DOI 10.1007/s00170-003-1956-8 O R I G I N A L A R T I C L E
Shu-Hsing Chung · Amy H.I. Lee · W.L. Pearn
Product Mix Optimization for Semiconductor Manufacturing Based on AHP
and ANP Analysis
Received: 21 June 2003 / Accepted: 15 September 2003 / Published online: 20 February 2004 Springer-Verlag London Limited 2004
Abstract In a competitive market, a company needs to utilize its available capacity efficiently in order to acquire high profit. The purpose of this paper is to present effective approaches to find a set of product mix optimal for the company to achieve the optimal manufacturing. Analytic hierarchy process (AHP) and analytic network process (ANP) approaches are taken to analyse multiple process inputs and outputs, incorporating experts’ opin-ion on their priority of importance, to obtain optimal product mixes for semiconductor production. The results provide guid-ance to the fabricator regarding strategies for accepting orders to maximize the manufacturing efficiency and the profit, while sim-ultaneously considering other important input and output factors for maintaining manufacturing smoothness.
Keywords Analytic hierarchy process · Analytic network process· Efficiency · Product mix · Semiconductor fabricator
1 Introduction
In order to survive in the competitive market, semiconductor manufacturing companies must increase or at least maintain their overall profit and return on investment. One basic task is to uti-lize existing capacity efficiently and effectively. The purpose of this paper is to present an effective approach to find a set of op-timal product mix that is feasible for production, which can be accepted by the semiconductor fabricator. The analysis is aimed at the strategic planning level and attempts to assess manufac-turing performance under various product mixes. A simulation model is first developed to collect the relevant performance out-comes under different product mixes. The AHP approach is first performed to evaluate various factors for different product mixes, to provide relative significance measures for each product mix. Then, the ANP, a general approach of AHP, is applied to analyse
S.-H. Chung (u) · A.H.I. Lee · W.L. Pearn Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, R.O.C. E-mail: t7533@cc.nctu.edu.tw
more complex interrelationships among the decision levels and attributes, and is also applied in our case study.
The rest of this paper is organized as follows. Section 2 gives an overall review of the manufacturing environment and descrip-tion of the product mix problem in semiconductor industry. Sec-tion 3 briefly reviews the decision-making tool AHP. SecSec-tion 4 describes the basics of ANP and presents a simplified version of the ANP approach. Section 5 discusses the process parameters selected for evaluation. Section 6 presents the system environ-ment for simulation. Sections 7 and 8 apply AHP and ANP to evaluate the data obtained from simulation. In Sect. 9, some con-clusion remarks are made.
2 Manufacturing environment and product mix
problem
Semiconductor industry today faces a very dynamic environ-ment. Product demand changes rapidly, while product life cycles are shortened. New process and machine technologies are de-veloped rapidly, and building new fabs and expanding fab cap-acity enter the already competitive market continuously. Like firms in other industries, semiconductor companies must meet customers’ ever-rising demands in order to survive. Failure to deliver product on time, with the right quality and quantity, can result in profit penalties or even losing customers. The length and variability of cycle times are important factors in the meas-urement of customer service. Semiconductor companies can be successful if they only focus on either of the two types: mass manufacturing with high volume and low cost, or a high level of product mix that is flexible [25]. In today’s semiconductor fabs, a wide range of logic and memory products are manufactured, and up to several dozen different product types with more than hundreds of derivatives can be processed simultaneously. High-volume, low-cost fabs produce only a few kinds of products, such as DRAM, in large quantity in order to have economies of scale; while high-mix, flexible fabs mainly produce customer specific chips (“make-to-order”) and aim to leverage economies of scope.
Modern fabs require a very high capital investment in plant and equipment, some 500 million US $ to 1 billion US $ each [13]. The tremendous amount of investment makes the manufacturers try every possibility for fully utilizing the facility. In the time of suppliers’ market, a small increase in produc-tion capacity often means large addiproduc-tional profits and higher market share. Efficient wafer fabrication often requires continu-ing phases of expansion, new knowledge and skill adoption and process improvement. Wafer manufacturing process is highly complex, with several hundred processing steps on a single wafer and a flow time of usually more than one month. Some machine groups may be used for the same operation more than once as successive circuit layers are added. Such operations of multiple visits to the same machine group is termed the re-entry prop-erty of the manufacturing flow [3]. The manufacturing flow of different products may differ significantly, and the processing time required of the machines for one type of product may be twice as much as that required for the other type of product [6]. Wafer fabs involve the most complex manufacturing system in the manufacturing world. The dynamics, varieties in processes, machines, and product demands often cause manufacturing bot-tlenecks to shift from one resource to another. All the factors discussed above are interrelated resulting in a complicated pro-duction planning problem for the semiconductor fabs.
Manufacturing planning involving product mix is a com-mon problem often encountered in manufacturing which used to be formulated classically, as a linear programming problem. The objective of such a problem is to maximize the profit from various product mix combinations, subject to constraints on dif-ferent resources [17]. Manufacturing a product requires a certain amount of resource of each type that is limited. The availability of each resource, therefore, is a constraint. In the past decade, the theory of constraints (TOC) has also been a popular approach for dealing with the product mix planning problem [9]. Based on the TOC, there exists at least one bottleneck resource in the sys-tem that critically impacts the syssys-tem performance [11]. Due to the mutually dependent characteristics of various manufacturing events, the output of the system is constrained by the bottle-neck resource. In order to acquire the highest profit attainable, the bottleneck resource must be fully utilized. It is known that the linear programming and TOC approaches are different in their implementation procedures but both methods are conceptu-ally equivalent and lead to the same solutions [15]. Two aspects most researchers have focussed on in finding the optimal prod-uct mix are cost accounting and manufacturing planning. Cost accounting involves estimating the manufacturing cost of each product type appropriately to find a product mix that maximizes the company’s profit. Malik [17] developed a mixed-integer pro-gramming model utilizing activity-based cost information to de-termine optimal product mix and product cost in a multi-product manufacturing environment. The second aspect, manufacturing planning, involves maximizing the efficiency of capacity alloca-tion across products. Chou et al. [6] formulated a mixed-integer linear programming to optimize the product mix, taking into consideration the requirement of manufacturing smoothing and machine backup.
