An application of stochastic programming in solving capacity allocation and migration planning problem under uncertainty

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1 An application of stochastic pogamming in solving capacity allocation and migation planning poblem unde uncetainty Yin-Yann Chen * and Hsiao-Yao Fan Depatment of Industial Management, National Fomosa Univesity, Yunlin 632, Taiwan *Coesponding autho. Abstact The semiconducto packaging and testing industy, which utilizes high-technology manufactuing pocesses and a vaiety of machines, is belongs to an uncetain make-to-ode (MTO) poduction envionment. Ode elease paticulaly oiginates fom custome demand; hence, demand fluctuation diectly affects capacity planning. Thus, managing capacity allocation is a difficult endeavo. This study aims to detemine the best capacity allocation with limited esouces to maximize the net pofit. Thee bottleneck stations in the semiconducto packaging and testing pocess ae mainly investigated, namely, die bond (DB), wie bond (WB), and molding (MD) stations. Deviating fom pevious studies that conside the deteministic pogamming model, custome demand in the cuent study is egaded as an uncetain paamete in fomulating a two-stage scenaio-based stochastic pogamming (SP) model. The SP model seeks to espond to shap demand fluctuations. Even if futue demand is uncetain, migation decision fo machines and tools will still obtain bette obust esults fo vaious demand scenaios. Moeove, two assessment indicatos, namely, the expected value of pefect infomation (EVPI) and the value of the stochastic solution (VSS), ae adopted to compae the solving esults of the deteministic planning model and stochastic pogamming model. Sensitivity analysis is pefomed to evaluate the effects of diffeent paametes on net pofit. Keywods: capacity planning, capacity allocation, stochastic pogamming, machine migation 1. Intoduction The semiconducto packaging and testing industy belongs to a flow line envionment. The opeating pocedue of this industy indicates that the die bond (DB), wie bond (WB), and molding (MD) stations ae the thee bottleneck stations in the manufactuing pocess. Consequently, capital expenditue on equipment fo the DB, WB, and MD stations esults in high puchasing cost. In machine utilization, the limitations of machine type and poduct categoy affect the output pe hou. Given the multi-factoy and multi-line envionment of this industy, demand uncetainty and inappopiate poduction planning esult in wasteful o insufficient machine capacity in all lines. Theefoe, this eseach investigates the DB, WB, and MD stations in the semiconducto packaging and testing industy as the objects of the study. The best esouce configuation and capacity allocation decision in the semiconducto packaging and testing industy can effectively utilize all esouces in all poduction lines, as well as assist plannes educe the eadjustment of the poduction scheduling to efficiently accomplish ode allocations as a esponse to substantial changes in demand. Such decision can also indiectly educe the migation cost of machines and tools. Theefoe, this study aims to investigate the capacity allocation and migation planning poblem by consideing demand uncetainty to solve the cuent poblems and challenges faced by the semiconducto packaging and testing industy.

2 The impotance of capacity planning to semiconducto packaging and testing industy is discoveed based on the afoementioned backgound and motivation. Thus, the cuent study aims to attempt capacity allocation and esouce configuation planning at the same time fo multi-peiod ode demands, to impove the cuent capacity plan being sepaately implemented by packaging and testing factoies and consideation of a single peiod, and to detemine the pope machine migation and capacity allocation decision. Demand in packaging and testing factoies fluctuates because of diffeent limitations in capacity planning and the make-to-ode (MTO) poduction envionment. Theefoe, deviating fom pevious studies that conside the deteministic pogamming model, the cuent study employs demand as an uncetain facto to solve the poblem though a two-stage stochastic pogamming (SP) model. The feasibility of the capacity planning method poposed in this study is also demonstated though a pactical case study. Moeove, based on the diffeent paametes change, sensitivity analysis fo the stochastic pogamming model is povided, to undestand if capacity planning will be affected by diffeent scenaios, demands, migation costs of machines and tools, sale pices of poducts, flexibility of capacity migation, and othe factos. This pape is oganized as follows. Section 2 eviews the elated liteatue. Section 3 establishes the definition of the capacity planning poblem of the semiconducto packaging and testing industy, as well as the development of a two-stage stochastic pogamming model and the poposed hybid appoach. Section 4 discusses the application and analysis of a cetain lage-scale semiconducto packaging and testing factoy case. Section 5 povides the conclusion of this study. 2. Liteatue eview 2.1 Oveview of the semiconducto packaging and testing industy chaacteistics Semiconducto poducts cuently compise fou categoies, namely, integated cicuit (IC), discete, senso, and optoelectonics. Howeve, this study aims to investigate the IC categoy. The IC manufactuing pocess with vetical integation featues has font- and back-end pocesses. The upsteam to downsteam pocesses of the taget industy coves the following five steps: IC design, mask making, IC making, chip packaging, and chip testing. The manufactuing pocess and poduction chaacteistics of the semiconducto packaging and testing industy ae descibed as follows. 1. Make to ode The semiconducto packaging and testing industy pepaes mateials based on odes placed by customes. Theeafte, poduction poceeds based on customes needs and sevices, theeby peventing the industy fom pedicting the demand in advance. Futhemoe, demand foecasting fo this industy cannot be leaned fom past expeience. Accodingly, this study consides the demand to be uncetain, and expesses fluctuations in custome needs though diffeent scenaios. This consideation will avoid insufficient o wasted capacity duing the poduction while addessing custome needs and maximizing net pofit. The poduction in the IC packaging and testing industy is custome-oiented. Poduct categoies ae divesified based on diffeent custome needs. Packaging types can be divided into lead fame package, ball gid aay, flip chip, system in package, and multi-chip packages. Moeove, each packaging type is divided into vaious poduct types because of the use of diffeent chip sizes o pin numbes. Poduct types also have diffeent capacity constaints. 2. Flow shop

