604
29303241
35403828
818012742
0192
1121
3038
4755
6371
7886
157223
286346
405461
516570
6231107
15381936
23082660
29853316
36253922
846212742
022
1221
3039
4755
6371
7987
159226
290351
410467
523578
6321125
15651971
23522713
30573387
37044010
872713272
0212
1221
3039
4756
6472
8087
161229
293355
415473
530586
6401141
15902005
23942763
31163454
37794094
897713772
0222
1221
3039
4856
6473
8088
162231
296359
420479
536593
6481157
16132036
24332810
31713517
38504172
921314244
0232
1221
3140
4857
6573
8189
164233
299363
424484
542599
6551172
16362066
24712855
32233576
39174247
943614691
0242
1222
3140
4957
6574
8290
165235
302366
429489
548606
6631186
16572094
25062897
32723633
39814317
964815115
0252
1222
3140
4958
6674
8290
166237
305370
432494
553612
6691199
16772121
25392938
33193686
40414385
984915517
0262
1222
3140
4958
6675
8391
168239
307373
436498
558618
6761212
16962146
25712976
33643738
40994448
1004115900
0272
1222
3240
5058
6775
8392
169241
310376
440502
563623
6821224
17152171
26023013
34073786
41534509
1022316265
0282
1222
3241
5059
6776
8492
170243
312379
443506
568628
6881236
17322194
26313047
34473833
42054567
1039716613
0292
1222
3241
5059
6876
8593
171244
314381
447510
572633
6931247
17492216
26583081
34863877
42554623
1056416946
032
1222
3241
5059
6877
8593
172246
316384
450514
577638
6981258
17642237
26853112
35233919
43034676
1072317263
0312
1222
3241
5160
6877
8594
173248
319387
453518
581643
7031268
17802258
27103143
35583960
43494726
1087517567
0322
1222
3242
5160
6977
8694
174249
320389
456521
585647
7081277
17942277
27353172
35923999
43924775
1102017859
0332
1223
3242
5160
6978
8695
175250
322391
459524
588651
7131287
18082296
27583200
36254036
44344821
1116018138
0342
1223
3342
5160
6978
8795
176252
324394
461527
592655
7171296
18222313
27803227
36564072
44744866
1129318851
0352
1223
3342
5261
7078
8796
177253
326396
464530
595659
7221304
18342331
28013252
36864106
45134909
1142218851
0363
1223
3342
5261
7079
8896
178254
328398
467534
599663
7271314
18492351
28273283
37234148
45614962
1160519077
0373
1223
3343
5261
7079
8897
179256
329400
469537
602667
7311322
18612367
28473307
37514180
45975003
1172619321
0383
1223
3343
5261
7079
8897
179257
331402
472539
606671
7341330
18732382
28673331
37784211
46325041
1184319556
0393
1223
3343
5261
7180
8997
180258
332404
474542
608674
7381337
18842397
28853352
38044241
46655079
1195619783
043
1223
3343
5362
7180
8998
181259
334406
476544
611677
7421345
18952412
29033375
38394270
46985115
1206420002
0413
1323
3343
5362
7180
8998
182260
335408
478547
614680
7451351
19052426
29213395
38544298
47295149
1216920213
0423
1323
3343
5362
7181
8998
182261
336409
480549
617683
7491358
19162439
29373416
38774324
47595183
1227020416
0433
1323
3443
5362
7281
9099
183262
338411
482551
619686
7521365
19252452
29533435
39004351
47885215
1236820613
0443
1323
3444
5363
7281
9099
184263
339412
484554
622689
7551371
19352464
29693454
39224375
48165247
1246220803
0453
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581377
19442476
29843472
39434399
48445277
1255320987
0463
1324
3444
5363
7282
91100
185265
341415
487558
627694
7611383
19522488
29993489
39634423
48705306
1264221165
0473
1324
3444
5463
7382
91100
185266
342417
489560
629697
7641388
19612499
30133506
39834445
48955334
1272721337
0483
1324
3444
5463
7382
91100
186266
343418
491562
631699
7661394
19692510
30263522
40024467
49205362
1280921503
0493
1324
3444
5463
7382
91100
186267
345419
492563
633702
7691399
19772511
30403538
40204488
49445388
1288921664
053
1324
3444
5464
7382
92101
187268
346421
494565
635704
7721404
19842531
30523554
40384509
49675414
1296621820
0513
1324
3444
5464
7383
92101
187269
347422
495567
637706
7741409
19922541
30653568
40564528
49895438
1304121971
0523
1324
3444
5464
7383
92101
188269
347423
497569
639708
7761414
19992550
30773583
40724548
50105462
1311322117
n
Figure 9 Continued
Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
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453767
10291256
14581641
18061957
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33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
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6877
8593
172246
317384
450514
577639
6991259
17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
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5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
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1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
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5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
