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Visualizing preferences on spheres for group decisions based on multiplicative preference relations
European Journal of Operational Research 203 (2010) 176–184
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Visualizing preferences on spheres for group decisions based on multiplicative preference relations Li-Ching Ma * Department of Information Management, National United University, 1, Lienda, Miaoli City 36003, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 28 June 2008 Accepted 14 July 2009 Available online 18 July 2009 Keywords: Visual interactive modeling Group decisions Decision support systems Decision Balls Multiplicative preference relations
a b s t r a c t Decision makers’ choices are often influenced by visual background information. One of the difficulties in group decision is that decision makers may bias their judgment in order to increase the possibility of a preferred result. Hence, the method used to provide visual aids in helping decision making teams both to observe the background context and to perceive outliers is an important issue to consider. This study proposes an extended Decision Ball model to visualize a group’s decisions. By observing the Decision Balls, each decision maker can: see individual ranking as well as similarities between alternatives, identify the differences between individual judgments and the group’s collective opinion, observe the clusters of alternatives as well as clusters of decision makers, and discover outliers. Thus, this method can help decision makers make a more objective judgment. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction There is strong evidence that decision makers’ choices are often influenced by visual background information (Simonson and Tversky, 1992; Seiford and Zhu, 2003). For example, in the context of marketing, consumer choice theories argue that context impacts the choices consumers make (Seiford and Zhu, 2003). Visual representations can simplify and aggregate complex information into meaningful patterns, assist people in comprehending their environment, and allow for simultaneous perception of parts of the whole as well as a perception of interrelations between parts (Maruyama, 1986; Meyer, 1991). Regarding group decisions, one of the major difficulties is that decision makers may bias their judgment in order to enhance the likelihood of a preferred result (Condon et al., 2003). Hence, the method used to provide visual aids in helping decision making teams both to observe the background context and to perceive outliers is an important issue. Several graphical methods have been developed to aid the decision-making process. For instance, Jank and Kannan (2005) proposed a spatial multinomial model of customer choice to assist firms in understanding how their online customers’ preferences and choices vary across geographic markets. Kiang (2001) extended a self-organizing map (SOM) (Kohonen, 1995) network to classify decision groups using neural network techniques. Condon et al. (2003) adopted the Sammon map, based on the concept of multidimensional scaling (MDS), to visualize group decisions in * Tel.: +886 37 381828; fax: +886 37 330776. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.07.008
the context of an analytic hierarchy process. MDS (Borg and Groenen, 1997; Cox and Cox, 2000) has been widely used in marketing and decision-making (Chintagunta, 1992; Andrews and Manrai, 1999; Condon et al., 2003) and is a classical technique used to provide a visual representation of similarities among a set of alternatives. All these methods, however, use a 2-dimensional geometric plane to illustrate the multidimensional data. A 2-dimensional plane model cannot properly illustrate three points that do not obey the triangular inequality (i.e. the total length of any two edges must be larger than the length of the third edge) neither can it display four points that are not on the same plane. For instance, consider the three points, A, B, C, as illustrated in Fig. 1, where the distance between AB, BC, and AC are 3, 1, and 6, respectively, as shown in Fig. 1b. It is impossible to show their relationship with three line segments on a 2-dimensional plane, see Fig. 1a. If there are four points, A, B, C and D, which are not on the same plane, as shown in Fig. 1c, it is quite impossible to present these four points on a 2-dimensional plane. An important advantage of a sphere model is that it involves no edges (Cox and Cox, 1991). All points on a sphere can be observed easily by rotating the sphere. For example, as shown in Fig. 1d, points B and C are placed in the front of the sphere; points A and D are on the back part of it. By using a sphere tool, developed by Li and Ma (2008), all points can be seen by rotating the sphere interactively. In addition, compared with a 3-dimensional cube model, a sphere model makes it easier for a decision maker to observe the relationships among alternatives, than does a 3-dimensional cube model because the former can exhibit points on the
L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
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2. Group decision making based on multiplicative preference relations
A
B 1
3
C
6
A 3 B
6
C
1
(b)
(a) D
D A B
A B
(c)
(e)
C
C
(d)
(f)
Fig. 1. Advantages of a sphere model (a) display line segments on a 2D plane (b) display curves on a sphere (c) display four points that are not on the same plane (d) display points that are on the different parts of the sphere (e) display points in a 3D cube (f) display points on the surface of a sphere.
surface of a sphere instead of inside the cube, as shown in Fig. 1e and f. A sphere model has been adopted for displaying nations’ trading data (Cox and Cox, 1991), visualizing even-swap decision processes (Li and Ma, 2008), and selecting portfolios for mutual funds (Ma, 2008), etc. A visualization method employed here which has not been used previously in visualizing preferences in group decisions is the Decision Ball. The Decision Ball model, proposed by Li and Ma (2008) used to assist a decision maker in ranking alternatives and visualizing the decision process, is an extension of MDS which allocates alternatives on the surface of a sphere. This study extends the Decision Ball model to visualize preferences in group decisions. By observing the Decision Balls, each decision maker can see individual ranking and similarities between alternatives, identify the differences between individual judgment and the group’s collective opinion, observe the clusters of alternatives as well as decision makers, and perceive outliers, thereby aiding decision makers in a more objective judgment of a situation. The proposed approach can be extensively applied in many fields. Possible applications are the selection of promotion plans in Marketing, evaluation of suppliers in Supply Chain Management, choice of store locations in Partnership, decision of capital investment in Finance, etc. This paper is organized as follows: Section 2 illustrates group decision making based on multiplicative preference relations; Section 3 develops extended Decision Ball models to allocate individual and group decisions around the surface of spheres, which formulates the main theoretical part of this paper; therefore, those readers only interested in the application of the proposed method can skip Section 3; Section 4 uses a numerical example to demonstrate how to visualize preferences in group decisions on Decision Balls; and Section 5 discusses preference adjustments and sensitivity analyses.