3 Wafe Incoming Quality Contol Patching Cystal cutting Die Bond Wie Bond Molding Pinting Electoplating Bumping Shaping Inspection Figue 1. Poduction flow of semiconducto packaging and testing factoy In an IC packaging and testing factoy, an ode is often divided into seveal wok odes. Figue 1 shows that each wok ode is manufactued based on the poduction flow. Meanwhile, IC packaging and testing factoies have multiple poduction lines. Fo example, a cetain lage-scale domestic packaging and testing factoy has appoximately 25 lines. This study only investigates the bottleneck stations in these factoies, namely the DB, WB, and MD stations. It is descibed as a flow shop poduction envionment. Thus, poducts will ente the WB station afte leaving the DB station. Afte the poducts ae pocessed in the WB station, they eventually ente the MD station. Theefoe, poducts have sequential dependencies in the factoy aea. 3. Unelated paallel machine Given the apid manne by which poducts ae updated, pocess changes and equipment upgade will compel semiconducto packaging and testing factoies to fequently puchase new o diffeent bands of machines to espond to maket changes. Having diffeent machine types and bands leads to the pesence of diffeent gades of machines in the poduction line, theeby esulting in the poduction patten of unelated paallel machines. When managements aange odes, they exet effot to satisfy custome needs and meet the maximal sevice level. In addition, they manufactue by utilizing machines with the highest capacity. Hence, they fist move the machines and allocate the pope machines in goups to avoid failue in poduction caused by inconsistencies in machine types. Most machines fo the semiconducto packaging and testing industy ae movable. Fo domestic IC packaging and testing factoies, the numbe of moving machines eaches appoximately 60 each month. 2.2 Capacity planning Kaabuk and Wu [1] indicate that capacity planning can be descibed as an iteative pocess between the following two main components: (1) capacity expansion, given pojected poduct demands, identify the equied manufactuing technologies and thei capacity levels to be physically expanded o outsouced though the planning peiod, and (2) capacity configuation, detemine which facility is to be configued with which technologies mix. The oveall objective is to meet a evenue model based on stategic demand planning (which blends demand foecasting and poactive maket development stategies). This objective can be viewed as meeting pojected demands with minimized total costs. Chen, Chen and Lu [2] pesent a capacity allocation and expansion poblem of thin film tansisto liquid cystal display (TFT-LCD) manufactuing in the multi-site envionment. The objective is to simultaneously seek an optimal capacity allocation plan and capacity expansion policy unde single-stage, multi-geneation and multi-site stuctues. Capacity allocation decides on pofitable poduct mixes and allocated poduction quantities of each poduct goup at each poduction

4 site. Capacity expansion is concened with detemining the timing, types, and sizes of capacity investments, especially in the acquisition of auxiliay tools. A mixed intege linea pogamming (MILP) is poposed, which consides many pactical chaacteistics. Finally, an industial case study modified fom a Taiwanese TFT-LCD manufactue is illustated and sensitivity analysis of some influential paametes is also addessed. The foundy is an industy whose demand vaies apidly and whose manufactuing pocess is quite complicated. Chen, Chen and Liou [3] exploe issues on mid-tem capacity planning fo an incement stategy of the numbe of auxiliay tools photo mask to incease the flexibility of poduction. The elated decisions include how to allocate appopiately the foecast demands of poducts among multiple sites and how to decide on the poduction quantities of poducts in each site afte eceiving custome-confimed odes. By constucting the mathematical pogamming model of capacity planning, the ates of capacity utilization and custome ode fulfillment ae found to be effectively enhanced by adding new masks to incease poduction flexibility. Lin, Wu, Chen and Shih [4] study stategic capacity planning poblems unde demand uncetainties in TFT-LCD industy. Demand foecasts ae usually inaccuate and vay apidly ove time. Thei eseach objective is to seek a capacity allocation and expansion policy that is obust to demand uncetainties. Special chaacteistics of TFT-LCD manufactuing systems ae consideed. A scenaio-based two-stage stochastic pogamming model fo stategic capacity planning unde demand uncetainties is peoposed. Compaing to the deteministic appoach, thei stochastic model significantly impove system obustness. Lin, Chen and Chu [5] efe to capacity planning as the pocess of simultaneously implementing a obust capacity allocation plan and capacity expansion policy acoss multiple sites against stochastic demand. Thei study constucts a stochastic dynamic pogamming (SDP) model with an embedded linea pogamming (LP) to geneate a capacity planning policy as the demand in each peiod is evealed and updated. Numeical esults ae illustated to pove the feasibility and obustness of the poposed SDP model. 2.3 Stochastic pogamming Given that demand uncetainty is consideed, this pape aims to fomulate a stochastic pogamming model fo solving capacity allocation and migation planning poblem. Dantzig [6] divided stochastic pogamming into two types: two-stage stochastic pogamming and multi-stage stochastic pogamming. Uibe, Cochan and Shunk [7] indicated that the decision vaiable of two-stage stochastic pogamming consists of two types: hee and now and wait and see. Hee and Now decision in the fist stage efes to decisions making when all infomation ae unknown. Wait and See decision in the second stage efes to decisions making afte all infomation have been fully evealed. Thus, decision vaiables fo the twostage stochastic pogamming ae a dependent issue, and the esults ae moe obust. Two-stage stochastic pogamming can be illustated though a scenaio tee. Figue 2 shows that the time point evealing uncetain factos is t=k. The time point is used to divide decisions into two stages. The fist-stage decision is fom t=1 to t=k. The esults affect the decisions afte t=k+1. Thus, they extend many banches. Moeove, each banch epesents a kind of scenaio and a goup of decision vaiables in the second stage.

