A multiple criteria decision for trading capacity between two semiconductor fabs

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A multiple criteria decision for trading capacity between

two semiconductor fabs

Muh-Cherng Wu

*

_{, Wen-Jen Chang}

1

Department of Industrial Engineering and Management, National Chiao Tung University, Hsin-chu, Taiwan, ROC

Abstract

This paper presents a multiple criteria decision approach for trading weekly tool capacity between two semiconductor fabs. Due to the high-cost characteristics of tools, a semiconductor company with multiple fabs (factories) may weekly trade their tool capacities. That is, a lowly utilized workstation in one fab may sell capacity to its highly utilized counterpart in the other fab. Wu and Chang [Wu, M. C., & Chang, W. J. (2007). A short-term capacity trading method for semiconductor fabs with partnership. Expert Systems with Application, 33(2), 476–483] have proposed a method for making weekly trading decisions between two wafer fabs. Compared with no trading, their method could effectively increase the two fabs’ throughput for a longer period such as 8 weeks. However, their trading decision-making is based on a single criterion—number of weekly produced operations, which may still leave a space for improving. We therefore proposed a multiple criteria trading decision approach in order to further increase the two fabs’ throughput. The three decision criteria are: number of operations, number of layers, and number of wafers. This research developed a method to find an optimal weighting vector for the three criteria. The method firstly used NN + GA (neural network + genetic algorithm) to find an optimal trading decision in each week, and then used DOE + RSM (design of experiment + response surface method) to find an optimal weighting vector for a longer period, say 10 weeks. Experiments indicated that the multiple criteria approach indeed outperformed the previous method in terms the fabs’ long-term throughput.

Keywords: Capacity trading; Semiconductor; Neural network; Genetic algorithm; Design of experiment; Response surface method

1. Introduction

The manufacturing of semiconductor products is a long-route process. A semiconductor product, called a wafer, involves hundreds of operations. These operations are gen-erally grouped into tens of layers. The tools for semicon-ductor manufacturing are capital-intensive. A tool may cost up to ten millions of dollars. A typical wafer fab (a semiconductor factory) involves hundreds of tools, which as a whole may cost over 1 billion dollars. Therefore, eﬀec-tive management of tool capacity has been a signiﬁcant research issue.

Traditionally, eﬀective management of tool capacity has been investigated from three perspectives—in terms of deci-sion horizon. From long-term perspective, some literature

addressed the tool planning problem (Wu, Erkoc, &

Kara-buk, 2005); that is, how to purchase tools to fulﬁll a long-term forecasted demand that typically ranges from one to

several years (Bard, Srinivasan, & Tirupati, 1999; C¸ atay,

Erengu¨c¸, & Vakharia, 2003; Christie & Wu, 2002; Connors, Feigin, & Yao, 1996; Grewal, Bruska, Wulf, & Robinson, 1998; Hood, Bermon, & Barahona, 2003; Swaminathan, 2000, 2002; Wu, Hsiung, & Hsu, 2005). From medium-term perspective (ranging from one to several quarters), research-ers addressed the product mix planning problems, which examined how to select customer orders to optimize

the use of tool capacity (Bermon & Hood, 1999; Chou &

Hong, 2000; Chung, Lee, & Pearn, 2003; Chung, Lee, & Pearn, 2005). From short-term perspective, the issues of

*

Corresponding author. Tel.: +886 3 5731913; fax: +886 3 5720610. E-mail addresses: mcwu@cc.nctu.edu.tw (M.-C. Wu), wjchang6@ ms39.hinet.net(W.-J. Chang).

1 _{Tel.: +886 5927700x2954; fax: +886 3 926848.}

www.elsevier.com/locate/eswa Expert Systems with Applications 35 (2008) 938–945

Expert Systems with Applications

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production planning (Odrey, Green, & Appello, 2001;

Chen, Chen, Lin, & Rau, 2005) and shop ﬂoor control (Chen & Chen, 1996; Kuroda, Tomita, & Maeda, 1999; Mo¨nch & Drießel, 2005) are addressed, which attempt to eﬀectively use the tool capacity to produce the committed orders. Most literature based on the three perspectives essen-tially dealt with the tool capacity problem of a single fab.