One common format for expressing preferences is the use of pairwise comparisons, an approach which forces one to directly choose one object over another when comparing two objects, rather than requiring the comparison all objects simultaneously (Cook et al., 2005). Pairwise comparison ranking problems can be provided with the magnitude of the degree of preference, intensity ranking or, in terms of ordinal preferences only, a preference ranking. These are sometimes referred to also as cardinal versus ordinal preferences (Hochbaum and Levin, 2006; Li and Ma, 2007). The multiplicative preference relation, as expressed in the Analytic Hierarchy Process (AHP) (Saaty, 1980), is one of the most common forms used to present intensity rankings. There are four common approaches that a group can assume in setting the priorities for multiplicative preference relations: consensus, vote or compromise, geometric mean of the individual judgment, and using separate models (Dyer and Forman, 1992; Lai et al., 2002). Using the first approach, a group of decision makers has to achieve a consensus on each entry in the pairwise comparison matrix. If a consensus cannot be reached, then a vote or compromise can be used to set the entry’s value. If the decision makers are unwilling to vote or to compromise, then a geometric mean of the individuals’ evaluations can be calculated. If the group members have significantly different objectives or a geometric mean cannot be obtained, each group member can make an evaluation separately. A collective model can then be built to aggregate individuals’ judgments. This study assumes that a consensus, through vote or a compromise can not be achieved, and that decision makers may create an incomplete pairwise comparison matrix because, pragmatically, decision makers are sometimes unable to make comparisons among certain criteria. Conventional AHP approaches (Saaty, 1980; Lai et al., 2002) cannot treat incomplete matrices because those methods identify the weights of criteria by computing the eigenvalues of pairwise comparison matrices. It is also difficult to calculate a geometric mean within a group when several pairwise comparison matrices are incomplete. Therefore, instead of adopting a geometric mean approach, this study develops a linear programming model to aggregate each group member’s judgment. Other motivations for building a linear programming model lie in the notion that this type of model is easily extended to a goal programming model which can solve decision problems within a range of preference formats (Yu, 2002; Fan et al., 2006). Given a set of alternatives A ¼ fA1 ; A2 ; . . . ; An g, which are compared with respect to each of the m criteria denoted as C 1 ; . . . ; C m . Let an evaluation group consist of h evaluators E1 ; . . . ; Eh . The pairwise comparison of C f over C g for an evaluator k, denoted as pkf;g , is the preference or importance specified by the evaluator as the ratio of the weights of C f to C g , for all f ; g ¼ 1; . . . ; m. Let pkf;g ¼ ðwkf =wkg Þ akf;g measure the relative dominance of C f over C g in terms of priority weighs wk1 > 0; . . . ; wkm > 0, where akf;g is a multiplicative term accounting for inconsistencies. Following Saaty (1980), the measure of pkf;g is described as a 1/9 to 9 ratio scale and pkf;g ¼ 1=pkg;f is assumed for all k, f, g. The weights of criteria for an evaluator k can be approximated as follows:
Min fwkf g
subject to
m X m wk X f k k pf ;g w g f ¼1 g>f m X
ð1Þ
wkf ¼ 1;
ð2Þ
wkf P 0 8f :
ð3Þ
f ¼1
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The objective here is to minimize the sum of deviations resulting from approximation. This model is a non-linear model because of the absolute term in the objective function. From the work of Li (1996), this model can be transformed into the following linear model: Model 1: Weight approximation model for an evaluator k
Min
Obj1 ¼
fwkf g
m X m X ðwkf pkf;g wkg þ 2qkf;g Þ
ð4Þ
f ¼1 g>f
subject to
wkf pkf;g wkg þ 2qkf;g P 0 8f ; g > f ;
ð5Þ
qkf;g
ð6Þ
P 0 8f ; g > f ;
ð2Þ; ð3Þ Constraints (5) and (6) are added into Model 1 to linearize the objective function. Let wf and wg be the collective weight of criteria k for the group. Considering the degree of importance regarding each evaluator, the degree of importance of an evaluator k, denoted P as the weight bk where hk¼1 bk ¼ 1 and bk P 0 for all k, is combined to form a new collective linear model as follows: Model 2: Weight approximation model for a group
Min
Obj2 ¼
fwf g
h X
bk
k¼1
m X m X ðwf pkf;g wg þ 2qkf;g Þ
! ð7Þ
f ¼1 g>f
subject to wf pkf;g wg þ 2qkf;g P 0 8k; f ; g > f ; qkf;g m X
ð8Þ
P 0 8k; f ; g > f ;
ð9Þ
wf ¼ 1;
ð10Þ
f ¼1
wf P 0 8f :
ð11Þ
The value of bk must be given in advance. However, if the value of bk is unknown, equal weight is assumed commonly. The smaller the value of the Obj2 is, the more consent the group may achieve. After the weights of the criteria are obtained, the individual and group decision can be visualized on Decision Balls. 3. Decision Ball models for group decision making Given: a set of alternatives A ¼ fA1 ; A2 ; . . . ; An g, which are compared with respect to each of the m criteria C 1 ; . . . ; C m . The values of criteria C 1 ; . . . ; C m for an alternative Ai are expressed as ci;f , for f ¼ 1; . . . ; m. All criterion values are assumed to be continuous data. Denote C f and C f as the lower and upper bounds of the criterion value of C f , respectively. After the weights of the criteria are obtained from Models 1 and 2, the score of alternatives for each evaluator as well as the group can be calculated. The score function in this study is assumed to be in an additive form because this is common practice and thereby more understandable with respect to the decision makers (Belton and Stewart, 2002). Denote Ski and Si as the score value of an alternative Ai for an evaluator k and the group, respectively. An additive score function of an alternative Ai ðci;1 ; ci;2 ; . . . ; ci;m Þ for an evaluator k is defined as below:
