Research Article Application of Two-Phase Fuzzy ...downloads.hindawi.com/journals/mpe/2015/780830.pdfResearch Article Application of Two-Phase Fuzzy Optimization Approach to Multiproduct

Research ArticleApplication of Two-Phase Fuzzy Optimization Approach toMultiproduct Multistage Integrated Production Planning withLinguistic Preference under Uncertainty

Shan Lu1 Hongye Su1 Lian Xiao2 and Li Zhu3

1State Key Laboratory of Industrial Control Technology Institute of Cyber-Systems and Control Zhejiang UniversityHangzhou 310027 China2The West Pipeline Company of CNPC Urumqi 830012 China3Liaoning Key Lab of Advanced Control Systems for Industry Equipment Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Hongye Su hysuiipczjueducn

Received 27 March 2015 Revised 24 July 2015 Accepted 6 August 2015

Academic Editor Sean Wu

Copyright copy 2015 Shan Lu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper tackles the challenges for a production planning problem with linguistic preference on the objectives in an uncertainmultiproduct multistage manufacturing environment The uncertain sources are modelled by fuzzy sets and involve those inducedby both the epistemic factors of process and external factors from customers and suppliers A fuzzy multiobjective mixed integerprogramming model with different objective priorities is proposed to address the problem which attempts to simultaneouslyminimize the relevant operations cost and maximize the average safety stock holding level and the average service level Theepistemic and external uncertainty is simultaneously considered and formulated as flexible constraints By defining the prioritylevels a two-phase fuzzy optimization approach is used to manage the preference extent and convert the original model into anauxiliary crisp one Then a novel interactive solution approach is proposed to solve this problem An industrial case originatingfrom a steel rolling plant is applied to implement the proposed approach The numerical results demonstrate the efficiency andfeasibility to handle the linguistic preference and provide a compromised solution in an uncertain environment

1 Introduction

Production planning in a multistage multiproduct produc-tion system concerns key production activities of eachinterconnected element Material flows involved start fromsuppliers to the manufacturing network and finally to thecustomers Coordination of lot sizes schedule for each ele-ment contributes to enhancing the utilization of activities thatproduce more values and thus attracts substantial interestsTypically production planning can be categorized into threeclasses in terms of time horizon for decision making long-term medium-term and short-term Long-term planningfocuses on identifying strategic decisions over a relativelylong time such as facility location and extent of additionalinvestments in processing networks Medium-term planninginvolves tactical decisions for the optimal use of variousresources and anticipated lot sizing for production stages

Models for medium-term planning comparatively accountfor the presence of production system topology and concerncontrolling the material flow inventory level transportssuppliers and so forth Short-term planning is related todetailed scheduling operations of a job such as hour-to-hour job sequencing The production planning problem inthis work focuses on a medium-term level which integratesactivities going across the interconnection from suppliers tocustomers

Normally for traditional manufacturers the materialflows within the production system present in two commonforms parallel and spread Parallel flows usually apply tosingle-product structure and switching product families arerather expensive In contrast spread flows are related todiversification in product mix and can provide services forcustomization However the dynamics in market and cus-tomers challenge the manufacturers to response fast and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 780830 20 pageshttpdxdoiorg1011552015780830

2 Mathematical Problems in Engineering

accurately It is usually difficult for the above two produc-tion forms to handle overachievement or underachievementfor the forecasted demand of specific products since theproduction stages within the system are either isolate orless correlate When the demand for a specific productexceeds the production capacity of its dedicated productionstages it may result in backorders or lost sales and on thecontrary it may also lead to a high level on inventory orexcess capacities Therefore many manufacturers nowadaysprefer to the production system in which products could bemanufactured in a flexible way

In this regard we consider a multistage productionsystem in which material and information flows may beinterconnected or interwoven with each other Other thanthe conventional parallel or spread topology structure theaddressed production system allows each production stage toschedule and allocate the items to be produced independentlyupon the information received from the customers In otherwords each product is produced through a designated routeconsisting of a series of production stages and each stagemayprocess items coming from one or more upstream stages

To deal with demand fluctuation from the customersseveral operations strategies have been developed and incor-porated in planning decisions such as backlogging lostsales or inventory holding [1ndash3] These techniques showseveral advantages on obtaining feasible solutions in a moreflexible way However in practical the dynamic natureof production environment and markets induces a highdegree of uncertainty thus increasing the risk for decision-makers Peidro et al [4] identified several metrics for distinctuncertain sources in supply chain planning and typicallyaddressed the uncertainty for supply process and demandIn this paper we consider three types of practical problemsencountered for decision-making (1) epistemic uncertaintythat originates from internal factors due to a lack of knowl-edge in production parameters or coefficients (2) externaluncertainty concerned with customers and suppliers and(3) trade-off among the conflicting objectives with decision-makersrsquo preference which is derived from the complicatedenvironment

Several techniques and researches attempted tomodel theuncertainty in the production planning among which fourmodelling approaches are identified conceptual analyticalartificial intelligence and simulation models [5] Among theresearches stochastic programming and probability distribu-tion show the applicability inmajor production categories [6ndash8] Stochastic programming formulates the uncertain sourcesbased on the random characteristics and applies probabilitydistributions that can be expressed in different forms suchas normal Beta and Poisson [9] Kira et al [10] proposeda stochastic linear programming model to formulate thehierarchical production planning under random demandFleten and Kristoffersen [11] presented a multistage mixed-integer linear stochastic programming model to address ashort-term production plan for a hydropower plant Zanjaniet al [12] developed a hybrid scenario tree with stationaryprobability distributions to integrate the uncertainty in thequality of raw materials and demand However as pointedout by Inuiguchi and Ramık [13] a stochastic programming

problem may not be solved easily since the computationalefforts would be enhanced dramatically with the problemscale Besides when there is a lack of evidence on historicalor statistical data the probabilistic reasoning modellingmethods are not always reliable and appropriate [14]

Alternatively fuzzy set theory and possibility theoryprovide an attractive tool to account for the uncertainty [15]There are several practical advantages of the fuzzy set theorywith respect to handling the uncertainty for productiondecisions (1) It is able to deal with the uncertainty that islack of data or evidence and is thus applicable to ill-definedscenarios (2) It allows the decision-makers to incorporatejudgements for improving the solution interactively (3)Thefuzzy model allows for development of more flexible andreliable decision tools that could be expressed linguisticallybased on human perception (4)The distinct fuzzy member-ship functions provide broader alternatives that are of highcomputational efficiency Bellman and Zadeh [16] originallyintroduced the concept of fuzzy aggregation operators uponwhich Zimmermann [17] presented a fuzzy linear program-ming approach to aggregating the fuzzy goals and constraints

Following the development of fuzzy optimization ap-proaches the fuzzy set theory shows its effectiveness andsuperiority to handle the uncertainty in a production plan-ning problem Fung et al [18] discussed amultiproduct aggre-gate production planning with fuzzy demand and capac-ity They formulated the dynamic balance constraints asfuzzysoft equations and solved the model using paramet-ric programming Wang and Liang [19] developed a fuzzymultiobjective linear programmingmodel for amultiproductaggregate production planning problem in a fuzzy environ-ment The fuzzy objectives were formulated using piecewiselinear membership functions and were aggregated by a maxoperator Wang and Liang [20] then addressed this problemby incorporating the imprecise forecasted demand operatingcost and capacityThey adopted the triangular fuzzy numberto represent the imprecise data and transform the fuzzy objec-tives by minimizing the three prominent points Vasant [21]proposed a fuzzy linear programmingmodel with a modifieds-curve membership function to solve a production planningproblem with vague parameters and objective coefficientsMula et al [22] studied a material resource planning withflexible constraints in a multiproduct multilevel manufactur-ing environment They applied a fuzzy linear programmingapproach with three kinds of aggregation schemes to decom-pose the original model into three fuzzy models Torabi andHassini [23] developed a novel multiobjective possibilisticmixed integer linear programming model for integratingprocurement production and distribution planning Themodel aimed to simultaneously minimize the total cost oflogistics and maximize the total value of purchasing Torabiet al [24] further studied a fuzzy hierarchical productionplanning problem The imprecise parameters along with thesoft constraints are introduced to provide required consis-tency between decisions of the adjacent levels Peidro et al [4]proposed a fuzzy model for a tactical supply chain planningwhich contemplates the different uncertain sources fromsupply demand and process They adopted a linear rankingfunction and triangular fuzzy numbers to transform the fuzzy

Mathematical Problems in Engineering 3

model into a crisp equivalent problem Taghizadeh et al[25] presented a fuzzy multiobjective linear programmingmodel for a multiperiod multiproduct production planningproblem that simultaneously minimizes production cost andmaximizes machine utilization The model is solved usingpiecewise linear membership functions through a nonsym-metric decision for the fuzzy objectives and constraints Alievet al [26] introduced a fuzzy-generic approach for solvingan aggregate production-distribution planning problem ina fuzzy environment Generally the above researches takeadvantage of various tools based on the concepts of fuzzy settheory that are applied in goal programming and softflexiblemathematical programming Other research studies relevantto fuzzy programming models include Hsu and Wang [27]Lan et al [28] Figueroa-Garcıa et al [29] Yaghin et al [30]and Su and Lin [31]

In real-world applications the production plan usuallyconcerns conflicting objectives regarding the performanceevaluated by various factors Preference on the objectivesimposed by the decision-makers which is a critical issue israther difficult to be represented in the conflicting objectivesproperly Fuzzy set theory provides an alternative to expressthe preference that is the more important the objectivethe higher the satisfying degree [32] With respect to theproduction planning in an uncertain environment howeverfew attentions are paid on the decision-makersrsquo preference inthe existing contributions in the literature Conventionallythe weighted additivemethods are used by assigning differentweights for the objectives [33 34] However the decision-makers still encounter the problems of weighing the relativeimportance of each objective since the weights are quitedependent on the preference and are not easy to be specifiedupon the subjective judgment With regard to the objectivepriorities Tiwari et al [35] constructed priority levels byspecifying themembership grades and solved the subproblemfor each level in sequence Chen and Tsai [36] used theconcept of desirable achievement degree to reflect the relativeimportance of goals explicitly that is themore important thegoal the higher the desirable achievement degree Howeverthese models are rigid in nature and may encounter thedifficulties in obtaining a feasible solution when acquiringhigh overall satisfying degrees or desirable preference extentbetween the objectives

In this paper a novel fuzzy multiobjective mixed integerlinear programmingmodel (FMO-MILP) is proposed for themultiproduct multistage production planning problem withlinguistic preference in a fuzzy environment The epistemicand external uncertainty is simultaneously formulated asflexible constraints and is handled by using a weightedaverage method and a fuzzy ranking method The modeldefines two key performance indicators as the objectivesalong with which the economic objectives are assessedin terms of relative importance The model attempts tominimize total cost of production overtime raw materi-als and inventory and in the meanwhile maximize theaverage safety stock holding level and the average servicelevel The production constraints include supply and pro-duction capacity warehouse space production route andproduct family inventory balance and backorders Linguistic

preference on the objectives imposed by the decision-makersis considered and constructed by priority levels To acquiredesirable solutions that fully reflect the preference a two-phase interactive satisfying optimization method is appliedthrough relaxing the overall satisfying degree The FMO-MILP model is transformed into an equivalent crisp modelby treating the objectives and constraints separately Then wedeveloped a novel interactive solution approach to solving theFMO-MILPmodel until a compromised solution is obtainedFinally the proposed approach is implemented on real-worldproduction planning in a steel rolling plant

In summary the main contributions of this work mainlylie in presenting (1) a novel multiproduct multistage produc-tion planning model with linguistic preference and variouspractical constraints simultaneously taking into accountthe epistemic and external uncertainty (2) definition andformulation of average safety stock holding level and averageservice level as fuzzy objectives for key indicators togetherwith fuzzy objectives for the operations cost where prioritylevels are constructed (3) a two-stage interactive solutionapproach to solving the fuzzy model which is able to treat notonly the fuzzy objectives with linguistic preference but alsothe epistemic and external uncertainty in a flexible way

The remainder of this work is organized as followsSection 2 analyses the problem details the assumptions andformulates the FMO-MILP model for the multiproduct mul-tistage production planning problem Subsequently Section 3presents the approach to handling the uncertainty and theobjectives with linguistic preference and develops the two-phase interactive solution approach Next an industrial caseis used to illustrate the applicability andpotential in Section 4Finally conclusions and remarks for further researches aregiven in Section 5

2 Model Formulation

21 Problem Descriptions and Assumptions The multiprod-uct multistage production planning problem addresses asmall or medium sized plant that manufactures varioustypes of products for the customers over a given planninghorizon Each product is processed stage-by-stage througha designated production route As the flexible nature ofthe production topology the production routes specifiedfor different products may be intersected thus complicatingthe modelling To model the production topology eachproduction stage is formulated regarding its successor andpredecessor stages Since the operations parameters or coef-ficients are often incomplete andor unobtainable epistemicuncertainty from the production-inventory system involvesthe available production capacity plannedmaintenance timeproduction efficiency warehouse space and safety stock levelThese uncertain sources are modelled by fuzzy numbersusing possibility theory External uncertainty from both thesuppliers and customers resulting volatile material and infor-mation flows is formulated as flexible constraints by fuzzysets The objectives evaluated by the decision-makers can bedivided into two different categories operations cost and keyperformance indicators concernedTheproposedmodel aimsto simultaneously minimize total cost related to production


and inventory operations and maximize the average safetystock holding level and the average service level In practicesuch complex and conflicting objectives along with differentconstraints imposed by suppliers production topology andcustomers really challenge the efforts on decision-makingThe assessment on each objective is vague when consideringpreference from the decision-makers The preference deci-sions are made in linguistic terms to distinguish the relativeimportance for the objectives such that ldquoaverage servicelevel is the most important during this planning horizonrdquo orldquoaverage safety stock holding level should be approximatelyhigher than certain value and is relatively more importantthan inventory costrdquo Therefore this work also defines andincorporates the relative importance of the objectives in afuzzy environment

