research article an application of stochastic programming

Research ArticleAn Application of Stochastic Programming in Solving CapacityAllocation and Migration Planning Problem under Uncertainty

Yin-Yann Chen and Hsiao-Yao Fan

Department of Industrial Management National Formosa University Yunlin 632 Taiwan

Correspondence should be addressed to Yin-Yann Chen rogeryycgmailcom

Received 31 August 2015 Revised 29 October 2015 Accepted 10 November 2015

Academic Editor Yan-Jun Liu

Copyright copy 2015 Y-Y Chen and H-Y Fan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The semiconductor packaging and testing industry which utilizes high-technology manufacturing processes and a variety ofmachines belongs to an uncertain make-to-order (MTO) production environment Order release particularly originates fromcustomer demand hence demand fluctuation directly affects capacity planning Thus managing capacity allocation is a difficultendeavor This study aims to determine the best capacity allocation with limited resources to maximize the net profit Threebottleneck stations in the semiconductor packaging and testing process are mainly investigated namely die bond (DB) wirebond (WB) and molding (MD) stations Deviating from previous studies that consider the deterministic programming modelcustomer demand in the current study is regarded as an uncertain parameter in formulating a two-stage scenario-based stochasticprogramming (SP) model The SP model seeks to respond to sharp demand fluctuations Even if future demand is uncertainmigration decision for machines and tools will still obtain better robust results for various demand scenarios A hybrid approachis proposed to solve the SP model Moreover two assessment indicators namely the expected value of perfect information (EVPI)and the value of the stochastic solution (VSS) are adopted to compare the solving results of the deterministic planning model andstochastic programming model Sensitivity analysis is performed to evaluate the effects of different parameters on net profit

1 Introduction

The semiconductor packaging and testing industry belongsto a flow line environment The operating procedure of thisindustry indicates that the die bond (DB) wire bond (WB)and molding (MD) stations are the three bottleneck stationsin themanufacturing process Consequently capital expendi-ture on equipment for theDBWB andMD stations results inhigh purchasing cost In machine utilization the limitationsof machine type and product category affect the output perhour Given the multifactory and multiline environment ofthis industry demand uncertainty and inappropriate pro-duction planning result in wasteful or insufficient machinecapacity in all lines Therefore this research investigates theDB WB and MD stations in the semiconductor packagingand testing industry as the objects of the study

The best resource configuration and capacity allocationdecision in the semiconductor packaging and testing industrycan effectively utilize all resources in all production lines

as well as assisting planners in reducing the readjustmentof the production scheduling to efficiently accomplish orderallocations as a response to substantial changes in demandSuch decision can also indirectly reduce themigration cost ofmachines and tools Therefore this study aims to investigatethe capacity allocation and migration planning problem byconsidering demand uncertainty to solve the current prob-lems and challenges faced by the semiconductor packagingand testing industry

The importance of capacity planning to semiconductorpackaging and testing industry is discovered based on theaforementioned background and motivation Thus the cur-rent study aims to attempt capacity allocation and resourceconfiguration planning at the same time for multiperiodorder demands to improve the current capacity plan beingseparately implemented by packaging and testing factoriesand consideration of a single period and to determine theproper machine migration and capacity allocation decision

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 741329 16 pageshttpdxdoiorg1011552015741329

2 Mathematical Problems in Engineering

Demand in packaging and testing factories fluctuatesbecause of different limitations in capacity planning and themake-to-order (MTO) production environment Thereforedeviating from previous studies that consider the determinis-tic programming model the current study employs demandas an uncertain factor to solve the problem through a two-stage stochastic programming (SP) model The feasibility ofthe capacity planning method proposed in this study is alsodemonstrated through a practical case studyMoreover basedon the different parameters change sensitivity analysis for thestochastic programming model is provided to understandif capacity planning will be affected by different scenariosdemandsmigration costs ofmachines and tools sale prices ofproducts flexibility of capacity migration and other factors

This paper is organized as follows Section 2 reviews therelated literature Section 3 establishes the definition of thecapacity planning problem of the semiconductor packagingand testing industry as well as the development of a two-stagestochastic programming model and the proposed hybridapproach Section 4 discusses the application and analysisof a certain large-scale semiconductor packaging and testingfactory case Section 5 provides the conclusion of this study

2 Literature Review

21 Overview of the Semiconductor Packaging and TestingIndustry Characteristics Semiconductor products currentlycomprise four categories namely integrated circuit (IC)discrete sensor and optoelectronics However this studyaims to investigate the IC category The IC manufacturingprocess with vertical integration features has front- and back-end processes The upstream to downstream processes of thetarget industry cover the following five steps IC design maskmaking IC making chip packaging and chip testing

The manufacturing process and production characteris-tics of the semiconductor packaging and testing industry aredescribed as follows

211 Make to Order The semiconductor packaging and test-ing industry prepares materials based on orders placed bycustomers Thereafter production proceeds based on cus-tomersrsquo needs and services thereby preventing the indus-try from predicting the demand in advance Furthermoredemand forecasting for this industry cannot be learnedfrom past experience Accordingly this study considers thedemand to be uncertain and expresses fluctuations in cus-tomer needs through different scenarios This considerationwill avoid insufficient or wasted capacity during the produc-tion while addressing customer needs and maximizing netprofit

The production in the IC packaging and testing industryis customer-oriented Product categories are diversified basedon different customer needs Packaging types can be dividedinto lead frame package ball grid array flip chip system inpackage and multichip packages Moreover each packagingtype is divided into various product types because of the useof different chip sizes or pin numbers Product types also havedifferent capacity constraints

212 Flow Shop In an IC packaging and testing factoryan order is often divided into several work orders Figure 1shows that each work order is manufactured based onthe production flow Meanwhile IC packaging and testingfactories have multiple production lines For example acertain large-scale domestic packaging and testing factoryhas approximately 25 lines This study only investigates thebottleneck stations in these factories namely the DB WBand MD stations It is described as a flow shop productionenvironment Thus products will enter the WB station afterleaving theDB station After the products are processed in theWB station they eventually enter the MD station Thereforeproducts have sequential dependencies in the factory area

213 Unrelated Parallel Machine Given the rapid manner bywhich products are updated process changes and equipmentupgrade will compel semiconductor packaging and testingfactories to frequently purchase new or different brands ofmachines to respond to market changes Having differentmachine types and brands leads to the presence of differentgrades of machines in the production line thereby resultingin the production pattern of unrelated parallel machinesWhen managements arrange orders they exert effort tosatisfy customer needs and meet the maximal service levelIn addition they manufacture by utilizing machines withthe highest capacity Hence they first move the machinesand allocate the proper machines in groups to avoid failurein production caused by inconsistencies in machine typesMost machines for the semiconductor packaging and testingindustry are movable For domestic IC packaging and testingfactories the number of moving machines reaches approxi-mately 60 each month

22 Capacity Planning Karabuk and Wu [1] indicate thatcapacity planning can be described as an iterative processbetween the following two main components (1) capacityexpansion given projected product demands identify therequired manufacturing technologies and their capacity lev-els to be physically expanded or outsourced through theplanning period (2) capacity configuration determinewhichfacility is to be configured with which technologies mix Theoverall objective is tomeet a revenuemodel based on strategicdemand planning (which blends demand forecasting andproactive market development strategies) This objective canbe viewed as meeting projected demands with minimizedtotal costs

Chen et al [2] present a capacity allocation and expan-sion problem of thin film transistor liquid crystal display(TFT-LCD) manufacturing in the multisite environmentThe objective is to simultaneously seek an optimal capacityallocation plan and capacity expansion policy under single-stage multigeneration and multisite structures Capacityallocation decides on profitable product mixes and allocatedproduction quantities of each product group at each produc-tion site Capacity expansion is concerned with determiningthe timing types and sizes of capacity investments especiallyin the acquisition of auxiliary tools A mixed integer linearprogramming (MILP) is proposed which considers manypractical characteristics Finally an industrial case study

Mathematical Problems in Engineering 3

Wafer Incoming quality control Patching Crystal cutting Die bond

Wire bond Molding PrintingBumping

Shaping InspectionElectroplating

Figure 1 Production flow of semiconductor packaging and testing factory

modified from a Taiwanese TFT-LCD manufacturer is illus-trated and sensitivity analysis of some influential parametersis also addressed

The foundry is an industry whose demand varies rapidlyandwhosemanufacturing process is quite complicated Chenet al [3] explore issues on midterm capacity planning for anincrement strategy of the number of auxiliary toolsmdashldquophotomaskrdquo to increase the flexibility of production The relateddecisions include how to allocate appropriately the forecastdemands of products amongmultiple sites and how to decideon the production quantities of products in each site afterreceiving customer-confirmed orders By constructing themathematical programming model of capacity planning therates of capacity utilization and customer order fulfillmentare found to be effectively enhanced by adding new masks toincrease production flexibility

Lin et al [4] study strategic capacity planning problemsunder demand uncertainties in TFT-LCD industry Demandforecasts are usually inaccurate and vary rapidly over timeTheir research objective is to seek a capacity allocation andexpansion policy that is robust to demand uncertainties Spe-cial characteristics of TFT-LCD manufacturing systems areconsidered A scenario-based two-stage stochastic program-ming model for strategic capacity planning under demanduncertainties is proposed Comparing to the deterministicapproach their stochastic model significantly improves sys-tem robustness

Lin et al [5] refer to capacity planning as the processof simultaneously implementing a robust capacity allocationplan and capacity expansion policy across multiple sitesagainst stochastic demandTheir study constructs a stochasticdynamic programming (SDP) model with an embeddedlinear programming (LP) to generate a capacity planningpolicy as the demand in each period is revealed and updatedNumerical results are illustrated to prove the feasibility androbustness of the proposed SDP model

23 Stochastic Programming Given that demand uncertaintyis considered this paper aims to formulate a stochasticprogramming model for solving capacity allocation andmigration planning problem Dantzig [6] divided stochasticprogramming into two types two-stage stochastic program-ming andmultistage stochastic programming Uribe et al [7]indicated that the decision variable of two-stage stochasticprogramming consists of two types ldquohere and nowrdquo and ldquowaitand seerdquo Here and now decision in the first stage refers todecision making when all information is unknown Wait and

see decision in the second stage refers to decision makingafter all information has been fully revealed Thus decisionvariables for the two-stage stochastic programming are adependent issue and the results are more robust

Two-stage stochastic programming can be illustratedthrough a scenario tree Figure 2 shows that the time pointrevealing uncertain factors is 119905 = 119896 The time point is usedto divide decisions into two stages The first-stage decision isfrom 119905 = 1 to 119905 = 119896 The results affect the decisions after119905 = 119896 + 1 Thus they extend many branches Moreover eachbranch represents a kind of scenario and a group of decisionvariables in the second stage

