Int. J. Production Economics 96 (2005) 15–36
Analytic networkprocess (ANP) approach for product mix
planning in semiconductor fabricator
Shu-Hsing Chung*, Amy H.I. Lee, W.L. Pearn
Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan Received 1 August 2003; accepted 9 February 2004
Abstract
This paper proposes an application of the analytic networkprocess (ANP) for the selection of product mix for efficient manufacturing in a semiconductor fabricator. In order to evaluate different product mixes, a hierarchical networkmodel based on various factors and the interactions of factors is presented. By incorporating experts’ opinion, a priority index can be calculated for each product mix studied, and a performance ranking of product mixes can be generated. The results provide guidance to a fab regarding strategies for accepting orders to maximize the manufacturing efficiency in considering the aspects of product, equipment efficiency and finance. The model can be easily understood and followed by administrators to determine the most efficient product mix for a fab.
r2004 Elsevier B.V. All rights reserved.
Keywords: Analytic networkprocess; Efficiency; Production performance; Product mix; Semiconductor fabricator
1. Introduction
Global competitiveness has become the biggest concern of semiconductor industry. How to increase the overall profit and the return on investment of a company is therefore very essential. The purpose of this paper is aimed at the strategic planning level and attempts to present an effective approach for product mix evaluation that allows for the consideration of various factors and important interactions among factors. The product mix selected can best represent a near-optimal utilization to the factory resources and a highest possible profit attained, and it can be a
reference for production planning and order acceptance.
Wafer fabs involve the most complex manufac-turing system in the manufacmanufac-turing world. Its manufacturing process is of high complexity, with several hundreds of processing steps on a single wafer and a flow time of usually more than 1 month. Different product mix only complicates the already-complex system. Depending on the types of products, the process plan of a product can range from very identical to being extremely distinctive, and the requirement of setups may also be different. The greater the difference, the more diverse the loading demand and batch difficulty on the factory. The actual throughput and cycle time under a given product mix thus depend on how badly the fab is bottlenecked, whether the bottleneckis shifted, and how many *Corresponding author. Tel.: 3-5731-638; fax:
+886-3-5722392.
E-mail address:t7533@cc.nctu.edu.tw (S.-H. Chung).
0925-5273/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2004.02.006
machine setups are needed because of product type conversions.
Many metrics can be applied for evaluating factory productivity. Leachman and Hodges (1996) evaluated semiconductor wafer fabrication plants around the world to quantify manufactur-ing performance and to establish comparative benchmarks in manufacturing technology, factory operations, organization, and management. The major technical metrics they used to measure manufacturing performance are cycle time per wafer layer, line yield, die yield, stepper produc-tivity, direct labor producproduc-tivity, total labor pro-ductivity and on-time delivery. Although their study provided a comprehensive performance evaluation and identified those practices that underlie top performance, there was no attempt to correlate the interactions of the metrics. There are at least three aspects that are necessary for measuring the overall effectiveness of a factory: production, utilization of assets, and costs (SEMI, 2002). SEMI provided a guideline for definition and calculation of overall factory efficiency (OFE) and other associated factory-level productivity metrics. The document focused on evaluating production; however, utilization of assets and costs were outside of its scope.
Organizing available data and providing a singular metric to compare performances is not an easy task. Chung et al. (2002), however, adopted a good nonlinear programming method called Data Envelopment Analysis (DEA) to deal with multiple inputs and outputs. Without pre-assigning weights, DEA can be used to measure multiple inputs and outputs for product mixes in a semiconductor fabricator, and an efficiency score for producing each product mix relative to other mixes can be obtained. The major advantage of DEA, without pre-assigning weights to any performance measure, can also be its drawback. Managers often have their own opinion on what performance measures are more important than others. In that case, analytic hierarchy process (AHP) and/or analytic networkprocess (ANP) can be a good alternative in evaluating production performance under different product mixes.
While AHP has been a popular research and application tool for multi-attribute
decision-mak-ing, the ANP technique so far has had only a few applications in literature. A matrix manipulation approach, developed by Saaty and Takizawa (1986), is applied to solve a network, which is very similar to a hierarchy but has dependence among criteria and dependence among alternatives with respect to each criterion.Lee and Kim (2000)
used the above-cited ANP approach within a zero-one goal-programming (ZOGP) model to suggest an information system project selection methodol-ogy, which can reflect interdependencies among evaluation criteria and candidate projects.Karsak et al. (2002)dealt with product planning in quality function deployment by also using a combined ANP and goal programming approach. Chung et al. (2004)adopted Saaty’s matrix manipulation concept and suggested a simplified ANP approach to analyze multiple process inputs and outputs, and with experts’ opinion on their priority of importance, to obtain optimal product mixes for semiconductor production. Sarkis (2002) pre-sented a systemic ANP model to evaluate environ-mental practices and programs in analyzing various projects, technological or business decision alternatives.Momoh and Zhu (1998)proposed an application of AHP and ANP to enhance the selection of generating power units for appropriate price allocation in a competitive power industry.
Meade and Sarkis (1999) suggested a decision methodology that applied ANP to evaluate alter-natives (e.g. projects) and to help organizations become more agile, with a specific objective of improving the manufacturing business processes.
Meade and Presley (2002) used ANP to support the selection of projects in a research and development (R&D) environment.
Suwignjo et al. (2000) and Bititci et al. (2001)
constructed an innovative frameworkand sup-porting system to let organizations incorporate and map performance measures in a hierarchical way. The quantitative model for performance measurement system (QMPMS) relies on AHP to quantify both tangible and intangible factors for performance. Bititci et al. further applied the QMPMS for manufacturing strategy evaluation and management in a dynamic environment.
Sarkis (2002) revisited the above works and applied ANP to the QMPMS process. Through
the utilization of the supermatrix approach, the combined effects of factors on organizational performance measures can be quantified, and the dynamic nature of strategic decisions can be evaluated.
Saaty suggested the use of AHP to solve the problem of independence on criteria and alter-natives and the use of ANP to solve the problem of dependence among criteria and/or alternatives (Saaty, 1996). The metrics for measuring manufac-turing performance, such as production through-put, cycle time, equipment utilization and WIP are highly interrelated. While some metrics are positively dependent, others may be negatively dependent. As a result, ANP is adopted in this research for determining the production performance of various product mixes.
Semiconductor companies as well as other industries need good problem solving methods that can be used in real practice. While ANP provides a good quantitative and qualitative tool to assist administrators, they may feel threatened by its complexity if they have no experience or a good understanding of the method. This paper can provide a good prototype for the users to conceptualize the process and to follow the procedure in the determination of a suitable product mix in manufacturing.
This paper is organized as follows. Section 2 discusses the process parameters selected for evaluation, and a multi-attribute selection frame-workrepresented as an ANP model is presented. Section 3 describes the system environment for simulation. Section 4 applies ANP to the evalua-tion of the efficiency under different product mixes. Some conclusion remarks are made in the last section, while the appendix briefly reviews the decision making tool ANP.