Manufacturing planning involving product mix in semicon-ductor industry is a complicated problem. In a semiconsemicon-ductor fab, machines are shared by a huge number of different products, resulting in a heavy load sharing the precious resource and con-sequently complex queuing may be present. Product mix level has considerable impact on production throughput, cycle time and the capability of meeting due dates, which are the most im-portant metrics for measuring manufacturing performance [8]. Production throughput, cycle time, machine utilization and work in process (WIP) inventory are highly interrelated. Some meas-urement criteria are positively dependent, and others may be (negatively dependent) trade-offs. For example, it is desirable to have both high throughput and low cycle time. But those two measures tend to conflict with each other. If we emphasize the increase of system throughput, the WIP level must be increased. Even though machines can be utilized more and high equip-ment utilization indicates high return from investequip-ment, which is favourable, the bottleneck machine may not have enough cap-acity to handle such demand, and so the WIP at the bottleneck must be at an excessive level. As a result, cycle time of prod-ucts will be significant, on-time deliveries will be decreased, and market responsiveness is diminished. Production throughput, the index we stress primarily, may even decrease ultimately. Under different product mixes, the manufacturing performance of the system will be different, and AHP/ANP will be used to evaluate which product mix would result in a more competitive manufac-turing performance and a better overall outcome for the wafer fab.
3 Review of the AHP
AHP, a mathematically based multi-criteria decision-making tool, is becoming popular with academic researchers for data analysis, model verifications to provide critical information for managers to make business decisions. AHP was originally in-troduced by Saaty [21–23] back in the early 1970s in response to the scarce resources allocation and planning needs for the military [21]. The AHP comprises six major steps [5, 21, 22]: 1. Define the unstructured problem. The problem should be
stated clearly and be put in broad context including the objec-tives and the outcomes.
2. Decompose the problem into a hierarchical structure. AHP decomposes a complex problem into a decision hierarchy much like a decision tree. The overall objective of the prob-lem, the top level of the hierarchy, can be decomposed into several criteria or attributes, a lower level of the hierarchy. Each next lower level represents the increased detail of these criteria or attributes and there can be several intermediate levels as required until no further decomposition is required. The bottom of the hierarchy usually represents alternatives or actions to be considered to solve the problem. A stan-dard format for AHP decision model is illustrated in Fig. 1. The structure of the hierarchy can be obtained by the opinion of experts or decision makers with a method such as brain-storming.
Fig. 1. A standard hierarchy for AHP
3. Employ pairwise comparisons. Decision elements at each hierarchy level are compared pairwise and assigned rela-tive scales. Decision makers will be asked to compare each of the paired elements in the matrices through question-naires. At the first level, they need to specify judgments about the relative importance of each criterion in terms of its contribution to achieving the overall objective. At the each next level, they need to indicate a priority for each de-tailed decision criterion in terms of how it contributes to each criterion. At the last level, a preference for each deci-sion alternative in terms of how it contributes to each sub-criterion must be made. Saaty [21] recommended the use of a nine-point scale to express preferences between options as equally, moderately, strongly, very strongly, or extremely preferred (with pairwise weights of 1, 3, 5, 7 and 9 respec-tively). Values of 2, 4, 6 and 8 are the intermediate values for the preference scale. Reciprocals are used for the in-verse comparisons. After each element has been compared, a paired comparison matrix is formed. If n objects, denoted by X1, X2, X3, . . . , Xn, are compared in pairs according to
their relative weights, denoted by w1, w2, w3, . . . , wn,
re-spectively, the pairwise comparisons can be represented in the form of a matrix [1, 21].
X1, X2, X3, . . . , Xn, are compared in pairs according to
their relative weights, denoted by w1, w2, w3, . . . , wn,
re-spectively, the pairwise comparisons can be represented in the form of a matrix [1, 21] .
A= w1 w1 w1 w2 · · · w1 wn w2 w1 w2 w2 · · · w2 wn ... ... ... ... wn w1 wn w2 · · · wn wn = a11 a12 · · · a1n a21 a22 · · · a2n ... ... ... ... an1 an2 · · · ann (1)
4. Find the maximum eigenvalues and eigenvectors in order to estimate the relative weights of the decision elements. After a comparison matrix has been formed, the priority of the element can be compared by the computation of eigenvalues and eigenvectors with the following formula, wherew is the eigenvector, the weight vector, of A, andλmaxis the largest
eigenvalue of A:
A· w = λmax· w (2)
5. Check the consistency property of the matrix. The quality of ultimate decision of AHP process is strongly related to the consistency of judgments that decision makers demonstrated during the series of pairwise comparisons [2]. Transitivity of preference implies that if A is preferred to B, and B is preferred to C, then A is preferred to C. This consistency property in the pairwise comparison is examined by the con-sistency ratio. The concon-sistency index (CI) and concon-sistency ratio (CR) are defined as [21]:
CI =λmax− n
n− 1 (3)
CR= CI
RI , (4)
where n is the number of items being compared in the ma-trix, and RI is a random index, the average consistency in-dex of randomly generated pairwise comparison matrices of similar size, as shown in Table 1. Satty [22] has set the ac-ceptable CR values for different matrix sizes. The threshold
CR values are 0.05 for a 3×3 matrix, 0.08 for a 4×4 matrix,
and 0.10 for larger matrices. When the calculated CR values exceed the threshold, it is an indication of inconsistent judg-ment. In such cases, the decision makers would need to revise the original values in the pairwise comparison matrix. 6. Aggregate the relative priorities of the decision elements to
obtain an overall rating for decision alternatives. If there is only one decision maker, an overall priority ranking of the decision alternatives can be obtained by combining the crite-rion priorities and priorities of each decision alternative rela-tive to each criterion. The results are normalized and summed to 1. The alternative with a higher rating is considered to be preferable. In the case of a group with more than one decision maker, an overall priority ranking will be generated for each person first, and a weighted average method will be applied to summarize individual rankings to determine the finalized preferential ranking of alternatives.
