Multidimensional data in multidimensional scaling using the analytic network process

(1)

Multidimensional data in multidimensional scaling using

the analytic network process

Jih-Jeng Huang

a

, Gwo-Hshiung Tzeng

b,c,*

, Chorng-Shyong Ong

a

Department of Information Management, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 106, Taiwan b

Institute of Management of Technology and Institute of Traﬃc and Transportation College of Management, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan

c_{Kainan University, No. 1, Kainan Road, Luchu, Taoyuan 338, Taiwan}

Received 24 December 2003; received in revised form 24 August 2004 Available online 2 November 2004

Abstract

Multidimensional scaling (MDS) is a statistical tool for constructing a low-dimensional configuration to represent the relationships between the objects. Although MDS has been widely used in various fields, it is difficult to evaluate similarity/ dissimilarity between the complex systems by human judgment. Even though we can divide a complex system into subsystems, which can be more easily evaluated, the relative weights of the subsystems are also a crucial problem. Because of these subsystems usually exist interdependence and feedback, the weights of the subsystems are hard to obtain. This paper proposes a method which combines the methods of the interpretive structural modeling (ISM) and the analytic network process (ANP) procedures to deal with the problem of the subsystemsÕ interdependence and feedback. In addition, we also provide a numerical example to illustrate the proposed method.

Keywords: Multidimensional scaling (MDS); Interdependence; Feedback; Interpretive structural modeling (ISM); Analytic network process (ANP)

1. Introduction

Multidimensional scaling (MDS) is used to rep-resent the relationships between the objects by constructing a conﬁguration of n points in low-dimension from the pairwise comparisons of the dissimilarities among a set of n objects (Arabic et al., 1987; Young and Hamer, 1987; Schiﬀman et al., 1981). The dissimilarities can be calculated

*

Corresponding author. Address: Institute of Management of Technology and Institute of Traﬃc and Transportation College of Management, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan. Tel.: +886 3 571212157505; fax: +886 3 5753926.

E-mail address:[email protected](G.-H. Tzeng).

(2)

by Euclidean distance or other weighted Euclidean distance (Kruskal and Wish, 1978) to form the proximity matrix. General speaking, MDS can be classified into metric MDS and non-metric MDS based on the type of the input data or whether or not the distance matrix satisfies Euclidean dis-tance. The goal of metric MDS and non-metric MDS is to visually reflect the empirical relation-ships in the data. More detailed discussions about MDS can refer to Mead (1992).

Although MDS has been widely used in various ﬁelds, such as psychophysics, sensor analysis or marketing, the problems of the featureÕs weights have received little attention in social science. In conventional MDS, the proximity matrix is given subjectively by asking such question as ‘‘How many grades of dissimilarity do you think there are between object 1 and object 2?’’ However it is diﬃcult for human to evaluate the dissimilarities in the complex systems. Even though we can divide a complex system into many subsystems which can be easily evaluated, the weights of the subsystems also is a hard problem because of existing interde-pendence and feedback relationships.

This paper proposes a method which combines the methods of the interpretive structural modeling (ISM) and the analytic network process (ANP) procedures to deal with the problem of the subsys-temsÕ interdependence and feedback. We also illus-trate a numerical example to show the steps of the proposed method. On the basis of the results, we can conclude that the weights play a key for the MDS analysis. In addition, the proposed method can provide the more informative and accurate re-sults than the conventional MDS analysis.

The rest of this paper is organized as follows. Section 2 states the problems in judging the prox-imity matrix. Section 3 describes the ANP method and ISM is presented in Section 4. A numerical example is illustrated in Section 5. Discussions and conclusions are presented in Section 6 and Section 7, respectively.

2. Problem descriptions

The problem of measurement can be described as follows. If we want to analyze the relationships

between the countries (such as the USA, the UK, Japan, and so on) using MDS analysis, we can construct the proximity matrix by our objective/ subjective judgment. However the problem arises of what are the criteria used to judge these grades of dissimilarity between the countries. It is clear that with the diﬀerent criteria, the results are vari-ous. However, in the conventional MDS method, we do not understand these criteria or their rela-tionships even though the information is

impor-tant for decision-makers or researchers. In

practice, we usually have some information which could be used to make the results more accurate and objective in MDS analysis.