5 Figue 2. Illustation of two-stage stochastic pogamming Listeş and Dekke [8] pesent a stochastic pogamming based appoach by which a deteministic location model fo poduct ecovey netwok design may be extended to explicitly account fo the uncetainties. They apply the stochastic models to a epesentative eal case study on ecycling sand fom demolition waste in The Nethelands. In Salema, Babosa- Povoa and Novais [9] wok, the design of a evese distibution netwok is studied. A genealized model is poposed. It contemplates the design of a geneic evese logistics netwok whee capacity limits, multi-poduct management and uncetainty on poduct demands and etuns ae consideed. A mixed intege fomulation is developed which is solved using standad B&B techniques. The model is applied to an illustative case. Lee, Dong and Bian [10] popose a stochastic pogamming based appoach to account fo the design of sustainable logistics netwok unde uncetainty. A solution appoach integating the sample aveage appoximation scheme with an impotance sampling stategy is developed. A case study involving a lage-scale sustainable logistics netwok in Asia Pacific egion is pesented to demonstate the significance of the developed stochastic model. Cadona-Valdés, Álvaez and Ozdemi [11] conside the design of a two-echelon poduction distibution netwok with multiple manufactuing plants, customes and a set of candidate distibution centes. The main contibution of the study is to extend the existing liteatue by incopoating the demand uncetainty of customes within the distibution cente location and tanspotation mode allocation decisions, as well as poviding a netwok design satisfying the both economical and sevice quality objectives of the decision make. In Kaa and Onut [12] study, a two-stage stochastic evenue-maximization model is pesented to detemine a longtem stategy unde uncetainty fo a lage-scale eal-wold pape ecycling company. This netwok-design poblem includes optimal ecycling cente locations and optimal flow amounts between the nodes in the multi-facility envionment. The poposed model is fomulated with two-stage stochastic mixed-intege and obust pogamming appoaches. Pishvaee, Jolai and Razmi [13] develop a stochastic pogamming model fo an integated fowad/evese logistics netwok design unde uncetainty. An efficient deteministic mixed intege linea pogamming model is developed fo integated logistics netwok design to avoid the sub-optimality caused by the sepaate design of the fowad and evese netwoks. Then the stochastic countepat of the poposed MILP model is developed by using scenaio-based stochastic appoach. Numeical esults show the powe of the poposed stochastic model in handling data uncetainty. In Amin and Zhang [14], a closed-loop supply chain netwok is investigated which includes multiple plants, collection centes, demand makets, and poducts. A mixed-intege linea pogamming (MILP) model is poposed that minimizes

6 the total cost. The model is extended to conside envionmental factos by weighed sums and ε-constaint methods. In addition, the impact of demand and etun uncetainties on the netwok configuation is analyzed by scenaio-based stochastic pogamming. Computational esults show that the model can handle demand and etun uncetainties, simultaneously. Ramezani, Bashii and Tavakkoli-Moghaddam [15] pesent a stochastic multi-objective model fo fowad/evese logistic netwok design unde a uncetain envionment including thee echelons in fowad diection (i.e., supplies, plants, and distibution centes) and two echelons in backwad diection (i.e., collection centes and disposal centes). The authos demonstate a method to evaluate the systematic supply chain configuation maximizing the pofit, custome esponsiveness, and quality as objectives of the logistic netwok. Mohammadi Bidhandi and Mohd Yusuff [16] popose an integated model and a modified solution method fo solving supply chain netwok design poblems unde uncetainty. The stochastic supply chain netwok design model is povided as a two-stage stochastic pogamming. The main uncetain paametes ae the opeational costs, the custome demand, and capacity of the facilities. In the impoved solution method, the sample aveage appoximation technique is integated with the acceleated Bendes decomposition appoach to impovement of the mixed intege linea pogamming solution phase. Sazva, Mizapou Al-e-hashem, Baboli and Akbai Joka [17] develop a stochastic mathematical model and popose a new eplenishment policy in a centalized supply chain fo deteioating items. In this model, they conside inventoy and tanspotation costs, as well as the envionmental impacts unde uncetain demand. The best tanspotation vehicles and inventoy policy ae detemined by finding a balance between financial and envionmental citeia. A linea mathematical model is developed and a numeical example fom the eal wold is pesented to demonstate its applicability and effectiveness. Lin, Chen and Chu [5] constuct a stochastic dynamic pogamming model with an embedded linea pogamming to geneate a capacity planning policy as the demand in each peiod is evealed and updated. Using the backwad induction algoithm, the model consides seveal capacity expansion and budget constaints to detemine a obust and dynamic capacity expansion policy in esponse to newly available demand infomation. Numeical esults ae also illustated to pove the feasibility and obustness of the poposed SDP model compaed to the taditional deteministic capacity planning model cuently applied by the industy. A distibuted enegy system is a multi-input and multi-output enegy system with substantial enegy, economic and envionmental benefits. The optimal design of such a complex system unde enegy demand and supply uncetainty poses significant challenges. Zhou, Zhang, Liu, Li, Geogiadis and Pistikopoulos [18] popose a two-stage stochastic pogamming model fo the optimal design of distibuted enegy systems. A two-stage decomposition based solution stategy is used to solve the optimization poblem with genetic algoithm pefoming the seach on the fist stage vaiables and a Monte Calo method dealing with uncetainty in the second stage. Detailed computational esults ae pesented and compaed with those geneated by a deteministic model. One of the most challenging issues fo the semiconducto testing industy is how to deal with capacity planning and esouce allocation simultaneously unde demand and technology uncetainty. In addition, capacity plannes equie a tadeoff among the costs of esouces with diffeent pocessing technologies, while simultaneously consideing esouces to manufactue poducts. The study of Wang and Wang [19] focuses on the decisions petaining to (i) the simultaneous esouce potfolio/investment and allocation plan, (ii) the most pofitable odes fom pending ones in each time bucket unde demand and technology uncetainty, (iii) the algoithm to efficiently solve the stochastic and mixed intege