2. Process parameters for evaluation
Different product mix has a different impact to the production performance, and production performance is a result of the interaction among equipment set availability, control rules and loading condition, just to name a few. In addition, there is no optimum production performance since
different people are interested in different perfor-mance indicators. For example, finance people may be interested in the final profit a fab can make, while industrial engineers want to generate the maximum throughput and maintain produc-tion smoothness. Furthermore, some performance indicators are positively related, but others may be counter-active. Therefore, based on the perfor-mance indicator selected, the perforperfor-mance evalua-tion under a specific product mix will be different. Which performance measures should be consid-ered and the importance of each measure, are essential in the analysis. Based on the ANP method, senior managers and experts from the Science-Based Industrial Parkin Taiwan are interviewed first to decide the major criteria and the subsequent detailed criteria for evaluating performance under different product mixes. The hierarchical networkstructure, which com-poses of three major criteria and numerous detailed criteria, is established as shown in
Fig. 1. Criteria are assumed to interrelate to each other; in consequence, detailed criteria are all interrelated even if they have different upper level criteria.
The three major criteria and detailed criteria for measuring manufacturing performance of a semi-conductor fab are defined as follows:
(1) Product: How products are manufactured in a fab.
WIP gives the number of lots that have been released into the wafer fab but have not yet been finished processing through all of their manufacturing steps.
Throughput (TP) shows the number of lots that pass through the final operation step in a period.
Total layers (TL) accounts the number of layers the bottleneckresource processed in a period of time.
Total cycle time (CT) measures the dura-tion of time, expressed in hours, con-sumed by a unit of production from the time of release into the fab until time of exit from the fab. It is a weighted average cycle time, where the weights are the ratio of product mix.
(2) Equipment efficiency: How effective the equip-ment is used in manufacturing, a measure of equipment performance.
Bottleneck utilization (BU) shows average utilization rate of the bottleneckin the system for a period of time. Since the throughput of the entire manufacturing system is determined by the bottleneck workstation, BN utilization should be as high as possible.
Capacity constrained resource (CCR) uti-lization (CU) shows average utiuti-lization rate of the CCR in the system for a period of time. Although CCR is not a bottle-neck, it also has a substantially high utilization rate. When CCR utilization rate is too high, there is a chance of bottleneckshifting.
(3) Finance: The amount of money a wafer fab can make or needs to spend in the manufacturing process. All finished products are assumed sold. Product price is determined by the market according to its product type. Most manufacturing costs of a semiconductor fab-ricator are fixed; that is, no matter how many products are produced, the operating costs are not varied much.
Total revenue (TR) is obtained by sum-ming up revenue of each product type, while revenue of a product type is
calculated by multiplying the price of the product type with its throughput and the price for a product is set by its product type and the number of layers that product goes through.
Variable costs (VC) include two major parts: total variable manufacturing costs and total holding costs. Direct material cost, the cost of raw wafers, is the primary part of total variable manufacturing costs. Other variable manufacturing costs in-clude indirect material cost and is varied according to the manufacturing level. The holding cost is the time cost of carrying WIP in the manufacturing system. Senior managers of several semiconductor manufacturing companies in Science-Based Indus-trial Parkin Taiwan and academic experts were involved in evaluating the criteria and subcriteria and gave pairwise comparison values.
3. System input and simulation
The simulation model for this study has been developed using data from an existing wafer fabrication factory located on the Science-Based Industrial Parkin Taiwan. To simplify the complexity of the environment for our analysis,
Determine the efficient production performance Product Equipment Efficiency Finance WIP Throughput Total layers
Total cycle time
BN Utilization
CCR Utilization
Total Revenue
Total Variable Costs
Product mix 1: 9 2: 8 3: 7 4: 6 5: 5 6: 4 7: 3 8: 2 9: 1
the simulation model is built under several assumptions. The fab consists of 83 workstations (W1–W83), and each workstation consists of a given number of identical machines operated in parallel. There are a total of 235 machines, which can be grouped into two types: batch and serial. The machines are subject to downtimes due to preventive maintenance and failures. W46, a stepper in the photolithography area, is the bottleneck, and W38, with a high utilization rate that is next to that of bottleneck, is the CCR. The strategy of manufacturing is to maximize the utilization of bottleneck. Two types of products, L and M, are manufactured, and both products are normal; that is, there is only one priority level. Product L is a logic product, while product M is a memory product. Each product type follows a distinct route. Product L requires 276 operations and passes through the bottleneck16 times. That is, 17 layers are processed. Product M requires 330 operations and passes through the bottleneck21 times (22 layers processed).
The releasing batch size is set to six lots for effective use of many workstations, which have a maximum batch size (MBS) of six lots. Lots with different product types cannot be processed simultaneously. Wafer lots are released under CONWIP, a fixed work-in-process (WIP) policy. The dispatching rule is first-in, first-out (FIFO). Product price is generally determined by the supply and demand of the market according to the product type and the number of layers the product needs to be processed. Because product price can vary tremendously in different time
horizons, several cases of price variation are analyzed in this paper and will be covered in detail in the next section. Direct material cost is assumed to be $100 per wafer. Indirect material cost, such as photo-resist, special gas, chemical and quartz, is set to $7.5 per layer for product L and $8 per layer for product M. A holding cost is considered for the WIP at an annual rate of 10%. Because CONWIP is adopted, WIP level is consistent throughout the period. Material is the major variable cost of a product, and the holding cost of material of the WIP will be calculated as: Total material cost of the WIP holding rate for the period.
The simulation program used in this research is EM-PLANT (Tecnomatix, 2001). The planning horizon in the simulation is set to 168 working days. Each working day consists of 24 working hours. The first 84 days are a warm-up period; hence, only results belonging to the next 84 days are collected. The simulation model is run 15 times to get adequate statistical results under each product mix. The data obtained from running simulation for each product mix (represented by I–IX) with manufacturing ratio of L to M is shown inTable 1.
4. ANP for product mix determination
In this section, an ANP approach is used. A total of nine experts contributed their professional experience to identify criteria and subcriteria that influence the decision and constructed the network Table 1
Simulation results
Product mix WIP (lots) TP (lots) T L(#) CT (h) BU (%) CU (%) VC ($)
I: Mix (1:9) 249.05 540 11,124 309.92 0.99 0.9 3,679,743 II: Mix (2:8) 248.9 548 11,070 305.2 0.99 0.87 3,667,558 III: Mix (3:7) 248.95 556 11,009 300.85 0.99 0.85 3,653,426 IV: Mix (4:6) 249.09 562 10,903 296.76 0.99 0.82 3,624,447 V: Mix (5:5) 249.27 570 10,830 292.85 0.99 0.79 3,606,663 VI: Mix (6:4) 249.45 580 10,788 288.99 0.99 0.76 3,599,343 VII: Mix (7:3) 249.61 588 10,702 285.23 0.99 0.76 3,577,421 VIII: Mix (8:2) 249.74 596 10,609 281.6 0.99 0.7 3,553,551 IX: Mix (9:1) 252.18 608 10,579 278.78 0.99 0.67 3,551,096
of determining the efficient manufacturing perfor-mance as already shown inFig. 4. Six of them are senior managers of production planning and finance departments from three internationally well-known semiconductor manufacturing compa-nies in Science-Based Industrial Parkin Taiwan, and the other three are scholars in production management from three universities in Taiwan. Here we take into consideration the interrelation-ship among criteria with respect to the goal and also the interrelationship among detailed criteria with respect to an upper level criterion (i.e., product, equipment efficiency and finance).