It is clear that in a complex system or construct, it is difficult for human to quantify a precise value in the proximity matrix. However, we could divide a complex system into many elements or subsys-tems which can be easily judged to measure the grades of elements. Then, sum these grades to ob-tain the final proximity matrix. The advantage of this method is that we can more easily judge the differences between the elements and understand what criteria are based on.

Although it seems rational that we can sum the grades of these elements to calculate the proximity matrix among the elements, the weights between these elements may not be the same. It is clear that these elements may have interdependent or feed-back relationships. It may distort the actual results if we assume the weights of the elements are equal. In this paper, ISM and ANP are used here to over-come these problems. First, ISM is used to con-struct the network relationships between elements, and then the ANP is used to calculate the weights of the elements for solving above problems.

3. The analytic network process

The ANP was proposed in (Saaty, 1996; Saaty

and Vargas, 1998) to overcome the problem of interdependence and feedback between criteria or alternatives. The ANP is the general form of the analytic hierarchy process (AHP) (Saaty, 1980) which has been used in multicriteria decision mak-ing (MCDM) to release the restriction of hierar-chical structure, and has been applied to project

(3)

selection (Meade and Presley, 2002; Lee and Kim, 2000), product planning, strategic decision (Sarkis, 2003; Karsak et al., 2002), optimal scheduling (Momoh and Zhu, 2003) and so on.

The ﬁrst phase of the ANP is to compare the criteria in whole system to form the supermatrix. This is done through pairwise comparisons by ask-ing ‘‘How much importance does a criterion have compared to another criterion with respect to our interests or preferences?’’ The relative impor-tance value can be determined using a scale of 1 to 9 to represent equal importance to extreme importance (Saaty, 1980, 1996). The general form of the supermatrix can be described as follows:

where Cmdenotes the mth cluster, emndenotes the nth element in mth cluster, and Wijis the principal eigenvector of the inﬂuence of the elements com-pared in the jth cluster to the ith cluster. In addi-tion, if the jth cluster has no inﬂuence to the ith cluster, then Wij= 0.

Therefore, the form of the supermatrix depends much on the variety of the structure. There are sev-eral structures which were proposed by Saaty includ-ing hierarchy, holarchy, suparchy, intarchy, etc. to demonstrate how the structure is aﬀected by the supermatrix. Here, two simple cases, which both have three clusters, are used to show how to form the supermatrix based on the structures (Fig. 1).

The supermatrix can be formed as the following matrix:

InFig. 2, the second case is more complex than the ﬁrst case.

Then, the supermatrix of the second case can be expressed as

After forming the supermatrix, the weighted supermatrix is derived by transforming all columns sum to unity exactly. This step is much similar to the concept of Markov chain for ensuring the sum of these probabilities of all states equals to 1. Next, we raise the weighted supermatrix to lim-iting powers such as Eq.(1)to get the global prior-ity vectors or called weights.

lim

k!1W

k _ð1Þ

In addition, if the supermatrix has the eﬀect of cyc-licity, the limiting supermatrix is not the only one.

Cluster 2

Cluster 1

Cluster 3

Fig. 1. The structure of the case 1.

Cluster 2

Cluster 1

Cluster 3

(4)

There are two or more limiting supermatrices in this situation, and the Cesaro sum would be calculated to get the priority. The Cesaro sum is formulated as

lim k!1 1 N XN k¼1 Wk _ð2Þ

to calculate the average eﬀect of the limiting super-matrix (i.e. the average priority weights). Otherwise, the supermatrix would be raised to large powers to get the priority weights. The detailed discussion of the mathematical processes of the ANP can refer toSaaty (1996)andSekitani and Takahashi (2001). In order to show the concrete procedures of the ANP, a simple example of system development is demonstrated to derive the priority of each crite-rion. As we know, the key to develop a success-ful system depending on the match of human

and technology factors. Assume the human factor can be measured by the criteria of business culture (C), end-user demand (E) and management (M). On the other hand the technology factor can be measured by the criteria of employee ability (A), process (P) and resource (R). In addition, human and technology factors are aﬀected each other as like as the structure shown inFig. 3.