The proposed FMO-MILP model is formulated based onthe following assumptions

(1) Items produced can be stored to be used in certainperiods but within the planning horizon The itemsare shipped to either the successor production stagesor customers for delivery

(2) Backorders are allowed for multiple periods butshould be fulfilled within the planning horizon Thisstrategy provides more flexibility for the productionsystem to schedule the resources in a feasible way

(3) Supply capacity for each rawmaterial is estimated andlimited by the suppliersThe total procurement capac-ity for all the raw materials is limited by purchasingdepartment of the manufacturer

(4) Suppliers different production stages and ware-houses are assumed located in close proximity so thelead time among them is ignored

(5) Raw materials work-in-process and end-items arestored in separated warehouses and thus formulatedindependently

(6) Some items can be acted as both work-in-process andend-items according to their usage with regard to theinventory level

(7) Dynamic safety stock strategy is considered by pro-viding a buffer to meet fluctuating customer demand

(8) Priority levels are constructed for all the objectives toreflect decision-makersrsquo judgement and preference onthe relative importance of each objectiveThe prioritystructure may not be constant and is subject to thepreference

Indices sets parameters and decision variables for theproposed FMO-MILP model are defined as follows

Indices are as follows

119895 index for production stages1198951015840 index for the immediate successor productionstages of 119895119894 index for items including raw materials work-in-process and end items

1198941015840 index for the immediate downstream items of 119894119897 index for item families119905 index for time period119904 index for raw material suppliers1199051015840 index for the periods after 11990511990510158401015840 index for the periods before 119905

Sets are as follows

119873 set for all the items119869 set for all the production stages119879 set for time horizon119861 subset of119873 denoting raw materials119882 subset of119873 denoting work-in-process119875 subset of119873 denoting end items119861(119904) set for raw materials suppliers 119904119862(119894) set for the production stages that produce 119894119874(119894) set for the first production stage of 119894119878(119894) set for the suppliers that supply raw material 119894119879(119894) set for the terminal production stage of 119894119871(119895) set for item families produced by productionstage 119895119872(119895) set for the items produced by production stage119895119865(119897) set for items involved in family 119897Ω(119895 119894) set for the immediate predecessor productionstages of 119895 for producing 119894Π(119895 119894) set for the items produced from 119894 by produc-tion stage 119895

Deterministic parameters are as follows

119860119905 hours in period 119905

cb119894 cost of supplying one unit of raw material 119894

cba119894 cost of backlogging one unit of end item 119894 for one

time periodcib119894 cost of holding one unit of raw material 119894 in

warehouse for one time periodciw119894 cost of holding one unit of work-in-process 119894 in

warehouse for one time periodcip119894 cost of holding one unit of end item 119894 in

warehouse for one time periodco119905 overtime cost per hour in period 119905

cp119894119895 cost of producing one unit of 119894 at stage 119895

cs119894119895 setup cost of stage 119895 for producing 119894

119877 large positive number119879119860119895119905 planned regular production capacity of stage 119895

in period 119905 (in hours)


119879119872119895119905 planned maintenance time of stage 119895 in period

119905 (in hours)119879119878119897119895 setup time for family 119897 at stage 119895

120573119894 penalty coefficient for backorders that are fulfilled

in the next planning horizonV119894 warehouse capacity occupied by one unit quantity

of the item 119894

Fuzzy parameters are as follows

120578119894119895 production efficiency of stage 119895 for producing 119894

119872119863119894119905 demand for end item 119894 isin 119875 in period 119905

119880119861119894119904119905 supply capacity of raw material 119894 for supplier

119904 in period 119905

119880119877119861119904119905 raw material supply capacity for supplier 119904

in period 119905

119880119868119861119905 raw material warehouse capacity in period 119905

119880119868119875119905 end item warehouse capacity in period 119905

119880119868119882119905 work-in-process warehouse capacity in period

119905119878119868119875119894119905 end item safety stock in period 119905

Continuous variables are as follows

119868119861119894119905 inventory level for rawmaterial 119894 isin 119861 in period 119905

119868119875119894119905 inventory level for end item 119894 isin 119875 in period 119905

119868119875minus119894119905 end item inventory deviation below the safety

stock level in period 119905119868119882119894119905 inventory level for work-in-process 119894 isin 119882

119871119872119897119895119905 minimum production quantity of family 119897 at

stage 119895 in period 119905119872119861119860

1198941199051199051015840 backlogging quantity of end item 119894 isin 119875 in

period 119905 that will be fulfilled in period 1199051015840119872119871119894119905 quantity of end item 119894 isin 119875 that fails to meet the

order demand in period 119905119872119861119894119904119905 supply quantity of raw material 119894 isin 119861 by

supplier 119904 in period 119905119872119875119894119905 quantity of end item 119894 isin 119875 shipped to the

warehouse in period 119905119872119883119894119895119905 production quantity of item 119894 isin 119882cup119875 at stage

119895 in period 119905119872119883119865119897119895119905 production quantity of family 119897 at stage 119895 in

period 119905119879119874119895119905 overtime production capacity at stage 119895 in

period 119905119876119861119894119905101584010158401199051015840 quantity of raw material 119894 isin 119861 shipped to the

warehouse in period 11990510158401015840 and which will be stored to beused in period 1199051015840

119876119875119894119905101584010158401199051015840 quantity of end item 119894 isin 119875 shipped to the

warehouse in period 11990510158401015840 and which will be stored tobe used in period 1199051015840119876119882119894119905101584010158401199051015840 quantity of work-in-process 119894 isin 119882 shipped

to thewarehouse in period 11990510158401015840 andwhichwill be storedto be used in period 1199051015840119880119872119897119895119905 maximum production quantity of family 119897 at

stage 119895 in period 119905

Integer variables are as follows

119864119894119905 binary variable that indicates whether the end

items shipped in period 119905 will be stored to be used inthe future119866119894119905 binary variable that indicates whether the end

item demand in period 119905 will be backlogged119909119897119895119905 binary variable that indicates whether item family

119897 is produced at stage 119895 in period 119905

22 Fuzzy Programming Model

221 Objective Functions Two conflicting objective cate-gories are considered simultaneously in the FMO-MILPmodel the operations cost and the key indicators concerned

(1) Objectives for the Operations Cost The operations costincludes those of production overtime raw materials andinventory holding Accordingly the following describes theseobjective functions that are expected to be minimized

min 1199111cong sum119894isin119872(119895)

sum119895isin119869

119879

sum119905=1

cp119894119895sdot 119872119883119894119895119905

+ sum119897isin119871(119895)

sum119895isin119869

119879

sum119905=1

cs119897119895sdot 119883119897119895119905

(1)

min 1199112cong sum119895isin119869

119879

sum119905=1

co119905sdot 119879119874119895119905 (2)


sum119904isin119878(119894)

119879

sum119905=1

cb119894119904sdot 119872119861119894119904119905 (3)


119879

sum119905=1

cib119894sdot 119868119861119894119905+ sum119894isin119882

119879

sum119905=1

ciw119894sdot 119868119882119894119905

+sum119894isin119875

119879

sum119905=1

cip119894sdot 119868119875119894119905

(4)

The symbol ldquocongrdquo represents the fuzzified version of ldquo=rdquoand reads ldquoessentially equal tordquo The production cost in (1) isgenerated by processing and setup activities The processingcost is calculated specific to each product and productionstage independently Setup cost is only related to changeoversbetween products of a family where similar products aregrouped Overtime production is allowed but relatively will


incur high cost in labour as defined in (2) Raw materialsare purchased from diverse suppliers with different prices asspecified in (3) Additionally inventory holding cost in (4)involves warehouses for raw materials work-in-process andend-items

(2) Objectives for Key Indicators Decisions on operationsassessed by cost are those objectives that could be measuredby economic performance easily and immediately Howeversome objectives having indirect effects on the cost are notsuitable to evaluate them by economic performance such ascustomer satisfaction and planning nervousness [37] Con-ventional researches attempted to estimate the parameters bycost and unify the measuring unit so that the multiobjectiveoptimization problem can be managed through a simpleadditive method interactive methods or fuzzymethods [38ndash40] Typical examples include estimation on backloggingpenalty and lost sale penalty [41] These estimations are usu-ally empirical andmay not satisfy the dynamic and diversifiedrequirements since the production environment and marketare rather volatile With regard to these problems this workformulates the objectives independently and carries out twokey indicators as noneconomic measured objectives that arefrequently encountered in the real application average safetystock holding level and average service level To incorporatethe above indicators into the model the following definitionsare given

(i) Safety stock holding level (SHL) is as follows

SHL119894() =

119879

sum119905=1

(1 minus119868119875minus119894119905

119878119868119875119894119905

) sdot100

119879forall119894 isin 119875 (5)

(ii) Average safety stock holding level (ASHL) is asfollows

ASHL () = sum119894isin119875

SHL119894

120593 (119875) (6)

SHL evaluates deviation below the safety stock level foreach product Generally ASHL measures the average level ofsafety stock holding for all the products where 120593(119875) indicatesthe number of elements in set 119875

(iii) Service level (SL) is as follows

SL119894() = max0 [1

minussum119879minus1

119905=1sum119879

1199051015840gt119905(1199051015840 minus 119905) sdot 119872119861119860

1198941199051199051015840 + sum119879

119905=1120573119894sdot 119872119861119860

119894119905119879+1

sum119879

119905=1119872119863119894119905

]

sdot 100 forall119894 isin 119875

(7)

(iv) Average service level (ASL) is as follows

ASL () = sum119894isin119875

SL119894

120593 (119875) (8)

SL concerns customer satisfactory in terms of backlog-ging level for each product It should be noted that thebackorders to be fulfilled in the next planning horizonhave a greater impact on those backorders that are fulfilledduring the current planning horizon Thus a weight penaltycoefficient 120573

119894is added for this scenario Similarly ASL defines

the average level on customer satisfactory for all the productsFollowing the concepts of the above indicators the

decision-makers seek to enhance the service and keep thesafety stock level which are presented by the objectivefunctions

max 1199115cong ASHL (9)

max 1199116cong ASL (10)

222 Constraints

(1)Material Balance for RawMaterialThe rawmaterials fromvarious suppliers are stored in the warehouse to be usedfor further processing In other words material flow of rawmaterial is controlled by both suppliers and initial productionstages The following constraints are applied to these twoaspects for raw material balance

119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905minus 119876119861119894119905119905

forall119894 isin 119861 forall119905 = 1 119879 minus 1

(11)

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905

forall119894 isin 119861 119905 = 119879 (12)

119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum

1198941015840isinΠ(119895119894)

sum119895isin119874(119894)

1198721198831198941015840119895119905

119894 isin 119861 forall119905 (13)

Equations (11) and (12) ensure the material balance atraw material warehouse Due to the operations strategy rawmaterials are allowed to be stored for multiple periods withinthe planning horizon Therefore raw materials suppliedduring the last period of planning horizon 119905 = 119879 would onlybe used for processing as shown in (12) Equation (13) statesthat the raw materials stored in the warehouse are only usedat initial production stages

sum119894isin119861

119872119861119894119904119905le 119880119877119861

119904119905forall119904 119905 (14)

119872119861119894119904119905le 119880119861

119894119904119905forall119894 119904 119905 (15)

Equation (14) represents the procurement capacity limitfrom the perspective of manufactures This restriction maybe related to the shipment capacity or budget Equation (15)implies that the supply capacity for each supplier is alsolimited by anupper boundThese twobounds are often lack offully knowledge and thus are expressed as fuzzy coefficients

(2) Material Balance for Work-in-Process Similarly materialflow of work-in-process is controlled by its upstream and


downstreamflows especially by its successor and predecessorproduction stages

119879

sum

1199051015840gt119905

1198761198821198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

minus sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905minus119872119875

119894119905

forall119894 isin 119882 | 1198941015840 1198951015840notin forall119905 = 1 119879 minus 1

(16)

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

= sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905+119872119875

119894119905

forall119894 isin 119882 | 1198941015840 1198951015840notin 119905 = 119879

(17)

119872119875119894119905= 0 forall119894 isin 119882 minus 119875 forall119905 (18)

Equation (16) presents material balance of the work-in-process warehouse for period 119905 = 1 119879 minus 1 inwhich two adjacent production stages are modelled throughconstructing sets with respect to their logical relationshipsEquation (17) shows the material balance for the last period119905 = 119879 where the items shipped to the warehouse should beused during the current periodOn account of the productiontopology items produced by a certain production stage maybe either stored for further processing or shipped for deliveryin terms of requirements In this regard (18) specifies thatthose items only for further processing are not allowed to beshipped to the end item warehouse