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517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
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5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
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5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
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5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 17
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0533
1324
3444
5464
7483
92101
188270
348424
498570
641710
7791418
20062560
30883597
40884566
50315485
1318322259
0543
1324
3445
5464
7483
92101
189271
349425
499572
643712
7811423
20132569
30993610
41044584
50515508
1325122396
0553
1324
3445
5564
7483
93102
189271
350426
501573
644714
7831427
20192577
31103623
41194601
50715529
1331722529
0563
1324
3545
5564
7483
93102
190272
351427
502575
646716
7851431
20252586
31213636
41344618
50905551
1338122658
0573
1324
3545
5565
7484
93102
190273
352428
503576
648718
7871435
20322594
31313648
41484634
51085571
1347922783
0583
1324
3545
5565
7484
93102
190273
353429
504578
649720
7891439
20372602
31413660
41624650
51265591
1350222905
0593
1324
3545
5565
7484
93103
191274
353430
505579
651721
7911443
20432609
31503671
41754665
51435610
1356123022
063
1324
3545
5565
7584
93103
191274
354431
507580
652723
7931447
20492617
31593682
41884680
51605628
1361723136
0613
1324
3545
5565
7584
94103
192275
355432
508581
654725
7951450
20542624
31683693
42014695
51765646
1367123247
0623
1324
3545
5565
7584
94103
192275
355433
509583
655726
7961454
20592631
31773703
42134709
51925664
1372423354
0633
1324
3545
5565
7585
94103
192276
356434
510584
656728
7981457
20642637
31853713
42254722
52075681
1377523458
0643
1324
3545
5565
7585
94103
193276
357435
511585
658729
8001460
20692644
31943723
42364735
52215697
1384023559
0653
1324
3545
5665
7585
94104
193277
357436
512586
659731
8011464
20742650
32013733
42474748
52365713
1387323657
0663
1324
3545
5665
7585
94104
193277
358436
513587
660732
8031467
20782656
32093742
42584760
52495728
1392023752
0673
1324
3546
5666
7585
95104
193278
359437
513588
661733
8041470
20832662
32163751
42684772
52635743
1396523844
0683
1324
3546
5666
7585
95104
194278
359438
514589
662735
8061472
20872668
32343759
42784783
52765757
1400923933
0693
1324
3546
5666
7585
95104
194279
360438
515590
664736
8071475
20912673
32313768
42884794
52885771
1405124104
074
1324
3546
5666
7685
95105
194279
360439
516591
665737
8081478
20952679
32373776
42984805
53005785
1409324104
0714
1325
3546
5666
7685
95105
195279
361440
517592
666738
8101480
20992684
32443783
43074816
53125798
1413224186
0724
1325
3546
5666
7686
85106
195280
361440
517593
667739
8111483
21032689
32503792
43154826
53245810
1417124265
0734
1325
3546
5666
7686
96106
195280
362441
518594
668740
8121485
21072694
32563798
43244835
53355823
1420924342
0744
1325
3546
5666
7686
96106
195280
362442
519594
669742
8131488
21102699
32623805
43324845
53455834
1424524416
0754
1325
3546
5666
7686
96106
195281
363442
520595
670743
8141490
21132703
32683812
43404854
53565846
1428024488
0764
1325
3546
5666
7686
96106
196281
363443
520596
670744
8161492
21172708
32763819
43484863
53665857
1431424558
0774
1325
3546
5666
7686
96106
196281
364443
521597
671744
8171494
21202712
32793825
43564872
53755868
1434724626
0784
1325
3546
5666
7686
96106
196282
364444
522597
672745
8181496
21232716
32843832
43634880
53845878
1437824692
0794
1325
3646
5666
7686
96106
196282
364444
522598
673746
8181498
21262720
32893837
43704888
53945888
1441024755
084
1325
3646
5667
7686
96106
196282
365445
523599
674747
8201500
21292724
32933843
43764896
54025898
1444024817
0814
1325
3646
5667
7786
96106
197283
365445
523599
674748
8211502
21322727
32983849
43834903
54115907
1446924877
0824
1325
3646
5767
7786
96106
197283
365446
524600
675749
8211504
21342731
33033854
43894910
54195916
1449724934
0834
1325
3646
5767
7786
96106
197283
366446
524601
676750
8221506
21372734
33073859
43954917
54275925
1452425007
0844
1325
3646
5767
7787
97107
197283
366446
524601
676750
8231507
21392738
33113864
44014924
54345934
1455025045
0854
1325
3646
5767
7787
97107
197284
366447
525602
677751
8241509
21422741
33153869
44074930
54425942
1457525097
0864
1325
3646
5767
7787
97107
197284
367447
526602
678752
8251510
21442744
33193874
44124937
54495949
1460025148
0874
1325
3646
5767
7787
97107
198284
367448
526603
678752
8251512
21462747
33233878
44184943
54555957
1462325197
0884
1325
3646
5767
7787
97107
198284
367448
526603
679753
8261513
21492750
33263883
44234948
54625964
1464625245
0894
1325
3646
5767
7787
97107
198284
368448
527604
679754