The first step of the ANP is to compare the importance between each criterion. For example, the first matrix below is to ask the question ‘‘For the criterion of employee ability, how much the importance does one of the human criteria than an-other’’. The other matrices can easily be formed with the same procedures. The next step is to calculate the influence (i.e. calculate the principal eigenvector) of the elements (criterion) in each component (matrix).

Ability Culture End-user Management Eigenvector Normalization

Culture 1 3 4 0.634 0.634 End-user 1/3 1 1 0.192 0.192 Management 1/4 1 1 0.174 0.174 Process Culture 1 1 1/2 0.250 0.250 End-user 1 1 1/2 0.250 0.250 Management 2 2 1 0.500 0.500 Resource Culture 1 2 1 0.400 0.400 End-user 1/2 1 1/2 0.200 0.200 Management 1 2 1 0.400 0.400

Culture Ability Process Resource

Ability 1 5 3 0.637 0.637 Process 1/5 1 1/3 0.105 0.105 Resource 1/3 3 1 0.258 0.258 End-user Ability 1 5 2 0.582 0.582 Process 1/5 1 1/3 0.109 0.109 Resource 1/2 3 1 0.309 0.309 Management Ability 1 1/5 1/3 0.136 0.136 Process 5 1 3 0.654 0.654 Resource 3 1/3 1 0.210 0.210

(5)

Now, we can form the supermatrix based on the above eigenvectors and the structure in Fig. 3. Since the human factor can aﬀect the technology factor, and vise versa, the supermatrix is formed as follows:

Then, the weighted supermatrix is obtained by ensuring all columns sum to unity exactly.

Last, by calculating the limiting power of the weighted supermatrix, the limiting supermatrix is obtained as follows:

and

As we see, the supermatrix has the eﬀect of cyc-licity, and the Cesaro sum (i.e. add the two matri-ces and dividing by two) is used here to obtain the ﬁnal priorities as follows:

In this example, the criterion of culture has the highest priority (0.233) in system development and the criterion of end-user has the least priority (0.105).

In order to show the effect of the structure in the ANP, the other structure, which has the feedback effect on human factors, is considered as inFig. 4. There are two methods to deal with the self-feedback effect. The first method simply place 1 in diagonal elements and the other method per-forms a pairwise comparison of the criteria on each criterion. In this example, we use the first method. With the same steps above, the unweighted Culture End-User Management Ability Process Resource Human Technology

(6)

supermatrix, the weighted supermatrix, and the limiting supermatrix can be obtained as follows, respectively:

Since the effect of cyclicity does not exist in this example, the final priorities are directly obtained by limiting the power to converge. Although the criterion of culture also has the highest priority, the priority changes from 0.233 to 0.310. On the other hand, the least priority is resource (0.084) instead of end-user. Compare to the priorities of the two examples, the struc-tures play the key to both the effects and the re-sults. In addition, it should be highlighted that when we raise the weighted matrix to limiting power, the weighted matrix should always be the stochastic matrix.

The advantage of the ANP is that it is not only appropriate for both quantitative and quali-tative data types, but it also can overcome the problem of interdependence and feedback be-tween all features. This is the reason why we adopt the ANP in our paper. However, it is clear that the key for the ANP is to determine the rela-tionship structure between all features in advance (Lee and Kim, 2000). In next section, we discuss the procedures of ISM to construct the interde-pendent structures based on the relationships of the features.

4. Interpretive structural modeling

Interpretive structural modeling (ISM),

pro-posed by Warﬁeld (1974a,b, 1976) is a

compu-ter-assisted methodology to construct and

understand the fundamental of the relationships of the elements in complex systems or situations. The theory of ISM is based on discrete mathe-matics, graph theory, social sciences, group deci-sion-making, and computer assistance (Warﬁeld, 1974a,b, 1976). The procedures of ISM are begun through individual or group mental mod-els to calculate binary matrices, also called rela-tion matrix, to present the relarela-tions of the elements.

A relation matrix can be formed by asking the question like ‘‘Does the feature ei inﬂect the fea-ture ej?’’ If the answer is ‘‘Yes’’ then pij= 1, other-wise pij= 0. The general form of the relation matrix can be presented as follows:

Culture End-User Management Ability Process Resource Human Technology

Fig. 4. The structure of system development with feedback eﬀects.

