A short-term capacity trading method for semiconductor fabs with partnership

A short-term capacity trading method for semiconductor

fabs with partnership

Muh-Cherng Wu

, Wen-Jen Chang

Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan

Abstract

This paper presents a capacity trading method for two semiconductor fabs that have established a capacity-sharing partnership. A fab that is predicted to have insufficient capacity at some workstations in a short-term period (e.g. one week) could purchase tool capacity from its partner fab. The population of such a capacity-trading portfolio may be quite huge. The proposed method involves three mod-ules. We first use discrete-event simulation to identify the trading population. Secondly, some randomly sampled trading portfolios with their performance measured by simulation are used to develop a neural network, which can efficiently evaluate the performance of a trading portfolio. Thirdly, a genetic algorithm (GA) embedded with the developed neural network is used to find a near-optimal trading portfolio from the huge trading population. Experiment results indicate that the proposed trading method outperforms two other bench-marked methods in terms of number of completed operations, number of wafer outs, and mean cycle time.

 2006 Elsevier Ltd. All rights reserved.

Keywords: Semiconductor fab; Capacity planning; Capacity trading; Neural network; Genetic algorithm

1. Introduction

Semiconductor manufacturing is a capital intensive industry. An up-to-date semiconductor fab may cost a few billion dollars to build, and an individual tool may charge over several tens of million dollars. Due to tremen-dously high equipment cost, how to effectively manage tool capacity is very important to semiconductor fabs. Such decisions associated with tool capacity are generally called capacity planning problems (Christie & Wu, 2002). The objective of capacity planning is to well prepare and allo-cate tool capacity in order to maximize a semiconductor fabs’ profit. In terms of planning horizon, capacity plan-ning problems could be classified into three levels: strategic, tactical, and operational levels.

At strategic level, capacity planning means the invest-ment decisions on procuring new tools. Based on demand forecast, the decision (also called tool planning) is to opti-mally determine the type and number of tools needed for new or existing fabs. The tool planning decision is distinct in requiring large amount of expenditure and long-lead time (3–9 months) in tool procurement. The tool planning problem, a long-term decision, usually covers 1–2 years in the planning horizon, and the annual tool expenditure of a semiconductor company may be over billions of dollars. Much literature on tool planning has been published (Wu, Erkoc, & Karabuk, 2005a). Some addressed scenarios with demand uncertainty and developed mix-integer program-ming models (Barahona, Bermon, Gunluk, & Hood, 2001; C¸ atay, Erengu¨c¸, & Vakharia, 2003; Hood, Bermon, & Barahona, 2003; Swaminathan, 2000, 2002); some others were concerned with scenarios with constraints imposed by target cycle time and developed solution methods by the application of queuing network (Bard, Srinivasan, & Tirupati, 1999; Connors, Feigin, & Yao, 1996; Iwata, Taji, & Tamura, 2003; Wu, Hsiung, & Hsu, 2005b) or by

0957-4174/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2006.05.012

*

Corresponding author. Tel.: +886 3 5927700x2954; fax: +886 3 5926848.

E-mail addresses: mcwu@cc.nctu.edu.tw (M.-C. Wu), wjchang6@ ms39.hinet.net(W.-J. Chang).

1 _{Tel.: +886 3 5731913; fax: +886 3 5720610.}

www.elsevier.com/locate/eswa Expert Systems with Applications 33 (2007) 476–483

simulation models (Chen & Chen, 1996; Grewal, Bruska, Wulf, & Robinson, 1998; Mollaghsemi & Evans, 1994) or by simulation and queuing techniques (Chou, Wu, Kao, & Hsieh, 2001; Hopp, Spearman, Chayet, Donohue, & Gel, 2002).

At tactical level, capacity planning refers to the decisions of product-mix and production planning. The product-mix decision is to find an optimal product mix to maximize the profit of a fab, equipped with a given tool portfolio. The production planning decision is to determine the release schedule of each fab given a corporate demand output schedule (Chen, Chen, Lin, & Rau, 2005). These two deci-sions can be regarded as medium-term decideci-sions because they usually cover several months in the planning horizon. Various techniques for solving the semiconductor product-mix and production planning problems have been devel-oped. These techniques involve the use of multi-objective programming (Chung, Lee, & Pearn, 2003, 2005), linear programming (Bermon & Hood, 1999), mix-integer pro-gramming (Chou & Hong, 2000).