A recent track on capacity planning turned to the mutual support of tools among diﬀerent fabs. A semicon-ductor company usually has its fabs built in a cluster; that is, the fabs are close in their physical locations. This pro-vides opportunities for tool capacity support among fabs.

Deboo (2000) and Toba et al. (2005)took several diﬀerent fabs as a single big one; a product before its completion may travel through more than one fab. They developed a dispatching algorithm for dynamically planning the route of a product. This approach implicitly assumed the adop-tion of a central management system. That is, each fab is not an autonomy unit; therefore, the performance of each individual fab may not be easily identiﬁed.

Managing each fab based on individual performance is undoubtedly important in motivating diligent work. A company may manage its multiple fabs by a de-centralized management system. That is, each fab operates in an autonomy manner and supports each other by trading tool

capacity periodically. Wu and Chang (2007) proposed a

method to ﬁnd a weekly trading portfolio in order to max-imize the two fabs’ throughput in a longer period such as 8 weeks. They used total number of weekly completed operations as the criterion for ﬁnding the trading portfolio. Experiments indicated that such a weekly trading criterion indeed would increase the longer-period fab throughput.

However, our analysis indicates that the capacity

trad-ing criterion proposed by Wu and Chang (2007) still left

a space for improving. Typically, a semiconductor product involves hundreds of operations, grouped into tens of lay-ers. The productivity of a fab could be measured in three indices—number of operations, number of layers, and number of products (known as throughput). Intuitively, maximizing the output volume of operations would maxi-mize that of layers, in turn that of products. However, a layer is a combination of speciﬁc operations; therefore, producing a higher number of operations may not lead to producing a higher number of layers if many operations are unable to be aggregated into a layer. This implies that the longer-period fabs’ throughput could be possibly increased by adopting a new capacity trading criterion.

This paper proposes the use of a multiple criteria in the decision of weekly capacity trading between two fabs. Three trading criteria are considered: number of operations, num-ber of layers, and numnum-ber of products. To integrate these three criteria, a weighting for each has to be determined. Different weighting assignments lead to different trading outcomes, in turn different longer-period throughputs. We therefore attempt to identify an optimal weighting vector for the three trading criteria in order to maximize total profit of the two fabs of a longer period, such as 10 weeks.

The research framework involves two modules. The ﬁrst module, for given a weighting vector of criteria, aims to determine its optimal weekly trading portfolio. That is, we attempt to ﬁnd a weekly trading portfolio that would

maximize a1Æ O+ b1ÆL+ c1Æ W, where [a1, b1, c1] denotes

the given weighs of criteria and O, L, and W respectively denotes the number of operations, layers, and wafers in that week. The second module attempts to ﬁnd an optimal

weighting vector [a*_{, b}*_{, c}*_{] in order to maximize the total}

proﬁt of the two fabs—which would yield after the two fabs have traded capacity for a longer period, say 10 weeks. These two modules have been implemented and tested by numerical examples. Experiments indicated that the pro-pose approach indeed outperform the previous work. 2. Module 1—ﬁnding weekly trading portfolio

The first module is to find a weekly trading portfolio that would maximize the fabs’ weekly aggregated output. The aggregated output involves three components: number of operations, number of layers, and number of wafers, with the weights for integrating the three components being given. The development of this module involves three steps: (1) how to define the solution space of trading portfolios, (2) how to evaluate the performance of each trading port-folio, and (3) how to find a near-optimal solution.

2.1. Deﬁne solution space

In practice, a semiconductor fab comprises dozens of workstations; each workstation involves a number of func-tionally identical tools. For any two workstations in diﬀer-ent fabs, we could trade their capacity if the two are functionally identical. Such a pair of workstations is called a tradable pair of workstations. Consider a case where the two fabs have m tradable pairs. The solution space of

trad-ing portfolios would include N ¼Qm_i¼1ðBiþ 1Þ elements,

where Bi denotes the maximum number of trading units

for a tradable pair i; that is, its trade option could range

from 0 to Biunits.

For a tradable pair i, we determine that Bi¼ round up

ð1

2ujqs;i qb;ij Qs;i T Þ, where u denotes the basic trading

units; qs,i denotes the utilization of the workstation that

sells capacity; qb,idenotes the utilization of the workstation

that buys capacity; Qs,idenotes the number of tools in the

workstation that sells capacity; and T denotes the number of working hours per week. Consider a case with u = 20 h,

qs, i= 60%, qb,i= 90%, Qs,i= 5 tools, and T = 168 h. Then

Bi= round_up(6.3) = 7 trading units. That is, the two

workstations can trade at most 140 h.