AHP has been widely employed in decision-making analysis in various fields such as political, social, economic and manage-ment sciences. In the field of manufacturing, AHP has been used in technology selection [20], the semiconductor facility layout design process [19, 27], plant location selection [28] and justifi-cation of flexible manufacturing systems [4], just to name a few. AHP combines both qualitative and quantitative approaches [5]. In the qualitative sense, it decomposes an unstructured problem
Table 1. Random index (RI) [22]
Order of 2 3 4 5 6 7 8
matrix (n)
into a systematic decision hierarchy. It then uses a quantitative way to employ pairwise comparison to determine the local and global priority weights and the overall ranking of the alterna-tives. We will use AHP as our tool to measure the manufacturing performance of the system under different product mixes.
4 Review of the ANP
The analytic network process (ANP) generalizes AHP by replac-ing hierarchies with networks [12]. It is the extension of AHP and is a more general form of AHP. ANP still involves the repre-sentation of relationships hierarchically, but it does not require as strict a hierarchical structure as AHP [18]. A major assumption of AHP is that each element in the hierarchy is supposed to be independent; however, in many cases, there is interdependence among criteria and alternatives [23]. Many researchers simply disregard this assumption of independence and still adopt AHP for their analysis. The use of AHP in this way means that the models lack the consideration of important interactions among and between decision-making levels [18]. Nonetheless, we can-not tell how good our results are unless we avoid oversimplifying and deal with the issue of dependence properly [24]. Figure 2 depicts the structural difference between a hierarchy and a net-work. A node represents a component (or cluster) with elements inside it, and an arc denotes the interaction between two com-ponents [12] . The direction of an arc represents dependence; a two-way arrow indicates interdependency between two compo-nents; and a loop shows the inner dependence of elements within a component. Figure 2a is the same hierarchy as shown in Fig. 1 and is a simple case of a network.
Saaty [21, 23] introduces the “supermatrix”, similar to Markov chains, to handle the interdependence characteristics among elements and components. Let the components (clus-ters) of a decision system be Ch, h= 1, . . . , n, and let each
component h have mh elements, denoted by eh1, eh2, . . . , ehmn. The influence of a set of elements belonging to a component, on any element from another component, can be represented
Fig. 2. Structural difference between a hierarchy and a network
as a priority vector by applying pairwise comparisons in the same way as the AHP. These priority vectors 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 follows [23]. C1 C2 · · · Cn e11e12· · · e1m1e21e22· · · e2m1· · · e31e32· · · e3m1 W= C1 C2 ... Cn e11 e12 ... e1m1 e21 e22 ... e2m2 ... en1 en2 ... enmn W11 W12 · · · W1n W21 W22 · · · W2n ... ... ... ... Wn1 Wn2 · · · Wnn
If the supermatrix is column stochastic, that is, if we can as-sume that its components have been weighted according to their contribution to the system, we can simply raise the superma-trix to powers to obtain the answer [23]. Otherwise, we need to generate a weighted supermatrix first, and then raise it to pow-ers. When a network is not too complicated, a simplified method may be employed. Saaty et al. [24] devised a matrix manip-ulation approach to solve a network which is very similar to a hierarchy, with the only difference of dependence among cri-teria and dependence among alternatives with respect to each criterion. Lee et al. [14] suggested an information system project selection methodology, which reflects interdependencies among evaluation criteria and candidate projects using the above cited ANP approach within a zero-one goal-programming (ZOGP) model. Karsak et al. [12] also used a combined ANP and goal-programming approach for product planning in quality function deployment. In this paper, we adopt Saaty’s [24] matrix manip-ulation concept and suggest an approach that is suitable for our problem with a network as shown in Fig. 3. The procedures are as follows:
1. Determine the importance of each decision criteria with re-spect to attaining the overall objective. To compare the de-cision criteria, experts need to answer the question: ”which decision criteria should be emphasized more in attaining the overall objective, and how much more?”. By pairwise com-parison of criteria with respect to the overall objective, we can obtain a matrix (W1) and an eigenvector (w1) of the
ma-Fig. 3. A standard network for this paper
trix for the criteria:
C1 C2 · · · Cn W1= C1 C2 ... Cn c11 c12 · · · c1n c21 c22 · · · c2n ... ... ... ... cn1 cn2 · · · cnn and w1= C1 C2 ... Cn c1 c2 ... cn ,
where n is the number of criteria.