Ski ðwk Þ ¼
m X
wkf
f ¼1
Pm
k f ¼1 wf
ci;f C f Cf Cf
ð12Þ
;
wkf
Ski
where ¼ 1 and P 0 for all k. It is obvious that 0 6 6 1, for all k. In order to make sure that all the weights of the criteria and the scores of the alternatives are positive, a criterion value ci;f with cost feature (it should be noted that a criterion is better kept as small as possible) is transferred from ci;f to ðC f ci;f Þ in advance. Following the score function, the dissimilarity function of reflecting the dissimilarities between alternatives Ai and Aj for an evaluator k is defined as
m X
dki;j ðwk Þ ¼
wkf
jci;f cj;f j
f ¼1
Cf Cf
ð13Þ
;
where 0 6 dki;j 6 1 and dki;j ¼ dkj;i for all k. Clearly, if ci;f ¼ cj;f for all f then dki;j ¼ 0 for all k. A Decision Ball model can then be constructed to display all alternatives on the surface of a hemisphere for either a specific evaluator or a group. The arc length between two alternatives is used to represent the dissimilarity between them, i.e., the larger the difference, the longer the arc length. However, because the arc length is monotonically related to the Euclidean distance between two points and both approximation methods make little difference to the resulting configuration (Cox and Cox, 1991), the Euclidean distance is used here for simplification. In addition, the alternative with a higher score is designed to be closer to the North Pole so that alternatives are located on the concentric circles in the order of score from a top view. For the purpose of comparison, an ideal alternative A is defined, where A ¼ A ðC 1 ; C 2 ; . . . ; C m Þ and S ¼ 1: A is designed to be located at the North Pole with coordinate ðx ; y ; z Þ ¼ ð0; 1; 0Þ. k Denote di;j as the Euclidean distance between Ai and Aj for an pffiffiffi k k evaluator k. Let di;j ¼ 2dki;j , such that if dki;j ¼ 0 then di;j ¼ 0 and if pffiffiffi pffiffiffi k k di;j ¼ 1 then di;j ¼ 2, where 2 is used because the distance bepffiffiffi tween the North Pole and equator is 2 when the radius = 1. Denote the coordinates of an alternative Ai on a ball as ðxki ; yki ; zki Þ for an evaluator k. The relationship between dki; and Ski is expressed as:
dki; ¼ 1 Ski
ð14Þ
because:
dki; ¼
m X
wkf
Cf Cf
¼
m X
wkf
ðC f C f Þ ðci;f C f Þ
Cf Cf ! ðC f C f Þ ðci;f C f Þ k k ¼ 1 Ski : wf wf Cf Cf Cf Cf f ¼1 f ¼1
f ¼1
¼
jci;f C f j
f ¼1
m X
m X
The coordinates of Aki and A are ðxki ; yki ; zki Þ and (0, 1, 0), respectively. pffiffiffi k k Since di;j ¼ 2dki;j and dki; ¼ 1 Ski ; ðdi; Þ2 ¼ ðxki 0Þ2 þ ðyki 1Þ2 þ pffiffiffi 2 ðzki 0Þ2 ¼ 2dki; ¼ 2ð1 Ski Þ2 . The relationship between yki and Ski is then expressed as:
yki ¼ 2Ski ðSki Þ2 :
ð15Þ Ski
Ski
yki
yki
It is clear that if ¼ 1 then ¼ 1; if ¼ 0, then ¼ 0. Based on the non-metric multidimensional scaling technique, ^k as a monotonic transformation of dk satisfying following denote d i;j
i;j
^k < d ^k . The coordinates ðxk ; yk ; zk Þ of condition: if dki;j < dku;v , then d u;v i;j i i i all alternatives Ai for an evaluator k can be calculated by the following model: Model 3: (A Decision Ball model for an evaluator k)
Min
fxki ;yki ;zki g
Obj3 ¼
n X n X k ^ k Þ2 ðdi;j d i;j i¼1
j>i
subject to yki ¼ 2Ski ðSki Þ2
8i;
ð16Þ
^k 6 d ^k e 8dk < dk ; d i;j u;v i;j u;v
ð17Þ
k
ðdi;j Þ2 ¼ ðxki xkj Þ2 þ ðyki ykj Þ2 þ ðzki zkj Þ2 ðxki Þ2
þ
ðyki Þ2
16
xki ; zki
þ
ðzki Þ2
6 1;
¼ 1 8i; 06
yki
e is a tolerable error:
8i; j;
ð18Þ ð19Þ
6 1 8i; ð20Þ
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L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
The objective of Model 3 is to minimize the sum of squared difk ^k . Expression (16) is from Eq. (15), ferences between di;j and d i;j where the alternative with a higher score is designed to be closer to the North Pole. Expression (17) is the monotonic transformation ^k . Expression (18) is from the definition of Euclidean from dki;j to d i;j distance. All alternatives are graphed on the surface of the northern hemisphere as restricted by (19) and (20). The Decision Ball model for a group decision can be similarly formulated. Model 3 is a non-linear model, which can be solved using some commercial optimization software, such as Global Solver of Lingo 9.0, to obtain an optimum solution. This model performs well when the number of alternatives is less than 10. However, in cases of alternatives number more than 10, some classification techniques, such as K-means (MacQueen, 1967) for instance, can be used to reduce the solving time by dividing the alternatives into several groups. The coordinates of group centers are calculated first; then, those group centers are treated as anchor points. The coordinates of alternatives can be obtained by calculating dissimilarities between the alternatives and the anchor points. Thus, all alternatives can be displayed on the Decision Ball within a reasonable amount of time. The sphere tool, developed by Li and Ma (2008), is adopted here to present spatial locations of alternatives. Decision makers can change viewpoints easily by rotating the sphere interactively.