Notice that the qs,iand qb,irefer to the workstation

uti-lizations in the coming week while there is no capacity trad-ing applied. The workstation utilizations are estimated by carrying out a discrete-event simulation. The simulation is a deterministic model; that is, the daily uptime of each tool is assumed to be a constant rather than stochastic. The adoption of this assumption is based on the simulation

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ﬁndings by Kim, Shim, Choi, and Hwang (2003) whose experiment results advocated the use of deterministic simu-lation model in predicting a fab’s short-term behavior such as one week. The deterministic simulation program is

abbreviated Det_Sim(Si, Tk), where Si denotes the initial

status of the two fabs at week i, and Tkdenotes the trading

portfolio applied at week i. In evaluating qs,i and qb,i, we

use Det_Sim(Si,T0), where T0 denotes a no-trading

decision.

2.2. Evaluate performance of trading portfolios

The performance of a trading portfolio is evaluated by

the fabs’ aggregated output I = a1Æ O+ b1Æ L+ c1Æ W

where O, L, and W respectively denotes the total number of operations, layers, and wafers produced by the two fabs;

and [a1, b1, c1] denotes the predeﬁned weights for the three

components. The higher the value of I, the more preferable is the trading portfolio.

As stated, the performance of a trading portfolio can be estimated by carrying out a deterministic simulation pro-gram. According to our experiments, one such estimation by using a typical personal computer (PC) takes about 40 s in computation. Consider a case where the fabs have four tradable workstations, each of which has at most 15 trading units. Then, the total number of trading portfolios

would be (15 + 1)4= 65,536, requiring about 30 days.

This research proposed two methods to reduce the com-putation time. Firstly, the neural network (NN) technique is used to emulate the function of simulation, which could yield results in a much speedy way. Experiments indicated that, by the NN technique, the computation time per esti-mation would be much less than one second. Secondly, we used the genetic algorithm (GA) technique to reduce the number of estimations; that is, trading portfolios are not exhaustively evaluated—only a limited number of ‘‘seem-ingly good’’ trading portfolios are tested.

Consider the simulation program as a transformation mechanism. Then, the input is a trading portfolio and the output involves three components (O, L and W). The application of NN technique is to construct an input/output mapping to emulate the function of the sim-ulation program. To do so, we ﬁrstly sample K trading portfolios and compute their performances by simulation. The obtained K pairs of input/output vectors are then used to construct a neural network for each fab by the

back-propagation algorithm (Fausett, 1994; McClelland

& Rumelhard, 1988; Rumelhart, Hinton, & Williams, 1986).

This algorithm would establish a non-linear mapping between the input/output vectors. That is, given an input vector, the neural network through the non-linear mapping could compute the output of each fab speedily. The neural network is called well-trained if its projected output vectors are close to those obtained from the simulation; typically their degree of discrepancies is measured by RMSE (root mean squared errors). A well-trained neural network could

then be interpreted as a ‘‘faster simulator’’. By constructing a neural network for each fab, we could then speedily com-pute the aggregated output of the two fabs.

2.3. Find a near-optimal solution

With a well-trained neural network, we used the GA technique to find a near-optimal trading portfolio from the huge solution space. The GA technique has been widely used in solving a large space search problem and has been justified in its effectiveness in finding a near-optimal

solu-tion (Mitchell, 1998). The GA used in this research is

brieﬂy described below.

Each trading portfolio is modeled as a string of numer-als, called chromosome. Each number in the string, called gene, represents a trading option for a particular type of workstation. The performance of a trading portfolio is called the ﬁtness of the chromosome. Changing the gene values of a existing chromosome leads to the creation of a new one. Methods for such a chromosome creation are called genetic operators.

The solution search mechanism of the GA is by evolving a population of chromosomes, which is limited in number during the evolution. An evolution generation means that the population has been updated once. In the population updating process, some chromosomes are replaced by ‘‘seemingly good’’ new ones, which are created by genetic operators. The genetic operators involve three types: repro-duction, crossover, and mutation. The population updat-ing process terminates when either the best solution cannot be improved further or the population has evolved over a predeﬁned number of generations. And the best chromosome in the ﬁnal population is the trading portfolio provided by the GA.