A semiconductor fab constantly faces some unexpected events such as machine breakdown, urgent orders, and hold lots. As known, the exact time when an unexpected event will occur cannot be predicted. Therefore, in the decisions of tool planning, product-mix planning and production planning, the stochastic effects of unexpected events are treated in a deterministic manner. For example, the effect of machine breakdown is modeled by giving each tool a sta-tic availability, even though the tool availability in fact changes stochastically and dynamically. Due to the stochas-tic behavior of unexpected events, the tool capacity of a fab that is available to deal with the planned demand in an upcoming short-term period (e.g. one week) may become excessive or insufficient for some workstations. We call this phenomenon a short-term capacity disequilibrium problem. At operational level, capacity planning refers to the deci-sion of solving the short-term capacity disequilibrium problem. A few studies proposed to solve the problem by leveraging the tool capacity of multiple fabs that belong to the same company. Taking these fabs as a single big fab and their operational control is centralized, these stud-ies aimed to develop real-time inter-fab dispatching rules for lots (Deboo, 2000; Toba, Izumi, Hatada, & Chiku-shima, 2005). However, when the operational control sys-tems of these fabs are not centralized, such real-time inter-fab dispatching rules may not be effectively imple-mented due to lack of real time shop-floor information. In practice, two fabs, not being equipped with a centralized operational control system, may still leverage their tool capacity by a weekly trading agreement. That is, based on a prediction of tool utilization, a fab with insufficient capacity at some tools would purchase tool capacity from its partner fab. The short-term off-line capacity trading decision is important but has been rarely investigated in literature.

This study aims to develop an effective method for mak-ing the short-term off-line capacity tradmak-ing decision to

max-imize the total number of completed operations of the two trading fabs. The proposed method involves three modules: (1) identifying a population of capacity trading portfolios, (2) evaluating the performance of a trading portfolio by the use of neural network together with discrete-event simula-tion techniques, (3) finding a near-optimal trading portfo-lio from the population by the use of a genetic algorithm (GA).

The remainder of the paper is organized as follows. Sec-tion2describes how to identify the population of capacity trading portfolios. Section3presents the method for eval-uating the performance of a trading portfolio. Section 4

discusses the GA technique for finding a near-optimal trad-ing portfolio. Section 5 shows a numerical example and concluding remarks are placed in Section6.

2. Population of capacity trading portfolios

There are two factors that determine the population of all possible capacity trading portfolios. The first factor refers to the number of workstations that would participate in capacity trading—herein called tradable workstations. The second factor refers to the upper bound of trading vol-ume for each tradable workstation. This section begins with the operational assumptions for the wafer fabs of interest, and presents the methods for identifying tradable worksta-tions and the population of capacity trading portfolios. 2.1. Operational assumptions of wafer fabs

In the decision problem, we assume that two semicon-ductor fabs have established a capacity-trading agreement and trade capacity weekly. The two fabs have n-pairs of common workstations by which capacity trading can be implemented. In each pair, the two workstations, physi-cally located in different fabs, are functionally identical and therefore can trade capacity. Some operational assumptions of the two fabs are described below.

(1) Each operation is only processed by a particular type of workstation.

(2) For an operation that can be processed by a pair of common workstation, its processing times at the two fabs are the same.

(3) The weekly tool capacity bought by a fab is uni-formly used; that is, the capacity used per day is a constant.

(4) A workstation with bought-in capacity is given a WIP threshold. If the WIP profile is higher than the threshold, wafer lots waiting before the workstation are continuously sent to the other fab until the daily amount of bought-in capacity has been used up. (5) Both fabs use the uniform policy in releasing lots.

That is, given a product mix, the number of lots released to each fab per day is a constant.

(6) The lot dispatching policy at each workstation in each fab is based on the First-In-First-Out rule.