2. Determine the importance of each detailed criteria with re-spect to its upper level criteria by assuming that there is no dependence among detailed criteria. To compare the detailed decision criteria, we need to know which detailed criteria should be emphasized more in determining their respective upper level criterion. The matrix and the eigenvector with re-spect to an upper level criterion (n) are as follows:
W2n= D1(n) D2(n) ... Dm(n) D1(n) D2(n) · · · Dm(n) d11(n) d12(n) · · · d1m(n) d21(n) d22(n) · · · d2m(n) ... ... ... ... dm1(n) dm2(n) · · · dmm(n) and w2n= D1(n) D2(n) ... Cm(n) d1(n) d2(n) ... dm(n) , for each n where m(n) is the number of detailed criteria with respect to an upper level n, and the total number of detailed criteria m is equal to the sum of all m(n), that is, m = m(1) + m(2) + . . . m(n).
3. The priorities of alternatives with respect to each of the de-tailed criterion are obtained and the general form of matrix and eigenvector are as follows:
Wbm= B1(m) B2(m) ... Br(m) B1(m) B2(m) · · · Br(m) b11(m) b12(m) · · · b1r(m) b21(m) b22(m) · · · b2r(m) ... ... ... ... br1(m) br2(m) · · · brr(m) and wbm= B1(m) B2(m) ... Br(m) b1(m) b2(m) ... br(m) , for each m , where r is the number of alternatives. We next combine the above eigenvectors with respect to the criterion n and obtain the following matrix:
W3n= B1 B2 ... Br D1(n) D2(n) · · · Dm(n) b1(1) b1(2) · · · b1(m) b2(1) b2(2) · · · b2(m) ... ... ... ... br(1) br(2) · · · br(m) for each n.
4. Analyse the interdependence among the detailed criteria. The inner dependence among detailed criteria under the same cri-terion are determined through analysing the impact of each detailed criterion on other detailed criteria with the same up-per level criterion. Pairwise comparisons through exup-perts’ opinion are applied in the literature [12, 14, 24]. In this pa-per, however, a correlation analysis is performed instead. We define the interdependence weight matrix of detailed criteria with the same upper level criterion as:
W4n= D1(n) D2(n) ... Dm(n) D1(n) D2(n) · · · Dm(n) v11(n) v12(n) · · · v1m(n) v21(n) v22(n) · · · v2m(n) ... ... ... ... vm1(n) vm2(n) · · · vmm(n) for each n.
5. Obtain the interdependence priorities,wDC(n), of the detailed
criteria by synthesizing the results from Step 2 and Step 4. wDC(n)= W4n× w2nfor each n.
6. The priorities of alternatives, w21(n), with respect to each of the three criteria are given by synthesizing the results
from Step 3 and Step 5 as follows:w21(n)= W3n× wDC(n)
for each n. We then define the matrix W21 by grouping to-gether the above three columns of w21(n)for all ns: W21=
w21(1), w21(2), . . . , w21(n)
7. The overall priorities for the alternatives are obtained by synthesizing the results from Step 1 and Step 6; that is, mul-tiplying W21byw1.wANP= W21× w1.
The ANP analysis results will be (alternative 1, alternative 2, . . . , alternative n). The above procedures are specifically de-signed to fit our problem and will be applied in the evaluation in Sect. 8.
5 Process parameters for evaluation
Depending on the comparison criterion used, some product mixes may perform better than the others. Without knowledge of the relative importance of these criteria, it is difficult to decide what product mix should be applied in a fab to achieve the op-timal manufacturing results. Based on the AHP method, experts are interviewed first to decide the major criteria and the subse-quent detailed criteria for evaluating performance under different product mixes for product A and B. A hierarchical structure, which consists of three major criteria and numerous detailed cri-teria, is established as shown in Fig. 2.
The three major criteria and the detailed criteria we used to measure manufacturing performance of a semiconductor fab are defined as follows:
1. Product How products are manufactured in a fab.
• WIP The number of lots of manufacturing that have been released into the wafer fab but have not yet finished being processing through all of their manufacturing steps. • Throughput The number of lots of manufacturing that
pass through the final operation step in a period. • Total layers The number of layers the bottleneck
pro-cessed in a period of time.
• Total cycle time The duration of time, expressed in hours, consumed by a unit of manufacturing 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 equipment is used in manufacturing, a measure of equipment performance.
• BN utilization shows average utilization rate of the bottle-neck in the system for a period of time. At the bottlebottle-neck workstation, equipment utilization should be as high as possible since it gates the throughput of the entire manu-facturing system.
• CCR utilization shows average utilization rate of the CCR in the system for a period of time. A CCR is a workstation which, although it is not a bottleneck, also has a substan-tially high utilization rate.
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. The price for a product is set by its prod-uct type and the number of layers that prodprod-uct goes through. Most manufacturing costs of a semiconductor fabricator are fixed; that is, no matter how many products are produced, the operating costs do not vary much.
• Total revenue is obtained by summing up revenue of each product type, while revenue of a product type is calcu-lated by multiplying the price of the product type by its throughput.
• Variable costs include two major parts: total variable manufacturing costs and total holding costs. Direct ma-terial cost, the cost of raw wafers, is the primary part of total variable manufacturing costs. Other variable manu-facturing cost includes indirect material cost and is varied according to the manufacturing level. The holding cost is the time cost of carrying WIP in the manufacturing sys-tem.
Senior managers of semiconductor manufacturing compa-nies located on the Science-Based Industrial Park in Taiwan and academic researches were involved in evaluating the criteria and sub-criteria and gave pairwise comparison values.
