24
Research Article Determining the Optimal Order Quantity with Compound Erlang Demand under (T,Q) Policy Aiping Jiang , 1 Kwok Leung Tam, 2 Yingzi Bao, 1 and Jialing Lu 1 1 SHU-UTS SILC Business School, Shanghai University, 20 Chengzhong Road, Jiading District, Shanghai 201899, China 2 School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia Correspondence should be addressed to Aiping Jiang; [email protected] Received 10 March 2018; Revised 25 June 2018; Accepted 12 July 2018; Published 19 August 2018 Academic Editor: Neale R. Smith Copyright © 2018 Aiping Jiang et al. is is an open access article distributed under the Creative Commons Attribution License, which 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 companies’ performance and profitability. Considering the critical role that electric power materials play in supporting maintenance operations and preventing equipment failure, it is essential to maintain an inventory to a reasonable level. However, a majority of these electric power materials exhibit an intermittent demand pattern characterized by random arrivals interspersed with time periods with no demand at all. ese characteristics cause additional difficulty for companies in managing these electric power material inventories. In response to the above problem, this paper, based on the electric power material demand data of Shanghai Electric Power Company, develops a new method to determine the optimal order quantity in a cost-oriented periodic review (, ) system, whereby unsatisfied demands are backordered and demand follows a compound Erlang distribution. corresponds to the value of that gives the minimum expected total inventory holding and backordering cost. Subsequently, an empirical investigation is conducted to compare this method with the Newsvendor model. Results verify its superiority in cost savings. Ultimately, considering the complicated calculation and low efficiency of that algorithm, this paper proposes an approximation and a heuristic algorithm which have a higher level of utility in a real industrial context. e approximation algorithm simplifies the calculation process by reducing iterative times while the heuristic algorithm achieves it by generalizing the relationship between the optimal order quantity and mean demand interarrival rate . 1. Introduction When managing electric power material inventories, compa- nies need to decide when to order and how much to order so as to maintain the appropriate inventory level. Considering that a majority of electric power materials exhibit an intermit- tent demand pattern, effective management of these electric power materials is essential to electric power companies. Intermittent demand appears at random, with time intervals during which no demand occurs. ese characteristics make inventory control a rather challenging task. ese electric power materials are usually characterized by low usage fre- quency, long intervals, and irregular demand. Furthermore, demand arrivals and sizes vary with usage and maintenance patterns , further adding to the degree of intermittence. Moreover, the historical data of electric power materials are limited, coupled with constraints on the forecasting method, demand forecast for these electric power materials is difficult. However, such electric power materials are usually essential components of expensive electric equipment. In the case of equipment failure, when electric power materials are unavailable and need to be ordered immediately, the cost of these urgent orders is higher than regular orders, significantly increasing the maintenance cost. In addition to this, the failure of a component in the electrical equipment is likely to lead to equipment failure, generating a high backordering cost and economic loss. On the other hand, excessive stock of electric power materials leads to backlogs and erosion, resulting in a significant waste of capital as well as the corresponding corporate cost. Some electric power materials stored in warehouses for a long time erode before they are put into use, while some are permanently idle due to the continuous changes and updates of electrical equipment. Since traditional inventory control is generally designed for regular demand, its application to intermittent demand is likely to lead to frequent stock-out events or excessive stock, Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 6085342, 23 pages https://doi.org/10.1155/2018/6085342

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Page 1: Determining the Optimal Order Quantity with Compound

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

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Lamda

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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

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

0

05

1

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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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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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8361533

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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

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Qlowast

Qlowast

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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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Page 2: Determining the Optimal Order Quantity with Compound

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

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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

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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

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

0

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x 104 Beta=60000

0

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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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Figure 9 Continued

Mathematical Problems in Engineering 17

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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

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700800

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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

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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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Page 3: Determining the Optimal Order Quantity with Compound

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

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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

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Beta=800

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

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x 104 Beta=60000

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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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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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14

1325

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199286

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55226032

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203292

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8361533

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44975035

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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

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58

1113

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1920

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1223

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7079

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326397

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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

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330401

470537

603668

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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

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Qlowast

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Qlowast

Qlowast

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Qlowast

Qlowast

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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