2.2. Identification of tradable workstations

A pair of common workstations is tradable if the tool utilization of one workstation in the upcoming week is lower than LB and that of the other is higher than UB,

where LB and UB are two predefined thresholds used to

define the intended extent of capacity difference in forming a trading pair. For a trading pair, the low-utilization work-station could sell tool capacity to the high-utilization one, which hereafter are respectively called a seller workstation and a buyer workstation.

We develop a discrete-event simulation program to esti-mate the tool utilization of the upcoming week. The simu-lation program is distinct in twofold.

First, the simulation program is a deterministic model, in which the breakdown of machine is modeled in an ‘‘aver-age’’ manner. That is, the tool breakdown is modeled by enlarging the processing time of an operation—through dividing the original processing time by the average tool availability. The reason for using a deterministic simulation model for estimating the tool utilization of the upcoming week is based on the findings of Kim, Shim, Choi, and Hwang (2003)on real-time scheduling. They claimed that in making a short-term scheduling decision the perfor-mance of using deterministic simulation is superior to that of using a single run of stochastic simulation. This reflects that a fab’s short-term behavior had better be predicted by using deterministic simulation. Using multiple runs of sto-chastic simulations may improve accuracy; however, it needs lengthy computation time.

Second, the simulation program must capture in detail the fab status at the decision point, which includes the pro-files of WIP, hold lots, breakdown of machines, and expected times for the recoveries of down-machines. Such information is important to the capacity trading decision because the time horizon for capacity trading is short— only one week. The ignorance of the initial fab status may seriously affect the performance of a capacity trading portfolio.

2.3. Characterizing the population of capacity trading portfolio

For a pair of tradable workstations, the possible amount of tradable capacity is determined by two factors: basic trading unit and upper bound of trading volume. We define the basic trading unit of capacity (u) as follows: u = c· max(Pij|i2 TW) where TW is a set consisting of all

tradable workstations, Pij is the processing time of

opera-tion j processed at workstaopera-tion i, and c P 1 is a predefined integer.

For a pair of tradable workstations i, its maximum num-ber of trading units can be defined as: Bi¼_2u1ðqSi qBiÞ

Q_i T , where qSi is the before-trading tool utilization of

the seller workstation and qBirepresents that of the buyer

workstation, Qi is the number of tools in the seller; and

T is the time horizon of the trading decision (i.e. one week).

The population of trading portfolio can be defined as follows: P = {[b1, b2, . . . , bm], bi2 Z, 0 6 bi6Bi}, where m

denotes the number of tradable workstations, and the num-ber of trading portfolios is NðP Þ ¼Qm_i¼1ðbiþ 1Þ. The value

of N(P) can be quite huge. Assume that the maximum trad-able units of each workstation are q units. Then, the sce-nario has qm trading portfolios; that is, if q = 20 and m = 4, there are totally 160,000 (204) capacity trading portfolios.

3. Performance evaluation of trading portfolios

For each trading portfolio, we can use the aforemen-tioned deterministic simulation program to estimate the performance of each fab. The performance refers to the total number of completed operations of the fab in the upcoming week and is briefly called move number hereafter. The sum of move numbers of the two fabs is regarded as the aggregated performance of the trading portfolio.

If it takes one minute to simulate a trading portfolio, an exhaustive evaluation of a typical trading population (with 204 trading portfolios) would take about 111 days. The required computation time is undoubtedly too long to be accepted in practice. To reduce the computation time for the performance evaluation of a trading portfolio, we attempt to construct a neural network to effectively and efficiently emulate the simulation of a fab.

3.1. Development of neural network

The basic idea of modeling the simulation of a fab by a neural network is as follows. First, we sample n trading portfolios from population P and evaluate the performance of each portfolio in the fab through simulation. After sim-ulation, the behavior of the fab in the upcoming week can be characterized by n sets of input/output vectors. An input vector refers to a trading portfolio and an output vector refers to the move number of the fab after capacity trading. Second, the n sets of input/output data are used to con-struct a neutral network for representing the simulation of the fab. The detailed procedure for constructing such a neural network is presented below.