7 pogamming poblem. The authos develop a constaint-satisfaction based genetic algoithm to esolve the above issues simultaneously. Dynamic pogamming appoach is a class of optimal design tools, such as einfocement leaning. Yan-Jun, Li, Shaocheng, Chen and Dong-Juan [20] poposed an online einfocement leaning algoithm fo a class of affine multiple input and multiple output (MIMO) nonlinea discete-time systems with unknown functions and distubances. Liu, Gao, Tong and Li [21] addessed an adaptive fuzzy optimal contol design fo a class of unknown nonlinea discete-time systems. The contolled systems ae in a stict-feedback fame, and contain unknown functions and non-symmetic deadzone. Wang, Liu and Wei [22] developed a finite-hoizon neuo-optimal tacking contol stategy fo a class of discetetime nonlinea systems. Chen, Yan-Jun and Guo-Xing [23] studied an adaptive tacking contol fo a class of nonlinea stochastic systems with unknown functions. Shaocheng, Yue, Yongming and Yanjun [24] poposed two adaptive fuzzy output feedback contol appoaches fo a class of uncetain stochastic nonlinea stict-feedback systems without the measuements of the states. Pevious studies have suveyed about capacity planning issue, but only a few studies have focused on the capacity allocation poblem consideing machine/tool migation planning and demand uncetainty simultaneously. This pape aims to detemine the best capacity allocation with limited esouces to achieve net pofit maximization in the semiconducto packaging and testing industy. Custome demand is egaded as an uncetain paamete in fomulating a two-stage scenaio-based stochastic pogamming model. This model seeks to espond to shap demand fluctuations. Even if futue demand is uncetain, migation decision fo machines and tools will still obtain bette obust esults fo vaious demand scenaios. Sensitivity analysis is also pefomed to evaluate the effect of diffeent paametes on net pofit. 3. Capacity planning of the semiconducto packaging and testing industy 3.1 Chaacteistics of capacity planning of the semiconducto packaging and testing industy This study aims to detemine machine migation, tool migation in all poduction lines, esouce configuation, capacity allocation, and poduct flow unde demand uncetainty to achieve net pofit maximization. 1. Resouce configuation The manufactuing pocess entails that a poduct should sequentially go though the DB, WB, and MD stations fo assembly-line poduction. The poduct consides the machine type in esouce configuation duing the DB and WB stages. Howeve, thee esouces, namely, machine type, tool type, and mateial categoy, ae consideed in the MD stage. Figue 3 shows that poduct 1 is manufactued in machine k1 o k2 in the DB station. This poduct is then pocessed in machine k2 o k3 in the WB station. Theeafte, the poduct is manufactued in the MD station though k1+n1+m4 o k2+n2+m4.

8 Figue 3. Illustation of esouce configuation 2. Poduct flow This study disegads defective poducts and only consides poduction though the thee sequential stages. Moeove, poduct flow balance must be maintained in the poduction line. Hence, the total poduct input must equal the final total output. Fo example, the poduct input fo poduct 1 is 1,000 units. Futhemoe, 400 and 600 units ae poduced in lines 1 and 2, espectively. Afte poduction though the thee sequential stages, the final total output emains 1,000 units. 3. Capacity allocation The capacity planning of all eceived odes is executed based on the cuent existing esouces in all poduction stages. A poduct is not limited to the same poduction line duing the entie poduction pocess; that is, a poduct can be manufactued in the diffeent lines though thee poduction stages. Fo example, a company has two lines, if the input of poduct 1 is 1,000 units. Take line 1 fo explanation. Fistly, 400 units ae manufactued in the DB station using machine k1 and 600 units using machine k2. Theeafte, 400 units ae manufactued in the WB station using machine k2 and 200 units using machine k3. Finally, 200 units ae manufactued in the MD station using esouce configuation k1+n1+m4 and 300 units using k2+n2+m4. Thus, 500 units of poduct 1 can be made afte the thee poduction stages fo this poduct ae completed sequentially in line 1. The emaining 500 units ae allocated to all poduction stages in line 2 fo manufactuing. 4. Machine and tool migation The pesence of seveal poduction lines and machines with diffeent technological capability in a company will esult in vaiations in the poduction capacities of all lines. Machines can be moved to all lines in each poduction stage, and tools can be moved to all lines in the MD stage based on the total numbe of available machines and tools. 3.2 Mathematical pogamming of capacity planning poblem fo the semiconducto packaging and testing industy unde demand uncetainty

9 A mathematical model of two-stage scenaio-based stochastic pogamming is fomulated by consideing custome demand as an uncetain paamete. This study aims to espond to shap demand fluctuation. Even if futue demand is uncetain, machine and tool migation decisions ae obust esults fo all demand scenaios Definition and desciption of capacity planning poblem unde demand uncetainty This study uses a scenaio tee to illustate the uncetain facto (Figue 4). Machine and tool migation decisions ae deemed the decisions made in the fist stage. The esults of these decisions emain constant with the vaying custome demands. Moeove, the second-stage capacity allocation decisions must be made based on the fist-stage decision esults. The esults in the second stage change with the vaying custome demands. In this study, two-stage decisions should be optimally detemined to achieve net pofit maximization. 1. Fist-stage decision: Robust capacity migation decision that consides demand uncetainty. Given thee demand scenaios, each type of machine and tool is consideed to detemine when and what quantity of machines and tools ae migated between lines in the poduction stage. Hence, capacity migation decision must be made in advance to conside the obust decision unde demand uncetainty as being unelated to diffeent demand scenaios. 2. Second-stage decision: Capacity allocation decision afte all demand infomation have been completely evealed. The following factos ae detemined afte a cetain demand scenaio occus: (1) poduction quantity fo each poduct in each line in all poduction stages duing each peiod, (2) tanspotation quantity between the diffeent poduction stages, (3) sales volume of each poduct in each peiod fo each custome, and (4) custome sevice level. Theefoe, capacity allocation decision is closely elated to the demand scenaio. Accoding to the capacity migation esult in the fist stage, the optimal capacity allocation decision can be detemined once a specific demand scenaio occus. Figue 4. Diagammatic sketch of scenaio tee of the uncetain facto Two-stage stochastic pogamming model of capacity planning poblem To solve the capacity planning poblem unde demand uncetainty, this study uses two-stage stochastic pogamming to constuct a mathematical model. This section explains the indices, paametes, decision vaiables, objective function, and constaints. 1. Indices