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634703

7701401

19802524

30443543

40254493

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1287621713

0592

1324

3444

5464

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92101

187268

346421

495566

536705

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19882536

30583560

40454516

49755422

1296121891

063

1324

3444

5464

7383

92101

191270

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497497

640709

7771415

20002552

30793585

40754450

50135464

1310522063

0613

1324

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5464

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20092563

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1318822231

0623

1324

3445

5564

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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

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20402604

31443663

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1349822856

0663

1324

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193274

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506506

652723

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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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Page 4: Determining the Optimal Order Quantity with Compound

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

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12

14

Q05 1 150

Lamda

05 1 150Lamda

10

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05 1 150Lamda

5

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05 1 150Lamda

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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

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0

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0

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0

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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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Figure 9 Continued

Mathematical Problems in Engineering 17

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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

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700800

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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

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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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Page 5: Determining the Optimal Order Quantity with Compound

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

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12

14

Q05 1 150

Lamda

05 1 150Lamda

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05 1 150Lamda

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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

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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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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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8361533

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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

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Qlowast

Qlowast

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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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Page 6: Determining the Optimal Order Quantity with Compound

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

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Beta=30

0 05 1 15Lamda

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Beta=50

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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

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

0

05

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x 104 Beta=60000

0

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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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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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8361533

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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

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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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Page 7: Determining the Optimal Order Quantity with Compound

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

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Beta=10

0 05 1 15Lamda

Beta=20

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Beta=30

0 05 1 15Lamda

Beta=40

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Beta=50

6

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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

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Beta=1000

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

0

05

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x 104 Beta=60000

0

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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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Figure 9 Continued

Mathematical Problems in Engineering 17

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359438

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360438

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1324

3546

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665737

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20952679

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7685

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32793825

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682757

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370451

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21672775

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55196028

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14

1325

3647

5767

7787

97107

199286

370451

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684759

8331526

21682776

33603923

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55226032

1486025699

115

1425

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7888

98108

202291

376459

539618

696761

8361533

21782789

33753942

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55526064

1496925941

125

1425

3747

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99109

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377460

540619

697761

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33763944

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135

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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

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Qlowast

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0031

58

1113

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3946

5359

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315357

397436

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15631766

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1018

2633

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6270

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6776

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1323

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1323

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7281

9099

183262

337410

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1230020305

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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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Page 8: Determining the Optimal Order Quantity with Compound

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

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Beta=10

0 05 1 15Lamda

Beta=20

0 05 1 15Lamda

Beta=30

0 05 1 15Lamda

Beta=40

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Beta=50

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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

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

0

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x 104 Beta=60000

0

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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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Figure 9 Continued

Mathematical Problems in Engineering 17

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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

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700800

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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

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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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Page 9: Determining the Optimal Order Quantity with Compound

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

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0 05 1 15Lamda

Beta=6

0 05 1 15Lamda

Beta=7

0 05 1 15Lamda

Beta=8

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Beta=9

0 05 1 15Lamda

Beta=10

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0 05 1 15Lamda

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6

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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

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0 05 1 15Lamda

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0 05 1 15Lamda

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0

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

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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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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

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194278

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662735

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20872668

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52765757

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0693

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194279

360438

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32313768

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1324

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7686

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361440

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32503792

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1325

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7686

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43244835

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0744

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1325

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8271515

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681755

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0934

1325

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1325

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97107

199286

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55016008

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1480025570

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1325

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21642771

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5767

7787

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1325

3647

5767

7787

97107

199286

370451

530608

684758

8321526

21672775

33573920

44674999

55196028

1484625669

14

1325

3647

5767

7787

97107

199286

370451

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684759

8331526

21682776

33603923

44705002

55226032

1486025699

115

1425

3647

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7888

98108

202291

376459

539618

696761

8361533

21782789

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55526064

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125

1425

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99109

202292

377460

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8361533

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135

1425

3747

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202292

378460

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44965033

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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

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Qlowast

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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

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2932

3437

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92104

113121

128133

137136

820

00

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00

00

0061

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1721

2529

3236

3942

6093

112128

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167178

187235

230191

12844

00

00

00

0071

713

1822

2731

3539

4346

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128149

168185

200215

228317

353356

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230156

690

00

0081

814

1924

2933

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4650

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142166

188209

228246

263386

458495

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00

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814

2025

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4853

91124

154181

207230

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293446

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684675

653620

00

011

815

2126

3137

4146

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165194

222249

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627721

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00

0111

915

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0121

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105145

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10101102

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6710

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Mathematical Problems in Engineering 19