Assume that fabs A and B have m tradable worksta-tions. Each tradable workstation i has at most bi trading

alternatives. The trading population then has NðP Þ ¼ Qm

i¼1ðbiþ 1Þ trading portfolios and can be expressed as

P = {Xk|1 6 k 6 N(P)}, where Xk= (x1k, x2k, . . . , xmk)

rep-resents the kth trading portfolio and xikrefers to the

num-ber of trading units for workstation i. Herein, xik> 0

denotes that fab A buys in capacity from fab B, and vice versa. Define Yk=Xk, then Yk represents the capacity

that fab B purchases from fab A in the kth trading portfolio.

From trading population P, n trading portfolios are ran-domly sampled and put in a set ~P¼ fXjg. For each trading

portfolio in ~P, a simulation program is executed for calcu-lating the move number of each fab in the upcoming week

after capacity trading. As shown in Fig. 1, a functional relationship TAj= SimA(Xj) is used to represent the

simula-tion program executed for fab A, where TAjrepresents the

move number of fab A under trading portfolio Xj. The n

sampled trading portfolios by simulation will generate n pairs of input/output vectors for fab A, denoted by Data Set A¼ fðXj; TAiÞjXj2 ~Pg. Similarly, a functional

relationship TBj= SimB(Yj) is used to represent the

simula-tion program executed for fab B, and n pairs of input/out-put data Data Set B¼ fðYj; TBiÞjXj2 ~Pg can be likewise

generated. Notice that the two simulation programs SimA

and SimB are both based on a deterministic simulation

model because the short-term (one week) performance of a fab is concerned.

Based on Data_Set_A, we construct a neural network NetA for representing the simulation program SimA. Out

of the n pairs of input/output vectors in Data_Set_A, n1

pairs are randomly sampled and used to build a back-prop-agation neural network NetAand the other (n n1) pairs is

used to test the effectiveness of NetA. The n pairs of data is

called training data of NetA. This neural network model

is represented as ~TAk ¼ NetAðXkÞ, where ~TAk represents

the performance of fab A under trading portfolio Xk,

calcu-lated by the trained neural network. Likewise, Data_Set_B can be used to construct a neural network ~TBk ¼ NetBðYkÞ

for modeling the simulation program SimB.

3.2. Neural network algorithm

We construct NetAand NetBby using the

back-propa-gation neural network (BPN) technique (Fausett, 1994), which has been widely used and justified to be effective in various applications (Vellido, Lisboa, & Vaughan, 1999). The architecture of a BPN involves three layers of neurons (Fig. 2); the first layer represents the input, the last layer is the output, and the hidden layer models the transformation mechanism from input to output. Each neuron in a layer and that in its subsequent layer is connected by a link on which a weight (a real number) is to be found.

The development of a BPN, also called training, is to determine the weight on each link so that the BPN can well model the input/output mapping of the simulation pro-gram of concern. For example, given an input Xjin

Data_-Set_A, the output ~TAj computed by a well-trained BPN

would be fairly close to the target output TAj. A

well-trained BPN, NetA, can therefore be used to model the

sim-ulation program SimA.

The detailed algorithm for training a BPN can be referred to (Fausett, 1994). The algorithm is essentially based on the gradient descent technique, which changes the network weights iteratively by the formula:

wijkðn þ 1Þ ¼ wijkðnÞ þ g  wijkðnÞ þ a  wijkðn  1Þ

where i denotes a neuron in layer k, j denotes a neuron in the preceding layer (k 1), and wijk represents the weight

between the two neurons. The constant g (which lies in the range 0–1) is called the learning rate, which determines the speed of convergence and a (also in the range 0–1) is called the momentum constant.

The accuracy of the model is evaluated in terms of the root-mean-square error (RMSE), the prediction of RMSEx

for fab x is calculated by obtaining the square-root error between the neural network’s predicted value and the actual target value and is given by

RMSEx¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n 1 Xn i1 ðTxi ~TxiÞ 2 s

where n is the number of training/testing data, Txi is the

move number of fab x computed by the simulation pro-gram Simx, and ~Txiis the move number of fab x computed

by the neural network Netx. The training error is the

RMSE of the data used for the network training, and the prediction error is the RMSE of the data reserved for net-work testing.