10 c = custome (c = 1, 2,, C ). i = poduct type (i = 1, 2,, I ). l = poduction line (l = 1, 2,, L ). s = poduction stage (s = 1, 2,, S ). j = esouce configuation (j = 1, 2,, J ). m = mateial type (m = 1, 2,, M ). k = machine type (k = 1, 2,, K). n = tool type (n = 1, 2,, N ). t = time peiod (t = 1, 2,, T ). = scenaio numbe ( = 1, 2,, R ). 2. Paametes (1) Demand elated paametes de ict = the demand quantity of custome c fo poduct i in time t unde scenaio. p = pobability value occuing in scenaio ( p 1 ). p ict = sales pice of custome c fo poduct i in time t. (2) Machine elated paametes kl lsk = initial amount of machine k in line l at stage s. ku ls = maximum numbe of machines in line l at stage s. ks ijsk = equied wok hous of machine k used at stage s fo manufactuing a unit of poduct i with esouce configuation j. ka sk = available wok hous of machine k at stage s. kb ll s = machine migation capability fom line l to l' at stage s. (3) Tool elated paametes nl lsn = initial amount of tool n in line l at stage s. nu ls = maximum numbe of tools in line l at stage s. ns ijsn = equied wok hous of tool n used at stage s fo manufactuing a unit of poduct i with esouce configuation j. na sn = available wok hous of tool n at stage s. nb ll s = tool migation capability fom line l to l' at stage s. (4) Mateial elated paametes mq smt = total available quantity of mateial m at stage s in time t. ms ijsm = consumption atio of mateial m fo manufactuing a unit of poduct i at stage s with esouce configuation j. (5) Poduction capability elated paamete tf ijs = poduction capability of poduct i at stage s with esouce configuation j. (6) Tanspotation elated paamete

11 tb lsl (s+1) = tanspotation capability fom line l at stage s to line l' at stage s+1. (7) Cost paametes vc iljs = poduction cost fo manufactuing a unit of poduct i in line l at stage s with esouce configuation j. kc s = machine migation cost at stage s. nc s = tool migation cost at stage s. 3. Decision vaiables (1) Fist-stage decision vaiables: capacity migation decision KQ lskt = the numbe of machine k fo line l at stage s in time t. KM ll skt = the migation numbe of machine k fom line l to line l' at stage s in time t. NQ lsnt = the numbe of tool n fo line l at stage s in time t. NM ll snt = the migation numbe of tool n fom line l to line l' at stage s in time t. (2) Second-stage decision vaiables: capacity allocation decision and sevice level XQ iljst = poduction amounts of poduct i with esouce configuation j fo line l at stage s in time t unde scenaio. RQ = tanspotation amounts of poduct i fom line l with esouce configuation j at stage s to line l' with iljsl j ( s 1) t esouce configuation j' at stage (s+1) in time t unde scenaio. SQ ict = sales amounts of poduct i fo custome c in time t unde scenaio. SL c = sevice level fo custome c unde scenaio. 4. Objective Function Maximize p ( pict SQict ) ( vciljs XQiljst ) i c t i l j s t ( kc KM ) ( nc NM ) s llskt s llsnt l l s k t l l s n t (1) The above is the objective function of two-stage stochastic pogamming. It aims to obtain the optimal capacity planning decision to seek the maximization of net pofit, as Eq.(1), net pofit=(sales evenue-vaiable poduction cost) -machine migation cost-tool migation cost. 5. Constaints (1) Fist-stage constaints Machine migation balance constaints KQlsk 0 kllsk l, s, k. (2). (3) KQlskt KQlsk ( t1) KMll skt KMl lskt l, s, k, t l l KQ ku l, s, k, t. (4) lskt ls

12 KM M kb l, l, s, k, t. (5) llskt ll s Constaint (2) shows the initial amount of machines in lines at each poduction stage; and constaint (3) indicates the numbe of machines equied fo lines at poduction stages in evey peiod. This numbe of machines in the cuent peiod is equal to the numbe of machines in the pevious peiod minus the numbe of machines moving to othe lines plus the numbe of machines that migated fom othe lines to this line. The total initial numbe of machines within the company must be equal to the total numbe of machines afte being migated between lines without inceasing o educing the numbe of machines. Constaint (4) expesses that the allocated numbe of machines should not be moe than the available space in the shop-floo poduction line. In addition, constaint (5) consides if machines have capability to be migated between lines. kb ll s efes to a binay paamete. 1 means machines can be migated between poduction lines, and 0 means they cannot be migated. Tool migation balance constaints NQlsn0 nllsn l, s, n. (6). (7) NQlsnt NQlsn( t1) NM llsnt NM llsnt l, s, n, t l l NQ nu l, s, n, t. (8) lsnt ls NM M nb l, l, s, n, t. (9) llsnt ll s Constaint (6) shows the initial amount of tools in lines at each poduction stage; and constaint (7) indicates the numbe of tools equied fo lines at poduction stages in evey peiod. This numbe of tools in the cuent peiod is equal to the numbe of tools in the pevious peiod minus the numbe of tools moving to othe lines plus the numbe of tools that migated fom othe lines to this line. The total initial numbe of tools within the company must be equal to the total numbe of tools afte being migated between lines without inceasing o educing the numbe of tools. Constaint (8) expesses that the allocated numbe of tools should not be moe than the available space in the shop-floo poduction line. In addition, constaint (9) consides if tools have capability to be migated between lines. nb ll s efes to a binay paamete. 1 means tools can be migated between poduction lines, and 0 means they cannot be migated. Domain estiction fo fist-stage decision vaiables KQ, KM, NQ, NM intege l, s, k, n, t. (10) lskt ll skt lsnt ll snt Constaint (10) shows the domain of vaiables, which indicates the chaacteistics of its intege vaiables. (2) Second-stage constaints Poduction and tanspotation balance constaints. l j XQiljst RQiljsl j ( s 1) t i, l, j, s 1,... S 1, t, RQil j ( s1) ljst XQiljst i, l, j, s 2,... S, t,. l j (11) (12) Oveall poduction and tanspotation must satisfy line flow balance, as shown in constaints (11) and (12). The allocated poduction amounts in a cetain line at this stage should be equal to the total amounts that ae tanspoted fom