The first step is to construct the comparison matrices at each component for pairwise compar-ison of the factors inside the component. The Delphi method was performed to obtain a consensus among the people who were involved (Forgarty et al., 1989). The Delphi method consisted of a series of repeated interrogations through questionnaires of a group of experts and managers whose judgments were of interest in order to arrive at a group position regarding an issue. After the initial interrogation of each individual, each subsequent interrogation was accompanied by providing information of preced-ing round of replies. Individuals were encouraged to reconsider and change their previous reply with the consideration of the replies of other members of the group. The group position was finally determined after four rounds.
There are a total of 15 pairwise comparisons. The goal and criteria for each pairwise comparison are listed in Appendix B, and the details are explained as follows. First, the importance of each decision criteria with respect to achieving the
overall objective, the efficient production perfor-mance, is determined. The question, ‘‘which criteria should be emphasized more in determining efficient manufacturing, and how much more?’’ was asked, and a nine-point scale was used to do the pairwise comparison. An example of pairwise comparison is shown inTable 2, and the compar-ison matrix for comparing the criteria (product, equipment efficiency and finance) in terms of their contribution to achieving the overall objective is shown inTable 3.
The priorities for the criteria, w21; can be
obtained by the procedure stated in Step 2 in the previous section and is
w21¼ P E F 0:258 0:105 0:637 2 6 4 3 7 5: ð1Þ
The eigenvector shows the priority of the three criteria. In the opinion of the senior managers and experts, finance, with a weight of 0.637, is the major factor in determining the efficiency of manufacturing performance simply because profit-ability is the ultimate goal of a company. Product and equipment efficiency rankthe second and the
Table 2
Pairwise comparison
In order to achieve the most efficient manufacturing performance, which criteria should be emphasized more?
Absolute Very strong Strong WeakEqual WeakStrong Very strong Absolute 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 1:1 1:2 1:3 1:4 1:5 1:6 1:7 1:8 1:9
Product X Equipment
efficiency
Product X Finance
Equipment efficiency X Finance
Table 3
Comparison matrix for the criteria Product Equipment efficiency Finance Product 1 3 1/3 Equipment efficiency 1/3 1 1/5 Finance 3 5 1
third with weights of 0.258 and 0.105. The consistency of this combination of values in the matrix is also checked to make sure that the interviewees do not make inconsistent judgment.
Assume there is no interdependence among detailed criteria, which detailed criteria should be emphasized more in determining their respective upper level criterion? The comparison matrices of detailed criteria in accordance to their respective upper level criteria (product, equipment, finance) and their eigenvectors are obtained. The comparison matrix and eigenvector for product are shown in Table 4. Pairwise comparison among its detailed criteria, WIP, throughput, total layers and cycle time, shows that total layers, the total number of layers the bottleneckcan process in a period of time, is the most important factor with a weight of 0.499, followed by throughput with a weight of 0.284. For the criteria equipment efficiency, bottleneck utilization, with a weight of 0.9, is the main focus since it governs the total output of the manufacturing system. For the criteria finance, total revenue, with a weight of 0.857, is relatively much more important than variable costs. In the group’s judgment, fixed cost accounts for the majority part of the total manufacturing costs, and variable costs under different manufacturing environments generally do not vary too much either. Revenue, on the other hand, is the indication of the success of a company and thus has a much higher contribution.
The eigenvectors for product ðw32ðPÞÞ; equipment
efficiency ðw32ðEÞÞ and finance ðw32ðF ÞÞ are organized
into a matrix, W32; that represents the relative
importance of detailed criteria with respect to their
upper level criteria.
W32¼ P E F WIP TP TL CT BU CU TR VC 0:134 0 0 0:284 0 0 0:499 0 0 0:083 0 0 0 0:900 0 0 0:100 0 0 0 0:857 0 0 0:143 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : ð2Þ
As stated in the previous section, product price fluctuates in time and is determined by the market condition at the time of order commitment. Prices of different product types can also be very different. For instance, even though the operations of logic I.C. are less than memory I.C., it may require a special recipe and its process can be more complex. In addition, the quantity of logic I.C. demanded by the clients is usually smaller than that of memory I.C. Therefore, the price of logic I.C. is generally higher than memory I.C. Because finance is the major concern of the people interviewed, and revenue under finance is much more important than the other factor, variable costs, we will evaluate several cases of different product prices under our simulation model.
When the price of logic I.C. (PL) is higher than or equal to that of memory I.C. ðPMÞ; total
revenue has an increasing trend as the ratio of product L increases. This is because product L requires less of operations than product M, and the total throughput increases as a higher ratio of product L is manufactured. The dual effect of a higher price of PL and a higher throughput
increases the total revenue as more product L is manufactured. On the other hand, when PL is
lower than PM; total revenue may increase,
decrease, or indeterminate from manufacturing product mix (1:9) to (9:1). As just mentioned, the total throughput increases as a higher ratio of product L is manufactured. The revenue generated from a higher throughput of product L with a lower price and a lower throughput of product M with a higher price, thus, is a result of counter-effects. The trend of total revenue will be Table 4
Comparison matrix and eigenvector for product
Product WIP TP TL CT Relative importance weights (eigenvector, w32ðPÞ) WIP 1 1/3 1/3 2 0.134 TP 3 1 1/3 4 0.284 TL 3 3 1 4 0.499 CT 1/4 1/4 1/4 1 0.083 CR ¼ 0:07:
determined by the difference between PLand PM:
For instance, if PL is $US 100 lower than PM; as
more product L is manufactured, total revenue decreases when PLis less than or equal to $400 and
increases when PL is more than or equal to $900.
However, when PLis between $450 and $850, total
revenue may either increase or decrease. Table 5
andFig. 2show some results of total revenue when PL is $100 lower than PM:Fig. 3shows the total
revenue when PL is $150 lower than PM: Total
revenue decreases as more product L is manufac-tured when PL is less than or equal to $700, and
increases when PLis more than or equal to $1350.
InFig. 4, total revenues of different product mixes when PL is $200 lower than PM are analyzed. As
more product L is manufactured, total revenue decreases when PL is less than or equal to $900,
and increases when PL is more than or equal to
$1800. Note that the results of total revenue of PM PL¼ $200 are exactly double of the results
of PM PL¼ $100: For example, when PL¼
$500 and PM ¼ $600; Mix (1:9) has a total revenue
of $7,965,000. When PL¼ $1000 and PM ¼ $1200;
Mix (1:9) has a total revenue of $15,930,000; both prices and total revenue are a double of the previous case. The total revenue trend of PL¼
$1000 and PM¼ $1200 is exactly the same as that
of PL¼ $500 and PM¼ $600; only that the
amount is double. Same conclusions can be made for PM PL¼ $300 and PM PL¼ $150; just to
name a few.
Following are four representative cases that will be used for our ANP analysis, and the total revenues of different product mixes under the four cases are shown inTable 6.
Case I: Price of logic I.C. ðPLÞ is greater than or
equal to that of memory I.C. ðPMÞ: If we let PLbe
$US 1200 and PMbe $US 1000, the total revenues
has an increasing trend as the ratio of product L increases.