Listes and Dekker [8] present a stochastic programmingbased approach by which a deterministic location model forproduct recovery network design may be extended to explic-itly account for the uncertainties They apply the stochasticmodels to a representative real case study on recycling sandfrom demolition waste in Netherlands In Salema et al[9] work the design of a reverse distribution network isstudied A generalizedmodel is proposed It contemplates thedesign of a generic reverse logistics network where capacitylimits multiproduct management and uncertainty on prod-uct demands and returns are considered A mixed integerformulation is developed which is solved using standard BampBtechniques The model is applied to an illustrative case

Lee et al [10] propose a stochastic programming basedapproach to account for the design of sustainable logisticsnetwork under uncertainty A solution approach integratingthe sample average approximation scheme with an impor-tance sampling strategy is developed A case study involving alarge-scale sustainable logistics network inAsia Pacific regionis presented to demonstrate the significance of the developedstochastic model Cardona-Valdes et al [11] consider thedesign of a two-echelon production distribution networkwith multiple manufacturing plants customers and a set ofcandidate distribution centers The main contribution of thestudy is to extend the existing literature by incorporatingthe demand uncertainty of customers within the distributioncenter location and transportationmode allocation decisionsaswell as providing a network design satisfying both econom-ical and service quality objectives of the decision-maker

In Kara and Onut [12] study a two-stage stochasticrevenue-maximization model is presented to determine along-term strategy under uncertainty for a large-scale real-world paper recycling companyThis network design problemincludes optimal recycling center locations and optimal flowamounts between the nodes in themultifacility environment


Scenario 1

Scenario 2

Scenario 3

TimeFirst stage Second stage

t = 1 t = 2 t = k t = k + 1 t = k + 2 t = Tmiddot middot middot

Figure 2 Illustration of two-stage stochastic programming

The proposed model is formulated with two-stage stochasticmixed integer and robust programming approaches Pishvaeeet al [13] develop a stochastic programming model for anintegrated forwardreverse logistics network design underuncertainty An efficient deterministic mixed integer linearprogramming model is developed for integrated logisticsnetwork design to avoid the suboptimality caused by theseparate design of the forward and reverse networks Thenthe stochastic counterpart of the proposed MILP modelis developed by using scenario-based stochastic approachNumerical results show the power of the proposed stochasticmodel in handling data uncertainty

In Amin and Zhang [14] a closed-loop supply chainnetwork is investigated which includes multiple plants col-lection centers demand markets and products A mixedinteger linear programming (MILP) model is proposedthat minimizes the total cost The model is extended toconsider environmental factors by weighed sums and 120576-constraint methods In addition the impact of demand andreturn uncertainties on the network configuration is analyzedby scenario-based stochastic programming Computationalresults show that the model can handle demand and returnuncertainties simultaneously

Ramezani et al [15] present a stochastic multiobjectivemodel for forwardreverse logistic network design under anuncertain environment including three echelons in forwarddirection (ie suppliers plants and distribution centers)and two echelons in backward direction (ie collectioncenters and disposal centers) The authors demonstrate amethod to evaluate the systematic supply chain configurationmaximizing the profit customer responsiveness and qualityas objectives of the logistic network Mohammadi Bidhandiand Rosnah [16] propose an integratedmodel and amodifiedsolution method for solving supply chain network designproblems under uncertainty The stochastic supply chainnetwork design model is provided as a two-stage stochasticprogramming The main uncertain parameters are the oper-ational costs the customer demand and capacity of the facil-ities In the improved solution method the sample averageapproximation technique is integrated with the accelerated

Bendersrsquo decomposition approach to improvement of themixed integer linear programming solution phase

Sazvar et al [17] develop a stochasticmathematical modeland propose a new replenishment policy in a centralizedsupply chain for deteriorating items In this model theyconsider inventory and transportation costs as well as theenvironmental impacts under uncertain demand The besttransportation vehicles and inventory policy are determinedby finding a balance between financial and environmentalcriteria A linear mathematical model is developed and anumerical example from the real world is presented todemonstrate its applicability and effectiveness Lin et al [5]construct a stochastic dynamic programming model withan embedded linear programming to generate a capacityplanning policy as the demand in each period is revealed andupdated Using the backward induction algorithm themodelconsiders several capacity expansion and budget constraintsto determine a robust and dynamic capacity expansionpolicy in response to newly available demand informationNumerical results are also illustrated to prove the feasibilityand robustness of the proposed SDP model compared to thetraditional deterministic capacity planning model currentlyapplied by the industry

A distributed energy system is a multi-input and multi-output energy system with substantial energy and economicand environmental benefits The optimal design of such acomplex system under energy demand and supply uncer-tainty poses significant challenges Zhou et al [18] proposea two-stage stochastic programming model for the optimaldesign of distributed energy systems A two-stage decompo-sition based solution strategy is used to solve the optimizationproblem with genetic algorithm performing the search onthe first-stage variables and a Monte Carlo method dealingwith uncertainty in the second stage Detailed computationalresults are presented and compared with those generated bya deterministic model

One of themost challenging issues for the semiconductortesting industry is how to deal with capacity planning andresource allocation simultaneously under demand and tech-nology uncertainty In addition capacity planners require


a tradeoff among the costs of resourceswith different process-ing technologies while simultaneously considering resourcesto manufacture products The study of K-J Wang and S-M Wang [19] focuses on the decisions pertaining to (i) thesimultaneous resource portfolioinvestment and allocationplan (ii) the most profitable orders from pending ones ineach time bucket under demand and technology uncertaintyand (iii) the algorithm to efficiently solve the stochastic andmixed integer programming problem The authors develop aconstraint-satisfaction based genetic algorithm to resolve theabove issues simultaneously

Dynamic programming approach is a class of optimaldesign tools such as reinforcement learning Liu et al [20]proposed an online reinforcement learning algorithm for aclass of affine multiple input and multiple output (MIMO)nonlinear discrete-time systems with unknown functionsand disturbances Liu et al [21] addressed an adaptive fuzzyoptimal control design for a class of unknown nonlineardiscrete-time systems The controlled systems are in a strict-feedback frame and contain unknown functions and non-symmetric dead-zone Wang et al [22] developed a finite-horizon neurooptimal tracking control strategy for a class ofdiscrete-time nonlinear systems Chen et al [23] studied anadaptive tracking control for a class of nonlinear stochasticsystems with unknown functions Tong et al [24] proposedtwo adaptive fuzzy output feedback control approaches fora class of uncertain stochastic nonlinear strict-feedbacksystems without the measurements of the states

Previous studies have surveyed about capacity planningissue but only a few studies have focused on the capac-ity allocation problem considering machinetool migrationplanning and demand uncertainty simultaneouslyThis paperaims to determine the best capacity allocation with limitedresources to achieve net profit maximization in the semicon-ductor packaging and testing industry Customer demand isregarded as an uncertain parameter in formulating a two-stage scenario-based stochastic programming model Thismodel seeks to respond to sharp demand fluctuations Even iffuture demand is uncertain migration decision for machinesand tools will still obtain better robust results for variousdemand scenarios Sensitivity analysis is also performed toevaluate the effect of different parameters on net profit

3 Capacity Planning of the SemiconductorPackaging and Testing Industry

31 Characteristics of Capacity Planning of the SemiconductorPackaging and Testing Industry This study aims to determinemachine migration tool migration in all production linesresource configuration capacity allocation and product flowunder demand uncertainty to achieve net profit maximiza-tion

311 Resource Configuration The manufacturing processentails that a product should sequentially go through theDB WB and MD stations for assembly-line production Theproduct considers themachine type in resource configurationduring the DB and WB stages However three resources

Product 1

Wire bond MoldingDie bond

Product category

Production stage

Machine type

Tool type

Material typeResource configuration

k1 k2 k1 k2k2 k3

n1 n2

m4 m4

Figure 3 Illustration of resource configuration

namely machine type tool type and material category areconsidered in the MD stage Figure 3 shows that product 1 ismanufactured in machine 1198961 or 1198962 in the DB station Thisproduct is then processed in machine 1198962 or 1198963 in the WBstation Thereafter the product is manufactured in the MDstation through 1198961 + 1198991 + 1198984 or 1198962 + 1198992 + 1198984

312 Product Flow This study disregards defective productsand only considers production through the three sequentialstages Moreover product flow balance must be maintainedin the production line Hence the total product input mustequal the final total output For example the product inputfor product 1 is 1000 units Furthermore 400 and 600 unitsare produced in lines 1 and 2 respectively After productionthrough the three sequential stages the final total outputremains as 1000 units

313 Capacity Allocation The capacity planning of allreceived orders is executed based on the current existingresources in all production stages A product is not limitedto the same production line during the entire productionprocess that is a product can be manufactured in thedifferent lines through three production stages For examplea company has two lines if the input of product 1 is 1000units Take line 1 for explanation Firstly 400 units are manu-factured in the DB station using machine 1198961 and 600 unitsusing machine 1198962 Thereafter 400 units are manufacturedin the WB station using machine 1198962 and 200 units usingmachine 1198963 Finally 200 units are manufactured in the MDstation using resource configuration 1198961 + 1198991 + 1198984 and 300units using 1198962 + 1198992 + 1198984 Thus 500 units of product 1 canbe made after the three production stages for this product arecompleted sequentially in line 1 The remaining 500 units areallocated to all production stages in line 2 for manufacturing

314 Machine and Tool Migration The presence of severalproduction lines and machines with different technologicalcapability in a company will result in variations in the


production capacities of all lines Machines can be moved toall lines in each production stage and tools can be movedto all lines in the MD stage based on the total number ofavailable machines and tools

32Mathematical Programming of Capacity Planning Problemfor the Semiconductor Packaging and Testing Industry underDemand Uncertainty A mathematical model of two-stagescenario-based stochastic programming is formulated byconsidering customer demand as an uncertain parameterThis study aims to respond to sharp demandfluctuation Evenif future demand is uncertain machine and tool migrationdecisions are robust results for all demand scenarios

321 Definition andDescription of Capacity Planning Problemunder Demand Uncertainty This study uses a scenario treeto illustrate the uncertain factor (Figure 4) Machine and toolmigration decisions are deemed to be the decisions made inthe first stage The results of these decisions remain constantwith the varying customer demands Moreover the second-stage capacity allocation decisions must be made based onthe first-stage decision results The results in the second-stage change with the varying customer demands In thisstudy two-stage decisions should be optimally determined toachieve net profit maximization

(1) First-Stage Decision Robust Capacity Migration DecisionThat Considers Demand Uncertainty Given three demandscenarios each type of machine and tool is considered todetermine when and what quantity of machines and toolsare migrated between lines in the production stage Hencecapacity migration decision must be made in advance toconsider the robust decision under demand uncertainty asbeing unrelated to different demand scenarios

(2) Second-Stage Decision Capacity Allocation Decision afterAll Demand Information Has Been Completely Revealed Thefollowing factors are determined after a certain demandscenario occurs (1) production quantity for each productin each line in all production stages during each period(2) transportation quantity between the different productionstages (3) sales volume of each product in each period foreach customer and (4) customer service level Thereforecapacity allocation decision is closely related to the demandscenario According to the capacity migration result in thefirst stage the optimal capacity allocation decision can bedetermined once a specific demand scenario occurs