(3) Material Balance for End Items Consider

119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119894119905minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

cong

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1minus 119868119875minus

119894119905minus1minus

119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119894119905forall119894 isin 119875 forall119905 = 1 119879 minus 1

(19)

119878119868119875119894119905minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

cong

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1

minus 119868119875minus

119894119905minus1+119872119875

119894119905minus119872119863

119894119905

forall119894 isin 119875 119905 = 119879

(20)

119872119875119894119905= sum119895isin119862(119894)

119872119883119894119895119905

forall119894 isin 119875 minus119882 forall119905 (21)

Equations (19) and (20) formulate the material balance atthe end itemwarehouse for periods 119905 = 1 119879minus1 and 119905 = 119879respectively Safety stock is incorporated in the equations toprovide a buffer against uncertainty where the inventory levelis allowed to fluctuate around the safety stock level Hencedeviation below the safety stock is utilized to relax this restric-tion on the end item warehouse Equation (21) indicates therelation ofmaterial flow into the end itemwarehouse with theitems produced by terminal production stages In real-worldscenarios product demand along with the safety stock isusually forecasted and estimated upon the historical statisticsand cannot reflect the real market precisely Therefore thesetwo variables are treated as fuzziness

(4) Inventory Level Constraints Due to the fact that rawmate-rials work-in-process and end items are stored separatelyunder different inventory management strategies inventoryconstraints should be formulated independently upon thedynamic characteristics

119868119861119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119861119894119905101584010158401199051015840 sdot V119894


(22)

119868119861119894119905ge 0 forall119894 isin 119861 119905 = 119879 (23)

sum119894isin119861

119868119861119894119905le 119880119868119861

119905forall119905 (24)

Since themodel depicts exact usage of rawmaterials fromone period to another lower boundary of the raw materialinventory level for periods 119905 = 1 119879 minus 1 and 119905 = 119879 canbe obtained by (22) and (23) respectivelyThe inventory levelshould be essentially less than a limited storage capacity asspecified by (24)

119868119882119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119882119894119905101584010158401199051015840 sdot V119894


(25)

119868119882119894119905ge 0 forall119894 isin 119882 119905 = 119879 (26)

sum119894isin119882

119868119882119894119905le 119880119868119882

119905forall119905 (27)

Similarly (25)ndash(27) imply the upper limit and lower limitfor the work-in-process warehouse

119868119875minus

119894119905ge 119878119868119875119894119905+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


(28)

119868119875minus

119894119905ge 119878119868119875119894119905minus 119868119875119894119905forall119894 isin 119875 119905 = 119879 (29)

119868119875minus

119894119905le 119878119868119875119894119905forall119894 isin 119875 forall119905 (30)

sum119894isin119875

119868119875119894119905le 119880119868119875

119905forall119905 (31)


The inventory level for the end items should be appropri-ately controlled to satisfy the fluctuating demand by utilizingthe flexible strategy of safety stock Equation (28) ensures thatthe end item inventory level for period 119905 = 1 119879minus1 shouldcover both surplus parts of the safety stock and those enditems stored for future use and (29) states the same restrictionfor period 119905 = 119879 However the buffer capacity of safetystock is also limited by the setting level itself thus the lowerdeviation is restricted by (30) Equation (31) represents thestorage capacity for the end item warehouse

(5) Production Capacity Constraints Consider

119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905

forall119897 isin 119871 (119895) forall119895 forall119905 (32)

119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905


Various items are grouped in families by product andprocessing specification to reduce capacity consumptioncaused by frequent setup Equation (32) defines attribution setand quantity relation to the families with the items Equations(33) and (34) specify the logical relationship between binarysetup and production quantity of the families

sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905

forall119895 forall119905

(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905

forall119895 forall119905(36)

The production capacity is consumed by both regularprocessing and setup Equation (35) ensures that the totalcapacity occupied is not allowed to exceed the maximumlevel which involves planned regular capacity and overtimecapacity as well Since the capacity is measured by timethe upper bound given by (35) should be further restrictedby physical time and maintenance plans in a period asshown by (36) Typically (35) and (36) are modelled asflexible constraints due to the uncertainty existing in capacityallocation and efficiency

(6) Backlogging Constraints Consider

119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 forall119894 isin 119875 forall119905 = 1 119879 minus 1 (37)

119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905


119864119894119905+ 119866119894119905le 1 forall119894 isin 119875 forall119905 = 1 119879 minus 1 (39)

Although inventory holding and backorders may be sus-tained for several periods for each product they are notallowed to concurrent during a period Equations (37)ndash(39)

follow ldquothe best I can dordquo policy that is the decision-makers should make the most of inventory to satisfy thecustomers before considering backorders [42] In particular(37) and (38) convert the inventory holding and backordersinto two binary variables and then (39) can simply restrictthe concurrence It should be noted that according to theoperations strategy the end items shipped to the warehouseduring the last period of the planning horizon can be onlyused within the same period however backorders may befulfilled during the next planning horizon In other wordsthe concurrence would not exist during period 119905 = 119879 and canbe neglected

(7) Decision Variable Constraints Consider

119883119897119895119905isin 0 1 forall119897 isin 119871 (119895) forall119895 forall119905 (40)

119864119894119905 119866119894119905isin 0 1 forall119894 isin 119875 forall119905 (41)

119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)

Equations (40) and (41) indicate the integral constraintsOther decision variables default to nonnegativity as shown in(42)

23 Model Comparisons The proposed FMO-MILP modelfor the multiproduct multistage production planning prob-lem is novel which considers meaningful technical featuresThe following Table 1 shows the qualitative comparisons ofthe proposed approach with some representative techniquesfor themultiproductmultistage production planning includ-ing Torabi and Hassini [23] Peidro et al [43] and Jamal-nia and Soukhakian [44] Several significant findings andadvantages of the proposed approach can be summarized asfollows First the proposed model involves several practicaltechniques to formulate the meaningful characteristics inthe multiproduct multistage production planning problemsuch as constructing the production route using set theoryallowing backorders for multiple periods and holding thesafety stock level These technical factors provide more flexi-bility in dealing with the variants on production and marketdemand Second the proposed model defines two significantindicators ASHL and ASL as maximized objectives togetherwith the operations cost which are treated to be fuzzy Theconstruction of ASHL and ASL presents the requirementsfrom both manufactures and customers directly instead ofquantifying them by cost Third the multiple objectives havedifferent priorities described by linguistic terms which canfully express the preference from decision-makers The two-stage interactive solution approach provides a flexible tool todeal with the objective priority where the preference extentcan be quantified Finally the proposed model simultane-ously considers the epistemic and external uncertainty whichaffects such a multiproduct multistage production environ-ment reflecting the uncertainty from production-inventorysystem suppliers and customersDifferent strategies by fuzzy


Table 1 Comparison of different models for the multiproduct multistage production planning

Factor Torabi and Hassini Peidro et al Jamalnia and Soukhakian The proposed approachObjective function Multiple Single Multiple MultipleObjective for cost Min Min Min MinAdditional objectives Max total value for purchasing Not considered Max customer satisfaction Max ASHL and ASLObjective priority Not considered Not considered Considered ConsideredPreference extent Not considered Not considered Not considered ConsideredProduct family Not considered Considered Considered ConsideredEpistemic uncertainty Considered Considered Considered ConsideredExternal uncertainty Considered Considered Not considered ConsideredSafety stock Considered Not considered Not considered ConsideredBackorder Not considered Allowed for single period Allowed for single period Allowed for multiperiodProduction route Not considered Not considered Not considered ConsideredOvertime Not considered Not considered Considered Considered

set theory and possibility theory are jointly developed to dealwith the fuzzy constraints

3 Solution Strategy

In the proposed problem the uncertainty is expressed asfuzzy numbers to construct the FMO-MILP model In thissection the original FMO-MILPmodel can be converted intoan equivalent crispmultiobjectivemixed integer linearmodelby treating the constraints in different forms independentlyThenwe apply a two-phase fuzzymultiobjective optimizationapproach to handling the objectives taking into accountlinguistic preference

31 Handling the Uncertain Sources In the FMO-MILPmodel the fuzzy numbers represent the uncertainty on eitherone side or both sides of the constraintsThe constraints withfuzzy numbers on one side hold a strict equality since theconverted crisp constraints by defuzzing the uncertainty arestill rigid By contrast those constraints with fuzzy numberson both sides may be rather flexible due to the fact that thefuzzy numbers come from different sources

In the literature fuzzy numbers can be modelled as vari-ous membership functions in terms of application scenariossuch as triangular trapezoidal monotonic linear piecewiselinear and S-shape [45] Here we adopt triangular mem-bership function to construct the fuzzy numbers modelledin the constraints As stated by Liang [46] and Li et al[47] the triangular fuzzy number is commonly adopted dueto its ease in simplistic data acquisition and flexibility offuzzy arithmetic operations Decision-makers are easy toestimate the maximum and minimum deviations from themost likely value Generally a triangular fuzzy number can becharacterized based on the following three prominent values[45]

(i) The most likely value that defines the most belongingmember to the set of available values (membershipdegree = 1 if normalized)

(ii) The most pessimistic value that defines the leastbelongingmember to the set of available values on theminimum accepted value

(iii) The most optimistic value that defines the leastbelongingmember to the set of available values on themaximum accepted value

Considering the above definitions on prominent valuesthe membership functions for the triangular fuzzy numbersin the 119896th fuzzy constraint can be represented as follows

120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906

119896refer to the most likely most pessimistic

and most optimistic value for the 119896th fuzzy constraintrespectively and 120583

119888119896

means the membership function forfuzzy number 119888

119896 If inducing the minimal acceptable degree

120572 isin [0 1] the prominent values can be reformulated asfollows

119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906

119896120572denote the most likely most pes-

simistic and most optimistic value with the minimal accept-able degree respectivelyThen themembership functions canbe reconstructed with respect to these three values

Recalling the original model (14) (15) (24) (27) (28)(29) (30) and (31) are those constraints with fuzzy numberson one side Tanaka and Asai [48] handled the triangularfuzzy number by using the weighted average of the mostpessimistic value and most optimistic value Alternativelythe weighted average method is utilized to treat the fuzzynumbers and construct crisp constraints [25 49] Then


the above constraints can be converted into the followingequations

sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

minus 119868119875119894119905forall119894 isin 119875 119905 = 119879

119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906

119896represent the correspondingweights of

the most likely most pessimistic and most optimistic valuefor the 119888

119896th fuzzy number respectively The most likely value

with the highest membership should therefore be assigned tomoreweights than the othersWhen theweights for the abovethree prominent values are given the solution can be furthercompromised by adjusting the minimal acceptable degree 120572

On the other hand (19) (20) (35) and (36) representthe fuzzy numbers on both sides of the constraints In thiscontext we treat the constraints based on comparison of theircorresponding fuzzy numbers which is referred to as fuzzyranking In the literature several fuzzy ranking methodsfor comparing fuzzy numbers are given in terms of variousclassification schemes [50 51]We consider the concept of set-inclusion along with the fuzzy ranking method proposed byRamık and ımanek [52] Then (19) (20) (35) and (36) canbe converted into crisp constraints by ordering the left-hand

and right-hand side fuzzy numbers The equivalent auxiliaryconstraints are expressed as

119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897

119894119905minus1120572minus 119868119875minus


119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)

32 Treatment of the Objectives with Linguistic PreferenceThe objective functions in the FMO-MILP model concernboth the operations cost and the key indicators which areconflicting and noncommensurable In such a fuzzy envi-ronment decision-makers are even more difficult to obtainfeasible solutions as desired among the multiple objectivesMembership approaches and fuzzy set theory are efficienttools to reformulate the objective functions as commensu-rable units Conventionally the relative importance of theobjectives is either specifiedwith aweight vector orand given

satisfying degrees [23 53] or compared by the correspondingsatisfying degrees [44] The above methods may lead to lowcomputational efficiency and additional constraints may betoo strict to obtain satisfying even feasible solution Thus weapply the interactive satisfying method by Li and Hu [32] toovercome these shortcomings where the satisfying degree isregarded as optimization variables and relaxed using varying-domain optimization method [54] The trade-off among theconflicting objectives with linguistic preference can then beachieved by relaxing the overall satisfying degree through aninteractive procedure To do so we implement a two-phasefuzzy optimization approach to solving the proposed problemwith linguistic preference

321 Constructing the Priority Levels and Objective Member-ship Functions Decision-makers usually use linguistic termsto express the preference such that objective A is moreimportant than B Considering the fact that the objectivewith higher priority has higher desirable achievement theobjectives involved can be grouped into different prioritylevels according to the preference To quantify as prior-ity levels the FMO-MILP problem can be constructedas

min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)

where 119875119899| 1 le 119899 le 119871 119899 isin 119885 is a set for priority levels in

sequence of relative importance 119911119899ℎ119899

(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ

119899isin 119885 represents a set of objectives in the

119899th priority level and 119866 refers to system constraintsBymeans of fuzzy set theory the objective functions con-

cerned can be generally expressed in the form of 119911119892(119909) le 119911

119892or

119911119892(119909) ge 119911

119892 which means the objective 119911

119892(119909) is approximately

less than or greater than the aspiration level 119911119892 respectively

To describe the conflict in the objectives a monotonicallydecreasing linear membership function is used to normalizethe fuzzy objectives in (1)ndash(4)