8271515
21512753
33303887
44284954
54685971
1466825290
094
1325
3646
5767
7787
97107
198285
368449
527604
680754
8271516
21532755
33333891
44324959
54745978
1468925335
0914
1325
3646
5767
7787
97107
198285
368449
528605
680755
8281517
21542758
33363895
44374965
54805985
1470925378
0924
1325
3646
5767
7787
97107
198285
368449
528605
681755
8291518
21562760
33393898
44414969
54865991
1472925419
0934
1325
3647
5767
7787
97107
198285
369449
528606
681756
8291520
21582763
33423902
44454974
54915997
1474825459
0944
1325
3647
5767
7787
97107
199285
369450
529606
682756
8301521
21602765
33453905
44494979
54966002
1476625497
0954
1325
3647
5767
7787
97107
199286
369450
529606
682757
8301522
21612767
33483909
44534983
55016008
1478325534
0964
1325
3647
5767
7787
97107
199286
369450
529607
683757
8311523
21632769
33503912
44574987
55066013
1480025570
0974
1325
3647
5767
7787
97107
199286
369451
530607
683758
8311524
21642771
33533915
44574991
55106018
1481625604
0984
1325
3647
5767
7787
97107
199286
370451
530607
683758
8321525
21662773
33553918
44634995
55146023
1483125637
0994
1325
3647
5767
7787
97107
199286
370451
530608
684758
8321526
21672775
33573920
44674999
55196028
1484625669
14
1325
3647
5767
7787
97107
199286
370451
530608
684759
8331526
21682776
33603923
44705002
55226032
1486025699
115
1425
3647
5868
7888
98108
202291
376459
539618
696761
8361533
21782789
33753942
44925028
55526064
1496925941
125
1425
3747
5868
7889
99109
202292
377460
540619
697761
8361533
21792790
33763944
44945030
55556069
1499525990
135
1425
3747
5868
7989
99109
202292
378460
541620
698761
8361533
21792790
33773946
44965033
55586072
1500726074
1355
1425
3747
5868
7989
99109
203292
378461
542621
699762
8361533
21792791
33783946
44975035
55606074
1503326145
n
Figure 9 119876lowast under varied 120582 120573 when 119896 = 2
18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
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7180
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052
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7281
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184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
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1460025085
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54695972
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0873
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5767
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369450
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093
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370452
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0973
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0983
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5768
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373455
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33943963
45175055
55826097
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1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
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600
800
Q
0
200
400
600
800
Q
0
200
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600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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18 Mathematical Problems in Engineering
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0031
58
1113
1517
1920
2223
3235
3532
2821
145
00
00
00
00
00
00
0041
610
1316
1922
2427
2931
4859
6773
7678
7978
7612
00
00
00
00
00
0051
611
1519
2226
2932
3437
6078
92104
113121
128133
137136
820
00
00
00
00
0061
712
1721
2529
3236
3942
6093
112128
143156
167178
187235
230191
12844
00
00
00
0071
713
1822
2731
3539
4346
78105
128149
168185
200215
228317
353356
333290
230156
690
00
0081
814
1924
2933
3742
4650
85115
142166
188209
228246
263386
458495
507498
473434
382319
00
0091
814
2025
3035
4044
4853
91124
154181
207230
252273
293446
548615
657679
684675
653620
00
011
815
2126
3137
4146
5155
96132
165194
222249
273297
319449
627721
789837
869886
890884
00
0111
915
2127
3338
4348
5358
101139
174206
236265
292318
342546
697814
906977
10321072
11001117
460
0121
916
2228
3439
4550
5560
105145
182217
249279
309337
363588
759897
10101102
11781239
12881325
6710
0131
916
2229
3540
4651
5762
109151
190226
260293
324353
382625
816972
11031214
13091389
14561512
12330
0141
916
2329
3541
4753
5863
112156
197234
270304
337369
399659
8661040
11881316
14271524
16091682
17410
0151
917
2330
3642
4854
5965
115161
203242
279315
349382
414690
9131101
12651408
15351647
17471836
2203225
0161
917
2430
3743
4955
6166
118165
208249
288325
361395
428718
9551158
13351493
16331760
18741976