322 Two-Stage Stochastic Programming Model of CapacityPlanning Problem To solve the capacity planning problemunder demand uncertainty this study uses two-stage stochas-tic programming to construct a mathematical model Thissection explains the indices parameters decision variablesobjective function and constraints

(1) Indices119888 = customer (119888 = 1 2 119862)119894 = product type (119894 = 1 2 119868)119897 = production line (119897 = 1 2 119871)

Scenario 1

Scenario 2

Scenario 3

The uncertain factor is revealed

Capacity allocation decisionCapacity migration decisionFirst stage Second stage

Demand

Figure 4 Diagrammatic sketch of scenario tree of the uncertainfactor

119904 = production stage (119904 = 1 2 119878)119895 = resource configuration (119895 = 1 2 119869)119898 = material type (119898 = 1 2 119872)119896 = machine type (119896 = 1 2 119870)119899 = tool type (119899 = 1 2 119873)119905 = time period (119905 = 1 2 119879)119903 = scenario number (119903 = 1 2 119877)

(2) Parameters

(I) Demand Related Parameters

119889119890119903

119894119888119905= the demand quantity of customer 119888 for product

119894 in time 119905 under scenario 119903119901119903 = probability value occurring in scenario 119903

(sum119903119901119903

= 1)119901119903119894119888119905

= sales price of customer 119888 for product 119894 in time119905

(II) Machine Related Parameters

119896119897119897119904119896

= initial amount of machine 119896 in line 119897 at stage 119904119896119906119897119904=maximumnumber ofmachines in line 119897 at stage

119904119896119904119894119895119904119896

= required work hours of machine 119896 used atstage 119904 for manufacturing a unit of product 119894 withresource configuration 119895119896119886119904119896= available work hours of machine 119896 at stage 119904

1198961198871198971198971015840119904= machine migration capability from lines 119897 to 1198971015840

at stage 119904

(III) Tool Related Parameters

119899119897119897119904119899

= initial amount of tool 119899 in line 119897 at stage 119904119899119906119897119904= maximum number of tools in line 119897 at stage 119904

119899119904119894119895119904119899

= required work hours of tool 119899 used at stage 119904for manufacturing a unit of product 119894 with resourceconfiguration 119895119899119886119904119899= available work hours of tool 119899 at stage 119904

1198991198871198971198971015840119904= tool migration capability from lines 119897 to 1198971015840 at

stage 119904


(IV) Material Related Parameters

119898119902119904119898119905

= total available quantity of material119898 at stage119904 in time 119905

119898119904119894119895119904119898

= consumption ratio of material 119898 for manu-facturing a unit of product 119894 at stage 119904 with resourceconfiguration 119895

(V) Production Capability Related Parameter

119905119891119894119895119904=production capability of product 119894 at stage 119904with

resource configuration 119895

(VI) Transportation Related Parameter

1199051198871198971199041198971015840(119904+1)

= transportation capability from line 119897 at stage119904 to line 1198971015840 at stage 119904 + 1

(VII) Cost Parameters

V119888119894119897119895119904

= production cost for manufacturing a unit ofproduct 119894 in line 119897 at stage 119904 with resource configura-tion 119895

119896119888119904= machine migration cost at stage 119904

119899119888119904= tool migration cost at stage 119904

(3) Decision Variables

(I) First-Stage Decision Variables Capacity Migration Deci-sion

119870119876119897119904119896119905

= the number of machines 119896 for line 119897 at stage119904 in time 1199051198701198721198971198971015840119904119896119905

= the migration number of machines 119896 fromline 119897 to line 1198971015840 at stage 119904 in time 119905119873119876119897119904119899119905

= the number of tools 119899 for line 119897 at stage 119904 intime 1199051198731198721198971198971015840119904119899119905

= the migration number of tools 119899 from line119897 to line 1198971015840 at stage 119904 in time 119905

(II) Second-Stage Decision Variables Capacity AllocationDecision and Service Level

119883119876119903

119894119897119895119904119905= production amounts of product 119894 with

resource configuration 119895 for line 119897 at stage 119904 in time119905 under scenario 119903119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

= transportation amounts of product 119894from line 119897 with resource configuration 119895 at stage 119904 toline 1198971015840 with resource configuration 1198951015840 at stage (119904+1) intime 119905 under scenario 119903119878119876119903

119894119888119905= sales amounts of product 119894 for customer 119888 in

time 119905 under scenario 119903119878119871119903

119888= service level for customer 119888 under scenario 119903

(4) Objective Function Consider the following

Maximize

sum

119903

119901119903

sum

119894

sum

119888

sum

119905

(119901119903119894119888119905times 119878119876119903

119894119888119905) minussum

119894

sum

119897

sum

119895

sum

119904

sum

119905

(V119888119894119897119895119904times 119883119876

119903

119894119897119895119904119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119896

sum

119905

(119896119888119904times 119870119872

1198971198971015840119904119896119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119899

sum

119905

(119899119888119904times 119873119872

1198971198971015840119904119899119905)

(1)

The above is the objective function of two-stage stochasticprogramming It aims to obtain the optimal capacity planningdecision to seek the maximization of net profit as (1) netprofit = (sales revenue minus variable production cost) minusmachinemigration cost minus tool migration cost

(5) Constraints

(I) First-Stage Constraints

(a) Machine Migration Balance Constraints Consider thefollowing

1198701198761198971199041198960

= 119896119897119897119904119896

forall119897 119904 119896 (2)

119870119876119897119904119896119905= 119870119876

119897119904119896(119905minus1)minussum

1198971015840

1198701198721198971198971015840119904119896119905

+sum

1198971015840

1198701198721198971015840119897119904119896119905

forall119897 119904 119896 119905

(3)

119870119876119897119904119896119905le 119896119906119897119904

forall119897 119904 119896 119905 (4)

1198701198721198971198971015840119904119896119905le 119872 times 119896119887

1198971198971015840119904forall119897 1198971015840

119904 119896 119905 (5)

Constraint (2) shows the initial amount of machines inlines at each production stage and constraint (3) indicates thenumber ofmachines required for lines at production stages inevery periodThis number of machines in the current periodis equal to the number of machines in the previous periodminus the number of machines moving to other lines plus


the number of machines that migrated from other lines tothis line The total initial number of machines within thecompanymust be equal to the total number of machines afterbeing migrated between lines without increasing or reducingthe number of machines Constraint (4) expresses that theallocated number of machines should not be more than theavailable space in the shop-floor production line In additionconstraint (5) considers if machines have capability to bemigrated between lines 119896119887

1198971198971015840119904refers to a binary parameter 1

means machines can be migrated between production linesand 0 means they cannot be migrated

(b) Tool Migration Balance Constraints Consider the follow-ing

1198731198761198971199041198990

= 119899119897119897119904119899

forall119897 119904 119899 (6)

119873119876119897119904119899119905= 119873119876

119897119904119899(119905minus1)minussum

1198971015840

1198731198721198971198971015840119904119899119905

+sum

1198971015840

1198731198721198971015840119897119904119899119905

forall119897 119904 119899 119905

(7)

119873119876119897119904119899119905le 119899119906119897119904

forall119897 119904 119899 119905 (8)

1198731198721198971198971015840119904119899119905le 119872 times 119899119887

1198971198971015840119904forall119897 1198971015840

119904 119899 119905 (9)

Constraint (6) shows the initial amount of tools in linesat each production stage and constraint (7) indicates thenumber of tools required for lines at production stages inevery period This number of tools in the current period isequal to the number of tools in the previous period minusthe number of tools moving to other lines plus the numberof tools that migrated from other lines to this line The totalinitial number of tools within the company must be equalto the total number of tools after being migrated betweenlines without increasing or reducing the number of toolsConstraint (8) expresses that the allocated number of toolsshould not be more than the available space in the shop-floorproduction line In addition constraint (9) considers if toolshave capability to be migrated between lines 119899119887

1198971198971015840119904refers to

a binary parameter 1 means tools can be migrated betweenproduction lines and 0 means they cannot be migrated

(c) Domain Restriction for First-Stage Decision VariablesConsider the following

119870119876119897119904119896119905 1198701198721198971198971015840119904119896119905 119873119876119897119904119899119905 1198731198721198971198971015840119904119899119905isin integer

forall119897 119904 119896 119899 119905

(10)

Constraint (10) shows the domain of variables whichindicates the characteristics of its integer variables

(II) Second-Stage Constraints

(a) Production and Transportation Balance Constraints Con-sider the following

119883119876119903

119894119897119895119904119905= sum

1198971015840

sum

1198951015840

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

forall119894 119897 119895 119904 = 1 119878 minus 1 119905 119903

(11)

sum

1198971015840

sum

1198951015840

119877119876119903

11989411989710158401198951015840(119904minus1)119897119895119904119905

= 119883119876119903

119894119897119895119904119905forall119894 119897 119895 119904 = 2 119878 119905 119903 (12)

Overall production and transportation must satisfy lineflow balance as shown in constraints (11) and (12) Theallocated production amounts in a certain line at this stageshould be equal to the total amounts that are transportedfrom this line to all lines at the next stage On the contrarythe total amounts that are transported from all lines at theprevious stage to a certain line at the current stage should beequal to the allocated production amounts in this line

(b) Capacity Constraints Consider the following

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119896119904119894119895119904119896) le 119870119876

119897119904119896119905times 119896119886119904119896

forall119897 119904 119896 119905 119903 (13)

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119899119904119894119895119904119899) le 119873119876

119897119904119899119905times 119899119886119904119899

forall119897 119904 119899 119905 119903 (14)

For capacity constraints constraints (13) and (14) indicatethat the production amounts multiplied by work hours ofmachines or tools consumed should not exceed the numberof machines or tools multiplied by available work hours ofa unit of machine or tool In short the sum of work hoursrequired for each product in available machine or tool shouldnot be more than the total available resource limit of thecompany

(c) Material Constraint Consider the following

sum

119894

sum

119897

sum

119895

(119883119876119903

119894119897119895119904119905times 119898119904119894119895119904119898) le 119898119902

119904119898119905forall119904119898 119905 119903 (15)

For material constraint (15) generally speaking theamounts of materials to be consumed in the productionprocess should not be beyond the quantity restriction ofavailable materials With limited resources the productionamounts multiplied by the material consumption ratio perunit will be less or equal to the total available quantity of thematerial

(d) Production Capability Constraint Consider the following

119883119876119903

119894119897119895119904119905le 119872 times 119905119891

119894119895119904forall119894 119897 119895 119904 119905 119903 (16)

For production capability constraint (16) shows whetherresource configuration of a certain product is able to be usedfor manufacturing this product Due to different types ofmachines and tools in lines at each production stage not allresource configurations can be used for manufacturing allkinds of products If 119905119891

119894119895119904= 1 the resource configuration in

the line at this stage can be used for manufacturing this typeof product on the contrary if 119905119891