120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)

where 119911119897119892for forall119892 = 1 4 denotes the highest possible

value with the membership degree of 1 and 119911119906119892for forall119892 =

1 4 is the smallest possible value with the membershipdegree of 0 Accordingly the fuzzy objectives modelled by

(9) are identified by fuzzy sets having an increasing linearmembership function

120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)

where 119911119897119892for forall119892 = 5 6 denotes the highest possible value

with the membership degree of 0 and 119911119906119892for forall119892 = 5 6 is the

smallest possible value with the membership degree of 1

322 Phase I Model Phase I is to obtain the maximumoverall satisfying degree for all the objectives without linguis-tic preference Introduced by Bellman and Zadeh [16] theminmax operator can aggregate the objective membershipfunctions using a new variable 120582 isin [0 1] According to thetreatment on the fuzzy constraints and the objective mem-bership functions aforementioned the auxiliary formulationfor the FMO-MILP model regardless of linguistic preferencecan be given by


max 120582

st eqs (10)ndash(12) (15)ndash(17) (20)ndash(22) (24) (25) (31)ndash(33) (36)ndash(41) (44)-(45)

119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)

However the optimal solution solved by the logical minoperator in model (50) provides no preference judged by thedecision-makers When considering the relative importancebetween the objectives the subjective constraints inducedmay lead to a decline in the overall satisfying degree

323 Phase II Model In phase II the maximum overallsatisfying degree 120582lowast yielded in phase I will be relaxedto certain extent by the varying-domain method so thatthe linguistic preference could be involved By inducing arelaxation parameterΔ120575 isin [0 1] the overall satisfying degreecan be relaxed as shown by the following for the objectives1199111 119911

6

120582lowastminus Δ120575 le 120583

119911119892

forall119892 = 1 6 (51)

The extent of relaxation corresponds to the priorityimposed by 120583

119911119892

To compare the relative importance between

the objectives in terms of preference the varying-domainmethod is applied by using a relaxation parameter 120574 isin

[minus1 0] to extend differences among the objective prioritiesConcerning the priority levels defined in (47) the satisfyingdegree comparison can be stated by

120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899

le 120574 forall119899 ge 2 ℎ119899isin 1 119898

119899 (52)

where 120583119911119899

ℎ119899

means the satisfying degrees for the objectives inthe 119899th priority level Equation (52) ensures that the objectiveslocated in the higher priority level will hold higher satisfyingdegrees In this regard phase II also uses min operator toquantify the preference extent Then the FMO-MILP modelcan be further formulated by incorporating the priority levelsand represented as follows

min 120574


membership functions for 119911119892 forall119892 = 1 6


119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1

le 120574 forall1 le 119899 le 119871 ℎ119899isin 1 119898

119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1

minus 1 le 120574 le 0

(53)

In the above model (53) 120574 is considered to quantify theextent of the relative importance defined by the priority levelswhich is relatively dependent on relaxation parameter Δ120575When the relaxation on 120582lowast is not enough the preferenceexpressed by the objective priorities may not be achievedand in this case 120574 = 0 Therefore relaxation Δ120575 should bealtered incrementally until the preference extent is achievedThis criterion relies on decision-makerrsquos judgement since themore the preference extent the lower the overall satisfyingdegree

33 Interactive Solution Procedure The proposed approachconcerns treating both the fuzzy constraints in differ-ent forms and incorporating the linguistic preference inthe objectives Decision-makers can take fully advantageof the flexible parameters to achieve a preferred trade-off between the overall satisfying degree and objectivepriorities In summary the original fuzzy model withlinguistic preference is solved through a novel interac-tive method which incorporates decision-makersrsquo judge-ment


Step 1 Model the fuzzy numbers in the constraints usingtriangular possibility distributions by specifying the threeprominent values

Step 2 Given the value of minimal acceptable degree 120572convert the fuzzy constraints (14) (15) (24) (27) (28) (29)(30) and (31) into crisp ones using the weighted averagemethod as shown in (45) and simultaneously concert thefuzzy constraints (19) (20) (35) and (36) into crisp onesusing the fuzzy ranking method

Step 3 Model the fuzzy objectives 1199111 1199112 1199113 1199114 and 119911

5 1199116

using themonotonically decreasing linear membership func-tion in (48) and themonotonically increasing linearmember-ship function in (49) respectively

Step 4 Regardless of linguistic preference on the objectivesaggregate the fuzzy model using a min operator 120582 isin [0 1]into a crisp phase I model Then the corresponding overallsatisfying degree 120582lowast can be obtained by solving the model asshown in model (50)

Step 5 Construct the priority levels for the objective func-tions 119911

1 119911

6 in terms of linguistic preference

Step 6 Relax the overall satisfying degree by (51) and formu-late the priority comparison by (52)

Step 7 Assume that the relaxation parameter Δ120575 = 0 andaggregate the FMO-MILPmodel and objective priorities intoa crisp phase II model

Step 8 Solve the crisp model (53) and the minimal value ofpreference extent 120574 can be obtained

Step 9 Check whether the preference extent obtained inStep 7 satisfies the relative importance by the decision-makers If 120582 = 0 which means all the objectives are withthe same importance and dissatisfy the linguistic preferencethen go to Step 10 If 120582 lt 0 but the preference extent isnot satisfied by the decision-makers then go to Step 10 aswell Otherwise stop the interactive procedure and the resultobtained provides a compromise solution

Step 10 Relax the overall satisfying degree 120582lowast obtained inStep 4 by augmenting the value of parameter Δ120575 and thengo back to Step 8

4 Case Study

41 Case Description This study investigates a steel rollingplant as a real-world case to illustrate the applicability of theproposed fuzzy programming model and evaluate the per-formance of the solution approach The plant manufacturessmall and midsize steel plates classified in 10 types Theseproducts are generally processed from 3 types of steel slabs asthe rawmaterials going through different production routesThe production system consists of 6 major production stagesand the production route for each product is constructed by acollection of these production stages according to the process

Table 2 Raw material and product matching matrix

Raw material 119894 isin 119861 Product 119894 isin 1198751198751 1198752 1198753 1198754 1198755 1198756 1198757 1198758 1198759 11987510

1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0

Table 2 presents the raw material and product matchingmatrix in which pair (119894 isin 119861 119894 isin 119875) = 1 indicates thatthe product 119894 isin 119875 is manufactured by the raw material119894 isin 119861 and otherwise (119894 isin 119861 119894 isin 119875) = 0 Table 3depicts the production stage and product matching matrixin which pair (119895 119894 isin 119875) = 1 implies that the production ofproduct 119894 isin 119875 goes through stage 119895 and otherwise (119895 119894 isin119875) = 0 It should be noted that the product is processedin sequence of 119895 = 1 2 6 constituting a productionroute Further we can describe the work-in-process for eachproduct upon its specific production route In addition theoutput items processed by production stage 119895 are indicatedin terms of each product Then in Table 3 for example wecan observe that product 11987510 is manufactured through aproduction route 1198691-1198692-1198694-1198695-1198696 with corresponding outputwork-in-process 1198691119863 1198692119863 1198694119860 1198695119865 1198696119861 by each produc-tion stage Concerning the production families modelled in(32)-(33) Table 4 gives the family structure with regard toeach production stage and its output items

The case addresses a planning horizon of 7 periods Theuncertainty in the constraints modelled by the triangularfuzzy numbers is defined according to the most likely val-ues with their deviations Table 5 gives the relevant fuzzyparameters within a certain range of intervals in which thethree prominent values by (43) are generatedThe operationalstrategy implies that the production planning is made ina make-to-order and make-to-stock environment It meansthat the products are manufactured based on the ordersreceived and in the meantime to keep the safety stock Thedynamic demand is forecasted for each period prior to theplanning horizon and is given in Table 6

42Model Implementation Recalling the interactive solutionprocedures aforementioned the triangular fuzzy numbersin the constraints can be easily converted into equivalentcrisp ones using the given prominent values by the weightedaveragemethod and fuzzy rankingmethod In this regard themodel concerns two classes of important parameters mini-mal acceptable degree 120572 and weights for the three prominentvalues (119908119897

119896 119908119898119896 119908119906119896) Considering that the decision-makers

usually assignmoreweights to themost likely values this casespecifies (119908119897

119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06

Membership functions for all the fuzzy objectives involvetwo prominent values that are subjectively evaluated Sincethe objectives 119911

1 1199112 1199113 1199114 related to cost that is difficult to

be quantified directly this work simply uses the result of addi-tive method for treating 119911

1 1199112 1199113 1199114 without considering

1199115 1199116 as a benchmark to construct the membership func-

tions On the other hand the objectives 1199115 1199116 concerning


Table 3 Production stage and product matching matrix

Production stage 119895 Product 119894 isin 1198751198751 1198752 1198753 1198754 1198755 1198756 1198757 1198758 1198759 11987510

1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861

Table 4 Item family structure

Family 119897 Items grouped 119865(119897)1 1198691119860 11986911198612 1198691119862 11986911198633 1198691119864

4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861

Table 5 Range of fuzzy parameter values

Fuzzyparameters Range of values Fuzzy

parametersRange ofvalues

120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)

general operations indicators are easily formulated based onthe historical statistics Then the membership functions forthe case study are constructed as follows

(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)

Followed by the solution procedures the crisp phase Imodel is implementedwith the abovemembership functionsThe model is solved using LINGO 110 optimization softwareon an Inter 25 GHz CPU The maximum overall satisfyingdegree 120582lowast can be obtained and is equal to 0916 Then inphase II preference on the objectives is incorporated and thedecision-makers in the case use linguistic terms to categorizethe objectives into four levels very important somewhat


Table 6 Demand forecast for each product

Product 119894 isin 119875 Demand forecast (119872119863119897119894119905119872119863119898

119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)

important important and general And the priority levels areconstructed as follows

Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112

In the context of the above priority structure and mem-bership functions phase II model is then implemented asfollows

min 120574


membership functions for 120583119911119892

forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)

43 Results and PerformanceAnalysis It is obvious that phaseII model is MILP problem and is subject to the value ofrelaxation parameter Δ120575 Initially we solve phase II modelassuming Δ120575 = 0 by LINGO 110 and get 120574 = 0 The solutionis equal to that obtained by phase I model which conforms tothe consistency with respect to phase I and II models In thiscase all the objectives are considered coequally without anypriorities

To seek a compromised solution and investigate theeffects of relaxation phase IImodel is conducted interactivelyby increasing Δ120575 linearly Since Δ120575 is used to relax theoverall satisfying degree so that the objective priorities couldbe incorporated a relative large value is not desirable Thenumerical experiment tests phase II model with a maximumvalue of Δ120575 = 030 Then a list of feasible solutions regardingthe priority structure can be identified as shown in Table 7


Table 7 Optimal results of phase II model under different relaxation levels

Δ120575 120574 Individual satisfying degree Individual objectives003 minus00168 (09028 08860 09028 09028 09197 09365) (3485388 42850 4458303 154859 7759 9405)006 minus00336 (08896 08560 08896 08896 09231 09567) (3492949 43600 4466262 155522 7769 9435)009 minus00502 (08762 08260 08762 08762 09264 09767) (3500552 44350 4474265 156189 7779 9465)012 minus00668 (08628 07960 08628 08628 09295 09963) (3508227 45100 4482344 156862 7789 9494)015 minus00780 (08440 07660 08440 08440 09220 10000) (3518918 45850 4493598 157800 7766 9500)018 minus00880 (08240 07360 08240 08240 09120 10000) (3525426 46600 4505598 158800 7736 9500)021 minus00980 (08040 07060 08040 08040 09020 10000) (3541718 47350 4517598 159800 7706 9500)024 minus01080 (07840 06760 07840 07840 08920 10000) (3553118 48100 4529598 160800 7676 9500)027 minus01180 (07640 06460 07640 07640 08820 10000) (3564518 48850 4541598 161800 7646 9500)030 minus01280 (07440 06160 07440 07440 08720 10000) (3575918 49600 4553598 162800 7616 9500)

It is clear that in each test the satisfying degrees obtainedaccordwith priorities among the objectives As the parameterΔ120575 increases several significant implications with regard tothe proposed approach can be explored First the extent vari-able 120574 decreases monotonously but with a diminished rate Arelaxation on the overall satisfying degree would enlarge thepreference extent between the objectivemembership degreeshowever the effect decreaseswhen intensifying the relaxationSecond the satisfying degrees for 119911

1 1199112 1199113 1199114 decrease and

those for 1199116 increase The satisfying degrees for 119911

5 show

a trend of ascending followed by leveling off from the valueof Δ120575 = 012 approximately on which the satisfying degreefor 1199116 is maximized Third the minimum satisfying degree

among the objectives is equal to the overall satisfying degree120582lowastlowast in phase II For example when Δ120575 = 009 120582lowastlowast = 120582lowast minusΔ120575 = 0826 and min

119892(120583119911119892

) = 1205831199112

= 0826 It indicatesthe consistency between the overall satisfying degree andthe relaxation degree Fourth the objectives follow the sametrend as the satisfying degrees and are between the twoprominent values with the membership degree of 0 and1 Taking a comprehensive view of the above results thedecision-makers can select a preferred solution consideringthe preference extent and individual satisfying degree