Similarly, the weights of criteria and scores of alternatives for partners E2 and E3 can be obtained, with ðw21 ; w22 ; w23 ; w24 Þ ¼ ð0:35; 0:35; 0:18; 0:12Þ; ðw31 ; w32 ; w33 ; w34 Þ ¼ð0:09; 0:26;0:13; 0:52Þ;ðS21 ; S24 ; S25 ; S26 ; S27 ; S28 Þ ¼ ð0:41; 0:62; 0:44; 0:46; 0:18; 0:25;0:47; S22 ; S23 ; 0:55Þ, and ðS31 ; S32 ; S33 ; S34 ; S35 ; S36 ; S37 ; S38 Þ ¼ ð0:69; 0:48; 0:61; 0:27; 0:10; 0:33; 0:52; 0:66Þ. The Decision Balls for partners E2 and E3 are depicted in Figs. 3 and 4, with the ordering of eight alternatives A2 A8 A7 A4 A3 A1 A6 A5 and A1 A8 A3 A7 A2 A6 A4 A5 , respectively. Suppose that the importance degrees relative to the three partners are the same. That is, b1 ¼ b2 ¼ b3 ¼ 1=3. Applying Model 2 to
Table 1 Criteria values for 8 candidate locations.
A1 A2 A3 A4 A5 A6 A7 A8 Maximum Minimum
C1 Store size
C2 Population
C3 Traffic
C4 Rental fee
850 1600 600 1000 900 1000 1500 800 1600 600
960 960 1140 950 840 750 840 1260 1260 750
510 520 550 660 450 500 440 500 660 440
3200 4500 4000 5600 5500 4400 3800 4000 5600 3200
4. A numerical example The choice of a store location has a profound effect on the entire business life of a retail operation. Suppose there are three partners E1 ; E2 , and E3 , who would like to run a convenience store. After filtering out clearly undesired locations, eight alternative spots A ¼ fA1 ; . . . ; A8 g are proposed. The partners set four criteria to fulfill: ðC 1 Þ sufficient space, ðC 2 Þ high population density, ðC 3 Þ convenient accessibility, and ðC 4 Þ low cost. Store size is measured in square feet. The number of people who live within a one-mile radius is used to calculate population density. The average number of vehicle traffic passing the location per hour is used to evaluate the accessibility. Cost, measured by monthly rent, is the only criterion in this example which the partners would like to keep as small as possible. The criteria values of eight candidate locations are listed in Table 1. The last two rows of Table 1 are the maximum and minimum values of criteria. These three partners use pairwise comparisons to express preferences among pairs of criteria, as listed in Tables 2–5. For example, the relative dominance of C 1 over C 4 for the E1 partner is 2 (expressed as p11;4 = 2) in Table 2. Because E2 and E3 partners are unable to make comparisons among some criteria, the relationships p22;3 and p31;4 are left blank in Tables 3 and 4, respectively. Applying Model 1 to the preference values of partner E1 yields ðw11 ; w12 ; w13 ; w14 Þ ¼ ð0:13; 0:51; 0:26; 0:10Þ with an objective value of 0.17. From Eq. (12), the scores of alternatives are calculated as ðS11 ; S12 ; S13 ; S14 ; S15 ; S16 ; S17 ;S18 Þ ¼ ð0:43; 0:48; 0:59; 0:51; 0:14; 0:17; 0:28; 0:68Þ. The dissimilarities between alternatives can also be obtained from Eq. (13). Applying Model 3 to these scores and dissimilarities yields coordinates of alternatives for partner E1 . The resulting Decision Ball for E1 is displayed in Fig. 2. Because the alternative with a higher score is designed to be closer to the North Pole, the order of alternatives can be read along the latitudes of the alternative: the higher the latitude, the higher the score. The order of eight alternatives for the partner E1 is A8 A3 A4 A2 A1 A7 A6 A5 . Also, recall that the distance between two alternatives represents the dissimilarity between them: the longer the distance, the larger the dissimilarity. For instance, the dissimilarity between A8 and A2 ðd18;2 ¼ 0:45Þ is larger than that between A8 and A3 ðd18;3 ¼ 0:20Þ.