Network structure and training issues, such as the num-ber of layers, numnum-ber of neurons, the learning rate and the momentum constant are determined during the network development process. These values are selected such that after training, the outputs of the established neural net-work best match the experiment data.

4. Finding a near-optimal trading portfolio

Using the developed neural network to evaluate the per-formance of a trading portfolio indeed saves computation. Yet, applying an exhaustive search embedded with the neu-ral network technique to find an optimal trading portfolio may still need a lengthy computation time because the solu-tion space is quite huge. We therefore developed a genetic

Yj TBj = mB (Yj) Simulation program f or fab A Xj TAj= SimA(Xj) Simulation program f or fab B S

Fig. 1. The input/output relationship of the simulation program.

input layer hidden layer output layer

Fig. 2. Architecture of back-propagation neural network with one hidden layer.

algorithm (GA) to efficiently obtain a near-optimal solu-tion. The reason for using GA is that the technique has been widely applied and shown satisfactory results in vari-ous space-searching problems.

In the proposed GA, a trading portfolio Xj= (x1j,

x2j, . . . , xmj) is called a chromosome where xij is called a

gene. The performance of a trading portfolio, represented by F(Xj), is called the fitness of the chromosome Xj. The

higher the value of F(Xj), the higher quality is chromosome

Xj.

The procedure of the GA is briefly described as follows. An initial GA population G(t = 0) is generated by ran-domly sampling N chromosomes from the trading portfo-lio population P. Three genetic operators (reproduction, crossover and mutation) are used to manipulate the chro-mosomes in G(t) to form the next generation population G(t + 1). The population is iteratively updated until a ter-minating condition is reached.

Reproduction is a process by which a chromosome with higher fitness value has higher probability of being repro-duced so that a larger number of the chromosome copies may be included in the new population. We use the elitist roulette wheel selection method in the reproduction process (Mitchell, 1998). After reproduction, the survived chromo-somes are stored in a ‘‘mating pool’’ and await mutation and crossover operations. The crossover operation ran-domly takes two chromosomes and interchanges part of their genetic values to produce two new chromosomes based on a crossover rate (Cr). The mutation operation is

implemented by randomly changing a genetic value based on a specific mutation rate (Mr). The entire process of three

genetic operators is intended to create ‘‘more fit’’ chromo-somes in the next population. The successive updating of G(t) is terminated when the best solution in G(t) keeps unchanged for Nbgenerations or when maximum number

of iterations (Nf) is reached (i.e. t = Nf).

5. Numerical example

We use a numerical example to justify the effectiveness of the proposed method for making the capacity trading decision. The example scenario includes two fabs (Fab_A and Fab_B) that attempt to trade capacity on a weekly basis. Table 1 shows the tool numbers and the products of the two fabs, in which the 4P1M product is a memory

product while the other 1PXM products are logic products. Notice that the tools required for producing memory prod-ucts are significantly different from that for producing logic products. The tool portfolios of the two fabs are thus essentially different and may be able to supplement each other in tool capacity. The routing information of each example product is provided by a semiconductor company in Taiwan.

We use eM-plant (Tecnomatix Technologies Ltd., 2001) to establish three simulation programs. Each of these three is a variant of a particular simulation program, with dis-tinction in their simulation settings.

The first simulation program (called Sim_1), executed in a single-replicate and stochastic simulation model, was used to create and update the decision scenario before capacity trading. The output of the scenario-creation pro-gram provides the WIP profiles and the machine up-down status of the two fabs, which reflects the initial status of the two fabs before trading and is the input of the second sim-ulation program.