26241067
0171
1017
2431
3844
5056
6268
121169
213256
296334
371407
441743
9931209
13991570
17231863
19892105
30101838
0181
1017
2531
3845
5157
6369
123172
218261
303342
380417
453767
10291256
14581641
18061957
20952223
33642546
0191
1018
2532
3945
5258
6470
125176
223267
309350
389427
464789
10611300
15131706
18822044
21932331
36893197
021
1018
2532
3946
5359
6571
127179
227272
315357
397436
474809
10911340
15631766
19522124
22832431
39893797
0211
1018
2633
4046
5359
6672
129181
230276
321363
404444
483827
11191377
16091821
20172198
23662431
42664287
0221
1018
2633
4047
5460
6673
131184
234281
326369
411452
492844
11441411
16521872
20762266
24432609
45224868
0231
1018
2633
4047
5461
6773
133186
237285
331375
418459
500860
11681442
16911920
21322329
25142688
47595336
0242
1120
2837
4552
6067
7482
149211
269325
379432
483532
5811023
14121768
20982408
27012980
32463502
720110242
0252
1120
2937
4553
6068
7583
151214
273330
385439
491541
5911043
14121808
21482468
27723060
33373602
750210846
0262
1120
2937
4553
6169
7684
153217
277335
391446
498550
6011062
14421846
21962526
28383137
34233698
778911422
0272
1121
2938
4554
6269
7784
155219
281340
397452
506558
6101080
14711883
22422580
29023210
35053789
806311971
0282
1121
3038
4654
6270
7885
156222
284344
402458
513566
6181098
15251918
22852633
29633280
35833876
832412496
0292
1121
3038
4755
6371
7986
158224
288348
407464
519574
6271114
15501951
23272683
30223346
36583959
857412997
032
1121
3039
4755
6471
7987
160227
291352
412469
526581
6351130
15731983
23672730
30773410
37304039
881313478
0312
1221
3039
4856
6472
8088
161229
294356
416475
532588
6421146
15962013
24052776
31313471
37994115
904213938
0322
1221
3039
4856
6573
8188
163231
297360
421480
538594
6501160
16182043
24412820
31823530
38654189
926214379
0332
1221
3140
4857
6573
8189
164233
300363
425485
543601
6571174
16392071
24772862
32313586
39284259
947214803
0342
1222
3140
4957
6674
8290
165236
302367
429490
549607
6631188
16602098
25102903
32793640
39894326
967515211
0352
1222
3140
4958
6674
8290
167237
305370
433494
554612
6701201
16792124
25432942
33243692
40474391
987015558
0362
1222
3140
4958
6675
8391
168239
307373
437498
559618
6761213
16982149
25742979
33683742
41034454
1005715933
0372
1222
3141
5058
6775
8492
169241
310376
440503
564624
6821225
17162173
26043015
34103790
41584514
1023816294
0382
1222
3241
5059
6776
8492
170243
312379
444507
568629
6881237
17332196
26333050
34503836
42104575
1041116641
0392
1222
3241
5059
6876
8593
171245
314382
447511
573634
6941248
17502218
26613084
34893881
42604628
1057916976
042
1222
3241
5059
6877
8593
172246
317384
450514
577639
6991259
17662240
26883116
35273924
43084682
1074017299
0412
1222
3242
5160
6877
8694
173248
319387
453518
581643
7041269
17822261
27143147
35643966
43554734
1089617611
0422
1222
3242
5160
6977
8694
174249
321390
456522
585648
7091279
17972281
27393177
35994006
44004784
1104717977
0432
1223
3242
5160
6978
8695
175251
323392
459525
589652
7141289
18112300
27633206
36334044
44444832
1119218202
0442
1223
3342
5161
6978
8795
176252
324394
462528
593656
7191289
18252319
27873234
36654082
44864879
1133218555
0452
1223
3342
5261
7079
8796
177253
326397
465531
597660
7231307
18392337
28093237
36974118
45274924
1152118830
0462
1223
3342
5261
7079
8896
178255
328399
467534
600664
7281316
18522354
28313264
37274153
45664968
1159919096
0472
1223
3343
5261
7079
8897
179256
330401
470537
603668
7321324
18652371
28523313
37574187
46045010
1172619353
0482
1223
3343
5262
7180
8897
180257
331403
472540
607672
7361333
18772387
28733337
37864219
46415051
1184819602
0492
1223
3343
5262
7180
8897
181258
333405
475543
610675
7401341
18892403
28923361
38134251
46765090
1196719844
052
1223
3343
5362
7180
8998
181260
334407
477546
613679
7441348
19002419
29113384
38404282
47115129
1208120078
0512
1323
3343
5362
7180
8998
182261
336409
479548
616682
7471356
19112433
29303406
38664311
47445166
1219220525
0522
1323
3443
5362
7281
9099
183262
337410
481551
619685
7511363
19222448
29483428
38914340
47765201
1230020305
0532
1323
3444
5363
7281
9099
183263
339412
484553
621689
7541370
19332462
29653449
39154368
48075236
1240420738
0542
1323
3444
5363
7281
9099
184264
340414
486556
624692
7581376
19432476
29823454
39394394
48385270
1250420945
n
Figure 10 Continued
Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
489560
629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
40254493
49495393
1287621713
0592
1324
3444
5464
7383
92101
187268
346421
495566
536705
7731407
19882536
30583560
40454516
49755422
1296121891
063
1324
3444
5464
7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