6 System input and simulation
To obtain a set of product mix that is optimal for factory manu-facturing, actual data is collected from a wafer fabrication fac-tory located on the Science-Based Industrial Park in Taiwan. Simulation results are used to estimate the manufacturing per-formance. To simplify the complexity of the environment for our analysis, this paper is based on the following assumptions. The fab produces two types of products, A and B. The process of each product type is different and unique. Both products are logic products. There is only one priority level of products; that is, all products are normal. Product A requires 305 operations and passes through the bottleneck 17 times. That is, 18 layers are pro-cessed. Product B requires 345 operations and passes through the bottleneck 19 times (20 layers processed). There are two types of machines: batch and serial. A total of 83 workstations (W1 to W83) are in the manufacturing environment. Preventive mainte-nance is considered in estimating usable manufacturing capacity. WS46, a stepper in the photolithography area, is the bottleneck.
The CCR is a machine with a high utilization rate that is close to that of the bottleneck. The strategy of manufacturing is to maximize the utilization of the bottleneck. The dispatching rule is first-in, first-out (FIFO). The releasing batch size for nor-mal lots is six lots. Such a setting is for effective use of many workstations, which have a maximum batch size (MBS) of six lots. Lots with different product types cannot be processed sim-ultaneously. Product price is determined according to the number of layers the product processed
Wafer lots are released under CONWIP (CONstant WIP), a fixed work-in- process (WIP) policy. For production smooth-ing, that is, to minimize cycle time variation, constant WIP con-trol is adopted for setting the suitable system WIP level of wafer
lots, L [25]. Little’s Law [15] in queuing theory, L= λ × W, is applied, whereλ is the releasing rate and W is the production cycle time. By adopting constant WIP control policy, wafer lot(s) can be released to shop floor only when the same quantity of wafers are finished and transferred out. That is, no new job is allowed to enter the system until a job leaves.
Because different products go through different processes, the charged price is set to be $ 40 per pass through the bot-tleneck for product A, and $ 42 for product B. Direct mate-rial cost is assumed to be $ 100 per wafer. Indirect matemate-rial cost, such as photo-resist, special gas, chemical and quartz, is set to be $ 7.5 per layer for product A, and $ 8 per layer for product B. 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 ma-jor 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 data obtained from run-ning simulations for each product mix (represented by I to IX) with manufacturing ratio of A to B is shown in Table 2. The data is to be used in AHP and ANP analysis in the next two sections.
7 AHP for product mix determination
Senior managers and experts contributed their professional expe-rience to identify criteria and sub-criteria that influence the de-cision. The hierarchical form of determining the efficient manu-facturing performance has already been shown in Fig. 4. Note that each element in the hierarchy is assumed to be indepen-dent. The next step is to construct the comparison matrices at each level of the hierarchy for pairwise comparison of the fac-tors in that level. The Delphi method was performed to obtain a consensus among the people who were involved [9]. To ar-rive at a group position regarding an issue, the Delphi method consisted of a series of repeated interrogations through question-naires of a group of experts and managers whose judgments were of interest. After the initial interrogation of each individual, each subsequent interrogation was accompanied by information re-garding the preceding round of replies, and each individual was encouraged to reconsider and change his previous reply in light
Table 2. Simulation results
Product mix WIP Throughput Total layers Total cycle time BN utilization CCR utilization Total revenue Variable costs
(A:B) (lots) (TP) (lots) (TL) (CT) (hours) (BU) (%) (CU) (%) (TR) ($) (VC) ($)
I: Mix (1:9) 274.89 600 11,880 307.9 0.99 0.86 12,420,000 3,883,068 II: Mix (2:8) 277.22 612 11,995 304.41 0.99 0.85 12,484,800 3,922,333 III: Mix (3:7) 276.32 620 12,028 299.49 0.99 0.84 12,462,000 3,934,707 IV: Mix (4:6) 272.73 624 11,981 293.69 0.99 0.83 12,355,200 3,920,942 V: Mix (5:5) 271.22 630 11,970 286.79 0.99 0.81 12,285,000 3,919,118 VI: Mix (6:4) 272.83 640 12,032 286.46 0.99 0.8 12,288,000 3,941,173 VII: Mix (7:3) 273.16 648 12,053 283.27 0.99 0.79 12,247,200 3,949,782 VIII: Mix (8:2) 276.38 660 12,144 281.34 0.99 0.78 12,276,000 3,981,518 IX: Mix (9:1) 276.97 668 12,158 278.63 0.99 0.76 12,224,400 3,987,868
Fig. 4. The hierarchical framework of factors
of the replies of other members of the group. The group pos-ition was finally determined after several rounds. The question, “which criteria should be emphasized more in determining ef-ficient manufacturing, and how much more?” was asked, and a nine-point scale was used to do the pairwise comparison. The pairwise comparison results are shown in Table 3, and the com-parison matrix for comparing the criteria in level 2 in terms of their contribution to achieving the overall objective is shown in Table 4.
An eigenvector and an eigenvalue are calculated using the eigenvalue method by Equation Eq. 1.
w1= P E F 00.258.105 0.637 and λmax= 3.04 .
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 effi-ciency of manufacturing performance simply because profitabil-ity is the ultimate goal of a company. Product and equipment efficiency rank the second and the third with weights of 0.258 and 0.105. To check the consistency of this combination of values in
Table 3. Pairwise comparison
In order to achieve the most efficient manufacturing performance, which criteria should be emphasized more?
Absolute Very strong Strong Weak Equal Weak Strong 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
In order to achieve the best manufacturing, which factor should be emphasized more?