2.4. An integrated procedure

The aforementioned steps of the ﬁrst module can be summarized below by a procedure called Find_Weekly_

Trading([a,b, c],Si), where [a, b, c] is the given weighting

vector and Sidenotes the initial status of the two fabs at

week i—the particular week for making the trading deci-sion. The output of the procedure is denoted by

Q_iða; b; cÞ, which represents the optimal trading portfolio

at week i.

Procedure Find_Weekly_Trading([a, b, c],Si)

Step 1: Use Det_Sim(Si,T0) to estimate the utilization of

each tradable pair of workstations

Step 2: Deﬁne the solution space N of trading portfolios Step 3: Establish a neural network to emulate Det_Sim

• Randomly sample K trading portfolios

• For each of the K trading portfolios, use Det_

Sim(Si,Tk) to compute its weekly outputs of O,

L, W in each fab

• Construct a neural network for each fab • Aggregate the projected outputs of the two fabs

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Step 4: Use GA to search an optimal trading portfolio

Q_iða; b; cÞ from solution space N.

3. Module 2—select optimum weights for trading criteria The ultimate purpose of trading tool capacity between fabs is to make their long-term total profits as higher as possible. In the first module, we have presented how to find a weekly optimum capacity trading decision based on a particular weighting vector [a, b, c]. This implies that the use of different weighting vectors may yield different total profits. Therefore, how to find an optimum weighting vec-tor is important.

The second module therefore attempts to ﬁnd an optimal

weighting vector [a*_{, b}*_{, c}*_{] for aggregating the three trading}

decision criteria—number of operations (O), number of lay-ers (L), and number of waflay-ers (W) in order to maximize the total proﬁt of the two fabs, from a longer-period perspective (T weeks). That is, making trading decisions successively for

T = 10 weeks based on the vector [a*_{, b}*_{, c}*_{] would}

maxi-mize the 10 weeks total proﬁt of the two fabs.

This module involves two procedures. The first proce-dure is to compute the T weeks total profit of the two fab, while the weighting vector [a, b, c] has been given. The second procedure is to find an optimum weighting

vec-tor [a*_{, b}*_{, c}*_{] from the set S = {[a, b, c]ja + b + c = 1,}

0 6 a, b, c 6 1}.

3.1. Compute total proﬁt in T weeks

The procedure to compute the T weeks total profit for a given weighting vector [a, b, c], called Compute_Profit (a, b, c), is described below with its notation firstly introduced.

Notation

[a, b, c] the given weighting vector

Q_iða; b; cÞ ¼ ½w1; w2; . . . ; wk an optimum trading portfolio

at week i, where widenotes the number of trading

units for tradable workstation i and k denotes the number of tradable pairs

Si the initial status of the two fabs at week i, which

describes the WIPs before each workstation and the up-or-down status of each machine

Fi the ﬁnal status of the two fabs at week i, which

de-scribes the WIPs before each workstation and the

up-or-down status of each machine; i.e.,Fi= Si+1

Vi= [v1, v2, . . . , vm] total produced volumes of each

prod-uct in the two fabs at week i, where m is the num-ber of product types

Ri= [r1, r2, . . . , rm] contribution margin of each product at

week i, where m is the number of product types

Pi total proﬁt of the two fabs at week i

P(a, b, c) total proﬁt of the two fabs in T weeks

Procedure Compute_Proﬁt(a, b, c)

Step 0: Initialization, input [a, b, c] and S1

For i = 1, . . . , T

Step 1: Use the procedure Find_Weekly_Trading([a, b, c],

Si) to compute Qiða; b; cÞ

Step 2: Use a stochastic simulation program Sto SimðSi;

Q_iða; b; cÞÞ to estimate Si+1and Viafter

implement-ing the tradimplement-ing portfolio Q_iða; b; cÞ

Step 3: Compute total proﬁt Pi¼ Vi RTi at week i

Endfor

Step 4: Compute the total proﬁt of the T weeks Pða; b; cÞ ¼

PT

i¼1Pi

In Step 2 of the above procedure, the stochastic

simula-tion program Sto_Sim(Si, Qiða; b; cÞ) is used for predicting

the behavior of the two fabs after carrying out a capacity

trading portfolio; that is, it takes Q_iða; b; cÞ and Sias input

and generates Si+1and Vias output, which could be used to

obtain the total proﬁt P(a, b,c).

