Stochastic Optimization of Product-Machine Qualification in a Semiconductor Back-end Facility

Transcription

1 Stochastic Optiization of Product-Machine Qualification in a Seiconductor Back-end Facility Mengying Fu, Ronald Askin, John Fowler, Muhong Zhang School of Coputing, Inforatics, and Systes Engineering, Arizona State University, Tepe, AZ 85287, USA Abstract In order to process a product in a seiconductor back-end facility, a achine needs to be qualified first by having product-specific software installed and then running test wafers through it to verify that the achine is capable of perforing the process correctly. In general, not all achines are qualified to process all products due to the high achine qualification cost and tool set availability. The achine qualification decision affects future capacity allocation in the facility and subsequently affects daily production schedules. To balance the tradeoff between current achine qualification costs and future potential backorder costs due to not enough achines qualified with uncertain deand, a stochastic product-achine qualification optiization odel is proposed in this paper. The L-shaped ethod and acceleration techniques are proposed to solve the stochastic odel. Coputational results are provided to show the necessity of the stochastic odel and the perforance of different solution ethods. Key words: anufacturing; product-achine qualification; production planning and scheduling; stochastic prograing 1

2 1 Introduction The seiconductor anufacturing process consists of two ain parts: the front-end process and the back-end process. The front-end process, also known as wafer fabrication, typically has a sall nuber of products and very coplex reentrant product flow. In contrast, the back-end process, also known as assebly and test, typically has hundreds or thousands of different products and relatively linear product flow. The research presented in this paper focuses on the back-end process. In a seiconductor back-end facility, each achine has to be configured for each of the products it will process in the future. This configuration (achine qualification) process includes installing and testing a software progra for each product on the achine. Due to the wide product ix, if all achines were to be qualified for all products, the achine qualification process could take considerable tie and engineering resources, thus incurring a high achine qualification cost. Meanwhile, not all achines are technologically capable of being qualified for all products. Because of short product life cycles and fast developent of new products in the seiconductor industry, new achines ay need to be procured frequently for new products. As a result, achines that perfor the sae operation could belong to different achine types/generations, with each type/generation only being able to be qualified for a subset of products. In addition, the productachine qualification decision affects the capacity planning decision and subsequently the future daily production schedule. Poor product-achine qualification decisions could cause shortages by not qualifying enough achines for a given product, or achine utilization ibalance by qualifying too any products on a sall subset of achines. Overqualification ay also coplicate scheduling decisions and lead to isallocation of capacity. In this paper, a ixed integer linear prograing odel (MILP) is first proposed to iniize product-achine qualification cost while considering future production scheduling. As the last part of the seiconductor anufacturing syste, on tie delivery of custoer orders is generally the ost iportant goal for the back-end process. Hence the objective of the MILP is set to iniize the weighted product-achine qualification costs and future backorder costs with a higher weight on the latter. Due to coputational liitations and deand forecast data availability, the production scheduling horizon in the odel is set to be a ediu ter (e.g. several weeks). In addition, the product deand is represented by a rando distribution to reflect the uncertainty. 2

3 The reainder of the paper is organized as follows. Section 2 is a literature review about product-achine qualification. In Section 3, the proble is clearly defined and a ixed integer linear prograing odel (MILP) is proposed to optiize product-achine qualification in the seiconductor back-end facility. In Section 4, a stochastic MILP odel is presented to account for the deand uncertainty in the production scheduling process. The L-shaped ethod and acceleration techniques are proposed to solve the stochastic odel. This is followed by Section 5, in which coputational results are presented to copare the deterinistic and stochastic odels as well as different solution ethods of the stochastic odel. Finally, conclusions and future research directions are provided in Section 6. 2 Literature Review Product-achine or operation-achine qualification is a very coon feature in the odern seiconductor anufacturing process. A few papers consider this feature in their scheduling odels [9, 12, 5, 14, 17, 18], but none of the proposes to change or optiize the current achine qualification. There are also soe other papers that utilize short-ter achine dedication to schedule the production activities [6, 4]. An operation-achine qualification anageent syste is proposed by [11] for a seiconductor front-end facility, in which four flexibility easures are developed to evaluate different operation-achine qualifications. The ipacts of different operation-achine qualifications, with different scores according to the four flexibility easures, on production scheduling are shown through siulation. [1] present a ixed integer linear prograing odel (MILP) for the product-achine qualification optiization of parallel ulti-purpose achines. The objective is to iniize achine configuration costs while obtaining a load-balanced capacity allocation. The MILP forulation is proved to be strongly NP-hard but could be relaxed to a transportation proble under certain assuptions. [15] presents a robustness easure for the ulti-purpose achine configuration odel developed by [1]. Maxial disturbance of the deand that changes the optial configuration is used as the robustness easure. [10] proposes a binary optiization odel for the operation-achine qualification of photolithography achines in a wafer fabrication factory. The objective is to obtain a load-balanced schedule at inial achine qualification costs. The cycle tie in the factory is shown to be decreased using the binary optiization odel co- 3

4 pared to achine qualifications developed by heuristic or educated guess eans. In soewhat related work, [8] propose an integer prograing odel for long-ter eployee staffing based on qualification profiles. The objective is to accoplish all tasks with inial total eployent costs. Eployee scheduling could be another application area of the ethodologies developed for the achine qualification anageent in the factory. None of these papers integrates the future production planning and scheduling of a ulti-stage anufacturing syste explicitly in their achine qualification optiization odels. On the other hand, achine qualification decisions have a critical long-ter ipact on the future production planning and scheduling. Furtherore, the interaction between qualification decisions for different stages ipacts delivery perforance. In this paper, a stochastic ixed integer linear prograing odel is proposed to optiize product-achine qualification in a ulti-stage anufacturing syste while considering future production scheduling with deand uncertainty. In the following section, we define the proble first and then propose a deterinistic odel. 3 Proble Stateent The back-end facility has ultiple stages and parallel achines at each stage. Products are processed in lots with a product-specific nuber of units in each lot. Setup ties are typically sequence-dependent and not included in the lot processing tie. However, for siplicity and coputational purposes, in this paper, the setup ties are not considered explicitly in the odel. Instead, the setup ties are odeled by decreasing the achine capacity by a certain percentage based on historical achine utilization data. Product-achine qualification is deterined in the odel, and thus only qualified achines can process a given product at a given stage. Initial product-achine qualification in the odel could be epty or given by an existing configuration. In the seiconductor industry, once a achine is qualified for one product, it will not be de-qualified for extra cost. Therefore, in this odel, no de-qualification is allowed. The objective of the odel is to balance achine qualification costs and future backorder costs. The tie horizon of future production scheduling in the odel is liited to a ediu ter (i.e. a couple of weeks). The scheduling horizon is divided into sall tie buckets to odel the oveent of lots between stages. Meanwhile, the production quantity of each product on each achine will be scheduled for 4

5 each tie bucket and ay be partial lots due to the assuption of continuous production. We assue that all the achine qualifications are finished at the beginning of production periods. A ixed integer linear prograing (deterinistic) odel is proposed in this section. The definition and notation of the eleents for the deterinistic achine qualification optiization (D-MQO) odel are listed below. Notation: P: nuber of products, with index p N p : nuber of stages for product p, with index n M[n]: nuber of unrelated achines at stage n, with index T: nuber of tie periods in the production scheduling horizon, with index t C: capacity in inutes of a achine in each tie period (C n,,t if it is achine, stage, and tie period dependent) A: available percentage of achine capacity in each tie period (1 A percent of achine capacity is reserved for setup and downtie activities) B p,0 : initial back order quantity of product faily p I p,n,0 : initial inventory of product p at (after) stage n b p : backorder cost per lot per tie period for product p d p,t : deand quantity for product p at the end of tie period t in lots t p,n, : lot processing tie of product p on achine at stage n c p,n, : cost of qualifying achine at stage n for product p S Q : a set of (p,n,) s with achine at stage n initially qualified for product p S Q : the copleent of set S Q Decision Variables: X p,n,,t R + : production quantity for product p in tie period t on achine at stage n I p,n,t R + : inventory quantity of product p at the end of tie period t after stage n B p,t R + : back order quantity of the product p at the end of tie period t Q p,n, B: 1 if achine at stage n is recoended to be qualified for product p, 0 otherwise Deterinistic Machine Qualification Optiization Model (D-MQO) in c p,n, Q p,n, + b p B p,t (1) (p,n,) S Q p,t 5

6 s.t. I p,n,t 1 + X p,n,,t X p,n+1,,t = I p,n,t, p,n < N p,t (2) I p,np,t 1 B p,t 1 + X p,np,,t d p,t = I p,n,t B p,t, p,t (3) X p,n+1,,t I p,n,t 1, p,n < N p,t (4) t p,n, X p,n,,t C A, n,1 M[n],t (5) p t p,n, X p,n,,t CQ p,n,, p,n,,t (6) Q p,n, = 1, (p,n,) S Q (7) X p,n,,t,i p,n,t,b p,t R +, p,n,,t (8) Q p,n, B, p,n, (9) The objective (1) is to iniize the total achine qualification and backorder costs. Constraints (2) are the inventory balance constraints for every product at every stage, except for the last stage, in each tie period. They indicate that the inventory quantity at the end of period t ust be equal to the beginning inventory plus production at stage n in period t inus consuption at the next stage n + 1 in period t. Constraints (3) are the inventory balance constraints for every product at the last stage in each tie period. They are siilar to constraints (2) except that the consuption at the next stage n + 1 in period t is replaced by deand at the end of period t. Backorders are allowed but incur cuulative backorder costs as shown in the objective expression (1). Constraints (4) are the aterial availability constraints, which state that the production quantity at stage n in period t ust be less than the inventory quantity at the previous stage n 1 at the end of period t 1. If a lot can flow through ore than one stage in one tie period, the right hand sides of constraints (4) can be expanded to include production at one or ore prior stages. Constraints (5) are the capacity constraints for every achine in each tie period, which state that the total production tie over all products ust be less than the available achine capacity after setup and downtie reservations. Constraints (6) are the achine qualification constraints, which state that production quantity X p,n,,t is zero unless achine at stage n is recoended to be qualified for product p. Constraints (7) define the initial qualification for achine at stage n already qualified for product p. Constraints (8) and (9) are the positive and binary constraints for decision variables, respectively. 6

7 The odel could be easily extended to include different process routes for different products and aterial handling tie between stages by slightly odifying the subscripts. For exaple, instead of X p,n+1,,t, X p,n+2,,t should be used in constraints (2) and (4) if product p skips stage n+1. If there is ore than one operation perfored at one stage, the stage subscript n can be substituted by operation subscript o in constraints (2), (3), and (4). Then in constraints (5) and (6), all the operations that could be perfored on achine at stage n should be considered in the left hand side. The aterial handling tie for product p between stage n and stage n+1 is added on the subscript t of all X p,n+1,,t s in constraints (2) and (4). If only bottleneck stages are odeled in the above forulation, which is possible when there are too any non-bottleneck stages in the anufacturing syste, the aterial handling tie can be further extended to include product-dependent delay ties at non-bottleneck stages. In the objective function (1), the total achine qualification cost is a one-tie cost and the total backorder cost over the production scheduling period (e.g. a week) actually represents recurring costs. In addition, since our ost iportant goal is to satisfy all deand, with iniizing achine qualification costs being the secondary objective, the achine qualification cost rates c p,n, s are set to be very sall copared to the backorder cost rates b p s. In an alternative forulation, we ay liit the total backorder cost p,t b pb p,t to a constant in the constraints and iniize achine qualification cost. With the alternative forulation, we could generate the Pareto optial frontier between the total backorder cost liit and the total achine qualification cost. The ediu-ter production scheduling considered in the above forulation is a snapshot of future production scheduling. Therefore it should reflect a steady state of the production syste. If we start with an epty syste in the above forulation, the start-up effect could give us a nonoptial achine qualification for future steady state production scheduling. As a result, Little s law [13] is used to estiate initial inventory quantities in the above forulation in a steady state syste: I p,n,0 = t p,n d p, p,n (10) where I p,n,0 is the initial inventory of product p at (after) stage n, t p,n is the average lot processing tie of product p at stage n, and d p is the average deand rate of product p. Average waiting tie could be included in t p,n if desired. To keep the production syste in steady state, the ending 7

8 inventory quantities at all stages should be greater than or equal to the corresponding starting inventory quantities or otherwise defined iniu. Therefore the following constraints should be added to the forulation during realization. I p,n,t I p,n,0, p,n (11) In the above deterinistic odel, the deand quantities d p,t s used in the production scheduling are assued to be certain at the tie when the achine qualification decisions are ade. However, the deand quantities are usually based on forecasts and thus uncertain in real world. Therefore, a stochastic odel is proposed in the following section to consider deand uncertainty. 4 Stochastic Machine Qualification Optiization Model (S-MQO) Machine qualification is usually a long ter factory configuration decision which incurs nonnegligible tie and onetary costs. It affects capacity allocation and thus daily production schedules directly. In our odel, the achine qualification decisions are integrated with ediu ter production scheduling. The objective is to iniize the total achine qualification costs and backorder costs. Since the deand data used in the production scheduling are uncertain, a stochastic achine qualification optiization odel is proposed in this section with the objective of iniizing total achine qualification costs and expected backorder costs. The purpose of this stochastic odel is to find a robust product-achine qualification atrix at inial qualification cost. Cost paraeters need to be assigned to achine qualification operations executed now and backorders that occur during the future planning horizon. Those paraeters should be deterined carefully considering that iniizing backorders is the priary objective and iniizing qualification costs is the secondary objective. A two-stage stochastic achine qualification odel is presented below. The deand is represented by a rando vector ξ = (d 0,0,...,d P,T ) T, with d p,t being the deand quantity of product p in period t. The objective (12) is to iniize the suation of total achine qualification costs (p,n,) S Q c p,n, Q p,n, and expected total backorder costs E[O(X,I,B,ξ)] over all possible 8

9 deand scenarios. in c p,n, Q p,n, + E[O(X,I,B,ξ)] (12) (p,n,) S Q s.t. Q p,n, = 1, (p,n,) S Q (13) Q p,n, B, p,n, (14) O(X, I, B, ξ) is the optial value of the following production scheduling subproble given a achine qualification atrix Q and a deand scenario ξ s : in p,t b p B p,t (15) s.t. I p,n,t 1 + X p,n,,t X p,n+1,,t = I p,n,t, p,n < N p,t (16) I p,np,t 1 B p,t 1 + X p,np,,t d p,t (ξ s ) = I p,np,t B p,t, p,t (17) X p,n+1,,t I p,n,t 1, p,n < N p,t (18) I p,n,t I p,n,0, p,n (19) t p,n, X p,n,,t C A, n,,t (20) p t p,n, X p,n,,t CQ p,n,, p,n,,t (21) X p,n,,t,i p,n,t,b p,t R +, p,n,,t (22) The first-stage decision variables Q p,n, s are deterined before the realization of rando deand vector ξ. The second-stage decision variables X p,n,,t s, I p,n,t s, and B p,t s are deterined based on the first-stage decision and the realized deand vector ξ. For the ease of reading, we list the additional notation of the stochastic odels as follows. Additional Notation: ξ s : deand scenario, with index s; all the notations defined in the deterinistic odel depending on the scenario are represented as (ξ s ) Ep,n,: k cut coefficient of Q p,n, generated in the L-shape ethod for iteration k 1 e k : the constant ter of the cut generated in the L-shape ethod for iteration k 1 Additional Decision Variables θ: upper bound variable of backorder cost in the L-shape ethod 9

10 γ(ξ s ), µ(ξ s ), σ(ξ s ), ϕ(ξ s ), π(ξ s ), ρ(ξ s ): dual variables of the subprobles of scenario ξ s in the L-shape ethod 4.1 Deterinistic Equivalent Forulation If the rando deand vector ξ can be represented or approxiated by a discrete distribution with possible deand scenarios (ξ 1,...,ξ S ) and associated probabilities (P(ξ 1 ),...,P(ξ S )), the previous two-stage stochastic odel could be rewritten as the following deterinistic equivalent forulation. X p,n,,t (ξ s ) s, I p,n,t (ξ s ) s, B p,t (ξ s ) s are the second-stage decision variables for deand scenario ξ s. in c p,n, Q p,n, + P(ξ s )b p B p,t (ξ s ) (23) (p,n,) S Q p,t,s s.t. I p,n,t 1 (ξ s ) + X p,n,,t (ξ s ) X p,n+1,,t (ξ s ) = I p,n,t (ξ s ), p,n < N p,t,s (24) I p,np,t 1(ξ s ) B p,t 1 (ξ s ) + X p,np,,t(ξ s ) d p,t (ξ s ) = I p,np,t(ξ s ) B p,t (ξ s ), p,t,s (25) X p,n+1,,t (ξ s ) I p,n,t 1 (ξ s ), p,n < N p,t,s (26) I p,n,t (ξ s ) I p,n,0 (ξ s ), p,n,s (27) t p,n, X p,n,,t (ξ s ) C A, n,,t,s (28) p t p,n, X p,n,,t (ξ s ) CQ p,n,, p,n,,t,s (29) Q p,n, = 1, (p,n,) S Q (30) X p,n,,t (ξ s ),I p,n,t (ξ s ),B p,t (ξ s ) R +, p,n,,t,s (31) Q p,n, B, p,n, (32) By solving this deterinistic equivalent forulation, an optial solution to the two-stage stochastic optiization proble (S-MQO) can be obtained. The deterinistic equivalent forulation is a ixed integer linear progra. As a result, when there are a large nuber of deand scenarios, products, or achines, the deterinistic equivalent forulation can be very difficult to solve. The L-shaped ethod and acceleration techniques are thus proposed to solve the S-MQO odel for large proble instances. 10

11 4.2 L-Shaped Method The extensive for of the deterinistic equivalent forulation has a block structure. Taking the dual of the extensive for, we can obtain a dual block-angular structure. Therefore, it is natural to exploit Dantzig-Wolf decoposition [7] on the dual or Bender s decoposition [2] on the prial. [16] extend this ethod to take care of feasibility in stochastic prograing, which is now called the L-shaped ethod. The classic L-shaped ethod was first developed only for stochastic linear progras. A valid set of feasibility cuts and optiality cuts is known to exist in the continuous case, based on duality theory in linear prograing. This knowledge fors the basis of the classic L-shaped ethod. Those cuts can also be used in the case where only soe first-stage variables are integers, e.g. the S-MQO odel. The L-shaped ethod has been extended to stochastic integer progras. The integer L-shaped ethod is the integration of the classic L-shaped ethod and branch-and-bound, during which optiality and feasibility cuts are added to LP relaxations. Since the S-MQO has binary first-stage variables and continuous second-stage variables, the classic L- shaped decoposition algorith is chosen instead of the integer L-shaped ethod. The L-shaped ethod is briefly described below as it applies to our proble. Algorith: L-Shaped Method Step 0 Set lower bound LB = and upper bound UB =. Set the iteration count i = 0. Set δ. Step 1 Solve the aster proble for an optial solution Q i LB = in c p,n, Q p,n, + θ (p,n,) S Q s.t. Q p,n, = 1, (p,n,) S Q Q p,n, B, p,n, θ Ep,n,Q k p,n, + e k,k = 1,2,...,i p,n, Step 2 For s = 1,...,S, solve the following subproble corresponding to Q i and ξ s O(Q i,ξ s ) = in p,t b p B p,t Dual variables 11

12 s.t. I p,n,0 + X p,n,,1 X p,n+1,,1 = I p,n,1, p,n < N p (γ p,n (ξ s )) I p,n,t 1 + X p,n,,t X p,n+1,,t = I p,n,t, p,n < N p,1 < t T I p,np,t 1 B p,t 1 + X p,np,,t d p,t (ξ s ) = I p,np,t B p,t, p,t (µ p,t (ξ s )) X p,n+1,,1 I p,n,0, p,n < N p (σ p,n (ξ s )) X p,n+1,,t I p,n,t 1, p,n < N p,1 < t T I p,n,t I p,n,0, p,n (ϕ p,n (ξ s )) t p,n, X p,n,,t C A, n,,t (π n,,t (ξ s )) p t p,n, X p,n,,t CQ i p,n,, p,n,,t (ρ p,n,,t(ξ s )) X p,n,,t,i p,n,t,b p,t R +, p,n,,t If (p,n,) S Q c p,n, Q i p,n, + s P(ξ s)o(q i,ξ s ) < UB, update the upper bound. Step 3 If (UB LB)/LB < δ, stop and return Q = {Q i } as the optial solution and UB as the optial objective value. Step 4 For each s = 1, 2,..., S, copute the cut coefficients E i+1 p,n, = s P(ξ s )( t ρ p,n,,t (ξ s ) C n,,t ) and e i+1 = P(ξ s )[ I p,n,0 γ p,n (ξ s ) + I p,n,0 σ p,n (ξ s ) s p,n<n p p,n<n p p,t + I p,n,0 ϕ p,n (ξ s ) + µ p,1 (ξ s ) (d p,1 (ξ s ) I p,np,0) p,n p + p,t>1µ p,t (ξ s ) d p,t (ξ s ) + π n,,t (ξ s ) C A]. n,,t Update i = i + 1 and go to Step 1. In the L-shaped ethod, the aster proble solved in Step 1 provides a lower linear approxiation for the function s P(ξ s)o(q,ξ s ) through a continuous variable θ and optiality cuts θ p,n, Ek p,n,q p,n, + e k, and therefore a lower bound LB for the objective function (23). The optial solution Q i obtained through the aster progra corresponds to a feasible solution 12

13 for the stochastic progra. It should be noted that in the first iteration i = 0, neither θ nor any optiality cut is included in the aster proble. In Step 2, all S subprobles are solved using the optial Q i obtained fro the aster proble and corresponding deand scenario ξ s. These S linear progras are solved independently, allowing for a coputationally convenient decoposition or parallelization. If all S subprobles are feasible, which in our case is always true since backorders are allowed in all subprobles, these subproble solutions together with the aster proble solution yield a upper bound UB of the original proble. When the upper bound UB and the lower bound LB are sufficiently close within a preset relative error ter δ, we conclude optiality. Otherwise the dual optial solutions of the subprobles are used to construct an optiality cut added in the aster progra in the next iteration. Only dual variables corresponding to constraints with positive right-hand-side values or positive coefficients of first-stage variables (Q p,n, s) will affect the cut coefficients. Those dual variables are represented as the γ p,n s, µ p,t s, σ p,n s, π n,,t s, ϕ p,n s, and ρ p,n,,t s in the parentheses. It should be noted that the initial inventory quantities I p,np,0 s at/after the last stage are assued to be zero, because the deand quantities B p,t s can always be adjusted to ake I p,np,0 s zero. In Step 4, according to duality theory the optiality cut s P(ξ s)o(q,ξ s ) = E i+1 p,n, Q p,n, + e i+1 is exact for Q i and is a lower linear approxiate for all other feasible Q s. In the classic L-shaped ethod, two types of cuts are added to the aster proble: feasibility cuts and optiality cuts. Optiality cuts are coputed in the previous algorith in Step 4. Feasibility cuts are added if and only if the aster solution in Step 1 is infeasible for certain subprobles in Step 2. Since backorders are allowed in our odel, all feasible aster proble solutions are feasible for all the subprobles. As a result, no feasibility cut is added in this algorith. 4.3 Acceleration of The L-Shaped Method The nuber of iterations in the L-shaped ethod for real world proble instances can be very large. To iprove the convergence behavior of the L-shaped ethod, the following acceleration techniques are proposed. Cut Disaggregation In the standard L-shaped ethod, one optiality cut is added at each iteration, which approxiates 13

14 the expectation of the second-stage objective functions given the current first-stage solution. Instead of one cut, S optiality cuts could be added at each iteration to approxiate individual secondstage objective functions per scenario. The optiality cut corresponding to deand scenario ξ s at iteration i is represented by in which θ s p,n, E s,i p,n,q p,n, + e s,i, E s,i p,n, = t ρ i p,n,,t (ξ s) C n,,t and e s,i = I p,n,0 γp,n(ξ i s ) + I p,n,0 σp,n(ξ i s ) p,n<n p p,n<n p p,t + I p,n,0 ϕ i p,n(ξ s ) + µ i p,1(ξ s ) (d p,1 (ξ s ) I p,np,0) p,n p + p,t>1µ i p,t (ξ s) d p,t (ξ s ) + πn,,t i (ξ s) C A. n,,t In the (i + 1)th iteration, the aster proble takes the following for. in c p,n, Q p,n, + P(ξ s )θ s (p,n,) S Q s s.t. Q p,n, = 1, (p,n,) S Q Q p,n, B, p,n, θ s Ep,n,Q s,i p,n, + e s,i,k = 1,2,...,i,s = 1,2,...,S p,n, This approach is referred to as ulticut L-shaped algorith [3]. In the ulticut version, there is no inforation loss due to cut aggregation, thus providing a better approxiation of the expectation of second-stage objective functions. Consequently, there are fewer iterations in the ulticut L-shaped ethod. However, since ore cuts are added at each iteration, the cost of the ulticut algorith is to solve larger aster probles. Qualification Cuts In the early iterations of the standard L-shaped ethod there are very few cuts in the aster proble. As a result, a inial nuber of achines are qualified in the optial solutions of the 14

15 aster proble, which results in large backorder quantities at the second-stage subprobles and a large nuber of iterations. To avoid such poor aster proble solutions, inforation of the second-stage subprobles is integrated in the aster proble by adding additional qualification cuts. Qualification cuts are added to ipose a lower bound restriction on the nuber of achines to be qualified for each product at each stage. The following forulation is defined as the single-scenario qualification subproble for ξ s (1 s S). in c p,n, Q p,n, (ξ s ) + P(ξ s ) b p B p,t (ξ s ) (p,n,) S Q p,t s.t. I p,n,t 1 (ξ s ) + X p,n,,t (ξ s ) X p,n+1,,t (ξ s ) = I p,n,t (ξ s ), p,n < N p,t I p,np,t 1(ξ s ) B p,t 1 (ξ s ) + X p,np,,t(ξ s ) d p,t (ξ s ) = I p,np,t(ξ s ) B p,t (ξ s ), p,t X p,n+1,,t (ξ s ) I p,n,t 1 (ξ s ), p,n < N p,t I p,n,t (ξ s ) I p,n,0 (ξ s ), p,n t p,n, X p,n,,t (ξ s ) C A, n,,t p t p,n, X p,n,,t (ξ s ) CQ p,n, (ξ s ), p,n,,t Q p,n, = 1, (p,n,) S Q X p,n,,t (ξ s ),I p,n,t (ξ s ),B p,t (ξ s ) R +, p,n,,t Q p,n, B, p,n, Let the B s p,t(ξ s ) s be the optial backorder quantities obtained fro the single-scenario qualification subproble for ξ s (1 s S) and the B o p,t (ξ s) s be the optial backorder quantities obtained fro the S-MQO odel. When c p,n, << P(ξ s )b p ( p,n,) holds, they ust satisfy the following conditions: b p Bs p,t (ξ s ) = p,t p,t b p Bo p,t (ξ s ), s (33) Note that both P(ξ s ) p,t b p B p,t s (ξ s) and P(ξ s ) p,t b p B p,t o (ξ s) are equal to the inial total backorder cost in deand scenario ξ s given that every achine is qualified for every product. Therefore, if Q s (ξ s ) is the unique optial achine qualification atrix obtained fro the single-scenario qualification subproble for ξ s (1 s S) and Q o is an optial achine qualification atrix obtained 15

16 fro the S-MQO proble, they ust satisfy the following conditions: Q o p,n, Q s p,n, (ξ s), p,n,s (34) Conditions (34) hold only when the following two assuptions are both valid: c p,n, << P(ξ s )b p ( p,n,,s) and each single-scenario qualification subproble has a unique optial achine qualification atrix. The first assuption c p,n, << P(ξ s )b p ( p,n,,s) holds if the cost paraeters c p,n, s and b p s are carefully chosen. Because there are usually ultiple optial solutions for real world applications, the second assuption usually does not hold. As a result, adding inequalities (34) in the aster proble leads to a sub-optial solution for the original S-MQO proble. However, if the first assuption holds, the expected total backorder costs over all scenarios should still be the sae with or without inequalities (34). Adding inequalities (34) will decrease the nuber of iterations in the L-shaped ethod. Thus the tradeoff here is between the total achine qualification cost and the solution tie of L-shaped ethod. Inequalities (34) are referred to as qualification cuts in this paper. Relaxed Qualification Cuts When the proble size increases, even the single-scenario qualification subproble can be difficult to solve since it is a ixed integer linear progra. In such cases, we can solve the LP relaxation of the single-scenario qualification subproble for an optial continuous achine qualification atrix Q s. Then a binary achine qualification Qs can be obtained using the following rule: { Qs = 1, Qs > ǫ Q s = 0, Q s ǫ where ǫ is a preset value between 0 and 1. A set of qualification cuts siilar to inequalities (34) can be added using Qs instead of Qs. Those cuts are called relaxed qualification cuts. They require significantly less tie for solving the (relaxed) single-scenario qualification subprobles. On the other hand, both optial achine qualification cost and expected backorder cost with relaxed qualification cuts can be larger than those of the original S-MQO proble. Therefore, the tradeoff here is still between the solution quality and solution tie. 16

17 5 Coputational Experients In this section we will present a nuerical experient solving a 5-product proble instance with the proposed odels and solution ethods. First, the anufacturing syste and deand inforation are introduced. Then the efficiencies of the two different stochastic solution ethods for the S- MQO odel will be discussed and copared using different nubers of scenarios. At the end, the solution quality of stochastic and deterinistic odels will be evaluated and thus copared through an optiization based scheduling syste. 5.1 Data M31 M41 M42 M11 M32 M21 M43 M12 M33 M44 M34 Product Flow M45 Figure 1: Manufacturing syste description The 5-product proble instance is based on a real seiconductor back-end facility with 4 bottleneck stages as shown in Figure 1. Usually there are 20 to 30 processing stages in a back-end facility. However, including all those stages in the atheatical odel results in a significantly larger forulation size. Therefore all the non-bottleneck stages are odeled as constant delays between bottleneck stages, as stated in Section 3. The delay tie on a non-bottleneck stage is estiated by the average throughput tie at this stage. It is assued there are ultiple identical parallel achines at each stage, as shown in Table 1. Every achine can be qualified to process every product. The production scheduling horizon in the odel is chosen to be 1 week, which is divided into 84 2-hr tie buckets. All the ties used in the experients are in 2-hr units, e.g. processing tie of 1.5 per lot in the experient represents 3-hour per lot actual processing tie. Two processing tie distributions are used in the experient to siulate production systes with approxiately 60% and 90% achine utilizations. Processing ties of all products at the sae stage 17

18 Nuber of products 5 Nu. of bottleneck stages 4 Nu. of achines (2,1,4,5) Stage 1 Processing Tie U(1.00,2.00) Stage 2 Processing Tie U(0.10,0.20) Stage 3 Processing Tie U(2.00,4.00) Stage 4 Processing Tie U(2.00,4.50) Table 1: Manufacturing Syste Description. Table 2: Weekly Deand. 60% Utilization 90% Utilization Weekly deand U(5, 25) U(5, 35) Weekly deand average Weekly deand axial are randoly generated based on the sae distribution, as shown in Table 1. Although products are allowed to have different processing routes or skip certain stages in the proposed odels, all products are assued to go through all stages in the sae linear sequence in the experient. Custoer orders or product types can be assigned with different priorities through their backorder cost rates (per lot per 2-hr tie bucket), e.g. iportant orders or product types with higher backorder cost rates. However in the experient, all product types and lots are assued to have the sae priority for siplicity, therefore the sae backorder cost rate. The initial product-achine qualification atrix is assued to be epty, with no achine qualified for any product. The weekly deand for future production scheduling is uncertain and randoly generated fro a unifor distribution in the experients as shown in Table 2. A sall 5-product proble is design based on the real size 25-product proble to copare two solution ethods of the stochastic S-MQO odel. The production syste description and weekly deand inforation for the 5-product proble are shown in Table 1 and Table 2 respectively. The available percentage of achine capacity in each tie Table 3: Size of the deterinistic equivalent of the S-MQO proble. S Constraints Variables Equality Inequality Continuous Binary 1 1,680 7,328 7, ,400 36,640 35, ,800 73,280 71, , , ,

19 period A is set to be 80% in all cases. The WIP inventory in the syste is estiated using Little s Law I p,n,0 = t p,n d p, p,n. In the experients, t p,n is estiated by the expected processing tie of product p at stage n fro Table 1, and d p is estiated by the expected deand of product p fro Table 2 divided by the total nuber of periods (84) in a week. The sizes of the deterinistic equivalents of the S-MQO proble for different S values are given in Table 3. There is a positive linear relationship between the nuber of constraints and continuous variables and the nuber of possible scenarios S. Even for a sall proble instance with only 5 products and 12 achines, there are 60 binary variables in the forulation. For a typical test facility with 25 aggregated product failies and 50 bottleneck-stage achines, there will be 1250 binary variables, thus aking it very difficult to solve. 5.2 Perforance of Different Solution Methods In the experient, two different solution ethods for the S-MQO odel are tested. One is to solve the deterinistic equivalent forulation (DE). The other is the L-shaped ethod (Bender). Proposed acceleration techniques of the L-shaped ethod are also tested, including cut disaggregation (CD), qualification cuts (QC), and relaxed qualification cuts (RQC). Solution ties of all tested solution ethods are listed in Table 4 and ploted in Figure 2 for different S values and achine utilizations. More details about the solution ties of different ethods are shown in Table 4, including solution/decoposition (BD) tie, tie for adding qualification cuts before the decoposition (QC tie), nuber of iterations in the decoposition algorith, and optiality gap at the end of runtie liit (36000 sec). The L-shaped ethod with cut disaggregation and relaxed qualification cuts ( Bender + CD + RQC ) has the shortest solution ties and fewest nubers of iterations. The L-shaped ethod with cut disaggregation and qualification cuts ( Bender + CD + QC ) has relatively few iterations but unstable solution ties. It is also noted that the tie required for solving single-scenario qualification subprobles (QC tie) increases significantly when S increases. As a result, adding qualification cuts is not suitable for real size proble instances. The L-shaped ethod with cut disaggregation ( Bender + CD ) has relatively short solution ties 19

20 but relatively large nubers of iterations, which could ake it unsuitable for real size proble instances. All other solution ethods have both long solution ties and larger nuber of iterations. The fewer nuber of iterations eans that the cut disaggregation with qualification cuts or the relaxed qualification cuts work well by cutting off infeasible solutions. The use of relaxed qualification cuts sacrifice the solution quality to certain extend, while the coputational tie is uch iproved. As shown in Table 5, the total cost with relaxed qualification cuts are increased within 5%, while the coputational ties are iproved largely. The quality of solutions of different ethods are listed in Table 5 for different S values and achine utilizations. Optial solutions obtained with the first two ethods are also optial for the original S-MQO odel. However, optial solutions obtained with the last four ethods can be sub-optial to the original S-MQO odel, due to QC/RQC cuts. Both the total qualification costs and the expected total backorder costs are shown in the Q cost and B cost coluns respectively in Table 5. In the experient, c p,n, = 0.1 ( p,n,) and b p = 1 ( p). Fro Bender + CD to Bender + QC or Bender + CD + QC, the optial B cost does not increase, and the optial Q cost and Total cost increase slightly. For Bender + RQC and Bender + CD + RQC, the optial B cost, Q cost and Total cost all increase. This is consistent with the previous analysis. The increase in B cost for Bender + RQC and Bender + CD + RQC is significant when S is 20. The reason is that c p,n, < P(ξ s )b p does not hold anyore when S is 20. Therefore, c p,n, s and b p s should be chosen carefully to ake sure that c p,n, < P(ξ s )b p is valid if Bender + CD + QC is to be ipleented. Bender + CD + RQC and Bender + CD are recoended for large size proble instances because of short solution ties and sall nubers of iterations. If Bender + CD does not find the optial solution and Bender + CD + RQC finds one, thus providing an upper bound of the S-MQO odel, a lower bound can be estiated by the LP relaxation of the original S-MQO proble. At the end, the optial qualification atrices obtained using the L-shaped ethod with cut disaggregation for different S values and achine utilizations are evaluated using a different set of 20 deand scenarios generated according to the distributions in Table 2. Each deand scenario is given an equal probability of A production scheduling linear progra is solved for each deand scenario and each optial qualification atrix. The total qualification cost and expected 20

21 Table 4: Solution tie coparison of different acceleration ethods. S = 5 60% Utilization 90% Utilization BD tie QC tie Iterations Gap BD tie QC tie Iterations Gap (sec) (sec) (%) (sec) (sec) (%) Bender Bender + CD Bender + QC Bender + CD + QC Bender + RQC Bender + CD + RQC S = 10 60% Utilization 90% Utilization BD tie QC tie Iterations Gap BD tie QC tie Iterations Gap (sec) (sec) (%) (sec) (sec) (%) Bender Bender + CD Bender + QC Bender + CD + QC Bender + RQC Bender + CD + RQC S = 20 60% Utilization 90% Utilization BD tie QC tie Iterations Gap BD tie QC tie Iterations Gap (sec) (sec) (%) (sec) (sec) (%) Bender Bender + CD Bender + QC Bender + CD + QC Bender + RQC Bender + CD + RQC total backorder cost for each optial qualification atrix are listed in the Q cost and B cost coluns of Table 6. Optial qualification atrices fro the deterinistic odel using the average or axial deand are listed in the first and second row. For both the 60% and 90% achine utilization cases, the optial qualification atrices obtained fro the stochastic odel outperfor those obtained fro the deterinistic odel. Not surprisingly, for the stochastic odel, the optial qualification atrix obtained with ore deand scenarios also has better perforance, because a larger nuber of deand scenarios provides a better approxiation of the original continuous distribution. With the large nuber of scenarios, the Bender s approach with cut disaggregation and relaxed qualification cuts is preferred for the tradeoff of the coputational efforts and solution quality. 6 Conclusion In this paper, a stochastic ixed integer linear prograing odel (S-MQO) is proposed to optiize product-achine qualifications for a seiconductor back-end facility. Future production 21

22 Table 5: Solution quality coparison of different acceleration ethods. S = 5 60% Utilization 90% Utilization Q cost B cost Total cost Gap(%) Q cost B cost Total cost Gap(%) Bender Bender + CD Bender + QC Bender + CD + QC Bender + RQC Bender + CD + RQC S = 10 60% Utilization 90% Utilization Q cost B cost Total cost Gap(%) Q cost B cost Total cost Gap(%) Bender Bender + CD Bender + QC Bender + CD + QC Bender + RQC Bender + CD + RQC S = 20 60% Utilization 90% Utilization Q cost B cost Total cost Gap(%) Q cost B cost Total cost Gap(%) Bender Bender + CD Bender + QC Bender + CD + QC Bender + RQC Bender + CD + RQC Table 6: Evaluation of different qualification atrices. S 60% Utilization 90% Utilization Q cost B cost Q cost B cost 1 (avg) (ax)

23 scheduling in a ediu ter horizon with deand uncertainty is considered. Setup ties and downtie are odeled indirectly by using the achine utilization rate fro historical data. The odel proposed doesn t bias to the different setup sequences for the setup tie. Therefore, for the general optial solution, the setup tie has the sae distribution fro the historical data. Depending on the conservatis, the decision aker ay choose different confidence levels of the utilization rate to be used in the odel. The L-shaped ethod and several acceleration techniques are proposed to solve the stochastic odel. In the nuerical experients, a 5-product exaple is used to evaluate different solution ethods and their solutions. Bender s Decoposition with Cut Disaggregation and possibly Relaxed Qualification Cuts applied to the stochastic deand forulation are recoended for deterining a robust qualification schedule. This approach is shown to have advantaged over deterinistic proble forulations. In this paper, we assue product-achine qualification decisions are ade and ipleented now for a foreseeable future with stationary deand. The odels described in this paper could be readily expanded to include tie-phased qualification decisions. An interesting topic for future research will be a ulti-stage stochastic odel for tie-phased qualification decisions. 7 Acknowledgent Intel Corporation supported this work. We are thankful to Jeffrey Pettinato and Naiping keng at Intel for their helpful feedback. References [1] A. Aubry, A. Rossi, M.L. Espinouse, and M. Jacoino. Miniizing setup costs for parallel ulti-purpose achines under load-balancing constraint. European Journal of Operational Research, 187(3): , [2] J.F. Benders. Partitioning procedures for solving ixed-variables prograing probles. Nuerische Matheatik, 4(1): , [3] J. R. Birge and F. Louveaux. A ulticut algorith for two-stage stochastic linear progras. European Journal of Operational Research, 34(3): , [4] K.E. Bourland and L.K. Carl. Parallel-achine scheduling with fractional operator requireents. IIE Transactions, 26(5):56 65, [5] P. Brucker, B. Jurisch, and A. Kräer. Coplexity of scheduling probles with ulti-purpose achines. Annals of Operations Research, 70:57 73, [6] G.M. Capbell. Using short-ter dedication for scheduling ultiple products on parallel achines. Production and Operations Manageent, 1(3): ,

24 [7] G.B. Dantzig and P. Wolfe. Decoposition principle for linear progras. Operations research, 8(1): , [8] A. Drexl and M. Mundschenk. Long-ter staffing based on qualification profiles. Matheatical Methods of Operations Research, 68(1):21 47, [9] J. Hurink, B. Jurisch, and M. Thole. Tabu search for the job-shop scheduling proble with ulti-purpose achines. OR Spectru, 15(4): , [10] J.P. Ignizio. Cycle tie reduction via achine-to-operation qualification. International Journal of Production Research, 47(24): , [11] C. Johnzén, P. Vialletelle, S. Dauzère-Pérès, C. Yuga, and A. Derreuaux. Ipact of qualification anageent on scheduling in seiconductor anufacturing. In S.J. Mason, R.R. Hill, L. Monch, T. Jefferson, and J. Fowler, editors, Proceedings of the 40th Conference on Winter Siulation, pages Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, [12] B. Jurisch. Lower bounds for the job-shop scheduling proble on ulti-purpose achines* 1. Discrete Applied Matheatics, 58(2): , [13] J.D.C. Little. A proof of the queuing forula L=λW. Operations Research, 9(3): , [14] Y. Mati and X. Xie. The coplexity of two-job shop probles with ulti-purpose unrelated achines. European Journal of Operational Research, 152(1): , [15] A. Rossi. A robustness easure of the configuration of ulti-purpose achines. International journal of production research, 48(3-4): , [16] R.M. Van Slyke and R. Wets. L-shaped linear progras with applications to optial control and stochastic prograing. SIAM Journal on Applied Matheatics, pages , [17] M.C. Wu, YL Huang, YC Chang, and KF Yang. Dispatching in seiconductor fabs with achine-dedication features. The International Journal of Advanced Manufacturing Technology, 28(9): , [18] M.C. Wu, H. Jiang Jr, and W.J. Chang. Scheduling a hybrid MTO/MTS seiconductor fab with achine-dedication features. International Journal of Production Econoics, 112(1): ,

25 Solution Tie (sec) Bender DE Bender+QC Bender+CD+QC Bender+RQC Bender+CD Bender+CD+RQC Nuber of Scenarios (a) 60% Utilization Cases Solution Tie (sec) Bender DE Bender+QC Bender+CD+QC Bender+RQC Bender+CD Bender+CD+RQC Nuber of Scenarios (b) 90% Utilization Cases Figure 2: Solution ties of different solution ethods 25