Absolute Very strong Strong Weak Equal Weak Strong 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
WIP X Throughput
WIP X Total layers
WIP X Total cycle time
Throughput X Total layers
Throughput X Total cycle time
Total layers X Total cycle time
In order to best utilize the equipment, which factor should be emphasized more?
Absolute Very strong Strong Weak Equal Weak Strong 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
BN utilization X CCR utilization
In order to achieve the best performance in finance, which factor should be emphasized more?
Absolute Very strong Strong Weak Equal Weak Strong 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
Total revenue X Total variable costs
Table 4. Comparison matrix for the criteria
Equipment
Product efficiency Finance
Product 1 3 1/3
Equipment efficiency 1/3 1 1/5
Finance 3 5 1
the matrix,λmaxis substituted into Eq. 3 to obtain CI, and CR is calculated by Eq. 4. CI=λmax− n n− 1 = 3.04 − 3 3− 1 = 0.02 , CR=CI RI = 0.02 0.58= 0.03 .
Since CR is less than 0.5, the threshold for a 3× 3 matrix, the comparison matrix is consistent. The comparison matrices of de-tailed criteria in accordance to their respective upper level criteria (product, equipment, finance), their eigenvectors and consistent ratios are obtained and shown in Tables 5 to 7. For the criteria product, pairwise comparison among its detailed criteria, WIP, throughput, total layers and cycle time, shows that total layers, the total number of layers the bottleneck can process in a period of time, is the most important factor with a weight of 0.499, fol-lowed by throughput with a weight of 0.284. For the criteria equip-ment 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 ei-ther. Revenue, on the other hand, is the indication of the success of a company and thus has a much higher contribution.
The simulation data from Table 2 are used to form the com-parison matrices of alternatives (product mixes) with respect to each detailed criteria (WIP, throughput, etc.). Instead of asking senior managers and experts to identify the relative score of the al-ternatives with respect to each of the detailed criteria, simulation data, which indicate the manufacturing performance of a fab under different product mixes, are objective measures and are used to re-flect the efficiency of manufacturing. Because the unit of measure of simulation data can range from number of lots to hours and to dollars, we need to transform these quantitative data 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 fac-tor 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 at level 3 is as follows [7]:
um(x) =
X− X−m X+m− X−m
.
X+m: The best value of detailed criteria m at level 3.
X−m: The worst value of detailed criteria m at level 3.
Product WIP TP TL CT Relative importance weights (eigenvector,w2P) 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 C R= 0.07
Table 5. Comparison matrix and eigenvector for product
Table 6. Comparison matrix and eigenvector for equipment efficiency
Equipment BU CU Eigenvector,w2E
efficiency
BU 1 9 0.900
CU 1/9 1 0.100
C R= 0.00
Table 7. Comparison matrix and eigenvector for finance
Finance TR VC Eigenvector,w2F
TR 1 6 0.857
VC 1/6 1 0.143
C R= 0.00
In this paper, numerous simulations are run to collect suffi-cient data, and X+mand X−mare the values that are suitable to be the two extreme levels of performance. For some detailed cri-teria (throughput, total layers, BN utilization, CCR utilization and total revenue), we prefer their values to be as great as pos-sible, and thus the best factors have the greatest values, and vice versa. On the other hand, for other detailed criteria (WIP, total cycle time and variable costs), their values are preferred to be small, and therefore the best ones are those with the smallest values. The simulation results in Table 2 are transformed into utility indices as shown in Table 8. The utility indices are then transformed into weights by dividing each utility index by the total value of the column, and the results that have the same up-per level criteria are grouped together to be W3P, W3Eand W3E, as shown in Tables 9 to 11.
Finally, the overall priorities for the product mixes are ob-tained by the following formula:
wAHP= W 3P× w2P, W3E× w2E, W3F× w2F × w1 = 0.086177 0.090815 0.099202 0.110431 0.118908 0.120284 0.124043 0.123891 0.126098 0.1156 0.1146 0.1136 0.1126 0.1107 0.1097 0.1087 0.1077 0.1058 0.148153 0.156140 0.146995 0.120432 0.101293 0.096430 0.082287 0.082706 0.066564 × 00.258.105 0.637 = I II III IV V VI VII VIII IV 0.129 0.135 0.131 0.117 0.107 0.104 0.095 0.096 0.086
If the AHP approach is employed, alternative II, product mix (2:8), can lead to efficient manufacturing with a relative impor-tance value of 0.135, followed by alternative III, product mix (3:7), with a relative importance value of 0.131. The third most efficient product mix is alternative I, product mix (1:9), with a value of 0.129. However, there seems to be a trend of decreas-ing efficiency as the ratio of product A increases.
8 ANP for product mix determination
In this section, a simplified ANP approach, which is introduced in Sect. 4, is adopted. Here we take into consideration the inter-relationship among detailed criteria with respect to an upper level criterion (i.e., product, equipment efficiency and finance). Note that the basic pairwise comparisons and simulation results are the same as those of AHP. The procedures are as follows:
Step 1: Determine the importance of each decision criteria with respect to achieving the overall objective, the efficient manufacturing performance. The results are the same as shown in Table 4, and
w1= P E F 00.258.105 0.637 .
Step 2: Assuming there is no interdependence among detailed criteria, which detailed criteria should be emphasized more in determining their respective upper level crite-rion? The results are the same as in Tables 5 to 7, and the eigenvectorsw2P,w2Eandw2Fare:
w2P= WIP TP TL CT 0.134 0.284 0.499 0.083 , w2E=CUBU 0.900 0.100 and w2F=VCTR 0.857 0.143 .