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93102
193273
353430
505505
650720
7901441
20402604
31443663
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51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
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7921446
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31563677
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0673
1324
3545
5565
7584
94103
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508508
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41824692
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0683
1324
3545
5565
7584
94103
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509509
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51935664
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0693
1324
3545
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94103
194276
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657729
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073
1324
3545
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7585
94104
194277
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512512
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0713
1324
3545
5665
7585
94104
195277
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513513
661732
8031467
20792657
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42594761
52505728
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0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Mathematical Problems in Engineering 19
12
34
56
78
910
2030
4050
6070
8090
100200
300400
500600
700800
9001000
30006000
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
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Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
Qlowast
0552
1324
3444
5463
7282
91100
185265
341415
487558
627694
7611383
19522488
29983488
39614420
48675302
1260221146
0562
1324
3444
5463
7382
91100
185266
343416
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629697
7641389
19622500
30143507
39834446
48955333
1269621340
0572
1324
3444
5463
7382
91100
186267
344419
491561
632700
7671395
19712513
30293525
40054470
49225364
1278721529
0582
1324
3444
5464
7382
91100
187268
345420
493564
634703
7701401
19802524
30443543
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1287621713
0592
1324
3444
5464
7383
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187268
346421
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19882536
30583560
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49755422
1296121891
063
1324
3444
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7383
92101
191270
348423
497497
640709
7771415
20002552
30793585
40754450
50135464
1310522063
0613
1324
3445
5464
7483
92101
192270
349425
499499
642711
7801420
20092563
30923602
40944572
50385492
1318822231
0623
1324
3445
5564
7483
93102
192271
350426
500500
644714
7821426
20172574
31063618
41134594
50625519
1326922394
0633
1324
3545
5564
7483
93102
192272
351427
502502
646716
7851431
20252584
31193633
41314612
50855545
1334822553
0643
1324
3545
5565
7484
93102
193273
352429
503503
648718
7881436
20322594
31313648
41494635
51085570
1342422706
0653
1324
3545
5565
7484
93102
193273
353430
505505
650720
7901441
20402604
31443663
41494654
51305595
1349822856
0663
1324
3545
5565
7584
93103
193274
354431
506506
652723
7921446
20472614
31563677
41664673
51525619
1357023001
0673
1324
3545
5565
7584
94103
193275
355432
508508
654725
7951450
20542623
31673691
41824692
51725642
1364023142
0683
1324
3545
5565
7584
94103
194276
356433
509509
655727
7971455
20602632
31783705
41994710
51935664
1364023278
0693
1324
3545
5565
7585
94103
194276
357434
510510
657729
7991459
20672641
31893718
42304727
52125686
1377423411
073
1324
3545
5665
7585
94104
194277
357435
512512
659731
8011463
20732649
32003730
42444744
52315707
1383823540
0713
1324
3545
5665
7585
94104
195277
358437
513513
661732
8031467
20792657
32103743
42594761
52505728
1390023666
0723
1324
3546
5666
7585
95104
195278
359437
514514
662734
8051471
20852665
32203755
42734777
52685748
1396023788
0733
1324
3546
5666
7685
95104
195279
360438
515515
664736
8071475
20912673
32303766
42864762
52855767
1401923906
0743
1324
3546
5666
7685
95104
195279
361439
516516
665737
8091479
20972680
32393781
42994807
53025786
1407624021
0753
1324
3546
5666
7686
95105
195280
361440
517517
667739
8111483
21022688
32483778
43124822
53195804
1413124133
0763
1325
3546
5666
7686
95105
195280
362441
518518
668741
8131486
21072695
32573789
43254836
53345822
1418524241
0773
1325
3546
5666
7686
95105
195281
363442
519519
669742
8141490
21132702
32663799
43374849