Table 5
Total revenues of different product mixes when PM PL¼ $100 (US$)
Mix (L:M) PL¼ $450; PL¼ $500; PL¼ $600; PL¼ $850; PL¼ $900; PM¼ $550 PM¼ $600 PM¼ $700 PM¼ $950 PM¼ $1000 Mix(1:9) 7,290,000 7,965,000 9,315,000 12,690,000 13,365,000 Mix(2:8) 7,261,000 7,946,000 9,316,000 12,741,000 13,426,000 Mix(3:7) 7,228,000 7,923,000 9,313,000 12,788,000 13,483,000 Mix(4:6) 7,165,500 7,868,000 9,273,000 12,785,500 13,488,000 Mix(5:5) 7,125,000 7,837,500 9,262,500 12,825,000 13,537,500 Mix(6:4) 7,105,000 7,830,000 9,280,000 12,905,000 13,630,000 Mix(7:3) 7,056,000 7,791,000 9,261,000 12,936,000 13,671,000 Mix(8:2) 7,003,000 7,748,000 9,238,000 12,963,000 13,708,000 Mix(9:1) 6,992,000 7,752,000 9,272,000 13,072,000 13,832,000 Changes of revenue as product L increases
Decrease Decrease, then increase Increase and decrease alternately Increase, decrease and increase Increase 6,000,000 8,000,000 10,000,000 12,000,000 14,000,000 16,000,000
I II III IV V VI VII VIII IX Product Mix T o tal Re v enue PL=450, PM=550 PL=500, PM=600 PL=600, PM=700 PL=850, PM=950 PL=900, PM=1,000
Case II: Price of logic I.C. (PL) is lower than
that of memory I.C. (PM). Assume PM PL¼
$200; and PL¼ $800 and PM = $1000. Total
revenue has a decreasing trend as the ratio of product L increases.
Case III: Price of logic I.C. ðPLÞ is lower than that
of memory I.C. ðPMÞ: Assume PM PL¼ $200;
and PL¼ $1200 and PM¼ $1400: Total revenue
fluctuates as the ratio of product L increases.
Case IV: Price of logic I.C. ðPLÞ is lower than
that of memory I.C. ðPMÞ: Assume PM PL¼
$200; and PL¼ $1900 and PM¼ $2100: Total
revenue has an increasing trend as the ratio of product L increases.
Since there is no interdependence among alter-natives (product mix), the alteralter-natives are com-pared with respect to each detailed criterion yielding the column eigenvectors regarding each
10,000,000 12,000,000 14,000,000 16,000,000 18,000,000 20,000,000 22,000,000
I II III IV V VI VII VIII IX Product Mix T o tal Re v enue 700 v 850 750 v 900 900 v 1050 1300 v 1450 1350 v 1500
Fig. 3. Total revenues of different product mixes when PM PL¼ $150:
10,000,000 14,000,000 18,000,000 22,000,000 26,000,000 30,000,000
I II III IV V VI VII VIII IX
Product Mix T o tal Re v enue 900 v 1100 1000 v 1200 1200 v 1400 1700 v 1900 1800 v 2000
Fig. 4. Total revenues of different product mixes when PM PL¼ $200:
Table 6
Total revenues of different product mixes under the four cases ($US)
Product mix Case I: Case II: Case III: Case IV:
(L:M) PL¼ $1200 and PL¼ $800 and PL¼ $1200 and PL¼ $1900 and
PM¼ $1000 PM¼ $1000 PM¼ $1400 PM¼ $2100 I: Mix (1:9) 13,770,000 13,230,000 18,630,000 28,080,000 II: Mix (2:8) 14,248,000 13,152,000 18,632,000 28,222,000 III: Mix (3:7) 14,734,000 13,066,000 18,626,000 28,356,000 IV: Mix (4:6) 15,174,000 12,926,000 18,546,000 28,381,000 V: Mix (5:5) 15,675,000 12,825,000 18,525,000 28,500,000 VI: Mix (6:4) 16,240,000 12,760,000 18,560,000 28,710,000 VII: Mix (7:3) 16,758,000 12,642,000 18,522,000 28,812,000 VIII: Mix (8:2) 17,284,000 12,516,000 18,476,000 28,906,000 IX: Mix (9:1) 17,936,000 12,464,000 18,544,000 29,184,000
detailed criterion. The simulation data fromTable 1 and the different cases of total revenue from
Table 6 are used here to form the comparison matrices of alternatives (product mixes) with respect to each detailed criteria (WIP, throughput, etc.). The reason of not using the opinion of senior managers and experts in identifying the relative score of the alternatives with respect to each of the detailed criteria is because simulation data, which indicate the manufacturing performance of a fab under different product mixes, are objective measures to reflect the efficiency of manufacturing. Because the unit of measurement of simulation data is various and can range from number of layers to percentage and to dollars, these quanti-tative data need to be transformed into values between zero to one. The concept of utility function is adopted to obtain a utility index and to show the relative performance of a factor under each product mix. By assigning values of zero and one to the worst and best outcomes, the general formula of a utility linear function of detailed criteria m is as follows (Clemen, 1996):
umðxÞ ¼ X Xm Xþ m Xm ; ð3Þ where Xþ
m is the best value of detailed criteria m;
X
m the worst value of detailed criteria m; X the
value of detailed criteria m under a certain product mix.
In this paper, numerous simulations are run to collect sufficient data, and Xmþ and Xm are the values that are suitable to be the two extreme levels of performance. For some detailed criteria (throughput, total layers, BN utilization, CCR utilization and total revenue), their values are preferred to be as great as possible, and thus the best factors are those with the largest values, and vice versa. On the other hand, for other detailed criteria (WIP, total cycle time and variable costs), we prefer their values to be small, and therefore the best ones have the smallest values. The simulation results in Table 2 and total revenue data from Table 6 are transformed into utility indices as shown in Table 7.
The utility indices are then transformed into weights by dividing each utility index to the total value of the column so that each column sums up
to one. Table 8 shows the matrix, W43ð1Þ; that
indicates the utility weights of alternatives with respect to each detailed criterion for Case I. For other cases, the matrix can be obtained by replacing the column of total revenue (TR) by the column of the specific case listed inTable 9.
The inner dependence among the criteria is determined through analyzing the impact of each criterion on other criteria by using pairwise comparisons. We askquestions such as ‘‘What is the relative importance of equipment efficiency when compared to finance on controlling pro-duct?’’ The result is 2 as denoted inTable 10, which shows the inner dependence matrix of criteria with respect to product. The dependencies among criteria are depicted in Fig. 5. The resulting eigenvectors obtained from pairwise comparisons formed matrix, W22; and are shown in Table 11.
Note that zeros are assigned to the eigenvector weights of criteria that are independent.
The inner dependence among the detailed criteria is analyzed next. A similar procedure as for the inner dependence among criteria is adopted here, except that correlation analysis of the simulation results is performed first and the results are used as a reference for the experts. A possible question is as follows: ‘‘What is the relative importance of cycle time (CT) when compared to bottleneck utilization (BU) on controlling through-put (TP)?’’ The answer is 2 as shown inTable 12. The schematic representation of the relationship among detailed criteria is presented inFig. 6. The relative importance weights of the inner depen-dence among detailed criteria are represented by W33 inTable 13.
A supermatrix allows for the resolution of the effects of interdependence between the elements of the system. It is a partitioned matrix, where each submatrix is composed of the vectors obtained from the pairwise comparison. As discussed in the appendix and shown by the dotted bracket in
Fig. 10, the supermatrix in this paper covers all the components in the network. The generalized form of the supermatrix is shown inFig. 7.