The second simulation program (called Sim_2), executed in a deterministic simulation model, aimed to generate the training data set for establishing the neural network. We firstly used Sim_2 to identify tradable workstations to define the solution space. In the testing example, four types of workstations (E10, E27, E31, E55) are found to be trad-able (Table 2). Fab_A would buy-in tool capacity for workstations E10 and E55, while Fab_B would buy-in tool capacity for E27 and E31. Using 20 h as the basic unit for capacity trading, the solution space involves 103,488 trad-ing portfolios. Secondly, we randomly select 2000 sets of trading portfolios and obtain each of their performances by Sim_2. The simulation run for measuring the perfor-mance of each trading portfolio can be executed on a differ-ent computer. Therefore we use 20 personal computers (Pentium IV, 2.0 GHz and 256-MB memory) to execute the 2000 simulation runs in parallel; and it takes about 1.35 h to finish all the simulation runs. The 2000 set of trading portfolios and their performance were used to establish the neural network. The neural network com-bined with the GA algorithm can be used to determine a near-optimal trading portfolio. The parameters of neural network and GA are shown inTable 3.

Table 1

Tools and products for Fab_A and Fab_B FAB Number of workstations Total number of machines Product Total processing time (h) Total number of operations Fab_A 60 270 4P1M 400 358 1P7M 440 412 Fab_B 60 198 1P3M 318 276 1P8M 480 446 Table 2

Capacity utility estimation of buyer workstations in each fab and maximum buy-in volume at the first trading decision scenario

Tradable workstation E10 E55 E27 E31 Estimate of utilization in upcoming week at Fab_A (%) 100 99.4 42.4 32.4 Estimate of utilization in upcoming week at Fab_B (%) 30.0 49.8 96.6 95.2 Maximum amount of tradable tool capacity (hours) 220 320 280 840

The third simulation program (called Sim_3), executed in a stochastic simulation model with 20 replicates and the simulation time horizon is one week, was used to justify the effectiveness of the trading portfolio suggested by the proposed algorithm. The output of Sim_3 includes the move number of each fab in the upcoming week after capacity trading. The sum of move numbers of the two fabs is taken as the performance of the trading portfolio. The proposed buy-in decision and its performance are shown inTable 4.

In summary, the three simulation programs are inte-grated into a procedure to create the proposed trading deci-sion for a particular week and justify its effectiveness. The input/output relationships in the procedure are shown in

Fig. 3. Program Sim_1 was used to create S(t), the decision scenario or the initial status of week t. Program Sim_2 was used to generate data for establishing a neural network, by which we can use the GA to yield ^Xt, the proposed trading

decision for week t. Program Sim_3 was used to evaluate the performance of ^Xt. With ^Xtand S(t) as input, program

Table 3

Parameters of neural network and genetic algorithm

Item Value Item Value

Neural network Number of input nodes 4 Number of training data 1500 Number of hidden nodes 7 Number of testing data 500 Number of output nodes 1 Maximum iterations 50,000 Learning rate 0.10 Momentum 0.80 Genetic

algorithm

Population size 100 Maximum iteration, Nf

20,000 Cross over rate 0.80 Maximum

generation, Nb

1000 Mutation rate 0.05

Table 4

The proposed buy-in decision and the performance of no-trading and proposed method Buyer workstation Buy-in capacity Move number with no-trading

Move number with proposed method Fab_A E10 160 72,071 74,598 E55 220 Fab_B E27 180 50,126 51,665 E31 720 Sim_1 Stochastic model

Initial status of week t , S(t)

Sim_2

Deterministic model GA + NN

Sim_ 3

Stochastic model

Performance of X_t

Proposed trading portfolio for week t, Xˆ_t

ˆ

Fig. 3. The input/output relationship of the three simulation programs.

Table 5

Comparison of the weekly effectiveness of proposed trading portfolio with no-trading and max-trading methods