53505839
1423724346
0783
1325
3546
5666
7686
96105
196281
363443
520520
671744
8161493
21172708
32743809
43484863
53655866
1428724449
0793
1325
3646
5666
7686
96105
196282
364444
521521
672745
8181496
21222715
32823819
43594876
53795872
1433624548
083
1325
3646
5666
7686
96105
196282
365444
522522
673747
8191499
21272721
32903829
43704888
53935888
1438324644
0813
1325
3646
5667
7786
96106
197283
365445
523523
674748
8211502
21322727
32973838
43814900
54075903
1443024738
0823
1325
3646
5767
7786
96106
197283
365446
524524
676749
8221505
21362733
33053847
43914912
54205918
1447424828
0833
1325
3646
5767
7787
96106
197283
366447
525525
677751
8231508
21402739
33123856
44014923
54335934
1451824916
0843
1325
3646
5767
7787
96106
198284
367447
526526
678752
8251511
21442739
33193873
44114934
54465946
1456025002
0853
1325
3646
5767
7787
96106
198284
367448
527527
679753
8261513
21482744
33253881
44204945
54585959
1460025085
0863
1325
3646
5767
7787
97106
198285
368449
527527
680754
8271516
21522749
33323889
44294955
54695972
1464025165
0873
1325
3647
5767
7787
97106
198285
368449
528528
681755
8291518
21562759
33383896
44384965
54805984
1467825243
0883
1325
3647
5767
7787
97106
199285
369450
529529
682756
8301521
21592764
33443903
44464975
54915997
1471525318
0893
1325
3647
5767
7787
97107
199286
369450
529529
683757
8311523
21632769
33503910
44554984
55026008
1475125391
093
1325
3647
5767
7787
97107
199286
370451
530530
684758
8321525
21662773
33553917
44624993
55126020
1478525462
0913
1325
3647
5767
7787
97107
199286
370452
531531
685759
8331527
21702778
33613924
44705002
55226030
1481925531
0923
1325
3647
5767
7787
97107
199287
371452
531531
685760
8341530
21732782
33663930
44775010
55316041
1485125597
0933
1325
3647
5767
7887
97107
200287
371453
532532
686761
8351532
21762786
33713936
44875019
55416051
1488225661
0943
1325
3647
5768
7888
97107
200287
371453
533533
687762
8361534
21792790
33763942
44915027
55496061
1491325724
0953
1325
3647
5768
7888
97107
200288
372453
533533
688763
8371535
21812794
33813948
44985034
55586071
1494225784
0963
1325
3647
5768
7888
98107
200288
372454
534534
688764
8381537
21842797
33853953
45045042
55666080
1497025842
0973
1325
3647
5768
7888
98107
200288
373454
534534
689765
8391539
21872801
33903958
45115049
55746089
1499725898
0983
1325
3647
5768
7888
98107
201288
373455
535535
690765
8401541
21892804
33943963
45175055
55826097
1502325952
0993
1325
3647
5768
7888
98108
201289
373455
535535
690766
8411542
21922807
33983968
45225062
55126105
1504926106
13
1425
3647
5868
7888
98108
201289
373456
536536
691767
8411544
21942810
34023973
45285068
55226123
1507326360
114
1425
3647
5868
7888
98108
202290
376459
539539
696761
8471556
22122835
34324010
45715118
56526175
1529926465
124
1425
3647
5868
7989
99109
203291
377460
541541
698761
8491558
22192840
34434019
45805129
56696199
1538726624
134
1425
3647
5868
7989
99109
203292
378460
541541
699761
8511563
22232848
34504024
45915143
56856210
1540226755
1354
1425
3647
5868
7989
99109
203292
378461
542542
699761
8521565
22262853
34564038
46055157
56966225
1544826893
n
Figure 10 119876lowast under varied 120582 120573 when 119896 = 3
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
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Submit your manuscripts atwwwhindawicom
20 Mathematical Problems in Engineering
0 05 1 15Lamda
Beta=1
0 05 1 15Lamda
Beta=2
0 05 1 15Lamda
Beta=3
0 05 1 15Lamda
Beta=4
0 05 1 15Lamda
Beta=5
0 05 1 15Lamda
Beta=6
0 05 1 15Lamda
Beta=7
0 05 1 15Lamda
Beta=8
0 05 1 15Lamda
Beta=9
0 05 1 15Lamda
Beta=10
0 05 1 15Lamda
Beta=20
0 05 1 15Lamda
Beta=30
0 05 1 15Lamda
Beta=40
0 05 1 15Lamda
Beta=50
5
10
15
Q5
10
15
20
25
Q
10
20
30
40
Q
10
20
30
40
50
Q
0
20
40
60
Q
0
20
40
60
80
Q
0
20
40
60
80
Q
20
40
60
80
100
Q
20
40
60
80
100
Q
0
50
100
150
Q
0
100
200
300
Q
0
100
200
300
Q
0
100
200
300
400
Q
0
200
400
600
Q
Figure 11 Continued
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 21
0 05 1 15Lamda
Beta=80
0 05 1 15Lamda
Beta=90
0 05 1 15Lamda
Beta=100
0 05 1 15Lamda
Beta=200
0 05 1 15Lamda
Beta=300
0 05 1 15Lamda
Beta=400
0 05 1 15Lamda
Beta=500
0 05 1 15Lamda
Beta=600
0 05 1 15Lamda
Beta=60
0 05 1 15Lamda
Beta=70
0
1000
2000
3000
4000
Q
0
2000
4000
6000
Q
0
1000
2000
3000
Q
0
1000
2000
3000
Q
0
500
1000
1500
2000
Q
0
500
1000
Q
0
200
400
600
800
Q
0
200
400
600
800
Q
0
200
400
600
Q
0
200
400
600
Q
Figure 11 The relationships between 120582 and 119876lowast with respect to varied 120573 when 119896 = 3