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Figure 10 119876lowast under varied 120582 120573 when 119896 = 3

20 Mathematical Problems in Engineering

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Figure 11 Continued

Mathematical Problems in Engineering 21

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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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Mathematical Problems in Engineering

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Page 10: Determining the Optimal Order Quantity with Compound

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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Figure 9 Continued

Mathematical Problems in Engineering 17

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359438

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360438

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1324

3546

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665737

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20952679

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7685

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32793825

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682757

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370451

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21672775

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55196028

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14

1325

3647

5767

7787

97107

199286

370451

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684759

8331526

21682776

33603923

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55226032

1486025699

115

1425

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7888

98108

202291

376459

539618

696761

8361533

21782789

33753942

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55526064

1496925941

125

1425

3747

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99109

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377460

540619

697761

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33763944

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135

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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

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Qlowast

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0031

58

1113

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3946

5359

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315357

397436

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15631766

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1018

2633

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6270

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6776

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1323

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1323

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7281

9099

183262

337410

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1230020305

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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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Page 11: Determining the Optimal Order Quantity with Compound

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

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0 05 1 15Lamda

Beta=6

0 05 1 15Lamda

Beta=7

0 05 1 15Lamda

Beta=8

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Beta=9

0 05 1 15Lamda

Beta=10

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0 05 1 15Lamda

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6

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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

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0 05 1 15Lamda

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0 05 1 15Lamda

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0

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0 05 1 15Lamda

0 05 1 15Lamda

x 104 Beta=3000

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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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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

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194278

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662735

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20872668

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52765757

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0693

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194279

360438

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32313768

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1324

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7686

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361440

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32503792

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1325

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7686

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43244835

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0744

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1325

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8271515

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681755

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0934

1325

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1325

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97107

199286

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55016008

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1480025570

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1325

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21642771

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5767

7787

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1325

3647

5767

7787

97107

199286

370451

530608

684758

8321526

21672775

33573920

44674999

55196028

1484625669

14

1325

3647

5767

7787

97107

199286

370451

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684759

8331526

21682776

33603923

44705002

55226032

1486025699

115

1425

3647

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7888

98108

202291

376459

539618

696761

8361533

21782789

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55526064

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125

1425

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99109

202292

377460

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8361533

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135

1425

3747

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202292

378460

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44965033

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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

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Qlowast

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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

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2932

3437

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92104

113121

128133

137136

820

00

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00

00

0061

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1721

2529

3236

3942

6093

112128

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167178

187235

230191

12844

00

00

00

0071

713

1822

2731

3539

4346

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128149

168185

200215

228317

353356

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230156

690

00

0081

814

1924

2933

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4650

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142166

188209

228246

263386

458495

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00

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814

2025

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4853

91124

154181

207230

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293446

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684675

653620

00

011

815

2126

3137

4146

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165194

222249

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627721

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00

0111

915

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0121

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105145

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10101102

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6710

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Mathematical Problems in Engineering 19

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Figure 10 119876lowast under varied 120582 120573 when 119896 = 3

20 Mathematical Problems in Engineering

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Figure 11 Continued

Mathematical Problems in Engineering 21

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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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Page 12: Determining the Optimal Order Quantity with Compound

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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Figure 9 Continued

Mathematical Problems in Engineering 17

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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

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700800

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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

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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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Page 13: Determining the Optimal Order Quantity with Compound

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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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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8361533

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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

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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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Page 14: Determining the Optimal Order Quantity with Compound

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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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

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14

1325

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199286

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55226032

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203292

378461

542621

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8361533

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33783946

44975035

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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

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0031

58

1113

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0422

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0442

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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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Page 15: Determining the Optimal Order Quantity with Compound

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

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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

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3342

5261

7079

8896

178254

328398

467534

599663

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18492351

28273283

37234148

45614962

1160519077

0373

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3343

5261

7079

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329400

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602667

7311322

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28473307

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1172619321

0383

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7079

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0393

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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

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

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Qlowast

Qlowast

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125

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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

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7281

9099

184264

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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