(7)

where eiis the ith element in the system, pijdenotes the relation between ith and jth elements, D is the relation matrix.

After constructing the relation matrix, we can calculate the reachability matrix using Eqs. (3)

and(4)as follows

M¼ D þ I ð3Þ

M¼ Mk _{¼ M}kþ1 _{k >}₁ _ð4Þ

where I is the unit matrix, k denotes the powers, and M* is the reachability matrix. Note that the reachability matrix is under the operators of the Boolean multiplication and addition (i.e. 1· 1 = 1 and 1 + 1 = 1). For example

M ¼ 1 0 1 1 ; M2_¼ 1 0 1 1

Next we can calculate the reachability set and the priority set based on Eqs.(5) and (6), respectively, as the following equations

RðtiÞ ¼ feijmji¼ 1g ð5Þ

AðtiÞ ¼ feijmij¼ 1g ð6Þ

where mijdenotes the value of the ith row and the jth column.

Then, according to Eq.(7), the levels and rela-tionships between the elements can be determined and the structure of the elementsÕ relationships can also be expressed using the graph.

RðtiÞ \ AðtiÞ ¼ RðtiÞ ð7Þ

Next, we also use a simple example to show the steps of ISM for more understanding. Assume the ecosystem consist of water (W), ﬁsh (F), hydro-phytes (H), and ﬁsherman (M) and the relation-ships can be expressed as the following graph and relation matrix (Fig. 5).

Then, the relation matrix adds the identity ma-trix to form the M mama-trix as follows:

M¼ D þ I ¼ 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 1 2 6 6 6 4 3 7 7 7 5

Last, the reachability matrix are obtained by pow-ering the matrix, M, to satisfy the Eq.(4).

M¼ M2_{¼ M}2þn ¼ 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 1 2 6 6 6 4 3 7 7 7 5; n¼ 1; 2; . . . where the star (*) indicates the derivative relation which does not emerge in the original relation ma-trix. In order to determine the levels of the ele-ments in a hierarchical structure, the reachability set and the priority set are derived based on Eqs.

(4) and (5).

Then, the ﬁrst level can be derived according to Eq. (7) and is the ﬁsherman. The other levels can also be determined with the same procedures (Table 1).

The ﬁnal results of the relationships of the

ele-ments based on rechability matrix and Table 2

can be plotted as the following graph (Fig. 6). Note that the relationships between the ﬁsher-man and the hydrophytes are generated by the reachability matrix. In addition, since hierarchical structural analysis (HSA) and fuzzy ISM (FISM)

W F M H 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 W F H M W F H M D

(8)

(Ohuchi and Kaji, 1989; Wakabayashi et al., 1995; Ohuchi et al., 1988) have been proposed to extend ISM to the feedback structure and the fuzzy envi-ronment, we can determine the various network structures in practice.

For completeness, we discuss the whole proce-dures of ISM. Since we focus on the network struc-ture rather than the hierarchical strucstruc-ture in this paper, the procedures of determining the levels are ignored. So we can just plot the network struc-ture based on the rechability matrix and ignore the results of the reachability set and the priority set. Using the relation matrix and the reachability ma-trix, we can easily construct the relationships of the elements. Then, on the basis of the

relation-ships, we can obtain the weights using the ANP. In next section, a numerical example is used to illustrate the processes of the proposed method.

5. Numerical example

Here, we provide a numerical example to dem-onstrate the proposed method. In order to obtain our results, the ﬁrst step is to construct the net-work structure using the ISM procedures. The sec-ond step is to calculate the limited matrix and to obtain the featureÕs weights based on the network structure in the ANP. The last step is to obtain the ﬁnal weighted proximity matrix according to the featureÕs weights, and then to proceed with MDS analysis.

This numerical example examines the closeness of six products using MDS analysis. As we know, the closeness of the products is the very important information for firms. On the basis of the results, decision-makers can identify the directly competi-tive products and the product position in the mar-ket. In this example, we assume that there are five features to be evaluated, including the price, the package, the location, the function, and the manu-facturer for measuring the closeness of the prod-ucts. The proximity matrices of the five features are shown inAppendix A.