Table 2 Preference matrix for the partner E1 . E1
C1
C1 C2 C3 C4
C2
C3
C4
1/4
1/3 2
2 5 2
C2
C3
C4
1
2
3 2 3
C2
C3
C4
1/3
1/2 2
Table 3 Preference matrix for the partner E2 . E2
C1
C1 C2 C3 C4
Table 4 Preference matrix for the partner E3 . E3
C1
C1 C2 C3 C4
1/2 1/4
Table 5 Weight vectors for three partners and the group.
C1 C2 C3 C4
E1
E2
E3
Group
0.13 0.51 0.26 0.10
0.35 0.35 0.18 0.12
0.09 0.26 0.13 0.52
0.19 0.48 0.24 0.10
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L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
Fig. 2. A Decision Ball of E1 ðA8 A3 A4 A2 A1 A7 A6 A5 Þ.
the preference values listed in Tables 2–4 yields Obj2 ¼ 0:66; ðw1 ; w2 ; w3 ; w4 Þ ¼ ð0:19; 0:48; 0:24; 0:09Þ, and ðS1 ; S2 ; S3 ; S4 ; S5 ; S6 ; S7 ; S8 Þ ¼ ð0:41; 0:52; 0:55; 0:50;0:16; 0:19; 0:33; 0:64Þ. The corresponding Group Decision Ball is shown in Fig. 5, with A8 A3 A2 A4 A1 A7 A6 A5 . In this example, the order of alternatives for partner E1 is exactly the same as the final group decision. It is also important to note that the coordinates of alternatives will be shifted once the sphere is rotated. In order to compare the decisions among different partners and the group, the x coordinate of one arbitrary alternative could be fixed. In this example, the x coordinate of A8 is fixed, that is, a constraint x8 ¼ 0 is added into Model 3. Combined with certain classification techniques, the Decision Balls can also display the grouping results. Take a well known classification technique, the K-means method (MacQueen, 1967), for instance. Suppose the number of groups required is 3, the grouping results of each partner and the final group decision are displayed in Figs. 2–5, where each group is marked with a dash circle. The alternatives in the final group decision (Fig. 5) are classified as the following three groups: fA8 ; A3 g; fA2 ; A4 g; fA1 ; A7 ; A6 ; A5 g. The classification results of alternatives for partner E1 (Fig. 2) is the same as that of the final group decision (Fig. 5). The grouping of (Fig. 3) is fA8 ; A3 ; A1 g; alternatives for partner E2 fA2 ; A7 ; A4 g; fA6 ; A5 g, and for partner E3 (Fig. 4) is fA8 ; A3 ; A2 g; fA1 ; A7 g; fA6 ; A4 ; A5 g. All partners classify fA8 ; A3 g and fA6 ; A5 g in the same group. In addition to rankings of alternatives and dissimilarities among alternatives, partners can observe differences among partners and
Fig. 3. A Decision Ball of E2 ðA2 A8 A7 A4 A3 A1 A6 A5 Þ.
Fig. 4. A Decision Ball of E3 ðA1 A8 A3 A7 A2 A6 A4 A5 Þ.
Fig. 5. A group Decision Ball ðA8 A3 A2 A4 A1 A7 A6 A5 Þ.
the final group decision regarding a specific alternative. Take A8 for instance, the evaluation results of three partners (marked with solid triangles) on a group Decision Ball are displayed in Fig. 6. As can be seen in Fig. 6, partner E3 ’s evaluation is the closest to the final group decision for A8 ; whereas, E2 ’ evaluation is the most different in relation to the others. Fig. 7 shows another example for evaluations of A2 , where E2 ’s evaluation is also the most different in relation to the others’. By observing the Decision Ball, the clusters of decision makers, as well as outliers, can be detected. The differences between an individual partner’s evaluation and the final group decision can also be easily seen using Decision Balls. For instance, partner E1 ’s evaluation is depicted on the group Decision Ball in Fig. 8, where the triangle indicates E1 ’s evaluation and the circle represents the final group opinion. In addition, partner E2 ’s evaluation is displayed on the group Decision Ball in Fig. 9. Comparing Figs. 8 and 9, the overall evaluation results of E1 is much closer to the final group decision than those of E2 . This problem was solved by Global Solver of Lingo 9.0 on a Pentium 4 personal computer. The coordinates of alternatives for E1 ; E2 ; E3 , and the group are listed in Table 6. It is noteworthy that the points corresponding to alternatives may be placed on the back part of the ball since the value of xki and zki is between 1 and 1. Take E3 for instance, the points corresponding to A1 and A4 should be placed on the back part of the ball because z31 and z34 are negative. However, in order to
L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
Fig. 6. Comparisons of A8 on a group Decision Ball.
Fig. 7. Comparisons of A2 on a group Decision Ball.
clearly display spatial relationships among alternatives, all alternatives in this example are presented in the front of the ball by rotating the sphere tool developed by Li and Ma (2008).