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1377423411

073

1324

3545

5665

7585

94104

194277

357435

512512

659731

8011463

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52315707

1383823540

0713

1324

3545

5665

7585

94104

195277

358437

513513

661732

8031467

20792657

32103743

42594761

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1390023666

0723

1324

3546

5666

7585

95104

195278

359437

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662734

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1396023788

0733

1324

3546

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7685

95104

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664736

8071475

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32303766

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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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Page 16: Determining the Optimal Order Quantity with Compound

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

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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

Qlowast

Qlowast

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30883597

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92101

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349425

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30993610

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190273

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95104

194278

359438

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662735

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20872668

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194279

360438

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361440

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667739

8111483

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32503792

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1325

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7686

96106

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362441

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43244835

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0744

1325

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669742

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363442

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522597

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5767

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54625964

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0894

1325

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5767

7787

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8271515

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1325

3646

5767

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7787

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681755

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21562760

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0934

1325

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1325

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44494979

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1325

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5767

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97107

199286

369450

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682757

8301522

21612767

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44534983

55016008

1478325534

0964

1325

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369450

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1480025570

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1325

3647

5767

7787

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199286

369451

530607

683758

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21642771

33533915

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0984

1325

3647

5767

7787

97107

199286

370451

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683758

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21662773

33553918

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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

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8361533

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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

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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

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Figure 10 Continued

Mathematical Problems in Engineering 19

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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

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0 05 1 15Lamda

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5

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400

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Figure 11 Continued

Mathematical Problems in Engineering 21

0 05 1 15Lamda

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0 05 1 15Lamda

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0 05 1 15Lamda

Beta=300

0 05 1 15Lamda

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0 05 1 15Lamda

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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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Page 17: Determining the Optimal Order Quantity with Compound

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

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Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

Qlowast

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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

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20542624

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51765646

1367123247

0623

1324

3545

5565

7584

94103

192275

355433

509583

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20592631

31773703

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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

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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

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2228

3439

4550

5560

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182217

249279

309337

363588

759897

10101102

11781239

12881325

6710

0131

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2229

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4651

5762

109151

190226

260293

324353

382625

816972

11031214

13091389

14561512

12330

0141

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3541

4753

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197234

270304

337369

399659

8661040

11881316

14271524

16091682

17410

0151

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279315

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414690

9131101

12651408

15351647

17471836

2203225

0161

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2430

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4955

6166

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Mathematical Problems in Engineering 19

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Figure 10 119876lowast under varied 120582 120573 when 119896 = 3

20 Mathematical Problems in Engineering

0 05 1 15Lamda

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Figure 11 Continued

Mathematical Problems in Engineering 21

0 05 1 15Lamda

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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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Page 18: Determining the Optimal Order Quantity with Compound

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

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690

00

0081

814

1924

2933

3742

4650

85115

142166

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228246

263386

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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

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Qlowast

Qlowast

Qlowast

Qlowast

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Qlowast

Qlowast

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0552

1324

3444

5463

7282

91100

185265

341415

487558

627694

7611383

19522488

29983488

39614420

48675302

1260221146

0562

1324

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5463

7382

91100

185266

343416

489560

629697

7641389

19622500

30143507

39834446

48955333

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0572

1324

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91100

186267

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632700

7671395

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30293525

40054470

49225364

1278721529

0582

1324

3444

5464

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91100

187268

345420

493564

634703

7701401

19802524

30443543

40254493

49495393

1287621713

0592

1324

3444

5464

7383

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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

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500500

644714

7821426

20172574

31063618

41134594

50625519

1326922394

0633

1324

3545

5564

7483

93102

192272

351427

502502

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7851431

20252584

31193633

41314612

50855545

1334822553

0643

1324

3545

5565

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93102

193273

352429

503503

648718

7881436

20322594

31313648

41494635

51085570

1342422706

0653

1324

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93102

193273

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1324

3545

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654725

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1324

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1324

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073

1324

3545

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512512

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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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Page 19: Determining the Optimal Order Quantity with Compound

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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Submit your manuscripts atwwwhindawicom

Page 20: Determining the Optimal Order Quantity with Compound

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

Page 21: Determining the Optimal Order Quantity with Compound

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

Page 22: Determining the Optimal Order Quantity with Compound

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

Page 23: Determining the Optimal Order Quantity with Compound

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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Page 24: Determining the Optimal Order Quantity with Compound

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