In order to compare with the conventional MDS, we ﬁrst assume the weights of all features are equal. Then, we can calculate the proximity

matrix based on Tables A.1–A.5 and the results

can be shown as inTable 3.

On the basis of the proximity matrix, we can use the conventional MDS procedures to obtain the coordinates of two-dimensional conﬁgurations of the products as shown inTable 4.

W F M H Level 1 Level 2 Level 3 Level 4 Fig. 6. Hierarchical structure of the elements. Table 1

The reachability set and the priority set

ei R(ti) A(ti) R(ti)\ A(ti) 1 1, 4 1, 2, 3 1 2 1, 2, 4 2, 3 2 3 1, 2, 3, 4 4 4 4 4 1, 2, 3, 4 4 Table 2

Levels in the ecosystem

Level 1 Fisherman

Level 2 Water

Level 3 Fish

Level 4 Hydrophytes

Table 3

The conventional proximity matrix

D* A B C D E F A 0 4.6 4.4 4 5.8 5.6 B 4.6 0 2.6 5.8 5.2 3.2 C 4.4 2.6 0 4.8 3.8 4 D 4 5.8 4.8 0 4.8 7 E 5.8 5.2 3.8 4.8 0 3 F 5.6 3.2 4 7 3 0

(9)

Next, we use the two-dimensional mapping to visually show the relationships of the products as the following ﬁgure.

On the basis of Fig. 7, we can conclude that there are four segments in this market using the conventional MDS analysis. The product A and the product E belong to the same cluster (i.e. the di-rect competitor) and the product F and the product D belong to the same cluster. The product B or the product C belongs to the niche market.

Next, we use the proposed method to consider the eﬀects of the weights in the MDS analysis. First, we will judge the relationships of the fea-tures, which can be done by our knowledge or the expertÕs opinions to form the relation matrix, D, as shown in Table 5.

From the relation matrix, we can calculate the

reachability matrix, M*, based on Eqs. (3) and

(4) and as shown inTable 6. The reachability ma-trix presents the relationships of all features, and we can construct the network structure based on

Table 6.

Based onTable 6, the network relationships can be constructed as shown inFig. 8.

Now, based on the structure as shown inFig. 8, we can process the pairwise comparisons between the features (Appendix B) to form the following supermatrix.

Fig. 7. Two-dimensional relationship mapping. Table 4

Coordinates of two-dimensional conﬁgurations

Object Dimension 1 Dimension 2

A 0.02 0.83 B 1.58 0.50 C 1.75 0.73 D 0.10 0.95 E 1.06 1.02 F 0.80 1.12 Table 5

The relation matrix between features

D Price Package Location Function Manufacturer

Price 0 1 0 1 0 Package 1 0 0 0 0 Location 1 0 0 0 0 Function 1 0 0 0 0 Manufacturer 1 0 0 0 0 Table 6

The reachability matrix between features

M* Price Package Location Function Manufacturer

Price 1 1 0 1 0

Package 1 1 0 1* 0

Location 1 1* 1 1* 0

Function 1 1* 0 1 0

Manufacturer 1 1* 0 1* 1

(10)

Then, the supermatrix is raised to limiting pow-ers to obtain the weights of all features. The weight of each feature can be shown inTable 7based on the results of the limited matrix.

Using Eq.(8), we can obtain the weighted prox-imity matrix to consider the network relationships of the features.

D_ij¼X m

k¼1

wkDijk; D¼ ½Dij ð8Þ

where D* denotes the weighted proximity matrix, wk denotes the featureÕs weight, and Dij denotes the proximity matrix according to the ith feature and the jth object. The weighted proximity matrix of the six products is represented as shown in

Table 8.

Finally, with the same procedures as like as in the conventional MDS analysis, the two-dimen-sional conﬁgurations and the two-dimentwo-dimen-sional relationship mapping can be obtained as shown inTable 9andFig. 9, respectively.

As shown in Fig. 7, there are only three seg-ments in this market. The product {A, E}, {B, D}, or {F, C} has the similar position and be-long to the same cluster. Compare with the results of the conventional MDS analysis, we can con-clude that the results show the significant differ-ences when considering the effects of the weights. In next section, we provide the detailed discussions and comparisons between the conventional MDS and the proposed method.