Table 8. Utility index of each detailed criteria under different product mixes
Alternative WIP Throughput Total layers Total cycle BN utilization CCR utilization Total revenue Variable costs
(TP) (TL) time (CT) (BU) (CU) (TR) (VC)
I 0.511 0.5 0.44 0.0525 1 0.8 0.1290323 0.58466 II 0.278 0.56 0.4975 0.13975 1 0.75 0.1489401 0.388335 III 0.368 0.6 0.514 0.26275 1 0.7 0.1419355 0.326465 IV 0.727 0.62 0.4905 0.40775 1 0.65 0.1091244 0.39529 V 0.878 0.65 0.485 0.58025 1 0.55 0.0875576 0.40441 VI 0.717 0.7 0.516 0.5885 1 0.5 0.0884793 0.294135 VII 0.684 0.74 0.5265 0.66825 1 0.45 0.0759447 0.25109 VIII 0.362 0.8 0.572 0.7165 1 0.4 0.0847926 0.09241 IX 0.303 0.84 0.579 0.78425 1 0.3 0.0689401 0.06066 Sum of column 4.828 6.01 4.6205 4.2005 9 5.1 0.9347465 2.797455
Table 9. Utility weight of detailed criteria with respect to product under different product mixes, W3P
W3P WIP TP TL CT I 0.106 0.083 0.095 0.012 II 0.058 0.093 0.108 0.033 III 0.076 0.100 0.111 0.063 IV 0.151 0.103 0.106 0.097 V 0.182 0.108 0.105 0.138 VI 0.148 0.116 0.112 0.140 VII 0.142 0.123 0.114 0.159 VIII 0.075 0.133 0.124 0.171 IX 0.063 0.140 0.125 0.187
Step 3: Since there is no interdependence among alternatives (product mix), they are compared with respect to each detailed criterion yielding the column eigenvectors re-garding each detailed criterion. These data are obtained from the transformation of simulation results and the same as for the AHP analysis in Tables 9 to 11 in the previous section. Three matrices W3P, W3Eand W3Fare restated here. WP TP TL CT W3P= I II III IV V VI VII VIII IX 0.106 0.083 0.095 0.012 0.058 0.093 0.108 0.033 0.076 0.100 0.111 0.063 0.151 0.103 0.106 0.097 0.182 0.108 0.105 0.138 0.148 0.116 0.112 0.140 0.142 0.123 0.114 0.159 0.075 0.133 0.124 0.171 0.063 0.140 0.125 0.187 BU CU W3E= I II III IV V VI VII VIII IX 0.111 0.157 0.111 0.147 0.111 0.137 0.111 0.127 0.111 0.108 0.111 0.098 0.111 0.088 0.111 0.078 0.111 0.059 , TR VC
Table 10. Utility weight of detailed criteria with respect to equipment
effi-ciency under different product mixes, W3E
W3E BU CU I 0.111 0.157 II 0.111 0.147 III 0.111 0.137 IV 0.111 0.127 V 0.111 0.108 VI 0.111 0.098 VII 0.111 0.088 VIII 0.111 0.078 IX 0.111 0.059
Table 11. Utility weight of detailed criteria with respect to finance under different product mixes, W3F
W3F TR VC I 0.138 0.209 II 0.159 0.139 III 0.152 0.117 IV 0.117 0.141 V 0.094 0.145 VI 0.095 0.105 VII 0.081 0.09 VIII 0.091 0.033 IX 0.074 0.022 W3F= I II III IV V VI VII VIII IX 0.138 0.209 0.159 0.139 0.152 0.117 0.117 0.141 0.094 0.145 0.095 0.105 0.081 0.090 0.091 0.033 0.074 0.022 .
Step 4: Analyse the interdependence among the detailed crite-ria. Correlation analysis is performed, and the values are normalized so that the sum of each column is one. The interdependence weight matrices of detailed criteria with the same upper level criterion are as in Tables 12 to 13.
Step 5: Obtain the interdependence priorities of the detailed cri-teria by synthesizing the results from Step 2 and Step 4. wDC(n)= W4n× w2nfor n= P, E, F wDC(P)= W4P× w2P WP TP TL CT = WP TP TL CT 1.000 0.004 0.000 0.020 0.000 0.518 0.458 0.470 0.000 0.000 0.542 0.000 0.000 0.478 0.000 0.510 × WP TP TL CT 0.134 0.284 0.499 0.083 = WP TP TL CT 0.137 0.415 0.270 0.178 wDC(E)= W4E× w2E
BU CU =CUBU 1.000 0.458 0.000 0.542 ×CUBU 0.900 0.100 =CUBU 0.946 0.054 wDC(F)= W4F× w2F TR VC =TRVC 0.558 0.442 0.442 0.558 ×VCTR 0.857 0.143 =TRVC 00.541.459
Table 12. Interdependence matrix of detailed criteria with respect to product
W4P WIP TP TL CT WIP 1.000 0.004 0.000 0.020 TP 0.000 0.518 0.458 0.470 TL 0.000 0.000 0.542 0.000 CT 0.000 0.478 0.000 0.510 Sum 1.000 1.000 1.000 1.000
Table 13. Interdependence matrix of detailed criteria with respect to
equip-ment efficiency
W4E BU CU
BU 1 0.458
CU 0 0.542
Sum 1.000 1.000
Table 14. Interdependence matrix of detailed criteria with respect to finance
W4F TR VC
TR 0.558 0.442
VC 0.442 0.558
Sum 1.000 1.000
Step 6: The priorities of alternatives with respect to each of the three criteria are given by synthesizing the results from Step 3 and Step 5 as follows:w21(n)= W3n×wDC(n)for n= P, E, F w21(P)= W3P× wDC(P) WP TP TL CT = I II III IV V VI VII VIII IX 0.106 0.083 0.095 0.012 0.058 0.093 0.108 0.033 0.076 0.100 0.111 0.063 0.151 0.103 0.106 0.097 0.182 0.108 0.105 0.138 0.148 0.116 0.112 0.140 0.142 0.123 0.114 0.159 0.075 0.133 0.124 0.171 0.063 0.140 0.125 0.187 × WP TP TL CT 0.137 0.415 0.270 0.178 = I II III IV V VI VII VIII IX 0.077 0.082 0.093 0.109 0.123 0.124 0.130 0.129 0.134 w21(E)= W3E× wDC(E) BU CU = I II III IV V VI VII VIII IX 0.111 0.157 0.111 0.147 0.111 0.137 0.111 0.127 0.111 0.108 0.111 0.098 0.111 0.088 0.111 0.078 0.111 0.059 ×CUBU 0.954 0.046 = I II III IV V VI VII VIII IX 0.113 0.113 0.112 0.112 0.111 0.110 0.110 0.109 0.108