5. Preference adjustments and sensitivity analyses The proposed method can provide visual aids in helping decision making teams to observe the background context and to perceive outliers; thus it can reduce the possibility of decision makers’ bias in their judgments, in order to obtain a preferred result. The proposed method can also assist decision makers in adjusting their judgments to find a consensus. Take partner E2 in the store selection problem for example. As shown in Fig. 9, the scores of A7 and A2 are much higher than those of the final group opinion (since the latitudes of A2 and A7 marked with a ‘‘triangle” are much higher than those marked with a ‘‘circle”). That is, the scores of A7 and A2 evaluated by E2 are too high. As listed in Table 1, A2 and A7 have the highest values in store size ðC 1 Þ, which indicates E2 over weighted criterion C 1 . Suppose E2 adjusts judgments regarding criterion C 1 , from fp21;2 ¼ 1; p21;3 ¼ 2; p21;4 ¼ 3g to fp21;2 ¼ 1=3; p21;3 ¼ 1=2; p21;4 ¼ 2g, as listed in Table 7. Denote E2a as E2 after adjustment. The weights of criteria and 2a scores of alternatives for E2a can be obtained, with ðw2a 1 ; w2 ; 2a 2a 2a 2a 2a 2a 2a 2a 2a ; w Þ ¼ ð0:20; 0:27; 0:40; 0:13Þ; ðS ; S ; S ; S ; S ; S ; S ; S Þ ¼ w2a 3 4 1 2 3 4 5 6 7 8 ð0:42; 0:52; 0:49; 0:58; 0:13; 0:26; 0:33; 0:50Þ. The Decision Ball for
181
Fig. 8. Comparisons of E1 (triangle) and the group decision (circle).
Fig. 9. Comparisons of E2 (triangle) and the group decision (circle).
E2a is depicted in Fig. 10, with the ordering of eight alternatives: A4 A2 A8 A3 A1 A7 A6 A5 . Applying Model 2 to the preference values of E1 ; E2a , and E3 yields Obj2 = 0.44, ðw1 ; w2 ; w3 ; w4 Þ ¼ ð0:16; 0:49; 0:25; 0:10Þ, and ðS1 ; S2 ; S3 ; S4 ; S5 ; S6 ; S7 ; S8 Þ ¼ ð0:42; 0:50; 0:56; 0:50; 0:15; 0:18; 0:31; 0:66Þ. The corresponding group Decision Ball is shown in Fig. 11, with A8 A3 A4 A2 A1 A7 A6 A5 . The value of Obj2 is reduced from 0.66 to 0.44. Since the smaller the value of the Obj2 is, the more consent the group may achieve, it means the group achieves greater consent after E2 adjusts its preferences. Fig. 12 shows the evaluations of E1 ; E2a ; E3 , and the group for a specific alternative A2 . In comparison with Fig. 7, E2a ’s score (latitude) in Fig. 12 is much closer to the group than that of E2 in Fig. 7. E2a ’s evaluation is depicted on the adjusted group Decision Ball in Fig. 13, where the triangle indicates E2a ’s evaluation and the circle represents the adjusted final group opinion. The sensitivity of the proposed approach for adding/removing an evaluator or an alternative is analyzed as follows. Suppose a partner E4 joins the group decision with preference matrix as listed in Table 8. The weights of criteria and scores of alternatives for E4 can be obtained, with ðw41 ; w42 ; w43 ; w44 Þ ¼ ð0:32; 0:42; 0:21; 0:05Þ; ðS41 ; S42 ; S43 ; S44 ; S45 ; S46 ; S47 ; S48 Þ ¼ð0:37; 0:60; 0:46; 0:50; 0:18; 0:21; 0:40; 0:57Þ. The Decision Ball for E4 is depicted in Fig. 14, with the ordering of eight alternatives A2 A8 A4 A3 A7 A1 A6 A5 . Applying Model 3 to the preferences matrices of E1 ; E2a ;E3 , and E4 yields Obj2 = 0.63, ðw1 ;w2 ; w3 ;w4 Þ ¼ ð0:17;0:51;0:26;0:06Þ, and ðS1 ;S2 ;
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Table 6 Coordinates of alternatives for E1 ; E2 ; E3 , and the group. E1
A1 A2 A3 A4 A5 A6 A7 A8
E2
E3
Group
x
y
z
x
y
z
x
y
z
x
y
z
0.50 0.58 0.13 0.65 0.74 0.61 0.72 0.00
0.67 0.73 0.83 0.76 0.27 0.32 0.49 0.90
0.55 0.37 0.54 0.08 0.62 0.72 0.50 0.45
0.31 0.47 -0.01 0.47 0.56 0.64 0.68 0.00
0.65 0.85 0.68 0.70 0.33 0.43 0.72 0.80
0.69 0.23 0.73 0.53 0.76 0.63 0.17 0.60
0.41 0.53 0.31 0.88 0.95 0.65 0.61 0.00
0.91 0.73 0.85 0.46 0.19 0.55 0.77 0.89
-0.09 0.43 0.43 -0.07 0.24 0.53 0.21 0.46
0.45 0.54 0.08 0.65 0.63 0.75 0.70 0.00
0.66 0.77 0.79 0.75 0.29 0.34 0.55 0.87
0.60 0.35 0.60 0.10 0.72 0.57 0.45 0.49
Table 7 Adjusted preference matrix for the partner E2 .
S3 ;S4 ;S5 ;S6 ;S7 ;S8 Þ ¼ ð0:39;0:50;0:56;0:53; 0:16;0:17;0:29; 0:66Þ. The corresponding group Decision Ball and comparisons of A2 for E1 ;E2a ;E3 , and E4 are shown in Fig. 15. Because the preference matrices of the original partners are not affected by adding a new evaluator, the Decision Balls for E1 ; E2a , and E3 are not changed. However, since the weights of criteria for the final group decision are affected by adding a new evaluator, the final group Decision Ball is altered. Comparisons of group Decision Ball with three partners (marked with triangle) and 4 partners (marked with circle) are shown in Fig. 16. The sensitivity analysis for removing an existing evaluator is similar to that of adding one.