13 this line to all lines at the next stage. On the contay, the total amounts that ae tanspoted fom all lines at the pevious stage to a cetain line at the cuent stage should be equal to the allocated poduction amounts in this line. Capacity constaints ( XQiljst ksijsk ) KQlskt kask l, s, k, t,. i j ( XQiljst nsijsn) NQlsnt nasn l, s, n, t,. i j (13) (14) Fo capacity constaints, constaints (13) and (14) indicate that the poduction amounts multiplied by wok hous of machines o tools consumed should not exceed the numbe of machines o tools multiplied by available wok hous of a unit of machine o tool. In shot, the sum of wok hous equied fo each poduct in available machine o tool should not be moe than the total available esouce limit of the company. Mateial constaint ( XQiljst msijsm) mqsmt s, m, t,. i l j (15) Fo mateial constaint (15), geneally speaking, the amounts of mateials to be consumed in the poduction pocess should not be beyond the quantity estiction of available mateials. With limited esouces, the poduction amounts multiplied by the mateial consumption atio pe unit will be less o equal to the total available quantity of the mateial. Poduction capability constaint XQ M tf i,, l j,,, s t. (16) iljst ijs Fo poduction capability, constaint (16) shows whethe esouce configuation of a cetain poduct is able to be used fo manufactuing this poduct. Due to diffeent types of machines and tools in lines at each poduction stage, not all esouce configuations can be used fo manufactuing all kinds of poducts. If tf ijs =1, the esouce configuation in the line at this stage can be used fo manufactuing this type of poduct; on the contay, if tf ijs =0, they cannot be used. Tanspotation capability constaint RQiljsl j ( s1) t Mtblsl ( s1) il,, jsl,,, j,, t. (17) Fo tanspotation capability, constaint (17) expesses whethe thee is tanspotation capability to move poducts fom the cuent stage to the next stage. The poduction pocess is an assembly flow line envionment. Thus, poducts ae bound to go though each poduction stage in tun and cannot evet to a pevious stage. If tb lsl (s+1) =1, thee is tanspotation capability to move poducts between stages.; on the contay, if tb lsl (s+1) =0, it indicates that thee is no tanspotation capability. Demand fulfillment constaints XQiljst SQict i, s S, c, t,. l j (18) SQ de i, c, t,. (19) ict ict

14 Demand fulfillment is indicated by constaints (18) and (19) espectively. Constaint (18) shows that sales volume in each scenaio should be equal to the total poduction amounts with esouce configuations in all lines. Constaint (19) expesses that the sales volume must be less o equal to the demands equied by customes. Sevice level SL c i i SQ de ict ict c, t,. (20) Constaint (20) shows that the sales volume divided by custome demands is the sevice level. Domain estiction fo second-stage decision vaiables XQ, RQ, SQ, SL 0 i, l, l, j, j, s, t, c,. (21) iljst iljsl j ( s 1) t ict c Constaint (21) indicates vaiable domain estiction Capacity planning poblem unde demand cetainty Diffeent fom the uncetainty model, the deteministic model does not conside demand fluctuation and only consides an aveage demand scenaio. Appendix A shows the detailed mathematical pogamming model that is used to compae the diffeences in solving esults between the deteministic model and stochastic pogamming model. 3.3 Poposed hybid appoach As the scenaio numbe is inceased, solving the scenaio-based stochastic pogamming model becomes consideably difficult because of the computation complexity. Theefoe, a hybid appoach is developed to efficiently addess the poposed two-stage stochastic pogamming model. We apply the paticle swam optimization (PSO) method combined with the AIMMS optimal modeling softwae in a hybid mechanism. Fist, an initial solution was geneated to detemine the migation numbe of machines and tools among the poduction lines. This esult was enteed into the AIMMS optimal modeling softwae with the ILOG CPLEX 12.6 solve to geneate the optimal poduction amounts of poducts. The esults ae etuned to the PSO algoithm to calculate the net pofit and to detemine whethe the temination conditions have been satisfied. This study sets the temination condition as the numbe of geneations. The seach ends when the numbe of geneations eaches the pe-set numbe of geneations. If this numbe is eached, then the PSO algoithm is used to yield the optimal numbe of machines and tools of each line to the AIMMS optimal modeling softwae to geneate the optimal poduction amounts of poducts. Fitness values ae calculated duing each geneation. The PSO algoithm is epeated until the temination condition is satisfied. The PSO steps ae stated as follows. Step 1: Geneation of an initial population. This study uses PSO to detemine the migation numbe of machines and tools among the poduction lines. Given the initial numbe of machines and tools, an initial population is geneated by andomly selecting the value limited to the available maximum numbe of machines and tools in each line. Step 2: Calculation of the fitness values.

15 The fitness value in this study is net pofit. Step 3: Updating the speed and position of the paticle The Equations (22) and (23) ae used to update the speed and position, using the following symbols. t: Iteation index, t = 1, 2,..., T i: Paticle index, i = 1, 2,..., I d: Dimension index, d = 1, 2,, D c 1 : Pesonal best position acceleation constant c 2 : Global best position acceleation constant C(n) : the C of the n time w(t) : Inetia weight in the t th iteation X id (t) : Position of the i th paticle at the d th dimension in the t th iteation V id (t) : Velocity of the i th paticle at the d th dimension in the t th iteation pbest id (t) : Pesonal best position of the i th paticle at the d th dimension gbest d (t) : Global best position at the d th dimension The mathematical model is expessed as follow: V ( t 1) w( t) V ( t) c C( n) ( pbest ( t) X ( t)) c (1 C( n)) ( gbest ( t) X ( t)). (22) id id 1 id id 2 id id Xid ( t1) Xid ( t) Vid ( t 1). (23) The following steps ae used to update the individual speed and position of each dimension: (1). Set i = 1. (2). Set d = 1. (3). Update the d dimension speed ( Vid ( t 1) ) in paticle i using Equation (22). (4). Update the d dimension position in paticle i using Equation (23). (5). Detemine whethe d is equal to D. If so, then i = i + 1. If not, then d = d + 1 and n = n + 1, and etun to Step (3). (6). Detemine whethe i is lage than I. If it is, this indicates that the update has concluded. If not, etun to Step (2). Step 4: Updating the paticle best (pbest) Updating the pbest involves eplacing the best position fo cuent individual paticles when the cuent individual fitness values ae supeio to the pbest fitness values. Othewise, the eplacement is not pefomed and the execution is epeated until all paticles have been updated. Step 5: Updating the global best (gbest) Updating the gbest involves eplacing the optimal population paticles when the cuent optimal individual solution fitness values ae supeio to the gbest fitness values. Othewise, the eplacement is not pefomed. Step 6: Detemining whethe the temination conditions ae eached. The temination condition fo the PSO algoithm pesented in this study is detemined when the numbe of iteations exceeds the set maximum iteation times. Othewise, the pocess etuns to Step 2.