w21(F)= W3F× wDC(F) TR VC = I II III IV V VI VII VIII IX 0.138 0.209 0.159 0.139 0.152 0.117 0.117 0.141 0.094 0.145 0.095 0.105 0.081 0.090 0.091 0.033 0.074 0.022 ×VCTR 0.558 0.442 = I II III IV V VI VII VIII IX 0.169 0.150 0.137 0.128 0.117 0.099 0.085 0.065 0.051
We define the matrix W21 by grouping together the above three columns: W21=
w21(P), w21(E), w21(F) W21= I II III IV V VI VII VIII IX 0.077 0.113 0.169 0.082 0.113 0.150 0.093 0.112 0.137 0.109 0.112 0.128 0.123 0.111 0.117 0.124 0.110 0.099 0.130 0.110 0.085 0.129 0.109 0.065 0.134 0.108 0.051
Step 7: The overall priorities for the alternatives (product mixes) are obtained by synthesizing the results from Step 1 and Step 6. That is, they are calculated by multiplying W21byw1.
wAN P= W 21× w1 P E F = I II III IV V VI VII VIII IX 0.077 0.113 0.169 0.082 0.113 0.150 0.093 0.112 0.137 0.109 0.112 0.128 0.123 0.111 0.117 0.124 0.110 0.099 0.130 0.110 0.085 0.129 0.109 0.065 0.134 0.108 0.051 × P E F 00.258.105 0.637 = I II III IV V VI VII VIII IX 0.140 0.129 0.123 0.121 0.118 0.107 0.099 0.086 0.078
The ANP analysis results indicate that the most efficient manufacturing is alternative I, product mix (1:9), with a relative importance value of 0.140, which is slightly more important than alternative II, product mix (2:8), with a value of 0.129. The rank-ing of the product mixes is exactly the descendrank-ing order from al-ternative I to IV, that is, from product mix (1:9) to (9:1). This im-plies that Product B is highly recommended for manufacturing.
If the AHP approach is employed, alternative II appears as the most efficient product mix with a relative importance value of 0.135, followed by alternative III with a relative importance value of 0.131, and alternative I with a value of 0.129. However, when interdependencies of detailed criteria are incorporated into the analysis by applying the simplified ANP approach, alterna-tive II declines to the second place with relaalterna-tive value of 0.129 whereas alternative I, ranked third in AHP approach, ranks in the first place with a value of 0.140. Consequently, the ranking re-sults of ANP are different from those of AHP. Incorporating the interdependencies among factors, the ANP approach is preferred to the AHP approach in the determination of efficient product mix.
The ranking order of ANP results is exactly from alternative I to IV, and this implies that product B is strongly advised to pro-duce. Some trends can be found from the simulation results in Table 2 and the obtained eigenvectors. In the group’s opinion,
fi-nance, 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 increas-ing trend as a higher ratio of product B, which has a higher price than A, is produced. This applies until alternative I, with product mix (1:9), where total layers, throughput and cycle time are ad-versely affected. Variable costs also have a decreasing trend as a higher ratio of product B is produced. Since both throughput and total layers are less in the product mixes with a higher ratio of product B, variable costs are lower. With the above combina-tion, product mixes with a higher ratio of product B are preferred. Other criteria (product and equipment efficiency), even though having some contribution weights, are not powerful enough to have a strong impact on the final ranking.
9 Conclusions
The selection of an appropriate product mix in a wafer fab is essential to achieve optimal manufacturing. This paper applied both AHP and ANP methods to evaluate the manufacturing per-formance under different product mixes in a semiconductor fab-ricator. Although the results of AHP and ANP do not show great disparity, there are some difference in ranking and the fi-nal priority weights of the alternatives. ANP, which considers the interrelationship among factors, should be adopted if possible. The results provide guidance for a fab in accepting orders when its capacity cannot fully satisfy all the product demand. A sim-plified ANP approach is used in this paper; however, a more complicated network of efficient product mix can be present in semiconductor manufacturing. How to organize a network problem and use the comprehensive ANP will be our future
re-search direction. In addition, due to the intensive competition in the semiconductor industry, multiple priority orders, such as hot lots, rush lots and normal lots, are accepted in order to satisfy customer demand. As a result, the manufacturing environment can be very complex, and this can also be our future focus of research.
Acknowledgement The authors would like to thank Dr. He Yau Kang and Dr. Ling-Fen Hsieh for their valuable comments and suggestions.
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