Fig. 10. A Decision Ball of E2a ðA4 A2 A8 A3 A1 A7 A6 A5 Þ.
Fig. 12. Comparisons of A2 on an adjusted group Decision Ball.
Fig. 11. An adjusted group Decision Ball ðA8 A3 A4 A2 A1 A7 A6 A5 Þ.
Fig. 13. Comparisons of E2a (triangle) and the adjusted group decision (circle).
L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
183
Table 8 Preference matrix for a new partner E4 . E4 C1 C2 C3 C4
C1
C2
C3
C4
1/3
2 2
7 9 5
Next, the sensitivity of the proposed approach for adding an alternative is analyzed. Suppose an alternative A9 is added as a candidate spot with c9;1 ¼ 950; c9;2 ¼ 1000; c9;3 ¼ 480, and c9;4 ¼ 4800. According to Model 1 and Model 2, adding an alternative does not change the weights of the criteria for the partners and the group. If the c9;f ; f ¼ 1; . . . ; 4, do not alter the original upper and lower bounds of C f , the scores of, and dissimilarities among, alternatives will remain the same for A1 ; . . . ; A8 . Therefore, the latitudes of alternatives will stay the same. Because relationships among A9 and the original eight alternatives are included into the Decision Ball model, the relative distances among alternatives may be slightly changed. Take E1 for instance, the corresponding Decision Ball after adding A9 is shown in Fig. 17. The Decision Ball before (marked with circle) and after adding A9 (marked with triangle) are shown in Fig. 18.
Fig. 16. Comparisons of group Decision Ball with three partners (triangle) and 4 partners (circle).
Fig. 17. A Decision Ball of E1 after adding A9 .
Fig. 14. A Decision Ball of E4 ðA2 A8 A4 A3 A7 A1 A6 A5 Þ.
The analyses of other partners and the group for adding A9 are similar to that of partner E1 . Sensitivity of the proposed approach for removing an alternative is also similar to that of adding an alternative. 6. Conclusions
Fig. 15. Comparisons of A2 for E1 ; E2a ; E3 , and E4 on a group Decision Ball.
This study extends the Decision Ball model to visualize group decision making. The arc length between two alternatives is used to represent the dissimilarity between them, e.g., the larger the difference, the longer the arc length. The alternative with a higher score is designed to be closer to the North Pole. By observing the Decision Balls, each evaluator can see individual ranking and similarities among alternatives, identify the differences between individual judgment and the group’s collective opinion, observe the clusters of alternatives as well as decision makers, and perceive outliers. Because outliers can be found easily, it can reduce the possibility of decision makers’ bias in their judgments to get a preferred result, and thereby help decision makers to achieve a more objective judgment. The proposed method can also assist decision makers in adjusting their judgments to find a consensus. One restriction of this approach is the running time that may considerably increase when the number of alternatives becomes large. In future studies, the question of how to linearize this nonlinear model in order to deal with larger problems can be
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Fig. 18. Comparisons of E1 before (circle) and after (triangle) adding A9 .
addressed. Nevertheless, because the pairwise comparison preference relationships are good for smaller problems or the final stage of decision making, the proposed approach has been shown helpful in the cases where the number of alternatives is fewer than 10. Acknowledgement This research is supported by the National Science Council of the Republic of China under contract NSC 96-2416-H-239-002. References Andrews, R.L., Manrai, A.K., 1999. MDS maps for product attributes and market response: an application to scanner panel data. Marketing Science 18 (4), 584– 604. Belton, V., Stewart, T.J., 2002. Multiple criteria decision analysis. An Integrated Approach. Kluwer Academic Publishers, Norwell, MA. Borg, I., Groenen, P., 1997. Modern Multidimensional Scaling. Springer, New York. Chintagunta, P.K., 1992. Estimating a multinomial probit model of brand choice using the method of simulated moments. Marketing Science 11 (4), 386–407.