The supermatrix for Case I, inserted with respective vectors and matrices obtained before, is shown in Table 14. Because the supermatrix includes interactions between clusters, e.g. there is
inner dependence among criteria and among detailed criteria, not each of the columns sums to one. A weighted supermatrix for Case I is transformed first to be stochastic as shown by MWI inTable 15. The weighted supermatrix is then raised to limiting powers to be M2kþ1I (where k is an arbitrarily large number) to capture all the interactions and to obtain a steady-state outcome. In Case I, convergence is reached at M33I ; and the
limit supermatrix, which shows the long-term stable weighted values, is shown inTable 16.
The overall priorities for the alternatives (pro-duct mixes) for Case I is given by the bottom left corner, the (4,1) block, of M33I : The alternative with the largest priority index should be the one selected. Alternative IX (product mix (9:1)), with a relative importance value of 0.1639, is the most efficient manufacturing product mix, followed by Table 7
Utility index of each detailed criteria under different product mix
Alternative WIP TP TL CT BU CU VC I 0.547 0.267 0.562 0.002 1.000 1.000 0.101 II 0.555 0.320 0.535 0.120 1.000 0.900 0.162 III 0.553 0.373 0.505 0.229 1.000 0.833 0.233 IV 0.546 0.413 0.452 0.331 1.000 0.733 0.378 V 0.537 0.467 0.415 0.429 1.000 0.633 0.467 VI 0.528 0.533 0.394 0.525 1.000 0.533 0.503 VII 0.519 0.587 0.351 0.619 1.000 0.533 0.613 VIII 0.513 0.640 0.305 0.710 1.000 0.333 0.732 IX 0.391 0.720 0.290 0.781 1.000 0.233 0.745 Sum of column 4.688 4.320 3.807 3.746 9.000 5.733 3.934
Alternative TR (Case I) TR (Case II) TR (Case III) TR (Case IV)
I 0.154 0.615 0.630 0.040 II 0.250 0.576 0.632 0.111 III 0.347 0.533 0.626 0.178 IV 0.435 0.463 0.546 0.191 V 0.535 0.413 0.525 0.250 VI 0.648 0.380 0.560 0.355 VII 0.752 0.321 0.522 0.406 VIII 0.857 0.258 0.476 0.453 IX 0.987 0.232 0.544 0.592 Sum of column 4.964 3.791 5.061 2.576 Table 8
Utility weights of alternatives with respect to each detailed criterion for Case I, W43ðIÞ
Alternative WIP TP TL CT BU CU TR VC I 0.117 0.062 0.148 0.001 0.111 0.174 0.031 0.026 II 0.118 0.074 0.141 0.032 0.111 0.157 0.050 0.041 III 0.118 0.086 0.133 0.061 0.111 0.145 0.070 0.059 IV 0.116 0.096 0.119 0.088 0.111 0.128 0.088 0.096 V 0.114 0.108 0.109 0.114 0.111 0.110 0.108 0.119 VI 0.113 0.123 0.103 0.140 0.111 0.093 0.131 0.128 VII 0.111 0.136 0.092 0.165 0.111 0.093 0.151 0.156 VIII 0.109 0.148 0.080 0.190 0.111 0.058 0.173 0.186 IX 0.083 0.167 0.076 0.208 0.111 0.041 0.199 0.189
alternative VIII (product mix (8:2)) with a value of 0.1502. The ranking of the product mixes for Case I is exactly the descending order from alternative IX to I, that is, from product mix (9:1) to (1:9). This implies that product L is highly recommended for manufacturing. The outcome is not hard to explain from the simulation results and the matrices obtained before. In the group’s opinion, finance, with a weight of 0.637, is much more important than other criteria. In addition, under criteria finance, total revenue is a lot more important than variable costs. Total revenue has an increasing trend as a higher ratio of product L, which has a higher price than M, is produced. In addition, factors such as throughput, cycle time and variable costs, all favor in manufacturing product L.
The overall priorities of the alternatives under the other three cases are shown in Table 17. The ranking of the product mixes for Case II is in the order as the ratio of product L increases until meeting alternative IX, which has a higher overall priority than alternative VIII. For Case IV, the efficiency of the product mixes increases from
alternative I–IX. Case III is a good representative case which shows that total revenue is not the only determinant of the final ranking of product mixes.
Table 18shows the ranking of alternatives by total revenue versus by the ANP. The alternative with the highest total revenue is alternative II, followed by alternative I and III. However, alternative IX ranks the first under the ANP analysis, and the ranking is in the order from alternative IX–I. This indicates that the ANP ranking is not solely influenced by the total revenue, but also by the other factors and the interactions among the factors.
5. Conclusions
The ANP is presented in this paper as a valuable method to support the selection of product mix that is efficient for a wafer fab to manufacture.
Table 9
Utility weight of alternatives with respect to total revenue (TR) for Cases II, III and IV
TR (Case II) TR (Case III) TR (Case IV)
I 0.162 0.124 0.016 II 0.152 0.125 0.043 III 0.141 0.124 0.069 IV 0.122 0.108 0.074 V 0.109 0.104 0.097 VI 0.100 0.111 0.138 VII 0.085 0.103 0.158 VIII 0.068 0.094 0.176 IX 0.061 0.107 0.230 Table 10
Inner dependence matrix of criteria with respect to product
Product Product Equipment
efficiency
Finance Relative importance weights (eigenvector)
Product 1 2 3 0.529
Equipment efficiency 1/2 1 2 0.309
Finance 1/3 1/2 1 0.162
Equipment Product Finance
Fig. 5. Inner dependence among criteria.
Table 11
Inner dependence matrix of criteria, W22 Product Equipment efficiency Finance Product 0.529 0.25 0.100 Equipment efficiency 0.309 0.75 0 Finance 0.162 0 0.900
The relative prices of products affect the total revenue trend under different product mixes, and the ANP can be applied to determine the efficiency of product mix when product prices change. Because profitability is the key to success for an enterprise, relative prices of products and the total revenues generated from different product mixes can often determine the desirable product mix. This is shown by the trend of the ranking of alternatives under most cases studied in this paper. However, there are times when the ranking of alternatives is not exactly the same as the ranking of total revenue of alternatives. This means that factors other than total revenue do have impact on the efficiency evaluation of product mixes. In this paper, the approach adopts ANP to deal with interrelated factors and provides users with procedures to be followed for the determination of a suitable product mix for wafer fabrication.
When product mix is predictable or when all products use each facility equally, an overall production forecast is sufficient to determine equipment requirements. However, when various products follow distinct routes and have different loadings on machines, any deviation from the product mix target can make the workstation utilizations time-varying over the planning hor-izon. The time-varying utilizations will in turn cause time-varying cycle times. How product mix should be set in a longer term will be our future research direction. In addition, due to the intensive competition in semiconductor industry, a variety of products are usually required to be Table 12
Inner dependence matrix of detailed criteria with respect to throughput
TP WP TP TL CT BU CU TR VC Relative importance weights
(eigenvector) WP 1 1/2 0 1 2 0 1/2 0 0.168 TP 2 1 0 2 3 0 2 0 0.3358 TL 0 0 0 0 0 0 0 0 0 CT 1 1/2 0 1 2 0 1 0 0.1848 BU 1/2 1/3 0 1/2 1 0 1/2 0 0.0958 CU 0 0 0 0 0 0 0 0 0 TR 2 1/2 0 1 2 0 1 0 0.218 VC 0 0 0 0 0 0 0 0 0 WP TP TL CT BU CU TR VC
Fig. 6. Inner dependence among detailed criteria.