6. Discussions and comparisons

MDS analysis is a wildly useful tool to descript the relative locations between objectives. Price Package

Function

Manufacturer Location

Fig. 8. Network structure.

Table 7

The weights of the features using the ANP

Price Package Location Function Manufacturer wT 0.424 0.000 0.000 0.127 0.449

Table 8

The weighted proximity matrix

D* A B C D E F A 0 2.771 4.821 3.254 5.467 6.415 B 2.771 0 2.322 5.17 5.179 4.075 C 4.821 2.322 0 5.204 4.424 2.737 D 3.254 5.17 5.204 0 5.696 6.424 E 5.467 5.179 4.424 5.696 0 2.728 F 6.415 4.075 2.737 6.424 2.728 0 Table 9

Coordinates of two-dimensional conﬁgurations using the ANP

Object Dimension 1 Dimension 2

A 1.62 0.47 B 1.08 0.94 C 0.54 1.33 D 1.46 0.44 E 1.21 0.53 F 0.25 0.94

(11)

In the conventional MDS, the weights of the fea-tures usually are ignored. However, it may be a big problem when the feature has the interde-pendent relationships. In our numerical example, it is clear that the results of the conventional MDS and the proposed method have the signiﬁ-cant diﬀerences.

The problem of our example is to identify the similar degrees of the six products using the MDS procedures. The proximate scores are given based on the five features (the price, the package, the location, the function, and the manufacturer). In the conventional MDS, all features are treated with the same weights. However, it is clear that the package and the location features do not affect the similar degree of the products. Using the ANP method, the weights are more close to the real world situation. The significant differences can also be found in our numerical example.

Another advantage of the proposed method is that we can understand the relationships between all features according to the results of the ISM procedures. Understanding the relationship struc-ture among feastruc-tures is also important for deci-sion makers. Therefore, the proposed method can provide the more informative and accurate results.

7. Conclusions

Although MDS is a useful statistical tool to understand the relationships between the objects visually, it is diﬃcult for humans to judge the proximity matrix in a complex system or situation. Even through we can divide a complex system into subsystems, in practice, the features or criteria in these subsystems usually exist interdependence or feedback, so it is hard to deal with the conven-tional statistical tools.

The ANP is used to overcome these problems, and has been widely used in various ﬁelds. By cal-culating the supermatrix, we can obtain the global priority vectors. However, the results are various with the diﬀerent network structures. In this paper, the ISM procedures are used to overcome the problem by calculating the relation matrix and reachability matrix to construct the network rela-tionships between the features. Then, the weighted matrix is obtained using the ANP analysis based on the results of ISM. Finally, the weighted prox-imity matrix is obtained by the ANP and is used for the MDS analysis. On the basis of the numerical results, we can conclude that the pro-posed method can provide the more informative and accurate results than the conventional MDS analysis.

Appendix A

The proximity matrices of price, package, loca-tion, function and manufacturer are given using 9 scaling by expertÕs opinions and are presented in

Tables A.1–A.5.

Fig. 9. Two-dimensional relationship mapping using the ANP.

Table A.1

The proximity matrix by the price feature

Price A B C D E F A 0 2 2 3 7 5 B 2 0 2 6 8 1 C 2 2 0 7 5 3 D 3 6 7 0 8 7 E 7 8 5 8 0 1 F 5 1 3 7 1 0

(12)

Appendix B

The comparison of the features can be described as in Table B.1.

References

Arabic, P., Carroll, J.D., DeSarbo, W.S., 1987. Three-Way Scaling and Clustering, Sage University Paper Series on Quantitative Application in the Social Sciences. Sage Publications, Beverly Hills and London.

Karsak, E.E., Sozer, S., Alptekin, S.E., 2002. Product planning in quality function deployment using a combined analytic network process and goal programming approach. Comput. Ind. Eng. 44 (1), 171–190.

Kruskal, J.B., Wish, M., 1978. Multidimensional Scaling, Sage University Paper Series on Quantitative Application in the Social Sciences. Sage Publications, Beverly Hills and London.

Lee, J.W., Kim, S.H., 2000. Using analytic network process and goal programming for interdependent information system project selection. Comput. Oper. Res. 27 (4), 367–382. Mead, A., 1992. Review of the development of

multidimen-sional scaling methods. Statistician 41 (1), 27–39.