16 4. Analysis and discussion on the semiconducto packaging and testing industy case 4.1 Intoduction to the case backgound This study aims to conduct a capacity allocation and migation planning fo custome demands by consideing a cetain lage-scale semiconducto packaging and testing factoy as the case study. Thee customes, eight types of poducts, and two poduction lines ae involved in this case. The manufactuing pocess is divided into thee bottleneck poduction stages, namely, the DB, WB, and MD stations in tun. Futhemoe, the factoy has thee types of machines, fou types of tools, and fou categoies of mateials. The planning hoizon coves fou peiods. Fo esouce configuation, the DB and WB stations have thee configuations consisting of machines. The MD station has seven kinds of configuations consisting of machines, tools, and mateials. Appendix B shows the elated infomation necessay fo this case study. 4.2 Capacity planning esults The case poblem is handled unde demand uncetainty. The maximum net pofit is $77,557, fo the stochastic pogamming model. Table 1 shows the numbe of machines fo the lines in the poduction stages in each time peiod. Table 2 pesents the migation numbe of machines between lines in each poduction stage in each time peiod. Table 3 indicates the numbe of tools fo the lines in the MD stage in each time peiod. Table 4 pesents the migation numbe of tools between lines in the MD stage in each time peiod. Table 5 expesses the sales amounts of poducts fo each custome in each time peiod unde diffeent scenaios. Table 1. The numbe of machines fo lines at poduction stages in each time peiod (KQ lskt ) Types of machine k1 k2 k3 time (month) time (month) time (month) Line Poduction stage DB WB MD DB WB MD Table 2. The migation numbe of machines between lines at each poduction stage in each time peiod (KM ll skt ) Poduction stage DB WB MD Types of machine Types of machine Types of machine k1 k2 k3 k1 k2 k3 k1 k2 k3 time time time time time time time time time Line Move to line

17 Table 3. The numbe of tools fo lines at MD stage in each time peiod (NQ lsnt ) Types of tool n1 n2 n3 n4 time (month) time (month) time (month) time (month) Line Poduction stage MD MD Table 4. The migation numbe of tools between lines at MD stage in each time peiod (NM ll snt ) Types of tool n1 n2 n3 n4 time time time time Line Move to line Table 5. The sales amounts of poducts fo each custome in each time peiod unde diffeent scenaios ( SQ ) Time peiod (month) Scenaio Poduct Custome scenaio1 i1 c1 45,955 80,375 11,400 37,666 scenaio1 i2 c1 137,866 40,188 72,154 0 scenaio1 i3 c1 99,999 21, ,030 scenaio1 i4 c2 91,911 60,281 54,115 0 scenaio1 i5 c2 22, ,563 45,096 15,066 scenaio1 i6 c3 99,999 48, ,379 scenaio1 i7 c3 53,614 24,113 53,175 33,899 scenaio1 i8 c3 199,998 21, ,300 scenaio2 i1 c1 48,000 96,000 13,500 60,000 scenaio2 i2 c1 144,000 48,000 96,000 0 scenaio2 i3 c1 99,999 22, ,428 scenaio2 i4 c2 96,000 72,000 72,000 0 scenaio2 i5 c2 24, ,000 60,000 24,000 scenaio2 i6 c3 99,999 54,307 48,647 0 scenaio2 i7 c3 56,000 28,800 68,192 54,000 scenaio2 i8 c3 199,998 26, ,000 scenaio3 i1 c1 50, , ,334 scenaio3 i2 c1 150,134 55, ,115 0 scenaio3 i3 c1 58, ,999 scenaio3 i4 c2 100,089 83,719 89,885 0 scenaio3 i5 c2 25, ,437 74,904 32,934 scenaio3 i6 c3 99,999 58, ,055 scenaio3 i7 c3 58,386 33,487 82,200 74,101 scenaio3 i8 c3 199,998 30, ,700 ict 4.3 Expected value of pefect infomation (EVPI) and value of the stochastic solution (VSS) WS stands fo wait and see ; thus the decision-make must wait fo all infomation to be evealed befoe making a decision. The objective is to maximize the net pofit. The solution obtained though the deteministic model with aveage demand is called the expected value (EV) solution. Though the EV solution, the individual objective values of all demand scenaios can be obtained. Theeafte, these objective values ae multiplied by the occuing pobability of the coesponding scenaio to obtain the expected value, namely, the expected esult of using the EV solution (EEV). The