Period Proposed trading portfolio Without any trading Maximum trading portfolio

Fab_A Fab_B Aggregate Fab_A Fab_B Aggregate Fab_A Fab_B Aggregate

1 74,598 (103.5%) 51,665 (103.1%) 126,263 (103.3%) 72,071 (100.0%) 50,126 (100.0%) 122,197 (100.0%) 73,756 (102.3%) 51,183 (102.1%) 124,939 (102.2%) 2 73,981 (103.2%) 50,545 (102.3%) 124,526 (102.9%) 71,675 (100.0%) 49,404 (100.0%) 121,079 (100.0%) 74,071 (103.3%) 49,124 (99.4%) 123,195 (101.7%) 3 74,797 (103.2%) 51,957 (103.4%) 126,754 (103.3%) 72,463 (100.0%) 50,254 (100.0%) 122,717 (100.0%) 73,026 (100.8%) 50,265 (100.0%) 123,291 (100.5%) 4 72,833 (103.1%) 50,693 (102.1%) 123,463 (102.7%) 70,664 (100.0%) 49,678 (100.0%) 120,342 (100.0%) 71,426 (101.1%) 51,036 (100.7%) 122,462 (101.8%) 5 73,978 (103.2%) 49,944 (103.3%) 123,922 (103.2%) 71,722 (100.0%) 48,369 (100.0%) 120,091 (100.0%) 72,738 (101.4%) 48,927 (101.1%) 121,665 (101.3%) 6 74,852 (105.7%) 49,896 (101.1%) 124,748 (103.8%) 70,849 (100.0%) 49,369 (100.0%) 120,218 (100.0%) 73,580 (103.8%) 49,808 (100.9%) 123,388 (102.6%) 7 74,065 (101.6%) 51,147 (104.4%) 125,212 (102.7%) 72,918 (100.0%) 48,975 (100.0%) 121,893 (100.0%) 74,213 (101.7%) 48,755 (99.5%) 122,968 (100.9%) 8 73,998 (103.7%) 51,268 (102.8%) 125,266 (103.3%) 71,394 (100.0%) 49,863 (100.0%) 121,257 (100.0%) 73,986 (103.6%) 49,861 (100.0%) 123,847 (102.1%) The move numbers are shown as a percentage of that of without any trading method in parenthesis.

Sim_1 was use to obtain S(t + 1), the decision scenario of week t + 1. The procedure can be repeatedly performed to obtain the results of consecutively implementing capac-ity trading for multiple weeks.

We justify the effectiveness of the proposed method by performing capacity trading for eight consecutive weeks. The proposed method is compared with two other meth-ods. The first one is without any trading. The second one, called the maximum trading method, requests each buyer workstation buys-in its maximum amount of trading units (Bi).Table 5shows that in each of the eight weeks the

proposed trading method outperforms the other two meth-ods (about 2.7–3.8% higher) in terms of the aggregate move number. This implies that the proposed trading method can effectively increase the aggregated number of com-pleted operations for the two fabs.

One may wonder such an increase in completed opera-tions could lead to an increase in completed products.Table 6 indicates that the proposed method outperforms the other trading methods in terms of throughput and mean cycle time. Herein, throughput refers to the total number of completed products during the eight trading weeks. This implies that the proposed capacity trading method could also effectively increase the aggregated fab throughput. 6. Concluding remarks

This paper develops a short-term off-line capacity trad-ing method for two semiconductor fabs. The method involves three major techniques. The first technique—dis-crete-event simulation is essentially used to evaluate the performance for a set of randomly sampled trading portfo-lios. The second technique—neural network is used to emulate the function of the simulation technique; that is, evaluating the performance of a trading portfolio at a much faster speed. The third technique—genetic algorithm is used to find a near-optimum trading portfolio in an effi-cient manner.

Two other trading methods (without-trading and maxi-mum-trading) are compared with the proposed one. Exper-iment results indicate that the proposed capacity trading method outperforms the two other methods in each of the three performance indices: aggregated number of com-pleted operations, aggregated throughput, and mean cycle time.

The implementation of the proposed method may take a significant amount of computation time in collecting the

simulation data for establishing the neural network. One possible extension to this study is developing methods for reducing the computation time for establishing the neural network.

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Table 6

The results after performing eight consecutive weeks of proposed trading portfolio and without any trading and maximum trading methods With proposed trading portfolio Without any trading Maximum trading

Fab_A Fab_B Fab_A Fab_B Fab_A Fab_B

4P1M 1P7M 1P3M 1P8M 4P1M 1P7M 1P3M 1P8M 4P1M 1P7M 1P3M 1P8M

Throughput 495 335 447 262 476 332 431 252 484 337 435 253

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