material inventory However its memoryless property isunrealistic The Erlang is an IFR distribution which bet-ter fits the nature of intermittent demand Therefore thispaper proposes an algorithm to determine the optimal orderquantity in a single echelon single electric power materialinventory system for (119879 119876) policy An empirical investigationwas conducted to collaborate the validity of this new methodby means of comparison with the classic Newsvendor modelThe findings indicate that this new method outperformsthe classic one in achieving cost reduction which makesit more economic and practical In addition consideringthe complexity of this new method an approximation and
a heuristic algorithm is proposed To sum up the paperdevelops three algorithms for the optimal order quantity eachof which has their own advantages and disadvantages
(1)The basic algorithm this method has great advantageswith regard to accuracy However it is time-consumingcomputation-intensive and not suitable for bulk operation
(2) The approximation algorithm it simplifies computa-tional complexity by reducing iterative numbers within anacceptable error range Such a method enables companies toenhance their efficiency of stock management with guaran-teed accuracy which is of great validity and utility in a real-world context
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
22 Mathematical Problems in Engineering
Table 4 Estimations of parameters 119888 and 119886 with respect to varied 120573119896 = 2 119896 = 3120573 c a MAPE 120573 c a MAPE
1 1248 368 1429 1 1208 308 14292 2296 345 800 2 2213 301 12003 3285 342 1081 3 3158 301 11114 4232 345 1064 4 4071 300 12775 5168 344 1034 5 4958 300 13796 6071 344 1029 6 5825 298 14717 6963 343 1139 7 6681 300 15198 7846 344 1236 8 7515 299 15739 8703 341 1212 9 8335 298 161610 9584 340 1193 10 9146 300 165120 17643 343 1330 20 17426 300 169130 25231 343 1370 30 24537 297 174050 39441 343 1453 50 38259 302 1788100 64337 340 1484 100 53304 299 1837500 302117 343 1512 500 298211 304 19861000 487352 344 1721 1000 463847 301 2135
(3) The heuristic algorithm it derives the optimal orderquantity by establishing a functional relation where theindependent variable is 120582 and the dependent variable is119876lowast This method is the simplest but has the least degree ofaccuracy among all algorithms
In addition these algorithms can also be applied to otherelectric power materials with intermittent demand withinother industries
Appendix
A Results of 119899 under Varied 120582 120573 119896See Figures 7 and 8
B Results of 119876lowast under Varied 120582 120573 119896See Figures 9 and 10
C The Relationships between 120582 and 119876lowast withrespect to Varied 120573When 119896 = 3
See Figure 11
Data Availability
The data used to support the findings of this study weresupplied by Shanghai Electric Power Company under licenseand so cannot be made freely available Requests for access tothese data should be made to Aiping Jiang Chengzhong RdJiaDing Dist Shanghai China
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper has been funded by theNational Science Foundation of China (Grant no 71302053)
References
[1] RMAdelson ldquoThe compoundPoisson distributionrdquo Journal ofthe Operational Research Society vol 17 no 1 pp 73ndash75 1966
[2] J K Friend ldquoStock control with random opportunities forreplenishmentrdquo Journal of the Operational Research Society vol11 no 3 pp 130ndash136 1960
[3] FHanssmannOperations Research in Production and InventoryControl John Wiley and Sons New York NY USA 1962
[4] J Xiao F L Lu and X Xiao ldquoStohastic newsboy inventorycontrol model and its solving on multivariate productd orderand pricingrdquo in Proceedings of the International Conference onInformation Computing and Applications pp 65ndash72 2010
[5] S Kebe N Sbihi and B Penz ldquoA Lagrangean heuristic for atwo-echelon storage capacitated lot-sizing problemrdquo Journal ofIntelligent Manufacturing vol 23 no 6 pp 2477ndash2483 2012
[6] P S S Uduman A Sulaiman and R Sathiyamoorthy ldquoNewsboy inventory model with demand satisfying SCBZ propertyrdquoBulletin of Pure and Applied Science vol 26 no 1 pp 145ndash1502007
[7] D Fathima P S SheikUduman and S Srinivasan ldquoGeneraliza-tion of newsboy problem with demand distribution satisfyingthe SCBZ propertyrdquo International Journal of ContemporaryMathematical Sciences vol 6 no 37-40 pp 1989ndash2000 2011
[8] D Fathima andP S S Uduman ldquoSingle period inventorymodelwith stochastic demand and partial backloggingrdquo InternationalJournal of Management vol 24 no 1 pp 37ndash59 2013
[9] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[10] Y-K Chen C-T Chen F-R Chiu and J-W Lian ldquoApplyingthe bootstrap method to newsvendor model incorporating