18 hee and now type indicates the maximized net pofit value of stochastic pogamming, which is called SP. Fo the capacity allocation and migation planning poblem in this study, the solving esult though SP unde uncetainty is compaed with the deteministic model. Two indicatos, namely, expected value of pefect infomation (EVPI) and value of the stochastic solution (VSS), ae used fo analysis. The optimal objective value of the stochastic pogamming model is compaed with the expected value of the WS solutions. The latte is calculated by detemining the optimal solution fo each possible ealization of the demand scenaios with cetainty. Clealy, it is bette to know the value of the futue actual demand befoe making a decision than having to make the decision befoe knowing. The diffeence between these two expected objective values is called EVPI. Futhemoe, EVPI measues the maximum amount a decision-make would be willing to pay in etun fo complete (and accuate) infomation about the futue to solve uncetainty. Thus, EVPI is defined as below (24). If EVPI is smalle, the stochastic pogamming esult is close to the esult obtained with complete infomation. By contast, if EVPI is lage, the influence of uncetain factos is geate and the pice paid fo obtaining complete infomation is consideably high. EVPI=WS-SP (24) VSS is used to measue the ability of the stochastic pogamming model to incease net pofit, with the attempt to solve uncetain factos. It is the diffeence between the solution of the SP model and the expected value of the objective function when fixing paametes to aveage values and using the coesponding optimal solution. Thus, VSS is defined as below (25). VSS conveys us how much we can gain moe if SP is used. If VSS is lage, the SP esult is bette than the expected esult when using the EV solution obtained by eplacing all possible demands with thei aveage values. VSS=SP-EEV (25) The elated measuements fo the case poblem in this study ae showed in Table 6. Table 6. The elated measuements fo the case poblem Net pofit WS 77,560, SP 77,557, EEV 77,439, EVPI VSS VSS 100(%) EEV 118, % Net pofit fluctuation unde diffeent combinations of pobability Diffeent pobability combinations ae designed to investigate whethe the occuing pobability of all demand scenaios affects the net pofit. The combined design individually povides significantly high pobability values to low, mean, and high demand scenaios. Table 7 shows that the capacity planning esults unde all pobability combinations indicate that net pofits using the SP model ae highe than those using the deteministic model. Moeove, if the occuing pobability of low demand scenaio is 0.8, then its net pofit is significantly lowe than that of the mean demand o high demand scenaio, which possesses an occuing pobability of 0.8. Theefoe, the occuing pobability of the scenaio is positively elated to the demand of each coesponding scenaio, that is, detemining the occuing pobability of scenaio is highly impotant when using the SP model.

19 Table 7. The elated measuements fo diffeent pobability combinations Pobability combination* WS SP EEV (0.8, 0.1, 0.1) 69,966, ,962, ,959, (0.1, 0.8, 0.1) 78,027, ,023, ,021, (0.1, 0.1, 0.8) 84,687, ,686, ,635, * means the occuing pobability of low demand, mean demand, and high demand scenaios, espectively Changes in EVPI and VSS unde diffeent pobability combinations The cuent study analyzes whethe the occuing pobabilities of all demand scenaios have an effect on EVPI and VSS. Accodingly, seveal pobability combinations of demand scenaios ae designed, including the pobability combination with consideably high occuing pobability of specific demand scenaio. EVPI and VSS unde diffeent pobability combinations ae shown in Table 8. Figue 5 shows that when the pobability combination is (0.1, 0.1, 0.8), the net pofit gap between the deteministic model and SP model is $50,569. Moeove, the decision-make is willing to pay $900 in etun fo the complete infomation on futue uncetainty. Hence, when the occuing pobability of high demand is highe, EVPI is lowe. Specifically, the solving esult of net pofit unde complete (pefect) infomation is close to the decision made by the SP model. Similaly, if VSS is highe, then the obtained benefit fom the SP model is bette. Table 8. EVPI and VSS unde diffeent pobability combinations Pobability combinations* EVPI VSS (0.8,0.1,0.1) 4,050 2,384 (0.3,0.5,0.2) 3,600 9,267 (0.3,0.4,0.3) 3,150 16,151 (0.333,0.333,0.333) 3,000 18,446 (0.2,0.3,0.5) 2,250 29,918 (0.1,0.1,0.8) ,569 * means the occuing pobability of low demand, mean demand, and high demand scenaios, espectively. Figue 5. The diagam fo EVPI and VSS unde diffeent pobability combinations Effect of demand vaiability on net pofit, EVPI, and VSS Thee types of demand vaiability ae designed in this study. Base Case aims to infe demands of all scenaios using the coefficient of vaiation. Small vaiation is equal to 90% of Base Case (middle vaiation), and lage vaiation is 110% of Base Case. Afte individually solving the thee diffeent vaiations, the net pofit in all vaiations unde the SP model

20 and deteministic model can be calculated (Table 9). It also can be found fom Figue 6, the gap in net pofit will incease with the incease of demand vaiation. Thus, the SP model consides demand uncetainty, and its esult is bette than that of the deteministic model, which only consides aveage demand. Table 9. Compaison of net pofit unde demand vaiability Demand vaiability EEV SP Gap Small vaiation 72,884,515 72,888,460 3,945 Middle vaiation 77,539,044 77,557,489 18,445 Lage vaiation 82,113,557 82,134,434 20,877 Figue 6. Gap in net pofit unde diffeent demand vaiability 4.4 Sensitivity analysis Effect of demand change on machine and tool migation and net pofit Demand change is the pimay poblem discussed in this study. The semiconducto packaging and testing industy cannot accuately foecast the actual demand of customes. If the demand change constantly shows positive gowth o a substantial negative eduction, then the two-stage SP model will significantly espond to consideable demand change than the deteministic model. Hence, when the actual demand is lowe, capacity waste can be educed. By contast, when the actual demand is highe, capacity shotage can be avoided. Fo the case company in this study, the inceasing demand esults in the continuous impovement in net pofit because of the demand gowth. Howeve, the numbe of machine and tool migation is unaffected by demand change; as demand deceases, net pofit, and the numbe of machine and tool migation ae educed as demand is deceased. Doing so can avoid unnecessay migation costs, as shown in Table 10 and Table 11. Table 10. Changes in migation costs and net pofit unde positively gowing demand Demand gowth multiples Machine migation cost 13,000 13,000 13,000 13,000 13,000 Tool migation cost 4,000 4,000 4,000 4,000 4,000 Net pofit 110,214, ,961, ,816, ,421, ,009,670

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