In order to analyze the effectiveness of proposed approachin terms of handling linguistic preference for each objectivewe define three performance measure indicators (1) therange of satisfaction degree (RSD) (2)maximum preferenceextent between the objectives and (3)minimum PE betweenthe objectives RSD is used to measure the maximum differ-ence between the satisfying degrees for all the objectives andis defined as follows [23]

RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)

Here preference extent refers to the difference betweenthe satisfying degrees for the objectives of adjacent prioritylevels With reference to the definition of the priority levelsin (47) we define the maximum and minimum preferenceextents as follows

PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin

005 01 015 02 025 03 0350Relaxation parameter Δ120575

0

005

01

015

02

025

03

035

04

045

05

(deg

)

Figure 1 RSD versus preference extent

PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)

Since the minimum satisfying degree among the objec-tives equals the overall satisfying degree the indicator RSDalso implies the level of consistency between the preferenceextent and the satisfying degrees For example if the decision-makers consider a larger preference extent the higher RSDwill be desirable As seen in Figure 1 the RSD and preferenceextents are compared with differentΔ120575 Figure 1 presents thatthe maximum preference extent holds the same level as theminimumpreference extent and is equal to the obtained valueof the extent variable 120574 It means the degree of differencebetween the objectives of the adjacent priority levels keeps thesameTheRSD shows a trend of ascendingwith an increase inΔ120575 The larger the value of RSD is the larger the difference ofthe satisfying degrees between the objectives of the adjacentpriority levels is Noticeably the ascending rate for RSD slowsdown when Δ120575 goes around 015 on which the objective


Routine operations costASHL

76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106

Figure 2 Operations cost versus ASHL

Routine operations costASL

938

941

944

947

95

953

ASL

()

03501 015 02 025 030050Relaxation parameter Δ120575

8

81

82

83

84

85

Ope

ratio

ns co

st

times106

Figure 3 Operations cost versus ASL

1199116 in the highest priority level has reached its maximum

satisfying degree So the increase of RSD when Δ120575 ge 015 isoriginated from the decrease of the overall satisfying degree

Obviously obtaining a decision vector that complies withthe decision-makers is subject to the relaxation degree witha broad range of possible values though the solutions are allfeasible To gain insights into the performance of the aboveresults we evaluate the objectives in a more efficient way asan auxiliary tool for determining the relaxation degree Asthe objectives 119911

1 1199112 1199113 1199114 are measured in terms of cost

we define the operations cost which is the sum of all the costfor processing setup overtime raw material and inventoryFigures 2 and 3 respectively present the operations cost withregard to the two key indicators ASHL() and ASL() underdifferent relaxation degrees A higher relaxation degree ofthe overall satisfying degree implies an increased operations

cost Regarding the ASHL in Figure 2 decisions with Δ120575 le012 should be selected since the ASHL drops off whenΔ120575 gt 012 As seen in Figure 3 the ASL rises dramaticallywith relaxation and reaches maximum value at approximateΔ120575 = 015 Therefore decisions with Δ120575 le 015 aresuperior regarding the ASL Taking into account the abovetwo scenarios an increase of relaxation degree when Δ120575 gt015 will enhance the operations cost without improving theALS but deteriorate the ASHL in the meanwhile Rationallythe decision-makers should choose a compromised solutionwith Δ120575 le 015

44 Comparisons To further illustrate the effectiveness of theproposed approach we implement the addressed productionplanning problem using different approaches by which fourapplication scenarios are set as follows

Scenario I Implement the model by the proposed two-phasefuzzy optimization approach with Δ120575 = 013

Scenario II Implement the model applying the additivemethod by Chen and Tsai [36] to deal with the objectivepriorities

Scenario III Remove the objective priorities from the modeland implement it with an aggregating ldquomaxrdquo operator asshown in phase I model

Scenario IV Regard the uncertain conditions as their deter-ministic versions and remove the objective priorities from themodel

As seen in Table 8 the results deriving from the fourscenarios are compared and several implications can befound First from the view of the individual satisfying degreethe results obtained in Scenario I can fully present thepreference defined in the priority structure And the ScenarioI obtains the largest value of RSD which implies a high levelof preference difference among the objectives By contrastthe objective priorities obtained by Scenario II are actuallycategorized into two levels in terms of the satisfying degreeswhich cannot properly express the preference imposed by thedecision-makers The results of PEmax and PEmin also reflectthe above implication Since Scenarios III and IV ignore thepreference on the objectives the values of RSD are rathersmall Second the higher the level of the preference extentis the smaller the overall satisfying degree is In ScenarioI the results show the highest level of the preference extentbut obtain the smallest overall satisfying degree ThereforeScenarios III and IV regardless of preference gain largeroverall satisfying degrees Third there are trade-off relationsbetween the objectives for the operations cost and the twoindicators Scenario I gains higher values in ASHL andASL where greater preference is attached by the decision-makers and increases the operations cost as a compromise Incomparison Scenarios II III and IV obtain lower values ofthe two indicators and the operations cost as well It impliesthat the preference would affect the values of the objectivesaccording to the individual priority structure imposed Con-sequently the proposed approach as applied in Scenario I


Table 8 Result comparison under different scenarios

Scenario I Scenario II Scenario III Scenario IVOperations cost 8199399 8132932 8124575 8119202Processing 3511318 3496019 3477878 3479953Overtime 45350 42896 42100 42190Raw material 4485598 4440631 4450397 4442677Inventory 157133 153386 154200 154382ASHL 7786 7797 7748 7737ASL 9500 9398 9374 9369

Individual satisfying degree (08576 07860 0857308573 09287 10000)

(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)

Overall satisfying degree 07860 08842 09169 09124RSD 02140 00481 0 00165PEmax 00716 00481 mdash mdashPEmin 00713 0 mdash mdash

shows its efficiency in expressing the linguistic preference bythe decision-makers and in the meantime provides a flexibletool to deal with trade-off among the conflicting objectives

5 Conclusions

In industrial practice production planning in an integratedframework for a multistage multiproduct manufacturingsystem should concern coordinated decisions in an uncertainenvironment This work develops a novel fuzzy multiobjec-tive mixed integer linear programming model to considerthe uncertainty in various forms using fuzzy set theoryand simultaneously handles multiple imprecise conflictingobjectives with linguistic preference imposed by the decision-makers The proposed fuzzy model aims to minimize thetotal cost of processing setup overtime raw material andinventory and in the meanwhile maximize the average safetystock holding level and the average service level Further atwo-phase fuzzy optimization approach is used to obtain acompromised solution with respect to the linguistic prefer-ence for the objectivesThe fuzzy model is then implementedon an industrial case and solved through a novel interactivesolution procedure The numerical experiment shows theeffectiveness of making decisions on production planningin such an uncertain environment along with decision-makersrsquo preference The results also indicate that the pro-posed approach can quantify the preference extent betweenthe objective priorities in a flexibleway so that a compromisedsolution is more obtainable

Finally some suggestions or directions for furtherresearches can be considered One of the main concerns isto qualify or quantify the fuzzy objectives using linguisticterms since it is not easy for the decision-makers to specifythe prominent values for the fuzzy membership functionsIn addition this work assumes trapezoidal and triangularmembership functions to respectively formulate the fuzzynumbers in the constraints and the fuzzy objectives Futureresearches can express uncertain sources in different formsapplying specific fuzzy membership functions such as piece-wise linear or S-shape membership functions Moreover

the lead time which existed is usually uncertain in natureHow to coordinate the production regarding the uncertainlead time between the adjacent production stages is practi-cally important

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (NSFC 61320106009) and the 111 Project ofChina (B07031)

References

[1] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008

[2] J-P Kenne A Gharbi andM Beit ldquoAge-dependent productionplanning and maintenance strategies in unreliable manufac-turing systems with lost salerdquo European Journal of OperationalResearch vol 178 no 2 pp 408ndash420 2007

[3] S Helber F Sahling and K Schimmelpfeng ldquoDynamic capac-itated lot sizing with random demand and dynamic safetystocksrdquo OR Spectrum vol 35 no 1 pp 75ndash105 2013

[4] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[5] JMula R Poler J P Garcıa-Sabater and F C Lario ldquoModels forproduction planning under uncertainty a reviewrdquo InternationalJournal of Production Economics vol 103 no 1 pp 271ndash2852006

[6] A Gupta C D Maranas and C M McDonald ldquoMid-termsupply chain planning under demand uncertainty customerdemand satisfaction and inventory managementrdquo Computers ampChemical Engineering vol 24 no 12 pp 2613ndash2621 2000


[7] T Santoso S Ahmed M Goetschalckx and A Shapiro ldquoAstochastic programming approach for supply chain networkdesign under uncertaintyrdquo European Journal of OperationalResearch vol 167 no 1 pp 96ndash115 2005

[8] A Azaron K N Brown S A Tarim and M Modarres ldquoAmulti-objective stochastic programming approach for supplychain design considering riskrdquo International Journal of Produc-tion Economics vol 116 no 1 pp 129ndash138 2008

[9] J R Birge and F Louveaux Introduction to Stochastic Program-ming Springer Berlin Germany 2011

[10] D Kira M Kusy and I Rakita ldquoA stochastic linear program-ming approach to hierarchical production planningrdquo Journal ofthe Operational Research Society vol 48 no 2 pp 207ndash211 1997

[11] S-E Fleten and T K Kristoffersen ldquoShort-term hydropowerproduction planning by stochastic programmingrdquoComputersampOperations Research vol 35 no 8 pp 2656ndash2671 2008

[12] M K Zanjani M Nourelfath and D Ait-Kadi ldquoA multi-stagestochastic programming approach for production planningwith uncertainty in the quality of raw materials and demandrdquoInternational Journal of Production Research vol 48 no 16 pp4701ndash4723 2010

[13] M Inuiguchi and J Ramık ldquoPossibilistic linear programminga brief review of fuzzy mathematical programming and acomparison with stochastic programming in portfolio selectionproblemrdquo Fuzzy Sets and Systems vol 111 no 1 pp 3ndash28 2000

[14] D Petrovic R Roy and R Petrovic ldquoSupply chain modellingusing fuzzy setsrdquo International Journal of Production Economicsvol 59 no 1 pp 443ndash453 1999

[15] A L Guiffrida and R Nagi ldquoFuzzy set theory applications inproduction management research a literature surveyrdquo Journalof Intelligent Manufacturing vol 9 no 1 pp 39ndash56 1998

[16] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[17] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[18] R Y K Fung J Tang and D Wang ldquoMultiproduct aggregateproduction planning with fuzzy demands and fuzzy capacitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part ASystems and Humans vol 33 no 3 pp 302ndash313 2003

[19] R-C Wang and T-F Liang ldquoApplication of fuzzy multi-ob-jective linear programming to aggregate production planningrdquoComputers amp Industrial Engineering vol 46 no 1 pp 17ndash412004

[20] R-C Wang and T-F Liang ldquoApplying possibilistic linear pro-gramming to aggregate production planningrdquo InternationalJournal of Production Economics vol 98 no 3 pp 328ndash3412005

[21] P M Vasant ldquoFuzzy production planning and its application todecision makingrdquo Journal of Intelligent Manufacturing vol 17no 1 pp 5ndash12 2006

[22] JMula R Poler and J P Garcia ldquoMRPwith flexible constraintsa fuzzy mathematical programming approachrdquo Fuzzy Sets andSystems vol 157 no 1 pp 74ndash97 2006

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] S A Torabi M Ebadian and R Tanha ldquoFuzzy hierarchicalproduction planning (with a case study)rdquo Fuzzy Sets andSystems vol 161 no 11 pp 1511ndash1529 2010

[25] K Taghizadeh M Bagherpour and I Mahdavi ldquoApplicationof fuzzy multi-objective linear programming model in a multi-period multi-product production planning problemrdquo Interna-tional Journal of Computational Intelligence Systems vol 4 no2 pp 228ndash243 2011

[26] R A Aliev B Fazlollahi B G Guirimov and R R AlievldquoFuzzy-genetic approach to aggregate production-distributionplanning in supply chain managementrdquo Information Sciencesvol 177 no 20 pp 4241ndash4255 2007

[27] H-M Hsu andW-PWang ldquoPossibilistic programming in pro-duction planning of assemble-to-order environmentsrdquo FuzzySets and Systems vol 119 no 1 pp 59ndash70 2001

[28] Y-F Lan Y-K Liu and G-J Sun ldquoModeling fuzzy multi-period production planning and sourcing problem with cred-ibility service levelsrdquo Journal of Computational and AppliedMathematics vol 231 no 1 pp 208ndash221 2009

[29] J C Figueroa-Garcıa D Kalenatic and C A Lopez-BelloldquoMulti-period mixed production planning with uncertaindemands fuzzy and interval fuzzy sets approachrdquo Fuzzy Setsand Systems vol 206 pp 21ndash38 2012

[30] R G Yaghin S A Torabi and S M T F Ghomi ldquoIntegratedmarkdown pricing and aggregate production planning in atwo echelon supply chain a hybrid fuzzy multiple objectiveapproachrdquo Applied Mathematical Modelling vol 36 no 12 pp6011ndash6030 2012

[31] T-S Su and Y-F Lin ldquoFuzzy multi-objective procurementproduction planning decision problems for recoverable man-ufacturing systemsrdquo Journal of Manufacturing Systems 2014

[32] S Li and C Hu ldquoTwo-step interactive satisfactory method forfuzzy multiple objective optimization with preemptive priori-tiesrdquo IEEE Transactions on Fuzzy Systems vol 15 no 3 pp 417ndash425 2007