119894119895119904= 0 they cannot be used

(e) TransportationCapability Constraint Consider the follow-ing

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

le 119872 times 1199051198871198971199041198971015840(119904+1)

forall119894 119897 119895 119904 1198971015840

1198951015840

119905 119903 (17)

For transportation capability constraint (17) expresseswhether there is transportation capability to move products


from the current stage to the next stage The productionprocess is an assembly flow line environmentThus productsare bound to go through each production stage in turn andcannot revert to a previous stage If 119905119887

1198971199041198971015840(119904+1)

= 1 there istransportation capability to move products between stageson the contrary if 119905119887

1198971199041198971015840(119904+1)

= 0 it indicates that there is notransportation capability

(f) Demand Fulfillment Constraints Consider the following

sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)

Demand fulfillment is indicated by constraints (18) and(19) respectively Constraint (18) shows that sales volumein each scenario should be equal to the total productionamounts with resource configurations in all lines Constraint(19) expresses that the sales volume must be less or equal tothe demands required by customers

(g) Service Level Consider the following

119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)

Constraint (20) shows that the sales volume divided bycustomer demands is the service level

(h) Domain Restriction for Second-Stage Decision VariablesConsider the following

119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)

Constraint (21) indicates variable domain restriction

323 Capacity Planning Problem under Demand CertaintyDifferent from the uncertainty model the deterministicmodel does not consider demand fluctuation and only con-siders an average demand scenario Appendix A (see Supple-mentaryMaterial available online at httpdxdoiorg1011552015741329) shows the detailed mathematical programmingmodel that is used to compare the differences in solvingresults between the deterministic model and stochastic pro-gramming model

33 Proposed Hybrid Approach As the scenario number isincreased solving the scenario-based stochastic program-ming model becomes considerably difficult because of thecomputation complexity Therefore a hybrid approach isdeveloped to efficiently address the proposed two-stagestochastic programming model We apply the particle swarmoptimization (PSO) method combined with the AIMMSoptimal modeling software in a hybrid mechanism First aninitial solution was generated to determine the migrationnumber of machines and tools among the production linesThis result was entered into the AIMMS optimal modeling

software with the ILOG CPLEX 126 solver to generate theoptimal production amounts of products The results arereturned to the PSO algorithm to calculate the net profit andto determine whether the termination conditions have beensatisfied This study sets the termination condition as thenumber of generations The search ends when the numberof generations reaches the preset number of generations Ifthis number is reached then the PSO algorithm is used toyield the optimal number of machines and tools of eachline to the AIMMS optimal modeling software to generatethe optimal production amounts of products Fitness valuesare calculated during each generation The PSO algorithm isrepeated until the termination condition is satisfiedThe PSOsteps are stated as follows

Step 1 (generation of an initial population) This study usesPSO to determine the migration number of machines andtools among the production lines Given the initial numberof machines and tools an initial population is generatedby randomly selecting the value limited to the availablemaximum number of machines and tools in each line

Step 2 (calculation of the fitness values) The fitness value inthis study is net profit

Step 3 (updating the speed and position of the particle)Equations (22) and (23) are used to update the speed andposition using the following symbols

119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863

1198881 personal best position acceleration constant

1198882 global best position acceleration constant

119862119903(119899) the 119862119903 of the 119899 time

119908(119905) inertia weight in the 119905th iteration

119883119894119889(119905) position of the 119894th particle at the 119889th dimen-

sion in the 119905th iteration

119881119894119889(119905) velocity of the 119894th particle at the 119889th dimension

in the 119905th iteration

119901119887119890119904119905119894119889(119905) personal best position of the 119894th particle at

the 119889th dimension

119892119887119890119904119905119889(119905) global best position at the 119889th dimension

The mathematical model is expressed as follows

119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)


The following steps are used to update the individualspeed and position of each dimension

(1) Set 119894 = 1

(2) Set 119889 = 1

(3) Update the 119889 dimension speed (119881119894119889(119905 + 1)) in particle

119894 using (22)

(4) Update the 119889 dimension position in particle 119894 using(23)

(5) Determinewhether 119889 is equal to119863 If so then 119894 = 119894+1If not then 119889 = 119889+1 and 119899 = 119899+1 and return to Step(3)

(6) Determine whether 119894 is larger than 119868 If it is thisindicates that the update has concluded If not returnto Step (2)

Step 4 (updating the particle best (119901119887119890119904119905)) Updating the119901119887119890119904119905 involves replacing the best position for current indi-vidual particles when the current individual fitness valuesare superior to the 119901119887119890119904119905 fitness values Otherwise thereplacement is not performed and the execution is repeateduntil all particles have been updated

Step 5 (updating the global best (119892119887119890119904119905)) Updating the 119892119887119890119904119905involves replacing the optimal population particles when thecurrent optimal individual solution fitness values are superiorto the 119892119887119890119904119905 fitness values Otherwise the replacement is notperformed

Step 6 (determining whether the termination conditions arereached) The termination condition for the PSO algorithmpresented in this study is determined when the number ofiterations exceeds the set maximum iteration times Other-wise the process returns to Step 2

4 Analysis and Discussion onthe Semiconductor Packaging andTesting Industry Case

41 Introduction to the Case Background This study aimsto conduct a capacity allocation and migration planningfor customer demands by considering a certain large-scalesemiconductor packaging and testing factory as the casestudy Three customers eight types of products and twoproduction lines are involved in this caseThemanufacturingprocess is divided into three bottleneck production stagesnamely the DB WB and MD stations in turn Furthermorethe factory has three types of machines four types of toolsand four categories of materialsThe planning horizon coversfour periods For resource configuration the DB and WBstations have three configurations consisting of machinesThe MD station has seven kinds of configurations consistingof machines tools and materials Appendix B (see Supple-mentary Material) shows the related information necessaryfor this case study

42 Capacity Planning Results The case problem is handledunder demand uncertainty The maximum net profit is$7755748983 for the stochastic programmingmodel Table 1shows the number ofmachines for the lines in the productionstages in each time period Table 2 presents the migrationnumber of machines between lines in each production stagein each time period Table 3 indicates the number of toolsfor the lines in the MD stage in each time period Table 4presents the migration number of tools between lines in theMD stage in each time period Table 5 expresses the salesamounts of products for each customer in each time periodunder different scenarios

43 Expected Value of Perfect Information (EVPI) and Valueof the Stochastic Solution (VSS) WS stands for ldquowait andseerdquo thus the decision-maker must wait for all informationto be revealed before making a decision The objective isto maximize the net profit The solution obtained throughthe deterministic model with average demand is called theexpected value (EV) solution Through the EV solution theindividual objective values of all demand scenarios can beobtained Thereafter these objective values are multipliedby the occurring probability of the corresponding scenarioto obtain the expected value namely the expected resultof using the EV solution (EEV) The ldquohere and nowrdquo typeindicates the maximized net profit value of stochastic pro-gramming which is called SP For the capacity allocationand migration planning problem in this study the solvingresult through SP under uncertainty is compared with thedeterministic model Two indicators namely expected valueof perfect information (EVPI) and value of the stochasticsolution (VSS) are used for analysis

The optimal objective value of the stochastic program-ming model is compared with the expected value of theWS solutions The latter is calculated by determining theoptimal solution for each possible realization of the demandscenarios with certainty Clearly it is better to know thevalue of the future actual demand before making a decisionthan having to make the decision before knowing Thedifference between these two expected objective values iscalled EVPI Furthermore EVPI measures the maximumamount a decision-maker would be willing to pay in returnfor complete (and accurate) information about the future tosolve uncertainty Thus EVPI is defined in (24) If EVPI issmaller the stochastic programming result is closer to theresult obtained with complete information By contrast ifEVPI is larger the influence of uncertain factors is greaterand the price paid for obtaining complete information isconsiderably high

EVPI =WS minus SP (24)

VSS is used to measure the ability of the stochasticprogramming model to increase net profit with the attemptto solve uncertain factors It is the difference between thesolution of the SP model and the expected value of theobjective function when fixing parameters to average valuesand using the corresponding optimal solution Thus VSS isdefined in (25) VSS conveys to us how much we can gain


Table 1 The number of machines for lines at production stages in each time period (119870119876119897119904119896119905

)

Line Production stage

Types of machine1198961 1198962 1198963

Time (month) Time (month) Time (month)1 2 3 4 1 2 3 4 1 2 3 4

1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1

Table 2 The migration number of machines between lines at each production stage in each time period (1198701198721198971198971015840119904119896119905)

Line Move to line

Production stageDB WB MD

Types of machine Types of machine Types of machine1198961 1198962 1198963 1198961 1198962 1198963 1198961 1198962 1198963

Time Time Time Time Time Time Time Time Time1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0

Table 3 The number of tools for lines at MD stage in each time period (119873119876119897119904119899119905

)


Types of tool1198991 1198992 1198993 1198994

Time (month) Time (month) Time (month) Time (month)1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1

Table 4 The migration number of tools between lines at MD stage in each time period (1198731198721198971198971015840119904119899119905)

Line Move to line

Types of tool1198991 1198992 1198993 1198994

Time Time Time Time1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

more if SP is used If VSS is larger the SP result is better thanthe expected result when using the EV solution obtained byreplacing all possible demands with their average values

VSS = SP minus EEV (25)

The related measurements for the case problem in thisstudy are showed in Table 6

431 Net Profit Fluctuation under Different Combinations ofProbability Different probability combinations are designedto investigate whether the occurring probability of all

demand scenarios affects the net profitThe combined designindividually provides significantly high probability values tolowmean and high demand scenarios Table 7 shows that thecapacity planning results under all probability combinationsindicate that net profits using the SP model are higherthan those using the deterministic model Moreover if theoccurring probability of low demand scenario is 08 then itsnet profit is significantly lower than that of the mean demandor high demand scenario which possesses an occurringprobability of 08 Therefore the occurring probability ofthe scenario is positively related to the demand of each


Table 5 The sales amounts of products for each customer in each time period under different scenarios (119878119876119903119894119888119905)

Scenario Product Customer Time period (month)1 2 3 4

Scenario 1 1198941 1198881 45955 80375 11400 37666Scenario 1 1198942 1198881 137866 40188 72154 0Scenario 1 1198943 1198881 99999 21265 0 62030Scenario 1 1198944 1198882 91911 60281 54115 0Scenario 1 1198945 1198882 22978 120563 45096 15066Scenario 1 1198946 1198883 99999 48893 0 33379Scenario 1 1198947 1198883 53614 24113 53175 33899Scenario 1 1198948 1198883 199998 21768 0 11300Scenario 2 1198941 1198881 48000 96000 13500 60000Scenario 2 1198942 1198881 144000 48000 96000 0Scenario 2 1198943 1198881 99999 22857 0 96428Scenario 2 1198944 1198882 96000 72000 72000 0Scenario 2 1198945 1198882 24000 144000 60000 24000Scenario 2 1198946 1198883 99999 54307 48647 0Scenario 2 1198947 1198883 56000 28800 68192 54000Scenario 2 1198948 1198883 199998 26000 0 18000Scenario 3 1198941 1198881 50045 108987 0 82334Scenario 3 1198942 1198881 150134 55812 116115 0Scenario 3 1198943 1198881 58416 0 0 99999Scenario 3 1198944 1198882 100089 83719 89885 0Scenario 3 1198945 1198882 25022 167437 74904 32934Scenario 3 1198946 1198883 99999 58778 0 61055Scenario 3 1198947 1198883 58386 33487 82200 74101Scenario 3 1198948 1198883 199998 30232 0 24700