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 23
group buying for optimal price discount and order quantityrdquoKybernetes vol 46 no 10 pp 1692ndash1705 2017
[11] F Hnaien A Dolgui and D D Wu ldquoSingle-period inventorymodel for one-level assembly system with stochastic lead timesand demandrdquo International Journal of Production Research vol54 no 1 pp 186ndash203 2016
[12] S Priyan and R Uthayakumar ldquoTwo-echelon multi-productmulti-constraint product returns inventory model with per-missible delay in payments and variable lead timerdquo Journal ofManufacturing Systems vol 36 article no 304 pp 244ndash2622015
[13] M Keramatpour S T A Niaki and S H R Pasandideh ldquoA bi-objective two-level newsvendor problem with discount policiesand budget constraintrdquo Computers amp Industrial Engineeringvol 120 pp 192ndash205 2017
[14] S Axsater Inventory Control Springer 2015[15] P Matheus and L Gelders ldquoThe (R Q) inventory policy subject
to a compound Poisson demand patternrdquo International Journalof Production Economics vol 68 no 3 pp 307ndash317 2000
[16] M Z Babai Z Jemai and Y Dallery ldquoAnalysis of order-up-to-level inventory systems with compound Poisson demandrdquoEuropean Journal of Operational Research vol 210 no 3 pp552ndash558 2011
[17] W T Dunsmuir and R D Snyder ldquoControl of inventorieswith intermittent demandrdquo European Journal of OperationalResearch vol 40 no 1 pp 16ndash21 1989
[18] F Janssen R Heuts and T De Kok ldquoOn the (R s Q) inventorymodel when demand is modelled as a compound Bernoulliprocessrdquo European Journal of Operational Research vol 104 no3 pp 423ndash436 1998
[19] R H Teunter A A Syntetos and M Z Babai ldquoDeterminingorder-up-to levels under periodic review for compound bino-mial (intermittent) demandrdquo European Journal of OperationalResearch vol 203 no 3 pp 619ndash624 2010
[20] A K Gupta W-B Zeng and Y Wu Probability and StatisticalModels Foundations for Problems in Reliability and FinancialMathematics Springer Science and Business Media 2010
[21] M A Smith and R Dekker ldquoOn the (Sndash1 S) stock model forrenewal demand processes Poissonrsquos poisonrdquo Probability in theEngineering and Informational Sciences vol 11 no 3 pp 375ndash386 1997
[22] C Larsen andG P Kiesmullerr ldquoDeveloping a closed-form costexpression for an (R s nQ) policy where the demand processis compound generalized Erlangrdquo Operations Research Lettersvol 35 no 5 pp 567ndash572 2007
[23] C Larsen and A Thorstenson ldquoA comparison between theorder and the volume fill rate for a base-stock inventory controlsystem under a compound renewal demand processrdquo Journalof the Operational Research Society vol 59 no 6 pp 798ndash8042008
[24] C Larsen and A Thorstenson ldquoThe order and volume fill ratesin inventory control systemsrdquo International Journal of Produc-tion Economics vol 147 pp 13ndash19 2014
[25] S Saidane M Z Babai M S Aguir and O Korbaa ldquoSpareparts inventory systemsunder an increasing failure rate demandinterval distributionrdquo in Proceedings of the 41st InternationalConference on Computers and Industrial Engineering 2011 pp214ndash219 USA October 2011
[26] S Saidane M Z Babai M S Aguir and O Korbaa ldquoOn theperformance of the base-stock inventory system under a com-pound Erlang demand distributionrdquo Computers amp IndustrialEngineering vol 66 no 3 pp 548ndash554 2013
[27] A A Syntetos M Z Babai and S Luo ldquoForecasting of com-pound erlang demandrdquo Journal of the Operational ResearchSociety vol 66 no 12 pp 2061ndash2074 2015
[28] R Y K Fung X Ma and H C W Lau ldquo(T S) policy for coor-dinated inventory replenishment systems under compoundPoisson demandsrdquo Production Planning and Control vol 12 no6 pp 575ndash583 2001
[29] Y Zhao ldquoAnalysis and evaluation of an assemble-to-ordersystem with batch ordering policy and compound PoissondemandrdquoEuropean Journal ofOperational Research vol 198 no3 pp 800ndash809 2009
[30] E Topan and Z P Bayindir ldquoMulti-item two-echelon spareparts inventory control problem with batch ordering in thecentral warehouse under compound Poisson demandrdquo Journalof the Operational Research Society vol 63 no 8 pp 1143ndash11522012
[31] A H C Eaves and B G Kingsman ldquoForecasting for the order-ing and stock-holding of spare partsrdquo Journal of the OperationalResearch Society vol 55 no 4 pp 431ndash437 2004
[32] E A Trabka and E W Marchand ldquoThe mean and variance ofintervals between renewals of a type I counter with a censoredPoisson inputrdquo Biological Cybernetics vol 7 no 6 pp 224ndash2271970
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom