Tool capacity planning in semiconductor manufacturing

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1 Available online at Computers & Operations Research 30 (2003) Tool capacity planning in semiconductor manufacturing Bulent C atay a;,s. Selcuk Erenguc b, Asoo J. Vakharia b a Faculty of Engineering and Natural Sciences, Sabanc University, Tuzla, Istanbul, Turkey b Department of Decision and Information Sciences, Warrington College of Business, University of Florida, 351 Stuzin Hall, Gainesville, FL , USA Received 1 July 2001; received in revised form 1 March 2002 Abstract The demand for distinct wafer types in semiconductor manufacturing is an explicit function of the electronic components in which these wafers are used. Given that the component demands vary not only by the product type but also over time, it is obvious that wafer demands are also lumpy and time varying. In this paper, we discuss strategic level investment decisions on procuring new equipment and aggregate level capacity planning. In this context, we examine the problem of planning wafer production over multiple time periods within a single facility assuming that a demand forecast for each wafer type for each period is known. To address this problem, we develop a multi-period mixed-integer programming model to minimize the machine tool operating costs, new tool acquizition costs, and inventory holding costs. Given that production of wafers requires a large number of operations with multiple tools capable of performing each operation, tool operating costs are explicitly minimized by integrating the assignment of specic operations to tools in our model. Since our model is computationally intractable, we propose a Lagrangean-based relaxation heuristic to nd ecient tool procurement plans. Scope and purpose Semiconductor manufacturing companies are faced with important capital investment decisions for the procurement of new types of machine tools for their facilities. This paper is motivated by the machine tool planning issues faced by such a facility in the US that spends a few million dollars every quarter on procurement of new machine tools. Additional requirements for new tools arise primarily from the replacement of obsolete equipment and the growth in demand for the existing products as well as the introduction of new semiconductor products that require newer technologies. Since most of these tools are very expensive special purpose equipment even a slight enhancement in the management s decision-making process might lead to signicant nancial improvement in the manufacturer s performance. In this paper, we model a multi-period tool capacity planning problem for given demand forecasts and we present a Lagrangean-based heuristic Corresponding author. Tel.: ; fax: addresses: catay@sabanciuniv.edu (B. C atay), erenguc@dale.cba.u.edu (S.S. Erenguc), vakharaj@ notes.cba.u.edu (A.J. Vakharia) /03/$ - see front matter? 2002 Elsevier Science Ltd. All rights reserved. PII: S (02)

2 1350 B. C atay et al. / Computers & Operations Research 30 (2003) solution approach to obtain ecient procurement strategies. We also provide computational experiments to test the quality of our results.? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Production planning; Semiconductor manufacturing; Capacity planning; Mathematical programming; Lagrangean relaxation 1. Introduction Semiconductor manufacturing represents one of the most complicated manufacturing processes. The complexity of the production planning and scheduling in the semiconductor industry is mainly due to the following factors in the manufacturing process for building circuits on the wafers. First, wafers typically require over 400 operation steps at several work centres. Second, each wafer makes multiple visits to the same work centre at dierent points in the fabrication process since the same operation types are needed to build multiple layers on top of a wafer. Such a product ow is known as a reentrant product ow and is illustrated in Fig. 1 [1]. Third, uncertain yields and unpredictable equipment downtime further complicate the production control of semiconductor products. Fourth, there exist several dierent recipes and many sequences of processes using the same or similar equipment. Finally, data acquizition and maintenance in a wafer fabrication facility (wafer fab) is extremely dicult and time consuming because of the high volume of data emanating from hundreds of thousands of transactions each day. Most of the research in semiconductor manufacturing has focused on demand satisfaction, maximization of equipment utilization, minimization of production costs, and maximization of throughput Evaporation Multiprobe Epitaxy 4 Oxidation Lithography Etching 3 Diffusion Cleaning Sintering 6 Coating 5 Fig. 1. Simple silicon TTL integrated circuit process owchart.

3 B. C atay et al. / Computers & Operations Research 30 (2003) with some capacity constraints. Linear programming is often proposed as a tool for production planning and scheduling in semiconductor manufacturing. However, LP formulations can be very large for large organizations with complex production environments such as semiconductor industry. Sullivan and Fordyce [2] point out that these LPs may require a very long time to generate the input data les that need to be fed into mathematical planning software and huge amounts of memory and disk space to store the data. However, with the use of new ERP systems and powerful computers data acquizition and storage are no longer a burdensome task. Golovin [3] indicates the diculty of selecting an appropriate objective function in the planning of the semiconductor manufacturing and shows that a detailed formulation of the problem in an integrated manner based upon a mathematical programming model is intractable and requires data which cannot be obtained reliably. He proposes a hierarchical approach such as that suggested by Bitran et al. [4,5]. Leachman [6] gives an optimization-based corporate-level production planning system that includes multiple facilities and integrates the production processes in these facilities. His model generates capacity-feasible start and out schedules for each manufacturing facility in the company. Leachman and Carmon [7] analyze capacity of semiconductor production facilities in which manufacturing operations may be performed by alternative tool types. Such operations can be performed by dierent tool types with dierent processing times. They propose a modelling technique that greatly reduces the size of the LP problems. Hung and Wang [8] present an alternative model for formulating the alternative tool capacitated planning problem as an extension of the technique proposed in [7]. Their technique considerably reduces the size of the LP problem for bin allocation planning in semiconductor manufacturing, thus saving substantial amounts of computer resources required. The interested reader is referred to [9,10] for a comprehensive review of the research in the semiconductor production planning and scheduling. Tool capacity planning in a complex semiconductor manufacturing environment is a dicult task. Tools have dierent manufacturing characteristics depending on the process and product involved. While newer tools are normally capable of processing advanced products as well as older products, older tools usually can only process older products and could require longer processing times. Further, older tools may provide lower yields and lower utilization levels due to increasing maintenance requirements. Fordyce and Sullivan [11] point out the importance of tool procurement based on demand forecasts for the managers of wafer fabs. In recent years, tool procurement planning has become more challenging at wafer fabs that produce application specic integrated circuits [12]. In this paper, we address a multi-period tool capacity planning problem for wafer manufacturing. Our scope is strategic level investment decisions on procuring new equipment and aggregate capacity planning. By recognizing that each wafer type requires multiple operations, we explicitly focus on assigning each operation to tool types in each period. Further, by incorporating capacity limits onto individual tools in each period, we allow for demand to be met either through a build-up of inventories or through increases in capacity. Since the existing methodology in tool capacity planning in the industry is based on deterministic demand forecast, we assume the demand is given. A similar approach to production planning in semiconductor manufacturing has been recently proposed by Swaminathan [12,13]. In [12], Swaminathan studies the single period tool capacity planning problem under demand uncertainty and formulates it as a stochastic integer program. He provides two heuristics to nd ecient tool procurement plans and uses Lagrangean relaxation to develop lower bounds. In [13], he extends this problem to multiple periods using scenario-based demands

4 1352 B. C atay et al. / Computers & Operations Research 30 (2003) and proposes again two heuristics using Lagrangean relaxation as the lower bounding method. He models the future demand by a set of demand scenarios and assigns a probability of occurrence to each scenario. For each demand scenario, the procurement plan is developed in order to meet the demand as closely as possible, after the demand is realized. In both papers, Swaminathan formulates the objective function as the expected stock-out costs and restricts the procurement plan with a budget constraint. The remainder of the paper is organized as follows. In Section 2, we describe the problem and formulate a mathematical model to address it. In Section 3, we present a Lagrangean-based solution procedure to obtain a lower bound and a good feasible solution to the problem. Section 4 contains experimental analysis of the proposed solution procedure on randomly generated data. Finally, Section 5 summarizes the conclusions and gives directions for future research. 2. Problem statement and formulation Manufacturing a computer chip is a complex process involving hundreds of steps and requiring from a few days to up to 3 months of processing time to complete. The semiconductor manufacturing process consists of various operation types such as oxidation, deposition, lithography, etching, ion implantation, etc., performed on dierent batches of products (wafers). As discussed above, some of these operation types are repeated several times to build dierent layers on top of the wafer and wafers may require up to 400 operation steps at a number of work centres. Each wafer type follows a particular sequence throughout its fabrication process visiting dierent work centres. Furthermore, each wafer makes multiple visits to the same work centre at dierent points in the fabrication process. The common practice in the layout of work centres for dierent types of processes is to group similar operation types together: furnaces are grouped in one area, ion implanters in another, and so forth. This layout requires wafers moving back and forth between work centres but it allows the utilization of the same equipment to process wafers at dierent steps. For instance, all oxidation processes may be completed on the same tool or in the same work centre although the wafers are required to travel to other work centres between the oxidation process steps. Since the specications of the process performed on a wafer type may dier from one visit to another, each visit is referred to as a distinct step. Furthermore, dierent wafer types may require the same process with the same specications. In that case, those wafer types are pooled into a single batch. The wafer type=types at each step is=are given a specic name that indicates both the wafer type=types and the process to be performed on the wafer type=types. Throughout this paper, wafer type=types at a particular processing step is=are referred to as an operation. In this environment, we focus on the loading of operations to alternative tool types where tool duplication is allowed for certain tool groups. We consider a population of tools that are capable of performing a certain manufacturing process only, such as oxidation, deposition, or ion implantation, etc. Tools having the same characteristics are gathered into tool groups and each tool group represents a set of identical tools with the same processing capabilities. In this context, tool groups are classied as primary tool groups and secondary tool groups where primary tools are the most ecient tools to process the specied operations and secondary tools are the alternative tools in case additional tool capacity is needed. Only primary tools may be procured if further capacity is required to meet the production schedules.

5 B. C atay et al. / Computers & Operations Research 30 (2003) Table 1 Notation for MPCAP t time period index, where t =1;:::;T, s secondary tool group index, where s = P +1;:::;M, p primary tool group index, where p =1;:::;P, i tool group index, where i = 1;:::;M, j operation index, where j = 1;:::;J, N ti number of tools currently available in tool group i in period t, [P] capability matrix: the binary entry p ij is 1 if tool group i can process operation j and 0 otherwise, V tj daily scheduled production volume of operation j in period t, Y tij yield of tool group i with respect to operation j in period t, PC tij daily processing capacity of tool group i with respect to operation j in period t, U ti utilization of the tool group i in period t, ( = net production hours=24 h) tij Fractional number of tools of type i required to process each unit of operation j in period t, { 1 PC tij = tij Y tij if p ij =1; 0 otherwise; c tij discounted operating cost of each tool of type i to process operation j in period t, F tp discounted capitalized procurement cost of an additional tool of type p in period t, h tj discounted holding cost of one unit of operation j at the end of period t. As we mentioned above, the wafer fabrication process may take several months to complete. Since the wafers visit the same tool groups in dierent steps of their manufacturing process, it may take days or even weeks for a wafer to revisit the same tool group. On the other hand, tool capacities are allocated based on a snapshot of the current production quantities, i.e. all the processing steps that require the same tool group are assumed to be performed in the same period although there may be a sequencing time lag between those steps. Since our point of interest is strategic level investment decisions on procuring new tools and aggregate capacity planning, we note here that a detailed production schedule is necessary for operational decisions. The problem of modelling the capacity of alternative, non-identical resources arises in almost all manufacturing environments where process technology is evolving. In this section, we develop a mixed-integer programming formulation to minimize operating cost of tools, inventory holding cost, and procurement cost of primary tools. For each operation, there are allocation variables to spread the production quantities among alternative tool groups. Inventory variables and integer variables are used to represent holding of inventories and procurement of primary tools, respectively. Inventory balance constraints are used to guarantee consistency of production volumes between consecutive periods and resource constraints are introduced both to satisfy capacity requirements and to determine additional tool requirements. Given the number of tools in each tool group and capacity consumption of each operation in each tool group, our objective is to nd the optimal assignment scheme of operations to tool groups. Before we present our capacity allocation problem formulation, we introduce the notation in Table 1 and dene the decision variables in Table 2. Table 3 provides a set of sample data for a tool group consisting of ion implanters and four operations that can be processed on this tool group. As we have mentioned earlier, real-life MIP formulations in semiconductor manufacturing operations are

6 1354 B. C atay et al. / Computers & Operations Research 30 (2003) Table 2 Decision variables for MPCAP X tij number of units of operation j assigned to tool group i in period t, I tj number of units of operation j held in inventory at the end of period t, Q tp number of additional tools of type p needed in period t. Table 3 Illustrative data for four operations processed on ion implanters Tool type Quantity Net production Utilization available h=24 h Ion implanter Operation Scheduled Production % Yield Number of number volume capacity tools needed very large and computationally intractable. Considering the fact that the wafer fab that motivated this study utilizes approximately 1200 tools and that a wafer may require over 400 operation steps, the notation in Tables 1 and 2 may be an indicator of the size of a mathematical model for quarterly tool procurement planning over a planning horizon of a few years time-span. Some specic size characteristics for our formulation are provided with the computational experiments in Section 4. In what follows a comprehensive mathematical formulation of the multi-period alternative tool type capacity allocation problem with duplicated tools is presented. Problem MPCAP: M min c tij tij X tij + i=1 s:t: I (t 1)j + h tj I tj + p=1 P F tp Q tp (1) M X tij I tj = V tj t; j; (2) i=1 tsj X tsj 6 N ts U ts t; s; tpj X tpj 6 N tp U tp + Q p U tp t; p; (4) =1 (3) Q ti 0 and integer t; i; (5)

7 B. C atay et al. / Computers & Operations Research 30 (2003) X tij 6 Zp ij t; i; j; (6) I tj 0 t; j: The objective function of the formulation is to minimize operating cost of tools, inventory holding cost, and procurement cost of additional tools. Constraints (2) ensure that the demands at each period are satised. Constraints (3) express the capacity limits on the secondary tool groups and constraints (4) express the capacity limits on the primary tool groups where capacity may be increased by adding new tools. We prefer to express tool capacity restrictions under two sets of constraints to emphasize two dierent classes of tool groups. These two constraints may as well be formulated as one constraint set similar to (4). Constraints (5) impose integrality and non-negativity restrictions on the additional tool variables and constraints (6) express the usual non-negativity conditions on assignment variables and ensure that operations are assigned to tool groups that are capable of processing them. Here, Z is a suciently large number. Backorders are prevented by imposing the non-negativity constraints (7). For a given U ti we dene b tij = tij =U ti and replace constraints (3) and (4) with b tsj X tsj 6 N ts t; s; (8) b tpj X tpj 6 N tp + Q p t; p: (9) =1 Since this model is computationally intractable, we now proceed to discuss a Lagrangean relaxationbased solution procedure to obtain a good lower bound on the minimum value of problem MPCAP and a feasible upper bound. (7) 3. Description of the Lagrangean-based heuristic solution method Lagrangean relaxation has been successfully employed by numerous researchers for integer=mixed integer programming applications. The approach is based on the observation that many dicult integer programming problems can be viewed as they are composed of two types of constraints: nice constraints and complicating constraints. Lagrangean relaxation is formed by multiplying the complicating constraints with corresponding penalty costs (Lagrangean multipliers) and including them into the objective function. By dualizing those constraints, we obtain a problem that is easy to solve and whose optimal value gives a lower bound on the optimal value of the minimization problem. The reader is referred to [13,14] for a detailed overview of the Lagrangean relaxation technique. Our solution method is based on the Lagrangean relaxation of the capacity constraints (8) and (9), and a set of cumulative tool requirements constraints, which is discussed in the next section. An overview of the overall solution procedure for problem MPCAP is presented in Fig. 2. Through Lagrangean relaxation of the above mentioned constraints we obtain two subproblems in Step 2, one with assignment and inventory variables, the other with additional tool requirements variables. The rst problem further decomposes into several subproblems that are easily solved by inspection; the

8 1356 B. C atay et al. / Computers & Operations Research 30 (2003) Step 1 Step 2 Step 3 Step 4 Initialize Lagrangean multipliers. Update the lower bound on the minimum value of the objective function by solving relaxed model. Compute the tool groups' workloads and inventories from the current solution and update Lagrangean multipliers with respect to violations of the constraints. Update the upper bound on the minimum value of the objective function; if a stopping criteria is met, stop; otherwise go to Step2. Fig. 2. Outline of the heuristic procedure. second problem is also solved by inspection. The solutions of these problems are used to compute a lower bound on the minimum value of the objective function. In Step 3, the assignment scheme and inventory levels resulting from Step 1 are determined and the resource workloads are computed using constraints (2), (8), and (9). The violations in these constraints are employed to update the Lagrangean multipliers, using a subgradient optimization technique. Finally, the upper bound on the minimum value of the objective function is computed in Step 4. The violations of capacity constraints (8) are eliminated through an operation shifting procedure and then a tool reduction procedure is applied to constraints (9) Computation of the lower bound Before we proceed with the Lagrangean relaxation of the problem MPCAP we rst consider adding a set of valid inequalities representing lower bounds on the cumulative number of tool capacity needed to satisfy the cumulative demand from periods 1 through t, where t =1;:::;T: P Q zp K t p=1 =1 z=1 t: (10) These constraints may provide better bounds in the relaxed problem. K t is the minimum cumulative additional tool requirements to meet the demands of periods 1 through t and may be obtained by solving the following problem: Problem CATR: min s:t: P p=1 Q tp M X ij =1 i=1 =1 V j t; j and (5); (6); (8) and (9)

9 B. C atay et al. / Computers & Operations Research 30 (2003) and then setting P K t = p=1 =1 z=1 Q zp t: Solving problem CATR is as dicult a task as solving problem MPCAP. Therefore, we propose the following set of uncapacitated problems that can easily be solved by inspection as an alternative procedure to obtain K t. The solution to CTR t gives minimum tool capacity requirements to meet the cumulative demand from periods 1 through t. Then K t is simply the dierence between the cumulative capacity requirement and cumulative tool capacity available. Problem CTR t : min s:t: M N b ij X ij =1 i=1 M X tij =1 i=1 =1 X ij 0 i; j: V j j; Then, K t = max 0; =1 M b zij X zij N zi : z=1 i=1 The solution procedure for Problem CTR t, t =1;:::;T is as follows: for each period =1;:::;t assign the whole volume of operation j to the fastest tool group i where it consumes least amount of resources regardless of the tool group s capacity but subject to its capability. Identify the number of additional tools required in each tool group i. Round up the dierence between the total number of tools needed and total number of tools currently available. Proposition. Constraints (10) and P p=1 =1 are equivalent. (t +1)Q p K t t (11) The proof of this proposition is straightforward and is omitted. Our solution method is based on the Lagrangean relaxation of the capacity constraints (8) and (9) and cumulative tool lower bounding constraints (11). The formulation of the relaxed problem is as follows:

10 1358 B. C atay et al. / Computers & Operations Research 30 (2003) Problem MPCAP-LR(; ): min M (c tij tij + ti b tij )X tij + N h tj I tj + P F tp Q tp i=1 p=1 P tp Q p p=1 =1 s:t: (2); (5) (7); P t (t +1)Q p + C p=1 =1 (12) where ; 0 and C is constant: M C = ti N ti + i=1 t K t : The term involving Lagrangean multiplier tp and Q tp in the objective function can alternatively be expressed as follows: P tp Q p = p=1 =1 P p Q tp : p=1 =t Using a similar expression for the term with the Lagrangean multiplier, the objective function is reformulated as follows: M i=1 (c tij tij + ti b tij )X tij + p=1 [ P F tp h tj I tj ] ( p +( t +1) ) Q tp + C: (13) =t The Lagrangean problem MPCAP-LR(; ) nicely decomposes into two subproblems, one involving the X tij and I tj variables only and the other involving Q tp variables only. First problem further decomposes into J uncapacitated multi-period single-item production planning problems which can be easily solved by inspection. A procedure to solve this problem is given in Table 4. The second problem can also be solved by inspection: if the coecient of the integer variable Q tp is positive then Q tp is set to zero. Otherwise, Q tp would be innite. We impose an upper bound for Q tp to avoid an unbounded solution. We use a subgradient optimization procedure to update the Lagrangean multipliers. Given initial values of 0 and 0, we generate a sequence of Lagrangean multipliers k and k through the addition of a direction vector which is multiplied by a step size k where k is a positive scalar. Hence, the updating of Lagrangean multipliers of the constraint sets (8) and (9) is done according

11 B. C atay et al. / Computers & Operations Research 30 (2003) Table 4 Solution procedure for uncapacitated multi-period single-item production planning problems set all X tij;i tj to zero for t =1;:::;T do i = argmin i{c tij tij + tib tij} set X ti j = V tj end for for t = T;:::;2 do for =(t 1);:::;1 do nd i and k such that X ti j 0 and X k j 0 if {c ti j ti j + ti b ti j c k j k j + k b k j + t 1 z= else = 1 end for if ( 0) do for z = ;:::;t 1 do I zj = I zj + X ti j end for X k j = X k j + X ti j X ti j =0 I tj = I (t 1)j V tj end if end for hzj} forbreak to the equations: ts k = max tp k = max 0; k 1 ts 0; k 1 tp + k b tsj X tsj N ts + k b tpj X tpj N tp =1 t; s; Q p t; p and Lagrangean multipliers of the constraints (11) is updated according to the equation: t k = max 0; k 1 t + k K t P p=1 =1 (t +1)Q p The step size k is updated at each iteration k using the following equation: k = k (UBk 1 LB( k 1 ; k 1 )) ; t

12 1360 B. C atay et al. / Computers & Operations Research 30 (2003) where = + 2 S b tsj X tsj N ts + s=1 p=1 P b tpj X tpj N tp K t =1 Q p 2 : P p=1 =1 2 (t +1)Q p The step size k depends on the parameter k, on the gap between the current lower bound LB( k 1 ; k 1 ) and the estimated minimum value of the objective function of the relaxed problem, which is approximated by the upper bound UB obtained by applying a heuristic method, and on the Euclidean norm of the deviations in the relaxed constraints (8), (9), and (11). The sequence k is determined by setting 0 =0:025 and by dividing k by 1.5 whenever LB( k 1 ; k 1 ) does not increase after a xed number of iterations. We terminate this procedure when one of the following stopping criteria is met: 1. An iteration number limit, 2. Maximum gap between the lower and upper bounds, 3. A limit on the value of the Euclidean norm of the deviations. MPCAP-LR(; ) has the Integrality Property, that is the optimal solution does not change if we drop the integrality restriction on Q tp. Thus, Lagrangean relaxation cannot provide better bounds than LP relaxation. However, we nd it promising to use Lagrangean relaxation since the problems we consider are of very large scale and Lagrangean relaxation can provide lower bounds substantially faster than the standard LP relaxation Computation of the upper bound At each iteration of the Lagrangean relaxation the upper bound is computed by rst modifying the solution to obtain feasibility and then by applying a procedure to improve the bound. Since the Lagrangean relaxation gives a solution that is feasible with respect to the inventory balance constraints, we must ensure that the production quantities do not lead to over usage of the secondary tools. With respect to the resource constraints, a feasible production plan can easily be created by shifting operations from overloaded secondary tool groups to capacity exible primary tool groups, subject to their capability. The selection of operations to be shifted is done by sorting the tool group operation pairs at each period in non-decreasing order of the product of consumption and operating cost coecients and then choosing the candidate tool group operation pair following that sequence. Note here that since another secondary tool group may as well be underutilized in the original plan and therefore may be included in the candidate list. We continue performing this shifting procedure until the tool requirements in all overloaded secondary tool groups at all periods are at their respective capacity levels. An operation quantity may be partially reassigned if it is not necessary to shift the whole production volume to bring the total resource consumption in certain tool group to capacity.

13 B. C atay et al. / Computers & Operations Research 30 (2003) Table 5 Solution procedure to obtain the upper bound for t =1;:::;T do sequence {ctij tij} in non-decreasing order end for for t =1;:::;T do for s =1;:::;S do while (MachReqd ts AvailMach ts) do following the order of the sequence: shift X tsj (or a fraction of X tsj) from MachGroup s to MachGroup a i if capacity and process capability permit, update MachReqd ts and MachReqd ti endwhile end for end for for t =1;:::;T do for p =1;:::;P do while (MachReqd tp AvailMach tp + MachUB tp) do following the order of the sequence: shift X tpj (or a fraction of X tpj) from MachGroup p to MachGroup i period t, if capacity and process capability permit update MachReqd ti and MachReqd ti end while end for end for a i is the tool group index in the ordered {c tij tij}. in The procedure is described in Table 5. MachUB indicates an arbitrary positive upper bound on the number of additional tools imposed to avoid an unbounded solution. After a feasible production plan is constructed, in the second stage we try to improve the solution by means of operation transfers between primary tool groups within the same period or dierent periods. Since the number of tools are rounded up to integer value, there may exist underutilized tool group or groups with slack capacity. Our goal here is to attempt to balance the workload using the slack capacity of one or more tool groups and saving a tool from another tool group. If this is done between dierent periods, an operation must be transferred from a period to an earlier period to prevent backorders. The procedure is performed in a similar fashion using the above mentioned sequence of tool group operation pairs starting with the current period and then proceeding with the preceding periods in an attempt to avoid excess inventories. In the following section, we present a computational study of randomly generated data sets to evaluate the performance of our solution method. 4. Experimental analysis The heuristic solution procedure is coded in C programming language. A series of computational experiments was carried out on a PC with 300 MHz Pentium II processor using several sets of ran-

14 1362 B. C atay et al. / Computers & Operations Research 30 (2003) Table 6 Experimental design Parameters Values used Total Procurement cost (10; 11); (10; 14); (10; 17); (10; 20) 4 Operating cost a (0:1; 0:3); (0:1; 0:7); (0:1; 1:1); (0:1; 1:5); (0:1; 1:9) 5 Utilization (0:65; 0:75); (0:65; 0:85); (0:65; 0:95) 3 Number of periods 20; 40 2 Number of tool groups=operations 10=200 15=300 20=400 25=500 30=600 Tool group structures b 29 Total number of problems 3480 a Variable cost is a fraction of the procurement cost. b 5 for 10=200 and 15=300, 6 for 20=400 and 25=500, and 7 for 30=600. domly generated problems of diering sizes. The performance of the procedure is evaluated using various combinations of cost, utilization, and primary=secondary tool group structures. Procurement cost coecient F tp, operating cost coecient c tij, and utilization U ti are drawn from uniform distributions using the parameters shown in Table 6. The operating cost is specied as a percentage of the procurement cost, i.e. operating cost is on the average 20% of the procurement cost using the uniform distribution U(0:1; 0:3), 40% of the procurement cost using U(0:1; 0:7), etc. We use a yearly interest rate of 8% to calculate the inventory holding cost h tj. F tp and c tij are generated for the rst month and discounted for succeeding months. In addition to these cost parameters, we also consider three uniform distributions for capacity utilization ratios since these are likely to impact the number of additional tools that will be needed. U ti is also generated for the rst month and then decreased using a certain constant ratio for the following months considering decline in the utilization of tools over time. Production volumes and production capacities are drawn from uniform distributions U(1; 600) and U(50; 1500), respectively. The number of operation types repeated throughout the manufacturing of the wafer is also generated from a uniform distribution U(2; 6). Without loss of generality, we assume 100% yields and zero initial inventories. We tested our procedure on ten problem types over a wide range of values of number of tool groups and operations. Each problem type is solved for dierent primary=secondary tool group structures with the same input data. The smallest problem we consider consists of 44,340 constraints, 44,060 variables, of which 60 are integers and the largest problem has 747,160 constraints, 744,960 variables, of which 960 are integers. A summary of the results for all of the problems considered is presented in Table 7. In this table, we report the % gaps between the lower bound (obtained using Lagrangean relaxation) and the upper bound (obtained using the heuristic) to judge the quality of the solutions. As can be seen, the gap is 8:1% on the average in all 20-period problem instances and 5:5% on the average in all 40-period problem instances. For the larger problems we were able to obtain better solutions. This is due to the fact that as the problem gets larger LP becomes a better approximation, the tool procurement decisions become less crucial and do not drastically aect the solution value, hence the gap gets smaller. We were able to obtain fairly good solutions quickly for large problems. It should be noted that an attempt was made to solve some problem instances of 40 periods, 500 operations, and 25 tool groups using the CPLEX J optimization library, and it took more than 80 min to obtain the optimal solution to the LP relaxation of the problem.

15 B. C atay et al. / Computers & Operations Research 30 (2003) Table 7 Summary of results showing the gaps between upper and lower bounds Tool groups=operations 10=200 15=300 20=400 25=500 30= Periods Best Average Worst Periods Best Average Worst (We were unable to solve larger problems due to computer memory limitations.) The average CPU time of all problem instances of the same structure using our solution procedure is 337 s. We have not attempted to solve the mixed-integer programming problem, for it is not practically possible to obtain a solution within a reasonable computing time. More detailed results of the performance of the solution procedure focusing on the cost and utilization parameters are reported in Table 8. This table also indicates the average computing times in seconds. The primary reason for presenting these results is to examine the inuence of these parameter settings on the quality of the solutions. It should be observed from this table that as utilization increases, % gap decreases, regardless of the procurement and operating costs. This result can be explained by noting that a decrease in the tool utilization tends to increase the need for additional tools and thus, the solution quality declines since the impact of the procurement decision becomes more signicant. Regardless of the procurement costs and utilization levels, the table noticeably indicates that % gap decreases when the operating cost increases. This may also be related to the weakened eect of the procurement costs in the total cost function. Finally, no discernible impact of the procurement costs can be observed from these results. All computing times were found to be stable and well within the acceptable limits. These results also support our earlier observation on the improving quality of the solution as the problem becomes larger. 5. Conclusions and limitations In this paper, we presented a mathematical model for the multi-period tool capacity planning in semiconductor manufacturing. A Lagrangean-based heuristic solution procedure was also introduced. The computational experiments indicate that our solution procedure produces good feasible solutions with reasonable computation times. Our method may be applied to production and capacity planning problems of other industries where resource allocation and new equipment acquizition decisions are made simultaneously. Demand for new equipment arises primarily from two sources: replacement of existing aged equipment and additional requirements for new equipment due to the introduction of new products and the growth in demand for the existing products. The model discussed here can also be extended to consider all processes in the manufacture of a computer chip by including constraints between consecutive process steps to guarantee consistency of

16 1364 B. C atay et al. / Computers & Operations Research 30 (2003) Table 8 Mean gaps and CPU times of all problem types Operating cost (0:1; 0:3) (0:1; 0:7) (0:1; 1:1) (0:1; 1:5) (0:1; 1:9) Utilization Procurement Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean cost GAP CPU GAP CPU GAP CPU GAP CPU GAP CPU time time time time time (0:65; 0:75) (10; 11) (10; 14) (10; 17) (10; 20) (0:65; 0:85) (10; 11) (10; 14) (10; 17) (10; 20) (0:65; 0:95) (10; 11) (10; 14) (10; 17) (10; 20)

17 B. C atay et al. / Computers & Operations Research 30 (2003) production volumes. Furthermore, backorders may also be considered in that context. The approach could also be extended to investigate replacement of capacity as well as expansion and disposal of equipment to respond to arbitrary changes in demand. A limitation of the model discussed in this paper is that it assumes deterministic technological changes, i.e., all technologies available in the future are assumed to be known at the beginning. Other limitations include assumptions of space availability in the facilities for additional equipment and accurate forecasts of future costs and demands. References [1] Toktay BL, Uzsoy R. A capacity allocation problem with integer side constraints. European Journal of Operational Research 1998;109(1): [2] Sullivan G, Fordyce K. IBM Burlington s logistics management system. Interfaces 1990;20(1): [3] Golovin JJ. A total framework for semiconductor production planning and scheduling. Solid State Technology. 1986; [4] Bitran GR, Hass EA, Hax AC. Hierarchical production planning: a single stage system. Operations Research 1981;29(4): [5] Bitran GR, Hass EA, Hax AC. Hierarchical production planning: a two stage system. Operations Research 1982;30(2): [6] Leachman RC. Modelling techniques for automated production planning in the semiconductor industry. In: Ciriani TA, Leachman RC, editors. Optimization in industry. New York: Wiley, [7] Leachman RC, Carmon TF. On capacity modelling for production planning with alternative tool types. IIE Transactions 1992;24(4): [8] Hung YF, Wang QZ. A new formulation technique for alternative material planning an approach for semiconductor bin allocation planning. Computers and Industrial Engineering 1997;32(2): [9] Uzsoy R, Lee CY, Martin-Vega LA. A review of production planning and scheduling models in the semiconductor industry. Part I. system characteristics, performance evaluation and production planning. IIE Transactions 1992;24(4): [10] Uzsoy R, Lee CY, Martin-Vega LA. A review of production planning and scheduling models in the semiconductor industry. Part II. shop-oor control. IIE Transactions 1994;26(5): [11] Fordyce K, Sullivan G. A dynamically generated rapid response capacity planning model for semiconductor fabrication facilities. In: Nash SG, Sofer A, editors. Impact of emerging technologies on computer science and operations research. Dordrecht: Kluwer Publishing, p [12] Swaminathan JM. Tool capacity planning for semiconductor fabrication facilities under demand uncertainty. European Journal of Operational Research 2000;120(3): [13] Swaminathan JM. Tool procurement planning for wafer fabrication facilities: a scenario-based approach. IIE Transactions 2002;34: [14] Georion AM. Lagrangean relaxation for integer programming. Mathematical Programming Study 1974;2: Bulent C atay is Assistant Professor in the Faculty of Engineering and Natural Sciences at Sabanci University. He has a B.Sc. degree in Industrial Engineering from Istanbul Technical University and a Ph.D. in Production and Operations Management from the University of Florida. His current research interests include production and capacity planning, logistics and supply chain management. S. Selcuk Erenguc is PricewaterhouseCoopers Professor and Chairman of the Decision and Information Sciences Department in the Warrington College of Business Administration, at the University of Florida. He holds a DBA degree from Indiana University. His current research interests include project management and scheduling and supply chain management. Dr. Erenguc has served on several editorial boards. His publications have appeared in many journals including Computers and Operations Research, Decision Sciences, International Journal of Production Research, Journal of

18 1366 B. C atay et al. / Computers & Operations Research 30 (2003) Operations Management, Journal of Optimization Theory and Applications, IIE Transactions, Management Science, Naval Research Logistics, Operations Research, Operations Research Letters and Production and Operations Management. Asoo J. Vakharia is the Beall Professor of Supply Chain Management in the Department of Decision and Information Sciences in the Warrington College of Business Administration at the University of Florida. He has a Bachelor s degree in Accounting and Economics from Bombay University, an MBA from the University of Wisconsin-Whitewater and a Ph.D. in Operations Management from the University of Wisconsin-Madison. His current research focuses on coordinating marketing and operations decisions in supply chains, design and control of cellular manufacturing systems and the development of strategic and tactical models for call center operations. His prior research has been published in Annals of OR, Decision Sciences, European Journal of Operational Research, IIE Transactions, International Journal of Flexible Manufacturing Systems, International Journal of Production Research, Journal of Operations Management, and Naval Research Logistics. He serves as an Associate Editor for the International Journal of Flexible Manufacturing Systems and is on the Editorial Review Board for the Journal of Operations Management and the Production and Operations Management Journal.