Table 13
Inner dependence matrix of detailed criteria, W33
WIP TP TL CT BU CU TR VC WIP 0.667 0.168 0 0 0 0 0 0.085 TP 0 0.335 0.333 0.333 0.333 0 0.308 0 TL 0 0 0.667 0 0 0 0 0.327 CT 0.333 0.184 0 0.667 0 0 0 0 BU 0 0.095 0 0 0.667 0.333 0 0 CU 0 0 0 0 0 0.667 0 0 TR 0 0.218 0 0 0 0 0.615 0.196 VC 0 0 0 0 0 0 0.077 0.392 I W34 W33 W32 W22 Alternatives Detailed Criteria Criteria Goal Alternatives Detailed Criteria Criteria Goal w21 I
AR
TI
CL
E
IN
P
RE
S
S
Table 14The supermatrix for Case I, MI
Goal Product Equipment Finance WP TP TL CT BU CU TR VC I II III IV V VI VII VIII IX Goal 1 Product 0.258 0.529 0.25 0.100 Equipment 0.105 0.309 0.75 0 Finance 0.637 0.162 0 0.900 WP 0.134 0 0 0.667 0.168 0 0 0 0 0 0.085 TP 0.284 0 0 0 0.335 0.333 0.333 0.333 0 0.308 0 TL 0.499 0 0 0 0 0.667 0 0 0 0 0.327 CT 0.083 0 0 0.333 0.184 0 0.667 0 0 0 0 BU 0 0.9 0 0 0.095 0 0 0.667 0.333 0 0 CU 0 0.1 0 0 0 0 0 0 0.667 0 0 TR 0 0 0.857 0 0.218 0 0 0 0 0.615 0.196 VC 0 0 0.143 0 0 0 0 0 0 0.077 0.392 I 0.117 0.062 0.148 0.001 0.111 0.174 0.031 0.026 1 II 0.118 0.074 0.141 0.032 0.111 0.157 0.05 0.041 1 III 0.118 0.086 0.133 0.061 0.111 0.145 0.07 0.059 1 IV 0.116 0.096 0.119 0.088 0.111 0.128 0.088 0.096 1 V 0.114 0.108 0.109 0.114 0.111 0.11 0.108 0.119 1 VI 0.113 0.123 0.103 0.14 0.111 0.093 0.131 0.128 1 VII 0.111 0.136 0.092 0.165 0.111 0.093 0.151 0.156 1 VIII 0.109 0.148 0.08 0.19 0.111 0.058 0.173 0.186 1 IX 0.083 0.167 0.076 0.208 0.111 0.041 0.199 0.189 1 S.-H. Chung et al. / Int. J. Production Economic s 9 6 (2005) 15 – 3 6
AR
TI
CL
E
IN
P
RE
S
S
Goal 0.5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Product 0.1290 0.2645 0.1250 0.0500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Equipment 0.0525 0.1545 0.3750 0.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Finance 0.3185 0.0810 0.0000 0.4500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 WP 0 0.0670 0.0000 0.0000 0.3334 0.0838 0 0 0 0 0 0.0425 0 0 0 0 0 0 0 0 0 TP 0 0.1420 0.0000 0.0000 0 0.1676 0.1667 0.1667 0.1667 0 0.1539 0 0 0 0 0 0 0 0 0 0 TL 0 0.2495 0.0000 0.0000 0 0.0000 0.3334 0 0 0 0 0.1634 0 0 0 0 0 0 0 0 0 CT 0 0.0415 0.0000 0.0000 0.1667 0.0922 0 0.3334 0 0 0 0 0 0 0 0 0 0 0 0 0 BU 0 0.0000 0.4500 0.0000 0 0.0475 0 0 0.3334 0.1667 0 0 0 0 0 0 0 0 0 0 0 CU 0 0.0000 0.0500 0.0000 0 0 0 0 0 0.3334 0 0 0 0 0 0 0 0 0 0 0 TR 0 0.0000 0.0000 0.4285 0 0.1090 0 0 0 0 0.3077 0.0981 0 0 0 0 0 0 0 0 0 VC 0 0.0000 0.0000 0.0715 0 0 0 0 0 0 0.0385 0.1961 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0.0585 0.0310 0.0740 0.0005 0.0555 0.0870 0.0155 0.0130 1 0 0 0 0 0 0 0 0 II 0 0 0 0 0.0590 0.0370 0.0705 0.0160 0.0555 0.0785 0.0250 0.0205 0 1 0 0 0 0 0 0 0 III 0 0 0 0 0.0590 0.0430 0.0665 0.0305 0.0555 0.0725 0.0350 0.0295 0 0 1 0 0 0 0 0 0 IV 0 0 0 0 0.0580 0.0480 0.0595 0.0440 0.0555 0.0640 0.0440 0.0480 0 0 0 1 0 0 0 0 0 V 0 0 0 0 0.0570 0.0540 0.0545 0.0570 0.0555 0.0550 0.0540 0.0595 0 0 0 0 1 0 0 0 0 VI 0 0 0 0 0.0565 0.0615 0.0515 0.0700 0.0555 0.0465 0.0655 0.0640 0 0 0 0 0 1 0 0 0 VII 0 0 0 0 0.0555 0.0680 0.0460 0.0825 0.0555 0.0465 0.0755 0.0780 0 0 0 0 0 0 1 0 0 VIII 0 0 0 0 0.0545 0.0740 0.0400 0.0950 0.0555 0.0290 0.0865 0.0930 0 0 0 0 0 0 0 1 0 IX 0 0 0 0 0.0415 0.0835 0.0380 0.1040 0.0555 0.0205 0.0995 0.0945 0 0 0 0 0 0 0 0 1 Chung et al. / Int. J. Production Economic s 9 6 (2005) 15 – 3 6 29AR
TI
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S
Table 16The limit supermatrix for Case I, M33 I
Goal Product Equipment Finance WP TP TL CT BU CU TR VC I II III IV V VI VII VIII IX Goal 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Product 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Equipment 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Finance 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 WP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TL 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CT 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 BU 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CU 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 VC 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0.0629 0.0899 0.1015 0.0454 0.0925 0.0592 0.1258 0.0163 0.0988 0.1552 0.0391 0.0514 1 0 0 0 0 0 0 0 0 II 0.0745 0.0963 0.1043 0.0605 0.099 0.0723 0.1246 0.0421 0.1021 0.144 0.0557 0.0635 0 1 0 0 0 0 0 0 0 III 0.0869 0.1024 0.1072 0.0769 0.1055 0.0856 0.1219 0.0679 0.1054 0.1358 0.0738 0.0767 0 0 1 0 0 0 0 0 0 IV 0.0984 0.1058 0.1089 0.0934 0.1095 0.0968 0.1142 0.0902 0.1082 0.123 0.0906 0.0998 0 0 0 1 0 0 0 0 0 V 0.1106 0.1108 0.1112 0.1102 0.1137 0.1095 0.1099 0.1129 0.1114 0.1103 0.1088 0.1162 0 0 0 0 1 0 0 0 0 VI 0.1246 0.1183 0.1147 0.1285 0.1196 0.1254 0.1093 0.1363 0.1153 0.0993 0.1301 0.124 0 0 0 0 0 1 0 0 0 VII 0.1371 0.123 0.118 0.1456 0.1238 0.1381 0.1035 0.159 0.1185 0.1001 0.1484 0.1427 0 0 0 0 0 0 1 0 0 VIII 0.1502 0.1273 0.119 0.1641 0.1276 0.1508 0.0977 0.1802 0.1217 0.074 0.1682 0.1628 0 0 0 0 0 0 0 1 0 IX 0.1639 0.133 0.1221 0.1828 0.1124 0.1664 0.0986 0.1976 0.1256 0.0629 0.1907 0.1674 0 0 0 0 0 0 0 0 1 S.-H. Chung et al. / Int. J. Production Economic s 9 6 (2005) 15 – 3 6
manufactured in order to satisfy customer demand. Multiple priority orders, such as hot lots, rush lots and normal lots, are often encountered in fabs. As a result, the manufacturing environment can be even more complex.