Table 6 The related measurements for the case problem

Net profitWS 7756048983SP 7755748983EEV 7743904428EVPI 300000VSS 11844555VSSEEV

times 100 () 015

corresponding scenario that is determining the occurringprobability of scenario is highly important when using the SPmodel

432 Changes in EVPI and VSS under Different Probabil-ity Combinations The current study analyzes whether theoccurring probabilities of all demand scenarios have an effecton EVPI and VSS Accordingly several probability com-binations of demand scenarios are designed including theprobability combination with considerably high occurringprobability of specific demand scenario EVPI and VSS underdifferent probability combinations are shown in Table 8Figure 5 shows that when the probability combination is (0101 08) the net profit gap between the deterministic modeland SP model is $50569 Moreover the decision-maker is

Table 7 The related measurements for different probability combi-nations

Probabilitycombinationlowast WS SP EEV

(08 01 01) 6996636166 6996231166 6995992800(01 08 01) 7802771620 7802366620 7802128253(01 01 08) 8468739396 8468649396 8463592463lowastTheoccurring probability of low demand mean demand and high demandscenarios respectively

willing to pay $900 in return for the complete information onfuture uncertainty Hence when the occurring probability ofhigh demand is higher EVPI is lower Specifically the solvingresult of net profit under complete (perfect) information iscloser to the decision made by the SP model Similarly ifVSS is higher then the obtained benefit from the SP modelis better

433 Effect of Demand Variability on Net Profit EVPI andVSS Three types of demand variability are designed in thisstudy Base Case aims to infer demands of all scenarios usingthe coefficient of variation Small variation is equal to 90of Base Case (middle variation) and large variation is 110of Base Case After individually solving the three differentvariations the net profit in all variations under the SP model


Table 8 EVPI and VSS under different probability combinations

Probability combinationslowast EVPI VSS(080101) 4050 2384(030502) 3600 9267(030403) 3150 16151(033303330333) 3000 18446(020305) 2250 29918(010108) 900 50569lowastTheoccurring probability of low demand mean demand and high demandscenarios respectively

60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)

Probability combinations (lowmeanhigh demand)

EVPIVSS

Figure 5The diagram for EVPI andVSS under different probabilitycombinations

and deterministic model can be calculated (Table 9) It alsocan be found from Figure 6 that the gap in net profit willincrease with the increase of demand variation Thus the SPmodel considers demand uncertainty and its result is betterthan that of the deterministic model which only considersaverage demand

44 Sensitivity Analysis

441 Effect ofDemandChange onMachine andToolMigrationand Net Profit Demand change is the primary problemdiscussed in this study The semiconductor packaging andtesting industry cannot accurately forecast the actual demandof customers If the demand change constantly shows positivegrowth or a substantial negative reduction then the two-stage SP model will significantly respond to considerabledemand change compared to the deterministicmodelHencewhen the actual demand is lower capacity waste can bereduced By contrast when the actual demand is highercapacity shortage can be avoided For the case company inthis study the increasing demand results in the continuousimprovement in net profit because of the demand growthHowever the number of machine and tool migrations isunaffected by demand change as demand decreases netprofit and the number of machine and tool migrationsare reduced as demand is decreased Doing so can avoidunnecessary migration costs as shown in Tables 10 and 11

Table 9 Comparison of net profit under demand variability

Demand variability EEV SP GapSmall variation 72884515 72888460 3945Middle variation 77539044 77557489 18445Large variation 82113557 82134434 20877

25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit

Small variation Middle variation Large variation

Gap in net profit between EEV and SP

Figure 6 Gap in net profit under different demand variability

442Effect of Changes inUnitMigrationCost onMachineToolMigration and Net Profit The unit migration cost affectsmoving times When the unit migration cost is more expen-sive it significantly increases the total migration cost therebylowering the net profit When the unit migration cost isconsiderably inexpensive frequent machinetool migrationsand production amounts of products may increase therebyincreasing the net profit For the case company in this studywhen the unit migration cost starts to increase the net profitwill decrease and the number of machinetool migrationswill also decrease When the unit migration cost is down thenet profit will increase However the number ofmachinetoolmigrations remains constant as shown in Tables 12 and 13

443 Effect of Sales Price Fluctuation on Machine and ToolMigration and Net Profit The sales price of products affectsnet profit If sales price is higher then the net profit increasesBy contrast if sale price is down then the net profit decreasesWhen sales price is higher salesmen will attempt to addressthe customer needs and provide higher service level whensales price is lower they cannot completelymeet the customerpromise needs thereby resulting in the occurrence of shortsupply which lowers service level Thus a better balanceddecision must be determined between sales revenue andproductionmigration costs For the case company in thisstudy as shown in Tables 14 and 15 when sales price is raisedthe net profit increases andmachinetoolmigration decisionsare not affected on the contrary when the sales price islowered the net profit decreases andmachinetool migrationamounts are also reduced because of low sales price

444 Effect of Migration Capability on Machine and ToolMigration and Net Profit Given that capacity allocationdecisions are made several products may not be manu-factured because of the limited flexibility of machine andtool migration Production capacity cannot be allocated


Table 10 Changes in migration costs and net profit under positively growing demand

Demand growth multiples2 4 6 8 10

Machine migration cost 13000 13000 13000 13000 13000Tool migration cost 4000 4000 4000 4000 4000Net profit 110214963 136961643 141816636 145421413 146009670

Table 11 Changes in migration costs and net profit under negatively decreasing demand

Demand reduction multiples09 07 05 03 01


Table 12 Changes in migration decisions and net profit under the increased unit migration cost

Increased unit migration cost (multiple)2 5 10 50 100

Machine migration amount 5 4 4 3 3Tool migration amount 4 4 3 3 2Net profit 77543489 77505527 77454535 77073544 76645716

Table 13 Changes in migration decisions and net profit under the reduced unit migration cost

Reduced unit migration cost (multiple)09 07 05 03 01


flexibly between different production lines Without migra-tion capability limitation all machines and tools becomemovable which is advantageous to the adjustment of capacityBy contrast if the flexibility of migration is limited thenadjusting to a considerably high capacity level is difficultthereby decreasing net profit as shown in Table 16 Moreoverthe number of machine migrations increases as migrationflexibility opens

5 Conclusion

This study considers a certain large-scale semiconductorpackaging and testing factory to address capacity allocationand migration planning problems under demand uncer-taintyThe planning scope includes three bottleneck stationsnamely the DB WB and MD stations Moreover the two-stage stochastic programming approach is applied and itsmathematical model is formulated to solve this problemMachine and tool migration decisions are deemed to be thefirst-stage decision The second-stage decision is capacity

allocation which can be solved once the uncertain factorsare revealed Hence when demand is changed machine andtool migration decisions remain to be a better robust resultThe measuring indicators EVPI and VSS are applied toevaluate the SP model and the deterministic EEV modelSP obtains a better net profit than EEV the VSS valuesobtained are positiveThus the two-stage SPmodel proposedin this study can indeed improve the deficiencies of the tra-ditional deterministic model Furthermore decision-makerscan make good use of sensitivity analysis results as reference

This paper can assist the semiconductor packaging andtesting factory in simultaneously conducting capacity allo-cation and resource configuration planning with the useof existing resources Moreover the two-stage SP methoddetermines a robust machine and tool migration decisionin advance as a response to future fluctuating demand Thismodel can also obtain the optimal capacity allocation andmigration planning decision It is closer to actual industryapplication and reaches the economic target of semiconduc-tor packaging and testing industry namelymeeting customerneeds and maximizing net profit


Table 14 Changes in migration decisions and net profit under the increased sales price

Increased sales price (multiple)2 4 6 10 50


Table 15 Changes in migration decisions and net profit under the reduced sales price



Table 16 Changes in migration decisions and net profit underdifferent migration flexibility

Migration capabilityLimited Opened

Machine migration amount 3 6Tool migration amount 3 3Machine migration cost 6500 11500Tool migration cost 3000 3000Net profit 78227955 84600698

Conflict of Interests

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

References

[1] S Karabuk and S D Wu ldquoCoordinating strategic capacityplanning in the semiconductor industryrdquo Operations Researchvol 51 no 6 pp 839ndash849 2003

[2] T-L Chen Y-Y Chen and H-C Lu ldquoA capacity allocationand expansion model for TFT-LCD multi-site manufacturingrdquoJournal of Intelligent Manufacturing vol 24 no 4 pp 847ndash8722013

[3] Y-Y Chen T-L Chen and C-D Liou ldquoMedium-term multi-plant capacity planning problems considering auxiliary tools forthe semiconductor foundryrdquo International Journal of AdvancedManufacturing Technology vol 64 no 9-12 pp 1213ndash1230 2013

[4] J T Lin C-H Wu T-L Chen and S-H Shih ldquoA stochasticprogrammingmodel for strategic capacity planning in thin filmtransistor-liquid crystal display (TFT-LCD) industryrdquo Comput-ers and Operations Research vol 38 no 7 pp 992ndash1007 2011

[5] J T Lin T-L Chen and H-C Chu ldquoA stochastic dynamic pro-gramming approach for multi-site capacity planning in TFT-LCD manufacturing under demand uncertaintyrdquo InternationalJournal of Production Economics vol 148 pp 21ndash36 2014

[6] G B Dantzig ldquoLinear programming under uncertaintyrdquoMan-agement Science vol 1 pp 197ndash206 1955

[7] A M Uribe J K Cochran and D L Shunk ldquoTwo-stage simu-lation optimization for agile manufacturing capacity planningrdquo

International Journal of Production Research vol 41 no 6 pp1181ndash1197 2003

[8] O Listes and R Dekker ldquoA stochastic approach to a casestudy for product recovery network designrdquo European Journalof Operational Research vol 160 no 1 pp 268ndash287 2005

[9] M I G Salema A P Barbosa-Povoa and A Q Novais ldquoAnoptimization model for the design of a capacitated multi-product reverse logistics network with uncertaintyrdquo EuropeanJournal of Operational Research vol 179 no 3 pp 1063ndash10772007

[10] D-H Lee M Dong and W Bian ldquoThe design of sustainablelogistics network under uncertaintyrdquo International Journal ofProduction Economics vol 128 no 1 pp 159ndash166 2010

[11] Y Cardona-Valdes A Alvarez and D Ozdemir ldquoA bi-objectivesupply chain design problem with uncertaintyrdquo TransportationResearch Part C Emerging Technologies vol 19 no 5 pp 821ndash832 2011