[33] R N Tiwari S Dharmar and J R Rao ldquoFuzzy goal program-mingmdashan additive modelrdquo Fuzzy Sets and Systems vol 24 no1 pp 27ndash34 1987

[34] C-C Lin ldquoAweightedmaxndashminmodel for fuzzy goal program-mingrdquo Fuzzy Sets and Systems vol 142 no 3 pp 407ndash420 2004

[35] R N Tiwari S Dharmar and J R Rao ldquoPriority structure infuzzy goal programmingrdquo Fuzzy Sets and Systems vol 19 no 3pp 251ndash259 1986

[36] L-H Chen and F-C Tsai ldquoFuzzy goal programming withdifferent importance and prioritiesrdquo European Journal of Oper-ational Research vol 133 no 3 pp 548ndash556 2001

[37] J Mula D Peidro and R Poler ldquoThe effectiveness of afuzzy mathematical programming approach for supply chainproduction planning with fuzzy demandrdquo International Journalof Production Economics vol 128 no 1 pp 136ndash143 2010

[38] A P Kanyalkar and G K Adil ldquoAn integrated aggregateand detailed planning in a multi-site production environmentusing linear programmingrdquo International Journal of ProductionResearch vol 43 no 20 pp 4431ndash4454 2005

[39] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[40] S-P Chen andW-L Huang ldquoAmembership function approachfor aggregate production planning problems in fuzzy environ-mentsrdquo International Journal of Production Research vol 48 no23 pp 7003ndash7023 2010

[41] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability finite production and partial backordering and


lost salerdquo European Journal of Operational Research vol 144 no3 pp 677ndash685 2003

[42] E Perea-Lopez B E Ydstie and I E Grossmann ldquoA modelpredictive control strategy for supply chain optimizationrdquoComputers and Chemical Engineering vol 27 no 8-9 pp 1201ndash1218 2003

[43] D Peidro J Mula M Jimenez andM del Mar Botella ldquoA fuzzylinear programming based approach for tactical supply chainplanning in an uncertainty environmentrdquo European Journal ofOperational Research vol 205 no 1 pp 65ndash80 2010

[44] A Jamalnia and M A Soukhakian ldquoA hybrid fuzzy goal pro-gramming approach with different goal priorities to aggregateproduction planningrdquo Computers amp Industrial Engineering vol56 no 4 pp 1474ndash1486 2009

[45] H Rommelfanger ldquoFuzzy linear programming and applica-tionsrdquo European Journal of Operational Research vol 92 no 3pp 512ndash527 1996

[46] T-F Liang ldquoApplication of fuzzy sets to manufacturingdis-tribution planning decisions in supply chainsrdquo InformationSciences vol 181 no 4 pp 842ndash854 2011

[47] D F Li J X Nan andM J Zhang ldquoA ranking method of trian-gular intuitionistic fuzzy numbers and application to decisionmakingrdquo International Journal of Computational IntelligenceSystems vol 3 no 5 pp 522ndash530 2010

[48] H Tanaka and K Asai ldquoFuzzy linear programming problemswith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[49] Y-J Lai andC-LHwang ldquoAnew approach to somepossibilisticlinear programming problemsrdquo Fuzzy Sets and Systems vol 49no 2 pp 121ndash133 1992

[50] C-H Yeh and H Deng ldquoA practical approach to fuzzy utilitiescomparison in fuzzy multicriteria analysisrdquo International Jour-nal of Approximate Reasoning vol 35 no 2 pp 179ndash194 2004

[51] G Bortolan and R Degani ldquoA review of some methods forranking fuzzy subsetsrdquo Fuzzy Sets and Systems vol 15 no 1 pp1ndash19 1985

[52] J Ramık and J ımanek ldquoInequality relation between fuzzynumbers and its use in fuzzy optimizationrdquo Fuzzy Sets andSystems vol 16 no 2 pp 123ndash138 1985

[53] K-M Tsai S-Y You Y-H Lin and C-H Tsai ldquoA fuzzy goalprogramming approach with priority for channel allocationproblem in steel industryrdquoExpert Systems with Applications vol34 no 3 pp 1870ndash1876 2008

[54] C-F Hu C-J Teng and S-Y Li ldquoA fuzzy goal programmingapproach to multi-objective optimization problem with priori-tiesrdquo European Journal of Operational Research vol 176 no 3pp 1319ndash1333 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


accurately It is usually difficult for the above two produc-tion forms to handle overachievement or underachievementfor the forecasted demand of specific products since theproduction stages within the system are either isolate orless correlate When the demand for a specific productexceeds the production capacity of its dedicated productionstages it may result in backorders or lost sales and on thecontrary it may also lead to a high level on inventory orexcess capacities Therefore many manufacturers nowadaysprefer to the production system in which products could bemanufactured in a flexible way

In this regard we consider a multistage productionsystem in which material and information flows may beinterconnected or interwoven with each other Other thanthe conventional parallel or spread topology structure theaddressed production system allows each production stage toschedule and allocate the items to be produced independentlyupon the information received from the customers In otherwords each product is produced through a designated routeconsisting of a series of production stages and each stagemayprocess items coming from one or more upstream stages

To deal with demand fluctuation from the customersseveral operations strategies have been developed and incor-porated in planning decisions such as backlogging lostsales or inventory holding [1ndash3] These techniques showseveral advantages on obtaining feasible solutions in a moreflexible way However in practical the dynamic natureof production environment and markets induces a highdegree of uncertainty thus increasing the risk for decision-makers Peidro et al [4] identified several metrics for distinctuncertain sources in supply chain planning and typicallyaddressed the uncertainty for supply process and demandIn this paper we consider three types of practical problemsencountered for decision-making (1) epistemic uncertaintythat originates from internal factors due to a lack of knowl-edge in production parameters or coefficients (2) externaluncertainty concerned with customers and suppliers and(3) trade-off among the conflicting objectives with decision-makersrsquo preference which is derived from the complicatedenvironment

Several techniques and researches attempted tomodel theuncertainty in the production planning among which fourmodelling approaches are identified conceptual analyticalartificial intelligence and simulation models [5] Among theresearches stochastic programming and probability distribu-tion show the applicability inmajor production categories [6ndash8] Stochastic programming formulates the uncertain sourcesbased on the random characteristics and applies probabilitydistributions that can be expressed in different forms suchas normal Beta and Poisson [9] Kira et al [10] proposeda stochastic linear programming model to formulate thehierarchical production planning under random demandFleten and Kristoffersen [11] presented a multistage mixed-integer linear stochastic programming model to address ashort-term production plan for a hydropower plant Zanjaniet al [12] developed a hybrid scenario tree with stationaryprobability distributions to integrate the uncertainty in thequality of raw materials and demand However as pointedout by Inuiguchi and Ramık [13] a stochastic programming

problem may not be solved easily since the computationalefforts would be enhanced dramatically with the problemscale Besides when there is a lack of evidence on historicalor statistical data the probabilistic reasoning modellingmethods are not always reliable and appropriate [14]

Alternatively fuzzy set theory and possibility theoryprovide an attractive tool to account for the uncertainty [15]There are several practical advantages of the fuzzy set theorywith respect to handling the uncertainty for productiondecisions (1) It is able to deal with the uncertainty that islack of data or evidence and is thus applicable to ill-definedscenarios (2) It allows the decision-makers to incorporatejudgements for improving the solution interactively (3)Thefuzzy model allows for development of more flexible andreliable decision tools that could be expressed linguisticallybased on human perception (4)The distinct fuzzy member-ship functions provide broader alternatives that are of highcomputational efficiency Bellman and Zadeh [16] originallyintroduced the concept of fuzzy aggregation operators uponwhich Zimmermann [17] presented a fuzzy linear program-ming approach to aggregating the fuzzy goals and constraints

Following the development of fuzzy optimization ap-proaches the fuzzy set theory shows its effectiveness andsuperiority to handle the uncertainty in a production plan-ning problem Fung et al [18] discussed amultiproduct aggre-gate production planning with fuzzy demand and capac-ity They formulated the dynamic balance constraints asfuzzysoft equations and solved the model using paramet-ric programming Wang and Liang [19] developed a fuzzymultiobjective linear programmingmodel for amultiproductaggregate production planning problem in a fuzzy environ-ment The fuzzy objectives were formulated using piecewiselinear membership functions and were aggregated by a maxoperator Wang and Liang [20] then addressed this problemby incorporating the imprecise forecasted demand operatingcost and capacityThey adopted the triangular fuzzy numberto represent the imprecise data and transform the fuzzy objec-tives by minimizing the three prominent points Vasant [21]proposed a fuzzy linear programmingmodel with a modifieds-curve membership function to solve a production planningproblem with vague parameters and objective coefficientsMula et al [22] studied a material resource planning withflexible constraints in a multiproduct multilevel manufactur-ing environment They applied a fuzzy linear programmingapproach with three kinds of aggregation schemes to decom-pose the original model into three fuzzy models Torabi andHassini [23] developed a novel multiobjective possibilisticmixed integer linear programming model for integratingprocurement production and distribution planning Themodel aimed to simultaneously minimize the total cost oflogistics and maximize the total value of purchasing Torabiet al [24] further studied a fuzzy hierarchical productionplanning problem The imprecise parameters along with thesoft constraints are introduced to provide required consis-tency between decisions of the adjacent levels Peidro et al [4]proposed a fuzzy model for a tactical supply chain planningwhich contemplates the different uncertain sources fromsupply demand and process They adopted a linear rankingfunction and triangular fuzzy numbers to transform the fuzzy








2 Model Formulation

















Sets are as follows



119860119905 hours in period 119905
















of the item 119894





119904 in period 119905


in period 119905











stage 119895 in period 119905119872119861119860






















min 1199111cong sum119894isin119872(119895)

sum119895isin119869

119879

sum119905=1

cp119894119895sdot 119872119883119894119895119905

+ sum119897isin119871(119895)

sum119895isin119869

119879

sum119905=1

cs119897119895sdot 119883119897119895119905

(1)


119879

sum119905=1

co119905sdot 119879119874119895119905 (2)


sum119904isin119878(119894)

119879

sum119905=1

cb119894119904sdot 119872119861119894119904119905 (3)


119879

sum119905=1

cib119894sdot 119868119861119894119905+ sum119894isin119882

119879

sum119905=1

ciw119894sdot 119868119882119894119905

+sum119894isin119875

119879

sum119905=1

cip119894sdot 119868119875119894119905

(4)






SHL119894() =

119879

sum119905=1

(1 minus119868119875minus119894119905

119878119868119875119894119905

) sdot100

119879forall119894 isin 119875 (5)


ASHL () = sum119894isin119875

SHL119894

120593 (119875) (6)



SL119894() = max0 [1


119905=1sum119879

1199051015840gt119905(1199051015840 minus 119905) sdot 119872119861119860

1198941199051199051015840 + sum119879

119905=1120573119894sdot 119872119861119860

119894119905119879+1

sum119879

119905=1119872119863119894119905

]


(7)


ASL () = sum119894isin119875

SL119894

120593 (119875) (8)







222 Constraints


119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905minus 119876119861119894119905119905


(11)

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905

forall119894 isin 119861 119905 = 119879 (12)

119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum

1198941015840isinΠ(119895119894)

sum119895isin119874(119894)

1198721198831198941015840119895119905

119894 isin 119861 forall119905 (13)


sum119894isin119861

119872119861119894119904119905le 119880119877119861

119904119905forall119904 119905 (14)

119872119861119894119904119905le 119880119861

119894119904119905forall119894 119904 119905 (15)





119879

sum

1199051015840gt119905

1198761198821198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

minus sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905minus119872119875

119894119905


(16)

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

= sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905+119872119875

119894119905


(17)




119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119894119905minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

cong

119905minus1

sum

11990510158401015840=1



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863


(19)

119878119868119875119894119905minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

cong

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1


119894119905minus1+119872119875

119894119905minus119872119863

119894119905

forall119894 isin 119875 119905 = 119879

(20)

119872119875119894119905= sum119895isin119862(119894)

119872119883119894119895119905




119868119861119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119861119894119905101584010158401199051015840 sdot V119894


(22)

119868119861119894119905ge 0 forall119894 isin 119861 119905 = 119879 (23)

sum119894isin119861

119868119861119894119905le 119880119868119861

119905forall119905 (24)


119868119882119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119882119894119905101584010158401199051015840 sdot V119894


(25)

119868119882119894119905ge 0 forall119894 isin 119882 119905 = 119879 (26)

sum119894isin119882

119868119882119894119905le 119880119868119882

119905forall119905 (27)


119868119875minus

119894119905ge 119878119868119875119894119905+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


(28)

119868119875minus


119868119875minus


sum119894isin119875

119868119875119894119905le 119880119868119875

119905forall119905 (31)




119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905


119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905


(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905




119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905


119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905








119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)







3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















2 Model Formulation

















Sets are as follows



119860119905 hours in period 119905
















of the item 119894





119904 in period 119905


in period 119905











stage 119895 in period 119905119872119861119860






















min 1199111cong sum119894isin119872(119895)

sum119895isin119869

119879

sum119905=1

cp119894119895sdot 119872119883119894119895119905

+ sum119897isin119871(119895)

sum119895isin119869

119879

sum119905=1

cs119897119895sdot 119883119897119895119905

(1)