Green production is especially important for semiconductor manufacturing and will determine the sustainability of a company in the long term. Semiconductor fabs use a lot of hazardous chemicals in their manufacturing process. Differ-ent product mixes require differDiffer-ent chemicals and gases in the process, and the residual process gases and reactive by-products generally contain toxic and corrosive gases that may be harmful to the environment. As a result, the determination of a good product mix may not only include product, equipment efficiency and finance factors,
environ-mental issues may also need to be taken into consideration. This can also be our future focus of research.
Acknowledgements
The authors would like to thank the anonymous reviewers, Dr. He-Yau Kang and Dr. Ling-Fen Hsieh for their valuable comments and sugges-tions.
Appendix A. Review of the ANP
The analytic hierarchy process (AHP) was developed by Saaty in 1971, and its purpose was to structure a decision process in a scenario influenced by multiple independent factors (Saaty, 1980). A complex problem can be decomposed into several sub-problems in terms of hierarchical levels, where each level represents a set of criteria or attributes relative to each sub-problem. The top level of the hierarchy is the goal of the problem, and the intermediate levels represent the factors of respective upper level. The last level contains the alternatives or actions to be considered in the achievement of the goal. AHP allows the factors to be compared with the importance of each factor relative to its impact on the solution of the problem. Since its introduction, AHP has been widely used in decision-making, and numerous applications have been published in literature (Shim, 1989).
The ANP, also introduced by Saaty, is a generalization of the AHP (Saaty, 1996). Whereas AHP represents a frameworkwith a uni-direc-tional hierarchical relationship, ANP allows for more complex interrelationships among decision levels and attributes. The ANP feedbackapproach replaces hierarchies with networks, in which the relationships between levels are not easily repre-sented as higher or lower, dominated or being dominated, directly or indirectly (Meade and Sarkis, 1999). For instance, not only does the importance of the criteria determine the impor-tance of the alternatives as in a hierarchy, but also the importance of the alternatives may have Table 17
Overall priorities for the alternatives under different cases Product mix (L:M) Case I Case II Case III Case IV I: Mix (1:9) 0.0629 0.1175 0.1006 0.0562 II: Mix (2:8) 0.0745 0.1173 0.1049 0.072 III: Mix (3:7) 0.0869 0.1171 0.1084 0.0869 IV: Mix (4:6) 0.0984 0.1127 0.1081 0.0925 V: Mix (5:5) 0.1106 0.1115 0.1103 0.1065 VI: Mix (6:4) 0.1246 0.1112 0.1158 0.1271 VII: Mix (7:3) 0.1371 0.1094 0.1176 0.1397 VIII: Mix (8:2) 0.1502 0.1057 0.1182 0.151 IX: Mix (9:1) 0.1639 0.1059 0.1226 0.1764 Table 18
Ranking of the alternatives by total revenue and by ANP for Case III Product mix (L:M) Total revenue (US$) Ranking by total revenue Ranking by ANP I: Mix (1:9) 18,630,000 2 9 II: Mix (2:8) 18,632,000 1 8 III: Mix (3:7) 18,626,000 3 7 IV: Mix (4:6) 18,546,000 5 6 V: Mix (5:5) 18,525,000 7 5 VI: Mix (6:4) 18,560,000 4 4 VII: Mix (7:3) 18,522,000 8 3 VIII: Mix (8:2) 18,476,000 9 2 IX: Mix (9:1) 18,544,000 6 1
impact on the importance of the criteria (Saaty, 1996). Therefore, a hierarchical structure with a linear top-to-bottom form is not applicable for a complex system.
A system with feedbackcan be represented by a networkwhere nodes correspond to the levels or components (Saaty, 1980). The structural differ-ence between a hierarchy and a networkis depicted in Fig. 8. The elements in a node (or level) may influence some or all the elements of any other node. In a network, there can be source nodes, intermediate nodes and sinknodes. Rela-tionships in a networkare represented by arcs, and the directions of arcs signify dependence (Saaty, 1996). Interdependency between two nodes, termed outer dependence, is represented by a two-way arrow, and inner dependencies among elements in a node are represented by a looped arc (Sarkis, 2002).
The process of ANP comprises four major steps (Meade and Sarkis, 1999;Saaty, 1996).
Step 1: Model construction and problem structur-ing: The problem should be stated clearly and decomposed into a rational system like a network. The structure can be obtained by the opinion of decision makers through brainstorming or other appropriate methods. An example of the format of a networkis as shown inFig. 8(b).
Step 2: Pairwise comparisons matrices and priority vectors: In ANP, like AHP, decision elements at each component are compared pair-wise with respect to their importance towards their control criterion, and the components themselves are also compared pairwise with respect to their contribution to the goal. Decision makers are asked to respond to a series of pairwise
compar-isons where two elements or two components at a time will be compared in terms of how they contribute to their particular upper level criterion (Meade and Sarkis, 1999). In addition, if there are interdependencies among elements of a compo-nent, pairwise comparisons also need to be created, and an eigenvector can be obtained for each element to show the influence of other elements on it. The relative importance values are determined with a scale of 1 to 9, where a score of 1 represents equal importance between the two elements and a score of 9 indicates the extreme importance of one element (row component in the matrix) compared to the other one (column component in the matrix) (Meade and Sarkis, 1999). A reciprocal value is assigned to the inverse comparison; that is, aij¼ 1=aji; where aijðajiÞ
denotes the importance of the ith (jth) element compared to the jth (ith) element. Like AHP, pairwise comparison in ANP is made in the frameworkof a matrix, and a local priority vector can be derived as an estimate of relative impor-tance associated with the elements (or compo-nents) being compared by solving the following equation:
A w ¼ lmax w; ðA:1Þ
where A is the matrix of pairwise comparison, w is the eigenvector, and lmax is the largest eigenvalue
of A.Saaty (1980)proposes several algorithms for approximating w: In this paper, the following three-step procedure is used to synthesize priorities (Anderson et al., 1997; Meade and Presley, 2002;
Saaty, 1980).
1. Sum the values in each column of the pairwise comparison matrix.
2. Divide each element in a column by the sum of its respective column. The resultant matrix is referred to as the normalized pairwise compar-ison matrix.