[12] S S Kara and S Onut ldquoA two-stage stochastic and robustprogramming approach to strategic planning of a reverse supplynetwork the case of paper recyclingrdquo Expert Systems withApplications vol 37 no 9 pp 6129ndash6137 2010

[13] M S Pishvaee F Jolai and J Razmi ldquoA stochastic optimizationmodel for integrated forwardreverse logistics network designrdquoJournal of Manufacturing Systems vol 28 no 4 pp 107ndash1142009

[14] S H Amin and G Zhang ldquoA multi-objective facility locationmodel for closed-loop supply chain network under uncertaindemand and returnrdquo Applied Mathematical Modelling vol 37no 6 pp 4165ndash4176 2013

[15] M Ramezani M Bashiri and R Tavakkoli-Moghaddam ldquoAnew multi-objective stochastic model for a forwardreverselogistic network design with responsiveness and quality levelrdquoApplied Mathematical Modelling vol 37 no 1-2 pp 328ndash3442013

[16] HMohammadi Bidhandi andM Y Rosnah ldquoIntegrated supplychain planning under uncertainty using an improved stochasticapproachrdquo Applied Mathematical Modelling vol 35 no 6 pp2618ndash2630 2011

[17] Z Sazvar S M J M Al-E-Hashem A Baboli and M RA Jokar ldquoA bi-objective stochastic programming model for acentralized green supply chain with deteriorating productsrdquoInternational Journal of Production Economics vol 150 pp 140ndash154 2014


[18] Z Zhou J Zhang P Liu Z Li M C Georgiadis and EN Pistikopoulos ldquoA two-stage stochastic programming modelfor the optimal design of distributed energy systemsrdquo AppliedEnergy vol 103 pp 135ndash144 2013

[19] K-J Wang and S-M Wang ldquoSimultaneous resource portfo-lio planning under demand and technology uncertainty inthe semiconductor testing industryrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 5 pp 278ndash287 2013

[20] Y J Liu T Li S C Tong C L P Chen and D J Li ldquoRein-forcement learning design-based adaptive tracking control withless learning parameters for nonlinear discrete-time MIMOsystemsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 26 pp 165ndash176 2015

[21] Y-J Liu Y Gao S Tong and Y Li ldquoFuzzy approximation-basedadaptive backstepping optimal control for a class of nonlineardiscrete-time systems with dead-zonerdquo IEEE Transactions onFuzzy Systems 1 page 2015

[22] D Wang D Liu and Q Wei ldquoFinite-horizon neuro-optimaltracking control for a class of discrete-time nonlinear systemsusing adaptive dynamic programming approachrdquo Neurocom-puting vol 78 no 1 pp 14ndash22 2012

[23] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014

[24] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinearstrict-feedback systemsrdquo IEEE Transactions on Systems ManandCybernetics Part B Cybernetics vol 41 no 6 pp 1693ndash17042011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

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

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Demand in packaging and testing factories fluctuatesbecause of different limitations in capacity planning and themake-to-order (MTO) production environment Thereforedeviating from previous studies that consider the determinis-tic programming model the current study employs demandas an uncertain factor to solve the problem through a two-stage stochastic programming (SP) model The feasibility ofthe capacity planning method proposed in this study is alsodemonstrated through a practical case studyMoreover basedon the different parameters change sensitivity analysis for thestochastic programming model is provided to understandif capacity planning will be affected by different scenariosdemandsmigration costs ofmachines and tools sale prices ofproducts flexibility of capacity migration and other factors

This paper is organized as follows Section 2 reviews therelated literature Section 3 establishes the definition of thecapacity planning problem of the semiconductor packagingand testing industry as well as the development of a two-stagestochastic programming model and the proposed hybridapproach Section 4 discusses the application and analysisof a certain large-scale semiconductor packaging and testingfactory case Section 5 provides the conclusion of this study

2 Literature Review

21 Overview of the Semiconductor Packaging and TestingIndustry Characteristics Semiconductor products currentlycomprise four categories namely integrated circuit (IC)discrete sensor and optoelectronics However this studyaims to investigate the IC category The IC manufacturingprocess with vertical integration features has front- and back-end processes The upstream to downstream processes of thetarget industry cover the following five steps IC design maskmaking IC making chip packaging and chip testing

The manufacturing process and production characteris-tics of the semiconductor packaging and testing industry aredescribed as follows

211 Make to Order The semiconductor packaging and test-ing industry prepares materials based on orders placed bycustomers Thereafter production proceeds based on cus-tomersrsquo needs and services thereby preventing the indus-try from predicting the demand in advance Furthermoredemand forecasting for this industry cannot be learnedfrom past experience Accordingly this study considers thedemand to be uncertain and expresses fluctuations in cus-tomer needs through different scenarios This considerationwill avoid insufficient or wasted capacity during the produc-tion while addressing customer needs and maximizing netprofit

The production in the IC packaging and testing industryis customer-oriented Product categories are diversified basedon different customer needs Packaging types can be dividedinto lead frame package ball grid array flip chip system inpackage and multichip packages Moreover each packagingtype is divided into various product types because of the useof different chip sizes or pin numbers Product types also havedifferent capacity constraints

212 Flow Shop In an IC packaging and testing factoryan order is often divided into several work orders Figure 1shows that each work order is manufactured based onthe production flow Meanwhile IC packaging and testingfactories have multiple production lines For example acertain large-scale domestic packaging and testing factoryhas approximately 25 lines This study only investigates thebottleneck stations in these factories namely the DB WBand MD stations It is described as a flow shop productionenvironment Thus products will enter the WB station afterleaving theDB station After the products are processed in theWB station they eventually enter the MD station Thereforeproducts have sequential dependencies in the factory area

213 Unrelated Parallel Machine Given the rapid manner bywhich products are updated process changes and equipmentupgrade will compel semiconductor packaging and testingfactories to frequently purchase new or different brands ofmachines to respond to market changes Having differentmachine types and brands leads to the presence of differentgrades of machines in the production line thereby resultingin the production pattern of unrelated parallel machinesWhen managements arrange orders they exert effort tosatisfy customer needs and meet the maximal service levelIn addition they manufacture by utilizing machines withthe highest capacity Hence they first move the machinesand allocate the proper machines in groups to avoid failurein production caused by inconsistencies in machine typesMost machines for the semiconductor packaging and testingindustry are movable For domestic IC packaging and testingfactories the number of moving machines reaches approxi-mately 60 each month

22 Capacity Planning Karabuk and Wu [1] indicate thatcapacity planning can be described as an iterative processbetween the following two main components (1) capacityexpansion given projected product demands identify therequired manufacturing technologies and their capacity lev-els to be physically expanded or outsourced through theplanning period (2) capacity configuration determinewhichfacility is to be configured with which technologies mix Theoverall objective is tomeet a revenuemodel based on strategicdemand planning (which blends demand forecasting andproactive market development strategies) This objective canbe viewed as meeting projected demands with minimizedtotal costs

Chen et al [2] present a capacity allocation and expan-sion problem of thin film transistor liquid crystal display(TFT-LCD) manufacturing in the multisite environmentThe objective is to simultaneously seek an optimal capacityallocation plan and capacity expansion policy under single-stage multigeneration and multisite structures Capacityallocation decides on profitable product mixes and allocatedproduction quantities of each product group at each produc-tion site Capacity expansion is concerned with determiningthe timing types and sizes of capacity investments especiallyin the acquisition of auxiliary tools A mixed integer linearprogramming (MILP) is proposed which considers manypractical characteristics Finally an industrial case study

















Scenario 1

Scenario 2

Scenario 3


















Product 1


Product category

Production stage

Machine type

Tool type


k1 k2 k1 k2k2 k3

n1 n2

m4 m4














Scenario 1

Scenario 2

Scenario 3



Demand



(2) Parameters


119889119890119903



(sum119903119901119903

= 1)119901119903119894119888119905



119896119897119897119904119896


119904119896119904119894119895119904119896



at stage 119904


119899119897119897119904119899


119899119904119894119895119904119899



stage 119904



119898119902119904119898119905


119898119904119894119895119904119898






1199051198871198971199041198971015840(119904+1)



V119888119894119897119895119904






119870119876119897119904119896119905






119883119876119903



11989411989711989511990411989710158401198951015840(119904+1)119905






Maximize

sum

119903

119901119903

sum

119894

sum

119888

sum

119905

(119901119903119894119888119905times 119878119876119903

119894119888119905) minussum

119894

sum

119897

sum

119895

sum

119904

sum

119905

(V119888119894119897119895119904times 119883119876

119903

119894119897119895119904119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119896

sum

119905

(119896119888119904times 119870119872

1198971198971015840119904119896119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119899

sum

119905

(119899119888119904times 119873119872

1198971198971015840119904119899119905)

(1)


(5) Constraints



1198701198761198971199041198960

= 119896119897119897119904119896

forall119897 119904 119896 (2)

119870119876119897119904119896119905= 119870119876

119897119904119896(119905minus1)minussum

1198971015840

1198701198721198971198971015840119904119896119905

+sum

1198971015840

1198701198721198971015840119897119904119896119905

forall119897 119904 119896 119905

(3)

119870119876119897119904119896119905le 119896119906119897119904

forall119897 119904 119896 119905 (4)

1198701198721198971198971015840119904119896119905le 119872 times 119896119887

1198971198971015840119904forall119897 1198971015840

119904 119896 119905 (5)







1198731198761198971199041198990

= 119899119897119897119904119899

forall119897 119904 119899 (6)

119873119876119897119904119899119905= 119873119876

119897119904119899(119905minus1)minussum

1198971015840

1198731198721198971198971015840119904119899119905

+sum

1198971015840

1198731198721198971015840119897119904119899119905

forall119897 119904 119899 119905

(7)

119873119876119897119904119899119905le 119899119906119897119904

forall119897 119904 119899 119905 (8)

1198731198721198971198971015840119904119899119905le 119872 times 119899119887

1198971198971015840119904forall119897 1198971015840

119904 119899 119905 (9)


1198971198971015840119904refers to



119870119876119897119904119896119905 1198701198721198971198971015840119904119896119905 119873119876119897119904119899119905 1198731198721198971198971015840119904119899119905isin integer

forall119897 119904 119896 119899 119905

(10)




119883119876119903

119894119897119895119904119905= sum

1198971015840

sum

1198951015840

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

forall119894 119897 119895 119904 = 1 119878 minus 1 119905 119903

(11)

sum

1198971015840

sum

1198951015840

119877119876119903

11989411989710158401198951015840(119904minus1)119897119895119904119905

= 119883119876119903

119894119897119895119904119905forall119894 119897 119895 119904 = 2 119878 119905 119903 (12)



sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119896119904119894119895119904119896) le 119870119876

119897119904119896119905times 119896119886119904119896

forall119897 119904 119896 119905 119903 (13)

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119899119904119894119895119904119899) le 119873119876