Notice that in evaluating the longer-period (say, T = 10 weeks) performance of a trading portfolio, we have to use a stochastic simulation rather than a deterministic sim-ulation. This is to reﬂect the unpredictable nature of a real wafer fab and attempts to justify the robustness of a weekly optimum trading portfolio.

3.2. Find an optimum weighting vector

We now discuss how to ﬁnd an optimum weighting vector

[a*_{, b}*_{, c}*_{] from the set S = {[a,b, c]ja + b + c = 1, 0 6 a,}

b, c 6 1} in order to maximize the total profit of the T weeks. The techniques of mixture design of experiments (DOE) and response surface method (RSM) are used. That is, we sample a number of weighting vectors, compute their total profits, and use the obtained results to construct a response surface, which is polynomial function that could be used to quickly estimate the total profit for each [a, b, c] in S.

Using an appropriate experiment design model is very important in eﬀectively predicting the behavior of a system. In our experiment, the three factors are imposed by a con-straint a + b + c = 1 and are not independent. Therefore,

the simplex centroid design (Montgomery, 1991) is used for

the three mixture components. As shown inTable 1, 10 design

points are selected in the experiment; the total proﬁt P(a, b, c) for each design point could be computed by applying the pro-cedure Compute_Proﬁt(a, b, c). The 10 pairs of [(a, b, c),P] are then used to construct the response surface. With the response

surface, an optimum weighting vector [a*_{, b}*_{, c}*_{] could then be}

easily identiﬁed by analytical approaches such as the gradient

method (Myers & Montgomery, 1995).

3.3. Rationales for choosing optimization techniques Notice that in the ﬁrst module we use the technique of NN + GA in ﬁnding an optimal weekly trading portfolio.

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In contrast, in the second module, we use the technique of DOE + RSM to ﬁnd an optimal weighting vector. A ques-tion may arise: why two diﬀerent techniques are used in searching an optimum solution?

By its very nature, the technique of DOE + RSM requires much less number of sampled solutions (only 10 points in this application) in constructing the response sur-face. Therefore, it is more suitable for cases with a longer computation time per solution evaluation, whereas the technique of NN + GA is more suitable for cases with a shorter computation time per solution evaluation.

In this research, the second module attempts to evaluate the performance of a weighting vector [a, b, c]. That is, for a given weighting vector [a, b, c], we need to compute the optimum trading portfolio for each of the T weeks. This would become quite time-consuming if we attempt to eval-uate a large number of weight vectors [a, b, c]. To limit the number of sampled weighting vectors, the technique of DOE + RSM is thus used in the second module.

4. Numerical example

The proposed approach has been tested by a numerical

example. Table 2 shows the number of product mixes,

number of workstations, contribution margin of each product, and some other features of process routes in the two trading fabs, where a type of product is denoted by xPyM which speciﬁes that the product has x poly-layers

and y metal layers (Xiao, 2001). The contribution margin

of each product and the process routes are disguised from those provided by industry. And we assumed that only four types of workstations are tradable between the two fabs.

The two simulation programs Det_Sim and Sto_Sim are

coded by using eM-plant (Tecnomatix Technologies Ltd,

2001). The experiment design and response surface

meth-ods are carried out by using MINITAB (Minitab Inc.,

2003). The neural network and the GA programs are coded

by using C programming language.

In the example, the two fabs are designed to trade capac-ity from week 1 to week 10 (i.e., T = 10). The fabs’ initial

status (S1) is generated by Sto_Sim, which starts from an

empty fab until comes to a steady state by uniformly releas-ing wafers based on the given product mixes.