119879

sum119905=1

co119905sdot 119879119874119895119905 (2)


sum119904isin119878(119894)

119879

sum119905=1

cb119894119904sdot 119872119861119894119904119905 (3)


119879

sum119905=1

cib119894sdot 119868119861119894119905+ sum119894isin119882

119879

sum119905=1

ciw119894sdot 119868119882119894119905

+sum119894isin119875

119879

sum119905=1

cip119894sdot 119868119875119894119905

(4)






SHL119894() =

119879

sum119905=1

(1 minus119868119875minus119894119905

119878119868119875119894119905

) sdot100

119879forall119894 isin 119875 (5)


ASHL () = sum119894isin119875

SHL119894

120593 (119875) (6)



SL119894() = max0 [1


119905=1sum119879

1199051015840gt119905(1199051015840 minus 119905) sdot 119872119861119860

1198941199051199051015840 + sum119879

119905=1120573119894sdot 119872119861119860

119894119905119879+1

sum119879

119905=1119872119863119894119905

]


(7)


ASL () = sum119894isin119875

SL119894

120593 (119875) (8)







222 Constraints


119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905minus 119876119861119894119905119905


(11)

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905

forall119894 isin 119861 119905 = 119879 (12)

119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum

1198941015840isinΠ(119895119894)

sum119895isin119874(119894)

1198721198831198941015840119895119905

119894 isin 119861 forall119905 (13)


sum119894isin119861

119872119861119894119904119905le 119880119877119861

119904119905forall119904 119905 (14)

119872119861119894119904119905le 119880119861

119894119904119905forall119894 119904 119905 (15)





119879

sum

1199051015840gt119905

1198761198821198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

minus sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905minus119872119875

119894119905


(16)

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

= sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905+119872119875

119894119905


(17)




119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119894119905minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

cong

119905minus1

sum

11990510158401015840=1



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863


(19)

119878119868119875119894119905minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

cong

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1


119894119905minus1+119872119875

119894119905minus119872119863

119894119905

forall119894 isin 119875 119905 = 119879

(20)

119872119875119894119905= sum119895isin119862(119894)

119872119883119894119895119905




119868119861119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119861119894119905101584010158401199051015840 sdot V119894


(22)

119868119861119894119905ge 0 forall119894 isin 119861 119905 = 119879 (23)

sum119894isin119861

119868119861119894119905le 119880119868119861

119905forall119905 (24)


119868119882119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119882119894119905101584010158401199051015840 sdot V119894


(25)

119868119882119894119905ge 0 forall119894 isin 119882 119905 = 119879 (26)

sum119894isin119882

119868119882119894119905le 119880119868119882

119905forall119905 (27)


119868119875minus

119894119905ge 119878119868119875119894119905+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


(28)

119868119875minus


119868119875minus


sum119894isin119875

119868119875119894119905le 119880119868119875

119905forall119905 (31)




119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905


119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905


(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905




119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905


119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905








119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)







3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Sets are as follows



119860119905 hours in period 119905
















of the item 119894





119904 in period 119905


in period 119905











stage 119895 in period 119905119872119861119860






















min 1199111cong sum119894isin119872(119895)

sum119895isin119869

119879

sum119905=1

cp119894119895sdot 119872119883119894119895119905

+ sum119897isin119871(119895)

sum119895isin119869

119879

sum119905=1

cs119897119895sdot 119883119897119895119905

(1)


119879

sum119905=1

co119905sdot 119879119874119895119905 (2)


sum119904isin119878(119894)

119879

sum119905=1

cb119894119904sdot 119872119861119894119904119905 (3)


119879

sum119905=1

cib119894sdot 119868119861119894119905+ sum119894isin119882

119879

sum119905=1

ciw119894sdot 119868119882119894119905

+sum119894isin119875

119879

sum119905=1

cip119894sdot 119868119875119894119905

(4)






SHL119894() =

119879

sum119905=1

(1 minus119868119875minus119894119905

119878119868119875119894119905

) sdot100

119879forall119894 isin 119875 (5)


ASHL () = sum119894isin119875

SHL119894

120593 (119875) (6)



SL119894() = max0 [1


119905=1sum119879

1199051015840gt119905(1199051015840 minus 119905) sdot 119872119861119860

1198941199051199051015840 + sum119879

119905=1120573119894sdot 119872119861119860

119894119905119879+1

sum119879

119905=1119872119863119894119905

]


(7)


ASL () = sum119894isin119875

SL119894

120593 (119875) (8)







222 Constraints


119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905minus 119876119861119894119905119905


(11)

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905

forall119894 isin 119861 119905 = 119879 (12)

119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum

1198941015840isinΠ(119895119894)

sum119895isin119874(119894)

1198721198831198941015840119895119905

119894 isin 119861 forall119905 (13)


sum119894isin119861

119872119861119894119904119905le 119880119877119861

119904119905forall119904 119905 (14)

119872119861119894119904119905le 119880119861

119894119904119905forall119894 119904 119905 (15)





119879

sum

1199051015840gt119905

1198761198821198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

minus sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905minus119872119875

119894119905


(16)

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

= sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905+119872119875

119894119905


(17)




119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119894119905minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

cong

119905minus1

sum

11990510158401015840=1



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863


(19)

119878119868119875119894119905minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

cong

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1


119894119905minus1+119872119875

119894119905minus119872119863

119894119905

forall119894 isin 119875 119905 = 119879

(20)

119872119875119894119905= sum119895isin119862(119894)

119872119883119894119895119905




119868119861119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119861119894119905101584010158401199051015840 sdot V119894


(22)

119868119861119894119905ge 0 forall119894 isin 119861 119905 = 119879 (23)

sum119894isin119861

119868119861119894119905le 119880119868119861

119905forall119905 (24)


119868119882119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119882119894119905101584010158401199051015840 sdot V119894


(25)

119868119882119894119905ge 0 forall119894 isin 119882 119905 = 119879 (26)

sum119894isin119882

119868119882119894119905le 119880119868119882

119905forall119905 (27)


119868119875minus

119894119905ge 119878119868119875119894119905+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


(28)

119868119875minus


119868119875minus


sum119894isin119875

119868119875119894119905le 119880119868119875

119905forall119905 (31)




119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905


119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905


(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905




119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905


119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905








119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)







3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














of the item 119894





119904 in period 119905


in period 119905











stage 119895 in period 119905119872119861119860






















min 1199111cong sum119894isin119872(119895)

sum119895isin119869

119879

sum119905=1

cp119894119895sdot 119872119883119894119895119905

+ sum119897isin119871(119895)

sum119895isin119869

119879

sum119905=1

cs119897119895sdot 119883119897119895119905

(1)


119879

sum119905=1

co119905sdot 119879119874119895119905 (2)


sum119904isin119878(119894)

119879

sum119905=1

cb119894119904sdot 119872119861119894119904119905 (3)


119879

sum119905=1

cib119894sdot 119868119861119894119905+ sum119894isin119882

119879

sum119905=1

ciw119894sdot 119868119882119894119905

+sum119894isin119875

119879

sum119905=1

cip119894sdot 119868119875119894119905

(4)






SHL119894() =

119879

sum119905=1

(1 minus119868119875minus119894119905

119878119868119875119894119905

) sdot100

119879forall119894 isin 119875 (5)


ASHL () = sum119894isin119875

SHL119894

120593 (119875) (6)



SL119894() = max0 [1


119905=1sum119879

1199051015840gt119905(1199051015840 minus 119905) sdot 119872119861119860

1198941199051199051015840 + sum119879

119905=1120573119894sdot 119872119861119860

119894119905119879+1

sum119879

119905=1119872119863119894119905

]


(7)


ASL () = sum119894isin119875

SL119894

120593 (119875) (8)







222 Constraints


119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905minus 119876119861119894119905119905


(11)

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905

forall119894 isin 119861 119905 = 119879 (12)

119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum

1198941015840isinΠ(119895119894)

sum119895isin119874(119894)

1198721198831198941015840119895119905

119894 isin 119861 forall119905 (13)


sum119894isin119861

119872119861119894119904119905le 119880119877119861

119904119905forall119904 119905 (14)

119872119861119894119904119905le 119880119861

119894119904119905forall119894 119904 119905 (15)





119879

sum

1199051015840gt119905

1198761198821198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

minus sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905minus119872119875

119894119905


(16)

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

= sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905+119872119875

119894119905


(17)




119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119894119905minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

cong

119905minus1

sum

11990510158401015840=1



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863


(19)

119878119868119875119894119905minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

cong

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1


119894119905minus1+119872119875

119894119905minus119872119863

119894119905

forall119894 isin 119875 119905 = 119879

(20)

119872119875119894119905= sum119895isin119862(119894)

119872119883119894119895119905




119868119861119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119861119894119905101584010158401199051015840 sdot V119894


(22)

119868119861119894119905ge 0 forall119894 isin 119861 119905 = 119879 (23)

sum119894isin119861

119868119861119894119905le 119880119868119861

119905forall119905 (24)


119868119882119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119882119894119905101584010158401199051015840 sdot V119894


(25)

119868119882119894119905ge 0 forall119894 isin 119882 119905 = 119879 (26)

sum119894isin119882

119868119882119894119905le 119880119868119882

119905forall119905 (27)


119868119875minus

119894119905ge 119878119868119875119894119905+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


(28)

119868119875minus


119868119875minus


sum119894isin119875

119868119875119894119905le 119880119868119875

119905forall119905 (31)




119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905


119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905


(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905




119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905


119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905








119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)







3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













SHL119894() =

119879

sum119905=1

(1 minus119868119875minus119894119905

119878119868119875119894119905

) sdot100

119879forall119894 isin 119875 (5)


ASHL () = sum119894isin119875

SHL119894

120593 (119875) (6)



SL119894() = max0 [1


119905=1sum119879

1199051015840gt119905(1199051015840 minus 119905) sdot 119872119861119860

1198941199051199051015840 + sum119879

119905=1120573119894sdot 119872119861119860

119894119905119879+1

sum119879

119905=1119872119863119894119905

]


(7)


ASL () = sum119894isin119875

SL119894

120593 (119875) (8)







222 Constraints


119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905minus 119876119861119894119905119905


(11)

11987611986111989411990510158401015840119905= sum119904isin119878(119894)

119872119861119894119904119905

forall119894 isin 119861 119905 = 119879 (12)

119905

sum

11990510158401015840=1

11987611986111989411990510158401015840119905= sum

1198941015840isinΠ(119895119894)

sum119895isin119874(119894)

1198721198831198941015840119895119905

119894 isin 119861 forall119905 (13)


sum119894isin119861

119872119861119894119904119905le 119880119877119861

119904119905forall119904 119905 (14)

119872119861119894119904119905le 119880119861

119894119904119905forall119894 119904 119905 (15)





119879

sum

1199051015840gt119905

1198761198821198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

minus sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905minus119872119875

119894119905


(16)

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

= sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905+119872119875

119894119905


(17)




119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119894119905minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

cong

119905minus1

sum

11990510158401015840=1



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863


(19)

119878119868119875119894119905minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

cong

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1


119894119905minus1+119872119875

119894119905minus119872119863

119894119905

forall119894 isin 119875 119905 = 119879

(20)

119872119875119894119905= sum119895isin119862(119894)

119872119883119894119895119905




119868119861119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119861119894119905101584010158401199051015840 sdot V119894


(22)

119868119861119894119905ge 0 forall119894 isin 119861 119905 = 119879 (23)

sum119894isin119861

119868119861119894119905le 119880119868119861

119905forall119905 (24)


119868119882119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119882119894119905101584010158401199051015840 sdot V119894


(25)

119868119882119894119905ge 0 forall119894 isin 119882 119905 = 119879 (26)

sum119894isin119882

119868119882119894119905le 119880119868119882

119905forall119905 (27)


119868119875minus

119894119905ge 119878119868119875119894119905+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


(28)

119868119875minus


119868119875minus


sum119894isin119875

119868119875119894119905le 119880119868119875

119905forall119905 (31)




119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905


119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905


(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905




119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905


119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905








119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)







3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879

sum

1199051015840gt119905

1198761198821198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

minus sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905minus119872119875

119894119905


(16)

119905minus1

sum

11990510158401015840=1

11987611988211989411990510158401015840119905+ sum119895isin119862(119894)

119872119883119894119895119905

= sum

1198941015840isinΠ(1198951015840119894)

sum

1198951015840isinΩ(119895119894)

11987211988311989410158401198951015840119905+119872119875

119894119905


(17)




119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119894119905minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

cong

119905minus1

sum

11990510158401015840=1



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863


(19)

119878119868119875119894119905minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

cong

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875119894119905minus1


119894119905minus1+119872119875

119894119905minus119872119863

119894119905

forall119894 isin 119875 119905 = 119879

(20)

119872119875119894119905= sum119895isin119862(119894)

119872119883119894119895119905




119868119861119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119861119894119905101584010158401199051015840 sdot V119894


(22)

119868119861119894119905ge 0 forall119894 isin 119861 119905 = 119879 (23)

sum119894isin119861

119868119861119894119905le 119880119868119861

119905forall119905 (24)


119868119882119894119905ge

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119882119894119905101584010158401199051015840 sdot V119894


(25)

119868119882119894119905ge 0 forall119894 isin 119882 119905 = 119879 (26)

sum119894isin119882

119868119882119894119905le 119880119868119882

119905forall119905 (27)