119897119904119899119905times 119899119886119904119899

forall119897 119904 119899 119905 119903 (14)



sum

119894

sum

119897

sum

119895

(119883119876119903

119894119897119895119904119905times 119898119904119894119895119904119898) le 119898119902

119904119898119905forall119904119898 119905 119903 (15)



119883119876119903

119894119897119895119904119905le 119872 times 119905119891

119894119895119904forall119894 119897 119895 119904 119905 119903 (16)






119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

le 119872 times 1199051198871198971199041198971015840(119904+1)

forall119894 119897 119895 119904 1198971015840

1198951015840

119905 119903 (17)




1198971199041198971015840(119904+1)


1198971199041198971015840(119904+1)



sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)



119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)



119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)








119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863



119862119903(119899) the 119862119903 of the 119899 time










119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)



(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Scenario 1

Scenario 2

Scenario 3


















Product 1


Product category

Production stage

Machine type

Tool type


k1 k2 k1 k2k2 k3

n1 n2

m4 m4














Scenario 1

Scenario 2

Scenario 3



Demand



(2) Parameters


119889119890119903



(sum119903119901119903

= 1)119901119903119894119888119905



119896119897119897119904119896


119904119896119904119894119895119904119896



at stage 119904


119899119897119897119904119899


119899119904119894119895119904119899



stage 119904



119898119902119904119898119905


119898119904119894119895119904119898






1199051198871198971199041198971015840(119904+1)



V119888119894119897119895119904






119870119876119897119904119896119905






119883119876119903



11989411989711989511990411989710158401198951015840(119904+1)119905






Maximize

sum

119903

119901119903

sum

119894

sum

119888

sum

119905

(119901119903119894119888119905times 119878119876119903

119894119888119905) minussum

119894

sum

119897

sum

119895

sum

119904

sum

119905

(V119888119894119897119895119904times 119883119876

119903

119894119897119895119904119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119896

sum

119905

(119896119888119904times 119870119872

1198971198971015840119904119896119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119899

sum

119905

(119899119888119904times 119873119872

1198971198971015840119904119899119905)

(1)


(5) Constraints



1198701198761198971199041198960

= 119896119897119897119904119896

forall119897 119904 119896 (2)

119870119876119897119904119896119905= 119870119876

119897119904119896(119905minus1)minussum

1198971015840

1198701198721198971198971015840119904119896119905

+sum

1198971015840

1198701198721198971015840119897119904119896119905

forall119897 119904 119896 119905

(3)

119870119876119897119904119896119905le 119896119906119897119904

forall119897 119904 119896 119905 (4)

1198701198721198971198971015840119904119896119905le 119872 times 119896119887

1198971198971015840119904forall119897 1198971015840

119904 119896 119905 (5)







1198731198761198971199041198990

= 119899119897119897119904119899

forall119897 119904 119899 (6)

119873119876119897119904119899119905= 119873119876

119897119904119899(119905minus1)minussum

1198971015840

1198731198721198971198971015840119904119899119905

+sum

1198971015840

1198731198721198971015840119897119904119899119905

forall119897 119904 119899 119905

(7)

119873119876119897119904119899119905le 119899119906119897119904

forall119897 119904 119899 119905 (8)

1198731198721198971198971015840119904119899119905le 119872 times 119899119887

1198971198971015840119904forall119897 1198971015840

119904 119899 119905 (9)


1198971198971015840119904refers to



119870119876119897119904119896119905 1198701198721198971198971015840119904119896119905 119873119876119897119904119899119905 1198731198721198971198971015840119904119899119905isin integer

forall119897 119904 119896 119899 119905

(10)




119883119876119903

119894119897119895119904119905= sum

1198971015840

sum

1198951015840

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

forall119894 119897 119895 119904 = 1 119878 minus 1 119905 119903

(11)

sum

1198971015840

sum

1198951015840

119877119876119903

11989411989710158401198951015840(119904minus1)119897119895119904119905

= 119883119876119903

119894119897119895119904119905forall119894 119897 119895 119904 = 2 119878 119905 119903 (12)



sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119896119904119894119895119904119896) le 119870119876

119897119904119896119905times 119896119886119904119896

forall119897 119904 119896 119905 119903 (13)

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119899119904119894119895119904119899) le 119873119876

119897119904119899119905times 119899119886119904119899

forall119897 119904 119899 119905 119903 (14)



sum

119894

sum

119897

sum

119895

(119883119876119903

119894119897119895119904119905times 119898119904119894119895119904119898) le 119898119902

119904119898119905forall119904119898 119905 119903 (15)



119883119876119903

119894119897119895119904119905le 119872 times 119905119891

119894119895119904forall119894 119897 119895 119904 119905 119903 (16)






119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

le 119872 times 1199051198871198971199041198971015840(119904+1)

forall119894 119897 119895 119904 1198971015840

1198951015840

119905 119903 (17)




1198971199041198971015840(119904+1)


1198971199041198971015840(119904+1)



sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)



119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)



119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)








119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863



119862119903(119899) the 119862119903 of the 119899 time










119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)



(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Product 1


Product category

Production stage

Machine type

Tool type


k1 k2 k1 k2k2 k3

n1 n2

m4 m4














Scenario 1

Scenario 2

Scenario 3



Demand



(2) Parameters


119889119890119903



(sum119903119901119903

= 1)119901119903119894119888119905



119896119897119897119904119896


119904119896119904119894119895119904119896



at stage 119904


119899119897119897119904119899


119899119904119894119895119904119899



stage 119904



119898119902119904119898119905


119898119904119894119895119904119898






1199051198871198971199041198971015840(119904+1)



V119888119894119897119895119904






119870119876119897119904119896119905






119883119876119903



11989411989711989511990411989710158401198951015840(119904+1)119905






Maximize

sum

119903

119901119903

sum

119894

sum

119888

sum

119905

(119901119903119894119888119905times 119878119876119903

119894119888119905) minussum

119894

sum

119897

sum

119895

sum

119904

sum

119905

(V119888119894119897119895119904times 119883119876

119903

119894119897119895119904119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119896

sum

119905

(119896119888119904times 119870119872

1198971198971015840119904119896119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119899

sum

119905

(119899119888119904times 119873119872

1198971198971015840119904119899119905)

(1)


(5) Constraints



1198701198761198971199041198960

= 119896119897119897119904119896

forall119897 119904 119896 (2)

119870119876119897119904119896119905= 119870119876

119897119904119896(119905minus1)minussum

1198971015840

1198701198721198971198971015840119904119896119905

+sum

1198971015840

1198701198721198971015840119897119904119896119905

forall119897 119904 119896 119905

(3)

119870119876119897119904119896119905le 119896119906119897119904

forall119897 119904 119896 119905 (4)

1198701198721198971198971015840119904119896119905le 119872 times 119896119887

1198971198971015840119904forall119897 1198971015840

119904 119896 119905 (5)







1198731198761198971199041198990

= 119899119897119897119904119899

forall119897 119904 119899 (6)

119873119876119897119904119899119905= 119873119876

119897119904119899(119905minus1)minussum

1198971015840

1198731198721198971198971015840119904119899119905

+sum

1198971015840

1198731198721198971015840119897119904119899119905

forall119897 119904 119899 119905

(7)

119873119876119897119904119899119905le 119899119906119897119904

forall119897 119904 119899 119905 (8)

1198731198721198971198971015840119904119899119905le 119872 times 119899119887

1198971198971015840119904forall119897 1198971015840

119904 119899 119905 (9)


1198971198971015840119904refers to



119870119876119897119904119896119905 1198701198721198971198971015840119904119896119905 119873119876119897119904119899119905 1198731198721198971198971015840119904119899119905isin integer

forall119897 119904 119896 119899 119905

(10)




119883119876119903

119894119897119895119904119905= sum

1198971015840

sum

1198951015840

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

forall119894 119897 119895 119904 = 1 119878 minus 1 119905 119903

(11)

sum

1198971015840

sum

1198951015840

119877119876119903

11989411989710158401198951015840(119904minus1)119897119895119904119905

= 119883119876119903

119894119897119895119904119905forall119894 119897 119895 119904 = 2 119878 119905 119903 (12)



sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119896119904119894119895119904119896) le 119870119876

119897119904119896119905times 119896119886119904119896

forall119897 119904 119896 119905 119903 (13)

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119899119904119894119895119904119899) le 119873119876

119897119904119899119905times 119899119886119904119899

forall119897 119904 119899 119905 119903 (14)



sum

119894

sum

119897

sum

119895

(119883119876119903

119894119897119895119904119905times 119898119904119894119895119904119898) le 119898119902

119904119898119905forall119904119898 119905 119903 (15)



119883119876119903

119894119897119895119904119905le 119872 times 119905119891

119894119895119904forall119894 119897 119895 119904 119905 119903 (16)






119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

le 119872 times 1199051198871198971199041198971015840(119904+1)

forall119894 119897 119895 119904 1198971015840

1198951015840

119905 119903 (17)




1198971199041198971015840(119904+1)


1198971199041198971015840(119904+1)



sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)



119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)



119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)








119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863



119862119903(119899) the 119862119903 of the 119899 time










119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)



(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Scenario 1

Scenario 2

Scenario 3



Demand



(2) Parameters


119889119890119903



(sum119903119901119903

= 1)119901119903119894119888119905



119896119897119897119904119896


119904119896119904119894119895119904119896



at stage 119904


119899119897119897119904119899


119899119904119894119895119904119899



stage 119904



119898119902119904119898119905


119898119904119894119895119904119898






1199051198871198971199041198971015840(119904+1)



V119888119894119897119895119904






119870119876119897119904119896119905






119883119876119903



11989411989711989511990411989710158401198951015840(119904+1)119905






Maximize

sum

119903

119901119903

sum

119894

sum

119888

sum

119905

(119901119903119894119888119905times 119878119876119903

119894119888119905) minussum

119894

sum

119897

sum

119895

sum

119904

sum

119905

(V119888119894119897119895119904times 119883119876

119903

119894119897119895119904119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119896

sum

119905

(119896119888119904times 119870119872

1198971198971015840119904119896119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119899

sum

119905

(119899119888119904times 119873119872

1198971198971015840119904119899119905)

(1)


(5) Constraints



1198701198761198971199041198960

= 119896119897119897119904119896

forall119897 119904 119896 (2)

119870119876119897119904119896119905= 119870119876

119897119904119896(119905minus1)minussum

1198971015840

1198701198721198971198971015840119904119896119905

+sum

1198971015840

1198701198721198971015840119897119904119896119905

forall119897 119904 119896 119905

(3)

119870119876119897119904119896119905le 119896119906119897119904

forall119897 119904 119896 119905 (4)

1198701198721198971198971015840119904119896119905le 119872 times 119896119887

1198971198971015840119904forall119897 1198971015840

119904 119896 119905 (5)







1198731198761198971199041198990

= 119899119897119897119904119899

forall119897 119904 119899 (6)