In constructing an NN for modeling a fab’s behavior for a particular week, we sampled K = 2000 trading portfolios. Then we use Det_Sim to compute the weekly fab output of each trading portfolio. Subsequently, the input/output of the 2000 trading portfolios are used to construct a neural network for the week. By using a typical PC (personal com-puter), it takes about 40 s for carrying out the simulation once. We used 50 PCs in a lab to do the computation. That is, 40 simulation runs have to be performed on a PC and takes about 26.7 min. With the 2000 data being available, the computation time for constructing the neural network is much faster, about less 1.5 min by using only one PC. Likewise, the computation eﬀort for the GA to obtain

Q_iða; b; cÞ takes less than 1.5 min. With the Qiða; b; cÞ being

available, we used Sto_Sim to compute the initial fab status of week i + 1, which takes about 1.0 min in computation.

Therefore, the computation time for running simula-tions, constructing a neural network, carrying out the GA search, and updating the fabs’ initial status of next week takes about 30 min in total. That is, with a computation facility of 50 PCs, the two fabs can make their weekly trad-ing decision in 30 min, if the weighttrad-ing vector of the decision criteria [a, b, c] has been given. In practice, half an hour deci-sion-making is acceptable for a weekly trading decision.

In contrast, the computation time for ﬁnding an optimal

weighting vector [a*_{, b}*_{, c}*_{] is much more computationally}

extensive. For a given weighting vector [a, b, c], we have to compute the optimal weekly trading decisions for 10 weeks, which will take 5 h (0.5 h/week * 10 weeks) com-putation on a computing facility with 50 PCs. In the DOE, we have 10 weighting vectors to be evaluated, and this will take about 50 h computation—seemingly an astonishing value! However, the decision of an optimal weighting

vec-tor [a*_{, b}*_{, c}*_{] is not made weekly. A practical application}

maybe updates [a*_{, b}*_{, c}*_{] quarterly, which then takes 50 h}

of computation—seemingly acceptable to a semiconductor fab. This computation time surely can be reduced if more number of PCs is used.

Table 1

Simplex centroid design points for a mixture experiment

Design point a b c 1 1 0 0 2 0 1 0 3 0 0 1 4 1/2 1/2 0 5 1/2 0 1/2 6 0 1/2 1/2 7 1/3 1/3 1/3 8 2/3 1/3 1/3 9 1/3 2/3 1/3 10 1/3 1/3 2/3 Table 2

Tools and products for the two fabs

FAB Number of

workstations

Total number of tools

Product Total processing time (h) Total number of operations Contribution margin Fab_A 60 275 4P1M 400 358 65 1P7M 440 412 80 Fab_B 60 201 1P3M 318 276 50 1P8M 480 446 100

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Table 1shows the 10 weighting vectors selected for

con-structing the response surface. Table 3 shows the weekly

optimum trading portfolios for the 10 weeks while

[a, b, c] = [1, 0, 0].Table 4shows total proﬁt of the 10 weeks

for each of the 10 weighting vectors in the DOE. The

equa-tion of the response surface with R2= 0.852 is shown

below, where y denotes the total proﬁt divided 1000; that is, in unit of thousand dollars. The contour and

three-dimensional plots of the response are shown inFig. 1. A

flag in the contour plot identifies the region that maximizes the total profit of the two fabs

y¼ 209:53a þ 211:90b þ 213:19c þ 29:64ab þ 17:34bc

þ 25:32ac 25:32abc þ 17:30a2_b_17:30ab2

þ 59:25a2

c 59:25ac2

The embedded algorithm in MINITAB for searching an optimal solution from the response surface is then used to

ﬁnd [a*_{, b}*_{, c}*_{] = (0.63, 0.13, 0.24). As shown in}_{Table 4}_{, the}

total proﬁt obtained by using [a*_{, b}*_{, c}*_{] exceeds those}

obtained by the 10 sampled weighting vectors. In addition, the table also indicates that the decision of no trading is the

lowest in total proﬁt. This conﬁrmed the claim ofWu and

Chang (2007); that is, weekly trading tool capacity is

eﬀec-tive in generating more proﬁt. Compared toWu and Chang

(2007), this research further increases the total proﬁt by using an optimal weighting vector in weekly trading.