3. Sum the elements in each row of the normalized pairwise comparison matrix, and divide the sum by the n elements in the row. These final numbers provide an estimate of the relative priorities for the elements being compared with respect to its upper level criterion.
(a) (b)
Fig. 8. Structural difference between a hierarchy and a network: (a) a hierarchy; (b) a network.
Priority vectors must be derived for all comparison matrices.
Step 3: Supermatrix formation: The supermatrix concept is similar to the Markov chain process (Saaty, 1996). To obtain global priorities in a system with interdependent influences, the local priority vectors are entered in the appro-priate columns of a matrix, known as a super-matrix. As a result, a supermatrix is actually a partitioned matrix, where each matrix segment represents a relationship between two nodes (components or clusters) in a system (Meade and Sarkis, 1999). Let the components of a decision system be Ck; k ¼ 1; y; n; and each component k
has mkelements, denoted by ek1; ek2; y; ekmk: The local priority vectors obtained in Step 2 are grouped and located in appropriate positions in a supermatrix based on the flow of influence from a component to another component, or from a component to itself as in the loop. A standard form of a supermatrix is as in (A.2) (Saaty, 1996).
As an example, the supermatrix representation of a hierarchy with three levels as shown in
Fig. 9(a), is as follows (Saaty, 1996):
Wh¼ 0 0 0 w21 0 0 0 W32 I 2 6 4 3 7 5; ðA:3Þ
where w21 is a vector that represents the impact of
the goal on the criteria, W32 is a matrix that
represents the impact of criteria on each of the alternatives, I is the identity matrix, and entries of zeros corresponding to those elements that have no influence.
For the above example, if the criteria are interrelated among themselves, the hierarchy is replaced by a networkas shown inFig. 9(b). The (2, 2) entry of Wngiven by W22 would indicate the
interdependency, and the supermatrix would be (Saaty, 1996) Wn¼ 0 0 0 w21 W22 0 0 W32 I 2 6 4 3 7 5: ðA:4Þ
Note that any zero in the supermatrix can be replaced by a matrix if there is an interrelationship W ¼ C1 ? Ck ? Cn e11 e12 ? e1m1 ? ek1 ek2 ? ekmk ? en1 en2 ? enmn e11 C1 e12 ^ e1m1 ^ ^ ek1 ek2 Ck ^ ekmk ^ ^ en1 Cn en2 ^ enmn W11 ? W1k ? W1n ^ ^ ^ Wk1 ? Wkk ? Wkn ^ ^ ^ Wn1 ? Wnk ? Wnn 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ðA:2Þ
of the elements in a component or between two components. Since there usually is interdepen-dence among clusters in a network, the columns of a supermatrix usually sum to more than one. The supermatrix must be transformed first to make it stochastic, that is, each column of the matrix sums to unity. A recommended approach by Saaty (1996) is to determine the relative importance of the clusters in the supermatrix with the column cluster (block) as the controlling component (Meade and Sarkis, 1999). That is, the row components with nonzero entries for their blocks in that column block are compared according to their impact on the component of that column block(Saaty, 1996). With pairwise comparison matrix of the row components with respect to the column component, an eigenvector can be obtained. This process gives rise to an eigenvector for each column block. For each column block, the first entry of the respective eigenvector is multiplied by all the elements in the first blockof that column, the second by all the elements in the second blockof that column and so on. In this way, the blocks in each column of the supermatrix is weighted, and the result is known as the weighted supermatrix, which is stochastic.
Raising a matrix to powers gives the long-term relative influences of the elements on each other. To achieve a convergence on the importance weights, the weighted supermatrix is raised to the power of 2k þ 1; where k is an arbitrarily large number, and this new matrix is called
the limit supermatrix (Saaty, 1996). The limit supermatrix has the same form as the weighted supermatrix, but all the columns of the limit supermatrix are the same. By normalizing each blockof this supermatrix, the final priorities of all the elements in the matrix can be obtained.
Step 4: Selection of best alternatives: If the supermatrix formed in Step 3 covers the whole network, the priority weights of alternatives can be found in the column of alternatives in the normalized supermatrix. On the other hand, if a supermatrix only comprises of components that are interrelated, additional calculation must be made to obtain the overall priorities of the alternatives. The alternative with the largest overall priority should be the one selected. In this paper, the first method is applied, and a supermatrix that covers the whole networkas shown by the bracket in Fig. 10, is formed.
Appendix B. Pairwise comparisons
In this paper, a total of 15 pairwise compari-sons are involved. The goal and criteria for each pairwise comparison are listed as in
Table 19. w21 w21 W22 W32 W32 Goal Criteria Alternatives Goal Criteria Alternatives (a) (b)
Fig. 9. Hierarchy and Network(Momoh and Zhu, 1998): (a) a hierarchy; (b) a network. Criteria Alternatives Goal Detailed Criteria
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Saaty, T.L., Takizawa, M., 1986. Dependence and indepen-dence: From linear hierarchies to nonlinear networks. European Journal of Operational Research 26, 229–237. Sarkis, J., 2002. Quantitative models for performance
measure-ment systems—alternate considerations. International Jour-nal of Production Economics 86, 81–90.
Table 19
Pairwise comparisons considered
Goal Criteria
1 To achieve the most efficient manufacturing performance Product, equipment efficiency and finance 2 To achieve the best performance in product (assuming no
interdependence among factors)
WIP (WP), throughput (TP), total layers (TL) and total cycle time (CT)
3 To achieve the best equipment utilization (assuming no interdependence among factors)
Bottleneckutilization (BU) and CCR utilization (CU) 4 To achieve the best performance in finance (assuming no
interdependence among factors)
Total revenue (TR) and total variable costs (VC) 5 Inner dependence of criteria with respect to product Product, equipment efficiency and finance 6 Inner dependence of criteria with respect to equipment
efficiency
Product, equipment efficiency and finance 7 Inner dependence of criteria with respect to finance Product, equipment efficiency and finance 8 Inner dependence of detailed criteria with respect to WP WP, TP, TL, CT, BU, CU, TR, VC 9 Inner dependence of detailed criteria with respect to TP WP, TP, TL, CT, BU, CU, TR, VC 10 Inner dependence of detailed criteria with respect to TL WP, TP, TL, CT, BU, CU, TR, VC 11 Inner dependence of detailed criteria with respect to CT WP, TP, TL, CT, BU, CU, TR, VC 12 Inner dependence of detailed criteria with respect to BU WP, TP, TL, CT, BU, CU, TR, VC 13 Inner dependence of detailed criteria with respect to CU WP, TP, TL, CT, BU, CU, TR, VC 14 Inner dependence of detailed criteria with respect to TR WP, TP, TL, CT, BU, CU, TR, VC 15 Inner dependence of detailed criteria with respect to VC WP, TP, TL, CT, BU, CU, TR, VC
SEMI, 2002. Provisional guideline for definition and calculation of overall factory efficiency (OFE) and other associated factory-level productivity metrics. SEMI Draft Document 3454. Shim, J.P., 1989. Bibliographical research on the analytic hierarchy
process. Socio-Economic Planning Sciences 23, 161–167.
Suwignjo, P., Bititci, U.S., Carrie, A.S., 2000. Quantitative models for performance measurement system. International Journal of Production Economics 64, 231–241.
Tecnomatix Technologies Ltd., 2001. EM-PLANT Objects Manual. Tecnomatix Software Company, Germany.
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