119873119876119897119904119899119905= 119873119876

119897119904119899(119905minus1)minussum

1198971015840

1198731198721198971198971015840119904119899119905

+sum

1198971015840

1198731198721198971015840119897119904119899119905

forall119897 119904 119899 119905

(7)

119873119876119897119904119899119905le 119899119906119897119904

forall119897 119904 119899 119905 (8)

1198731198721198971198971015840119904119899119905le 119872 times 119899119887

1198971198971015840119904forall119897 1198971015840

119904 119899 119905 (9)


1198971198971015840119904refers to



119870119876119897119904119896119905 1198701198721198971198971015840119904119896119905 119873119876119897119904119899119905 1198731198721198971198971015840119904119899119905isin integer

forall119897 119904 119896 119899 119905

(10)




119883119876119903

119894119897119895119904119905= sum

1198971015840

sum

1198951015840

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

forall119894 119897 119895 119904 = 1 119878 minus 1 119905 119903

(11)

sum

1198971015840

sum

1198951015840

119877119876119903

11989411989710158401198951015840(119904minus1)119897119895119904119905

= 119883119876119903

119894119897119895119904119905forall119894 119897 119895 119904 = 2 119878 119905 119903 (12)



sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119896119904119894119895119904119896) le 119870119876

119897119904119896119905times 119896119886119904119896

forall119897 119904 119896 119905 119903 (13)

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119899119904119894119895119904119899) le 119873119876

119897119904119899119905times 119899119886119904119899

forall119897 119904 119899 119905 119903 (14)



sum

119894

sum

119897

sum

119895

(119883119876119903

119894119897119895119904119905times 119898119904119894119895119904119898) le 119898119902

119904119898119905forall119904119898 119905 119903 (15)



119883119876119903

119894119897119895119904119905le 119872 times 119905119891

119894119895119904forall119894 119897 119895 119904 119905 119903 (16)






119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

le 119872 times 1199051198871198971199041198971015840(119904+1)

forall119894 119897 119895 119904 1198971015840

1198951015840

119905 119903 (17)




1198971199041198971015840(119904+1)


1198971199041198971015840(119904+1)



sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)



119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)



119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)








119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863



119862119903(119899) the 119862119903 of the 119899 time










119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)



(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898119902119904119898119905


119898119904119894119895119904119898






1199051198871198971199041198971015840(119904+1)



V119888119894119897119895119904






119870119876119897119904119896119905






119883119876119903



11989411989711989511990411989710158401198951015840(119904+1)119905






Maximize

sum

119903

119901119903

sum

119894

sum

119888

sum

119905

(119901119903119894119888119905times 119878119876119903

119894119888119905) minussum

119894

sum

119897

sum

119895

sum

119904

sum

119905

(V119888119894119897119895119904times 119883119876

119903

119894119897119895119904119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119896

sum

119905

(119896119888119904times 119870119872

1198971198971015840119904119896119905)

minussum

119897

sum

1198971015840

sum

119904

sum

119899

sum

119905

(119899119888119904times 119873119872

1198971198971015840119904119899119905)

(1)


(5) Constraints



1198701198761198971199041198960

= 119896119897119897119904119896

forall119897 119904 119896 (2)

119870119876119897119904119896119905= 119870119876

119897119904119896(119905minus1)minussum

1198971015840

1198701198721198971198971015840119904119896119905

+sum

1198971015840

1198701198721198971015840119897119904119896119905

forall119897 119904 119896 119905

(3)

119870119876119897119904119896119905le 119896119906119897119904

forall119897 119904 119896 119905 (4)

1198701198721198971198971015840119904119896119905le 119872 times 119896119887

1198971198971015840119904forall119897 1198971015840

119904 119896 119905 (5)







1198731198761198971199041198990

= 119899119897119897119904119899

forall119897 119904 119899 (6)

119873119876119897119904119899119905= 119873119876

119897119904119899(119905minus1)minussum

1198971015840

1198731198721198971198971015840119904119899119905

+sum

1198971015840

1198731198721198971015840119897119904119899119905

forall119897 119904 119899 119905

(7)

119873119876119897119904119899119905le 119899119906119897119904

forall119897 119904 119899 119905 (8)

1198731198721198971198971015840119904119899119905le 119872 times 119899119887

1198971198971015840119904forall119897 1198971015840

119904 119899 119905 (9)


1198971198971015840119904refers to



119870119876119897119904119896119905 1198701198721198971198971015840119904119896119905 119873119876119897119904119899119905 1198731198721198971198971015840119904119899119905isin integer

forall119897 119904 119896 119899 119905

(10)




119883119876119903

119894119897119895119904119905= sum

1198971015840

sum

1198951015840

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

forall119894 119897 119895 119904 = 1 119878 minus 1 119905 119903

(11)

sum

1198971015840

sum

1198951015840

119877119876119903

11989411989710158401198951015840(119904minus1)119897119895119904119905

= 119883119876119903

119894119897119895119904119905forall119894 119897 119895 119904 = 2 119878 119905 119903 (12)



sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119896119904119894119895119904119896) le 119870119876

119897119904119896119905times 119896119886119904119896

forall119897 119904 119896 119905 119903 (13)

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119899119904119894119895119904119899) le 119873119876

119897119904119899119905times 119899119886119904119899

forall119897 119904 119899 119905 119903 (14)



sum

119894

sum

119897

sum

119895

(119883119876119903

119894119897119895119904119905times 119898119904119894119895119904119898) le 119898119902

119904119898119905forall119904119898 119905 119903 (15)



119883119876119903

119894119897119895119904119905le 119872 times 119905119891

119894119895119904forall119894 119897 119895 119904 119905 119903 (16)






119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

le 119872 times 1199051198871198971199041198971015840(119904+1)

forall119894 119897 119895 119904 1198971015840

1198951015840

119905 119903 (17)




1198971199041198971015840(119904+1)


1198971199041198971015840(119904+1)



sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)



119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)



119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)








119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863



119862119903(119899) the 119862119903 of the 119899 time










119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)



(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198731198761198971199041198990

= 119899119897119897119904119899

forall119897 119904 119899 (6)

119873119876119897119904119899119905= 119873119876

119897119904119899(119905minus1)minussum

1198971015840

1198731198721198971198971015840119904119899119905

+sum

1198971015840

1198731198721198971015840119897119904119899119905

forall119897 119904 119899 119905

(7)

119873119876119897119904119899119905le 119899119906119897119904

forall119897 119904 119899 119905 (8)

1198731198721198971198971015840119904119899119905le 119872 times 119899119887

1198971198971015840119904forall119897 1198971015840

119904 119899 119905 (9)


1198971198971015840119904refers to



119870119876119897119904119896119905 1198701198721198971198971015840119904119896119905 119873119876119897119904119899119905 1198731198721198971198971015840119904119899119905isin integer

forall119897 119904 119896 119899 119905

(10)




119883119876119903

119894119897119895119904119905= sum

1198971015840

sum

1198951015840

119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

forall119894 119897 119895 119904 = 1 119878 minus 1 119905 119903

(11)

sum

1198971015840

sum

1198951015840

119877119876119903

11989411989710158401198951015840(119904minus1)119897119895119904119905

= 119883119876119903

119894119897119895119904119905forall119894 119897 119895 119904 = 2 119878 119905 119903 (12)



sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119896119904119894119895119904119896) le 119870119876

119897119904119896119905times 119896119886119904119896

forall119897 119904 119896 119905 119903 (13)

sum

119894

sum

119895

(119883119876119903

119894119897119895119904119905times 119899119904119894119895119904119899) le 119873119876

119897119904119899119905times 119899119886119904119899

forall119897 119904 119899 119905 119903 (14)



sum

119894

sum

119897

sum

119895

(119883119876119903

119894119897119895119904119905times 119898119904119894119895119904119898) le 119898119902

119904119898119905forall119904119898 119905 119903 (15)



119883119876119903

119894119897119895119904119905le 119872 times 119905119891

119894119895119904forall119894 119897 119895 119904 119905 119903 (16)






119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

le 119872 times 1199051198871198971199041198971015840(119904+1)

forall119894 119897 119895 119904 1198971015840

1198951015840

119905 119903 (17)




1198971199041198971015840(119904+1)


1198971199041198971015840(119904+1)



sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)



119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)



119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)








119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863



119862119903(119899) the 119862119903 of the 119899 time










119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)



(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198971199041198971015840(119904+1)


1198971199041198971015840(119904+1)



sum

119897

sum

119895

119883119876119903

119894119897119895119904119905= 119878119876119903

119894119888119905forall119894 119904 = 119878 119888 119905 119903 (18)

119878119876119903

119894119888119905le 119889119890119903

119894119888119905forall119894 119888 119905 119903 (19)



119878119871119903

119888= [

sum119894119878119876119903

119894119888119905

sum119894119889119890119903

119894119888119905

] forall119888 119905 119903 (20)



119883119876119903

119894119897119895119904119905 119877119876119903

11989411989711989511990411989710158401198951015840(119904+1)119905

119878119876119903

119894119888119905 119878119871119903

119888ge 0

forall119894 119897 1198971015840

119895 1198951015840

119904 119905 119888 119903

(21)








119905 iteration index 119905 = 1 2 119879

119894 particle index 119894 = 1 2 119868

119889 dimension index 119889 = 1 2 119863



119862119903(119899) the 119862119903 of the 119899 time










119881119894119889(119905 + 1) = 119908 (119905) 119881

119894119889(119905)

+ 1198881119862119903 (119899) (119901119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

+ 1198882(1 minus 119862119903 (119899)) (119892119887119890119904119905

119894119889(119905) minus 119883

119894119889(119905))

(22)

119883119894119889(119905 + 1) = 119883

119894119889(119905) + 119881

119894119889(119905 + 1) (23)



(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1) Set 119894 = 1

(2) Set 119889 = 1


119894 using (22)
















)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











)


Types of machine1198961 1198962 1198963


1198971

DB 10 10 10 10 15 15 15 15 0 0 0 0WB 5 4 4 4 6 6 6 6 1 1 1 1MD 9 9 9 9 10 10 10 10 1 1 1 1

1198972

DB 0 0 0 0 5 5 5 5 6 6 6 6WB 0 0 0 0 10 10 10 10 8 8 8 8MD 2 2 2 2 5 5 5 5 6 6 6 1


Line Move to line




1198971 1198972 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 01198972 1198971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0


)


Types of tool1198991 1198992 1198993 1198994


1198971 MD 1 1 1 1 30 29 29 29 1 1 1 1 29 29 29 291198972 MD 29 29 29 29 0 1 1 1 19 19 19 19 1 1 1 1


Line Move to line

Types of tool1198991 1198992 1198993 1198994


1198971 1198972 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 01198972 1198971 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0












times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















times 100 () 015











60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












60000

50000

40000

30000

20000

10000

0

(080101

)

(030502

)

(030403

)

(033303330333

)

(020305

)

(010108

)


EVPIVSS







25000

20000

15000

10000

5000

0

Gap

in n

et p

rofit





















5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























5 Conclusion
















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an application of stochastic programming

Documents