The extraordinary outcome of the proposed approach has two important implications. Firstly, it advocates the use of multiple criteria in trading weekly capacity between fabs. Secondly, a weekly performance evaluation of a

Table 3

Ten weekly optimal trading portfolios for [a, b, ,c] = [1, 0, 0]

Week Trading portfolio

[a, b, c] = [1, 0, 0] 1 (180, 240, 960, 320) 2 (180, 220, 780, 300) 3 (160, 240, 640, 280) 4 (160, 240, 600, 260) 5 (180, 220, 620, 300) 6 (140, 220, 560, 300) 7 (160, 200, 600, 300) 8 (180, 180, 640, 280) 9 (160, 200, 600, 300) 10 (160, 180, 680, 240) Table 4

The responses of 10 design points

Design point Criteria weights Throughput of each product Total proﬁt Total output

a b c 4P1M 1P7M 1P3M 1P8M 1 1 0 0 993 628 787 554 209,535 2962 2 0 1 0 1001 638 792 562 211,905 2993 3 0 0 1 1013 645 799 558 213,195 3015 4 1/2 1/2 0 1024 661 820 577 218,140 3082 5 1/2 0 1/2 1041 648 816 574 217,705 3079 6 0 1/2 1/2 1031 651 812 572 216,895 3066 7 1/3 1/3 1/3 1027 659 823 581 218,725 3090 8 2/3 1/3 1/3 1085 627 833 585 220,835 3130 9 1/3 2/3 1/3 1024 655 808 568 216,160 3055 10 1/3 1/3 2/3 1025 641 810 557 214,105 3033 No. trading – – – 882 569 689 488 186,100 2628 Optimal trading 0.63 0.13 0.24 1088 675 841 591 225,870 3195 A 0 1 B1 0 C 1 0 220.8 219.6 218.4 217.2 216.0 214.8 214.8 213.6 212.4 A = 0.628186 B = 0.132103 C = 0.239711 Total profit = 221.207 Contour Plot of Totalprofit

A 1 .0 0 0. 00 0. 00 210 B 215 y 220 1.00 0.00 1.00 C

Surface Plot of Total profit

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semiconductor fab may better be evaluated from three perspectives—number of operations, number of layers, number of wafers. Due to the long cycle time characteris-tics of semiconductor manufacturing, managers in prac-tice tend to evaluate a fab’s weekly performance by the number of produced operations, and evaluate its quarterly performance by the number of wafers produced. To align the short-term (week) and relative long-term (quarter) objectives, the weekly performance evaluation may need a multiple criteria such as proposed.

5. Concluding remarks

This paper presents a method for determining an

opti-mal weighting vector [a*_{, b}*_{, c}*_{] for a multiple criteria}

deci-sion on weekly trading tool capacity between two semiconductor fabs. The three decision criteria involve number of operations (O), number of layers (L) and num-ber of wafers (W). That is, if the two fabs make weekly trading decision by maximizing the aggregated output

a*_{O + b}*_{L + c}*_{W, their resulting long-term proﬁts will}

reach a maximum.

In the proposed method, we used the techniques of NN + GA to obtain an optimal trading portfolio for each week while the weighting vector [a, b, c] is given. The data set for constructing a NN is generated by a deterministic simulation program Det_Sim. The total proﬁt in the case of using a weighting vector [a, b, c] is estimated by a sto-chastic simulation program Sto_Sim. Finally, the tech-niques of DOE + RSM are used to ﬁnd an optimal

weighting vector [a*_{, b}*_{, c}*_{], where a design point in the}

DOE denotes a sampled weight vector.

In terms of total proﬁt, experiments indicated that the proposed multiple criteria approach outperformed the previous study that uses only a single criterion—number of operations. This ﬁnding has two important implica-tions. Firstly, it advocates the use of multiple criteria in trading tool capacity. Secondly, a weekly performance evaluation of a semiconductor fab may better be evalu-ated from three perspectives—number of operations, number of layers, number of wafers. In practice, manag-ers tend to evaluate a fab’s weekly performance by the number of produced operations, and evaluate its quarterly performance by the number of wafers produced. To align the short-term (week) and long-term (quarter) objectives, the weekly performance evaluation may need a multiple criteria such as proposed.

Some possible enhancements of this research are being investigated. Firstly, how to reduce the computation eﬀorts in evaluating the performance of a weighting vector, which now takes about 5 h on 1 facility with 50 PCs. Secondly, we attempt to justify whether the adoption of a smaller/longer time bucket for making capacity trading decisions would improve the total proﬁts further. A smaller time bucket may denote that a trading decision is made every 3 days, while a longer time bucket may denote that a trading deci-sion is made every 2 weeks.

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