119868119875minus

119894119905ge 119878119868119875119894119905+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


(28)

119868119875minus


119868119875minus


sum119894isin119875

119868119875119894119905le 119880119868119875

119905forall119905 (31)




119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905


119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905


(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905




119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905


119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905








119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)







3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119872119883119865119897119895119905= sum119894isin119865(119897)

119872119883119894119895119905


119872119883119865119897119895119905ge 119871119872

119897119895119905sdot 119883119897119895119905


119872119883119865119897119895119905le 119880119872

119897119895119905sdot 119883119897119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119894119895

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119895119905+ 119879119874119895119905


(35)

119879119860119895119905+ 119879119874119895119905le 119860119905minus 119879

119895119905




119877 sdot 119864119894119905ge

119879

sum

1199051015840gt119905


119877 sdot 119866119894119905ge

119879

sum

1199051015840gt119905








119868119861119894119905 119868119875119894119905 119868119875minus

119894119905 119868119882119894119905 119871119872119897119895119905119872119861119860

1198941199051199051015840 119872119871

119894119905119872119861119894119904119905

119872119875119894119905119872119883119894119895119905119872119883119865

119897119895119905 119879119874119895119905 119876119861119894119905101584010158401199051015840 119876119875119894119905101584010158401199051015840

119876119882119894119905101584010158401199051015840 119880119872

119897119895119905ge 0 forall119894 119895 119897 119904 119905

(42)







3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













3 Solution Strategy








120583119888119896

(119909) =

119888119896(119909) minus 119888

119897

119896

119888119898119896minus 119888119897119896

if 119888119897119896le 119888119896(119909) lt 119888

119898

119896

119888119906

119896minus 119888119896(119909)

119888119906119896minus 119888119898119896

if 119888119898119896le 119888119896(119909) lt 119888

119906

119896

0 otherwise

(43)

where 119888119898119896 119888119897119896 and 119888119906



119888119896




119888119897

119896120572= 119888119897

119896+ 120572 (119888

119898

119896minus 119888119897

119896) forall119896

119888119898

119896120572= 119888119898

119896forall119896

119888119906

119896120572= 119888119898

119896+ (1 minus 120572) (119888

119906

119896minus 119888119898

119896) forall119896

(44)

where 119888119898119896120572 119888119897119896120572 and 119888119906






sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum119894isin119861

119872119861119894119904119905le 119908119897

1sdot 119880119877119872119861

119897

119904119905120572+ 119908119898

1sdot 119880119877119872119861

119898

119904119905120572+ 119908119906

1

sdot 119880119877119872119861119906

119904119905120572forall119904 119905

119872119861119894119904119905le 119908119897

2sdot 119880119872119861

119897

119894119904119905120572+ 119908119898

2sdot 119880119872119861

119898

119894119904119905120572+ 119908119906

2

sdot 119880119872119861119906

119894119904119905120572forall119894 119904 119905

sum119894isin119861

119868119861119894119905le 119908119897

3sdot 119880119868119861119897

119905120572+ 119908119898

3sdot 119880119868119861119898

119905120572+ 119908119906

3sdot 119880119868119861119906

119905120572

forall119905

sum119894isin119882

119868119882119894119905le 119908119897

4sdot 119880119868119882

119897

119905120572+ 119908119898

4sdot 119880119868119882

119898

119905120572+ 119908119906

4sdot 119880119868119882

119906

119905120572

forall119905

119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

+

119905

sum

11990510158401015840=1

119879

sum

1199051015840gt119905

119876119875119894119905101584010158401199051015840 sdot V119894minus 119868119875119894119905


119868119875minus

119894119905ge 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572


119868119875minus

119894119905le 119908119897

5sdot 119878119868119875119897

119894119905120572+ 119908119898

5sdot 119878119868119875119898

119894119905120572+ 119908119906

5sdot 119878119868119875119906

119894119905120572

forall119905

sum119894isin119875

119868119875119894119905le 119908119897

6sdot 119880119868119875119897

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572+ 119908119898

6sdot 119880119868119875119898

119905120572

forall119905

(45)

where119908119898119896119908119897119896 and119908119906







119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119897

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119906


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119898

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119898


119879

sum

1199051015840gt119905

1198761198751198941199051199051015840 + 119878119868119875

119906

119894119905120572minus 119868119875minus

119894119905minus

119879

sum

1199051015840gt119905

1198721198611198601198941199051199051015840

=

119905minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906



119905minus1

sum

11990510158401015840=1

11987211986111986011989411990510158401015840119905

+119872119875119894119905minus119872119863

119897


119878119868119875119897

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119897


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119906

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119898

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119898


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119898

119894119905forall119894 isin 119875 119905 = 119879

119878119868119875119906

119894119905120572minus 119868119875minus

119894119905minus119872119861119860

119894119905119905+1

=

119879minus1

sum

11990510158401015840=1

11987611987511989411990510158401015840119905+ 119878119868119875

119906


119894119905minus1minus 119905 +119872119875

119894119905

minus119872119863119897

119894119905forall119894 isin 119875 119905 = 119879

sum119894isin119872(119895)

(119872119883119894119895119905

120578119906119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119897

119895119905120572+ 119879119874119895119905


sum119894isin119872(119895)

(119872119883119894119895119905

120578119898119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119898

119895119905120572+ 119879119874119895119905



sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum119894isin119872(119895)

(119872119883119894119895119905

120578119897119894119895120572

) + sum119897isin119871(119895)

119879119878119897119895sdot 119883119897119895119905le 119879119860119906

119895119905120572+ 119879119874119895119905

forall119895 forall119905119879119860119897

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119897

119895119905120572forall119895 forall119905

119879119860119898

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119898

119895119905120572forall119895 forall119905

119879119860119906

119895119905120572+ 119879119874119895119905le 119860119905minus 119879119872

119906

119895119905120572forall119895 forall119905

(46)




min or max 1198751(1199111

1(119909) 119911

1

1198981

(119909)) 119875119871(119911119871

1(119909) 119911

119871

119898119871

(119909))

st 119909 isin 119866

(47)



(119909) | 1 le 119899 le 119871 1 le

ℎ119899le 119898119899 119899 isin 119885 ℎ




119892or

119911119892(119909) ge 119911





120583119911119892

(119909) =

1 if 119911119892(119909) lt 119911

119897

119892

119911119906119892minus 119911119892(119909)

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

0 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 1 4

(48)





120583119911119892

(119909) =

0 if 119911119892(119909) lt 119911

119897

119892

119911119892(119909) minus 119911

119897

119892

119911119906119892minus 119911119897119892

if 119911119897119892le 119911119892(119909) le 119911

119906

119892

1 if 119911119892(119909) gt 119911

119906

119892

forall119892 = 5 6

(49)






max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










max 120582


119911119892le 119911119906

119892minus 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 1 4

119911119892ge 119911119897

119892+ 120582 (119911

119906

119892minus 119911119897

119892) forall119892 = 5 6

0 le 120582 le 1

(50)



6


119911119892

forall119892 = 1 6 (51)


119911119892




120583119911119899minus1

ℎ119899minus1

minus 120583119911119899

ℎ119899


119899 (52)

where 120583119911119899

ℎ119899


min 120574




119911119892

forall119892 = 1 6

120583119911119899

ℎ119899

minus 120583119911119899+1

ℎ119899+1


119899 119911119899

ℎ119899

isin 1199111 119911

6

0 le Δ120575 le 1


(53)







5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













5 1199116




1 119911







4 Case Study




1198611 1 1 1 1 0 0 0 0 1 01198612 0 0 0 0 1 1 0 0 0 11198613 0 0 0 0 0 0 1 1 0 0






119896 119908119898119896 119908119906119896) = (0167 0666 0167) and 120572 = 06










1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198691 1 1198691119860 1 1198691119860 1 1198691119861 1 1198691119861 1 1198691119862 1 1198691119863 1 1198691119864 1 1198691119864 1 1198691119860 1 11986911198631198692 1 1198692119860 1 1198692119860 1 1198692119861 1 1198692119861 1 1198692119862 1 1198692119863 1 1198692119864 1 1198692119864 1 1198692119860 1 11986921198631198693 1 1198693119860 1 1198693119861 1 1198693119862 1 1198693119863 1 1198693119864 0 0 0 1 1198693119860 01198694 0 0 0 0 0 1 1198694119860 1 1198694119861 1 1198694119862 0 1 11986941198601198695 1 1198695119860 1 1198695119861 1 1198695119862 1 1198695119863 1 1198695119864 1 1198695119865 1 1198695119866 1 1198695119867 1 1198695119860 1 11986951198651198696 0 0 0 0 0 0 0 0 1 1198696119860 1 1198696119861



4 1198692119860 11986921198615 1198692119862 11986921198636 1198692119864

7 1198693119860 1198693119861 1198693119862 11986931198638 1198693119864

9 1198694119860

10 1198694119861 119869411986211 1198695119860 1198695119861 1198695119862 119869511986312 1198695119864

13 1198695119865

14 1198695119866 119869511986715 1198696119860

16 1198696119861




120578119894119895

(27 63) 119880119868119875119905

(136 170)119880119861

119894119904119905(120 230) 119880119868119882

119905(85 105)

119880119877119861119904119905

(300 500) 119878119868119875119894119905

(8 11)119880119868119861119905

(85 105) 119880119868119875119905

(136 170)


(119911119897

1 119911119906

1) = (3430000 4000000)

1205831199111

=

1 if 1199111lt 3430000

4000000 minus 1199111

570000if 3430000 le 119911

1le 4000000

0 if 1199111gt 4000000

(119911119897

2 119911119906

2) = (40000 65000)

1205831199112

=

1 if 1199112lt 40000

65000 minus 1199112

25000if 40000 le 119911

2le 65000

0 if 1199112gt 65000

(119911119897

3 119911119906

3) = (4400000 5000000)

1205831199113

=

1 if 1199113lt 4400000

5000000 minus 1199113

600000if 4400000 le 119911

3le 5000000

0 if 1199113gt 5000000

(119911119897

4 119911119906

4) = (150000 200000)

1205831199114

=

1 if 1199114lt 150000

200000 minus 1199114

50000if 150000 le 119911

4le 200000

0 if 1199114gt 200000

(119911119897

5 119911119906

5) = (50 80)

1205831199115

=

0 if 1199115lt 50

1199115minus 50

30if 50 le 119911

5le 80

1 if 1199115gt 80

(119911119897

6 119911119906

6) = (80 95)

1205831199116

=

0 if 1199116lt 80

1199116minus 80

15if 80 le 119911

6le 95

1 if 1199116gt 95

(54)





119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894119905119872119863119906

119894119905)

119905 = 1 119905 = 2 119905 = 3 119905 = 4 119905 = 5 119905 = 6 119905 = 7

1198751 (0 0 0) (0 0 0) (0 0 0) (120 130 135) (120 130 135) (120 130 135) (120 130 135)1198752 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (0 0 0) (0 0 0)1198753 (80 90 95) (80 90 95) (0 0 0) (0 0 0) (0 0 0) (80 90 95) (80 90 95)1198754 (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125) (110 120 125)1198755 (90 100 105) (90 100 105) (90 100 105) (90 100 105) (0 0 0) (0 0 0) (90 100 105)1198756 (100 110 115) (100 110 115) (100 110 115) (0 0 0) (0 0 0) (0 0 0) (0 0 0)1198757 (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85) (70 80 85)1198758 (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)1198759 (80 90 95) (80 90 95) (80 90 95) (80 90 95) (80 90 95) (0 0 0) (0 0 0)11987510 (0 0 0) (0 0 0) (100 110 115) (100 110 115) (100 110 115) (100 110 115) (100 110 115)


Level 1 1199116

Level 2 1199115

Level 3 1199111 1199113 1199114

Level 4 1199112


min 120574



forall119892 = 1 6

120583119911119892

ge 0916 minus Δ120575 forall119892 = 1 6

1205831199115

minus 1205831199116

le 120574

1205831199111

minus 1205831199115

le 120574

1205831199113

minus 1205831199115

le 120574

1205831199114

minus 1205831199115

le 120574

1205831199112

minus 1205831199111

le 120574

1205831199112

minus 1205831199113

le 120574

1205831199112

minus 1205831199114

le 120574

0 le Δ120575 le 1


(55)







1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1 1199112 1199113 1199114 decrease and


5 show



119892(120583119911119892

) = 1205831199112



RSD = max119892

(120583119911119892

) minusmin119892(120583119911119892

) forall119892 isin 119866 (56)


PEmax= max1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(57)

RSDPEmax

PEmin


0

005

01

015

02

025

03

035

04

045

05

(deg

)


PEmin= min1198921198921015840

100381610038161003816100381610038161003816120583119911119892

minus 1205831199111198921015840

100381610038161003816100381610038161003816

forall119911119892isin 119875119899 1199111198921015840 isin 119875119899+1 119892 119892

1015840isin 119866

(58)




76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











76

765

77

775

78

785

ASH

L (

)


8

81

82

83

84

85

Ope

ratio

ns co

st

times106



938

941

944

947

95

953

ASL

()


8

81

82

83

84

85

Ope

ratio

ns co

st

times106


















(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(08842 08842 0932309323 09323 09323)

(09160 09160 0916009160 09160 09160)

(09124 09124 0928909124 09124 09124)



5 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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