Meade, L.M., Presley, A., 2002. R&D project selection using the analytic network process. IEEE Trans. Eng. Manage. 49 (1), 59–66.

Momoh, J.A., Zhu, J., 2003. Optimal generation scheduling based on AHP/ANP. IEEE Trans. Systems Man Cyber-net.—Part B: Cybernet. 33 (3), 531–535.

Ohuchi, A., Kaji, I., 1989. Correction procedures for ﬂexible interpretive structural modeling. IEEE Trans. Systems Man Cybernet. 19 (1), 85–94.

Ohuchi, A., Kase, S., Kaji, I., 1988. MINDS: A ﬂexible interpretive structural modeling system. In: Proc. 1988 IEEE Internat. Conf. on Systems, Man, and Cybernetics, vol. 2, pp. 1326–1329.

Saaty, T.L., 1980. The Analytic Hierarchy Process. McGraw-Hill, New York.

Saaty, T.L., 1996. Decision Making with Dependence and Feedback: The Analytic Network Process. RWS Publica-tions, Pittsburgh.

Saaty, T.L., Vargas, L.G., 1998. Diagnosis with dependent symptoms: Bayes theorem and the analytic hierarchy process. Oper. Res. 46 (4), 491–502.

Table A.2

The proximity matrix by the package feature

Package A B C D E F A 0 8 3 5 9 7 B 8 0 5 6 1 1 C 3 5 0 3 2 7 D 5 6 3 0 4 9 E 9 1 2 4 0 1 F 7 1 7 9 1 0 Table A.3

The proximity matrix by the location feature

Location A B C D E F A 0 8 6 4 6 5 B 8 0 2 9 8 3 C 6 2 0 4 4 1 D 4 9 4 0 4 7 E 6 8 4 4 0 5 F 5 3 1 7 5 0 Table A.4

The proximity matrix by the function feature

Function A B C D E F A 0 1 3 5 2 2 B 0 1 3 7 4 C 0 7 4 8 D 0 4 6 E 0 4 F 0 Table A.5

The proximity matrix by the manufacturer feature

Manufacturer A B C D E F A 0 4 8 3 5 9 B 0 3 5 2 7 C 0 3 4 1 D 0 4 6 E 0 4 F 0 Table B.1

The comparison of the features

Criterion X For criterion, how much X is more important than Y?

Y

Price Function 7 Package

Location Function 1 Package

Location Function 1/3 Price

Location Package 1/5 Price

Manufacturer Function 7 Package

Manufacturer Function 1 Price

Manufacturer Package 1/9 Price

Function Package 1/5 Price

(13)

Sarkis, J., 2003. A strategic decision framework for Green supply chain management. J. Cleaner Production 11 (4), 397–409.

Schiﬀman, S.S., Reynolds, M.L., Young, F.W., 1981. Intro-duction to Multidimensional Scaling. Academic Press, New York.

Sekitani, K., Takahashi, I., 2001. A uniﬁed model and analysis for AHP and ANP. J. Oper. Res. Soc. Jpn. 44 (1), 67–89.

Wakabayashi, T., Itoh, L., Ohuchi, A., 1995. A method for constructing system models by fuzzy ﬂexible interpretive structural modeling fuzzy systems. In: Internat. Joint Conf. of the 4th IEEE Internat. Conf. on Fuzzy Systems and The

Second Internat. Fuzzy Engineering Symposium, Proc. 1995 IEEE Internat. Conf. on, Vol. 2, pp. 913–918.

Warﬁeld, J.N., 1974a. Toward interpretation of complex structural modeling. IEEE Trans. Systems Man Cybernet. 4 (5), 405–417.

Warﬁeld, J.N., 1974b. Developing Interconnection Matrices in Structural Modeling. IEEE Trans. Systems Man Cybernet. 4 (1), 81–87.

Warﬁeld, J.N., 1976. Societal Systems: Planning, Policy, and Complexity. Wiley Interscience, New York.

Young, F.W., Hamer, R.M., 1987. Multidimensional Scaling: History, Theory, and Applications. Lawrence Erlbaum Associates, Hillsdale, NJ.