Condon, E., Golden, B., Wasil, E., 2003. Visualizing group decisions in the analytic hierarchy process. Computers and Operations Research 30, 1435–1445. Cook, W.D., Golany, B., Kress, M., Penn, M., Raviv, T., 2005. Optimal allocation of proposals to reviewers to facilitate effective ranking. Management Science 51 (4), 655–661. Cox, T.F., Cox, M.A.A., 1991. Multidimensional scaling on a sphere. Communications on Statistics – Theory and Methods 20 (9), 2943–2953. Cox, T.F., Cox, M.A.A., 2000. Multidimensional Scaling. Chapman & Hall, London. Dyer, R.F., Forman, E.H., 1992. Group decision support with the analytic hierarchy process. Decision Support Systems 8, 99–124. Fan, Z.P., Ma, J., Jiang, Y.P., Sun, Y.H., Ma, L., 2006. A goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference relations. European Journal of Operational Research 174, 311–321. Hochbaum, D.S., Levin, A., 2006. Methodologies and algorithms for group-rankings decision. Management Science 52 (9), 1394–1408. Jank, W., Kannan, P.K., 2005. Understanding geographical markets of online firms using spatial models of customer choice. Marketing Science 24 (4), 623–634. Kiang, M.Y., 2001. Extending the Kohonen self-organizing map networks for clustering analysis. Computations Statistics and Data Analysis 38, 161–180. Kohonen, T., 1995. Self-organizing Maps. Springer, Berlin. Lai, V.S., Wong, B.K., Cheung, W., 2002. Group decision making in a multiple criteria environment: A case using the AHP in software selection. European Journal of Operational Research 137, 134–144. Li, H.L., 1996. An efficient method for solving linear goal programming problems. Journal of Optimization Theory and Applications 9 (2), 467–471. Li, H.L., Ma, L.C., 2007. Detecting and adjusting ordinal and cardinal inconsistencies through a graphical and optimal approach in AHP models. Computers and Operations Research 34, 780–798. Li, H.L., Ma, L.C., 2008. Visualizing decision processes on spheres based on the even swap concept. Decision Support Systems 45, 354–367. Ma, L.C., 2008. Selecting portfolios for mutual funds visually on decision balls. In: INFORS International Conference. MacQueen, J.B., 1967. Some methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1. University of California Press, Berkeley, pp. 281–297. Maruyama, M., 1986. Toward picture-coded information systems. Futures 18, 450– 452. Meyer, A., 1991. Visual data in organizational research. Organization Science 2 (2), 18–236. Saaty, T.L., 1980. The Analytic Hierarchy Process. McGraw-Hill, New York. Seiford, L.M., Zhu, J., 2003. Context-dependent data envelopment analysis – Measuring attractiveness and progress. OMEGA 31 (5), 397–408. Simonson, I., Tversky, A., 1992. Choice in context: tradeoff contrast and extremeness aversion. Journal of Marketing Research 29, 281–295. Yu, C.S., 2002. A GP-AHP method for solving group decision-making fuzzy AHP problems. Computers and Operations Research 29, 1969–2001.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Visualizing preferences on spheres for group decisions based on multiplicative preference relations Li-Ching Ma * Department of Information Management, National United University, 1, Lienda, Miaoli City 36003, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 28 June 2008 Accepted 14 July 2009 Available online 18 July 2009 Keywords: Visual interactive modeling Group decisions Decision support systems Decision Balls Multiplicative preference relations
a b s t r a c t Decision makers’ choices are often influenced by visual background information. One of the difficulties in group decision is that decision makers may bias their judgment in order to increase the possibility of a preferred result. Hence, the method used to provide visual aids in helping decision making teams both to observe the background context and to perceive outliers is an important issue to consider. This study proposes an extended Decision Ball model to visualize a group’s decisions. By observing the Decision Balls, each decision maker can: see individual ranking as well as similarities between alternatives, identify the differences between individual judgments and the group’s collective opinion, observe the clusters of alternatives as well as clusters of decision makers, and discover outliers. Thus, this method can help decision makers make a more objective judgment. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction There is strong evidence that decision makers’ choices are often influenced by visual background information (Simonson and Tversky, 1992; Seiford and Zhu, 2003). For example, in the context of marketing, consumer choice theories argue that context impacts the choices consumers make (Seiford and Zhu, 2003). Visual representations can simplify and aggregate complex information into meaningful patterns, assist people in comprehending their environment, and allow for simultaneous perception of parts of the whole as well as a perception of interrelations between parts (Maruyama, 1986; Meyer, 1991). Regarding group decisions, one of the major difficulties is that decision makers may bias their judgment in order to enhance the likelihood of a preferred result (Condon et al., 2003). Hence, the method used to provide visual aids in helping decision making teams both to observe the background context and to perceive outliers is an important issue. Several graphical methods have been developed to aid the decision-making process. For instance, Jank and Kannan (2005) proposed a spatial multinomial model of customer choice to assist firms in understanding how their online customers’ preferences and choices vary across geographic markets. Kiang (2001) extended a self-organizing map (SOM) (Kohonen, 1995) network to classify decision groups using neural network techniques. Condon et al. (2003) adopted the Sammon map, based on the concept of multidimensional scaling (MDS), to visualize group decisions in * Tel.: +886 37 381828; fax: +886 37 330776. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.07.008
the context of an analytic hierarchy process. MDS (Borg and Groenen, 1997; Cox and Cox, 2000) has been widely used in marketing and decision-making (Chintagunta, 1992; Andrews and Manrai, 1999; Condon et al., 2003) and is a classical technique used to provide a visual representation of similarities among a set of alternatives. All these methods, however, use a 2-dimensional geometric plane to illustrate the multidimensional data. A 2-dimensional plane model cannot properly illustrate three points that do not obey the triangular inequality (i.e. the total length of any two edges must be larger than the length of the third edge) neither can it display four points that are not on the same plane. For instance, consider the three points, A, B, C, as illustrated in Fig. 1, where the distance between AB, BC, and AC are 3, 1, and 6, respectively, as shown in Fig. 1b. It is impossible to show their relationship with three line segments on a 2-dimensional plane, see Fig. 1a. If there are four points, A, B, C and D, which are not on the same plane, as shown in Fig. 1c, it is quite impossible to present these four points on a 2-dimensional plane. An important advantage of a sphere model is that it involves no edges (Cox and Cox, 1991). All points on a sphere can be observed easily by rotating the sphere. For example, as shown in Fig. 1d, points B and C are placed in the front of the sphere; points A and D are on the back part of it. By using a sphere tool, developed by Li and Ma (2008), all points can be seen by rotating the sphere interactively. In addition, compared with a 3-dimensional cube model, a sphere model makes it easier for a decision maker to observe the relationships among alternatives, than does a 3-dimensional cube model because the former can exhibit points on the
L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
177
2. Group decision making based on multiplicative preference relations
A
B 1
3
C
6
A 3 B
6
C
1
(b)
(a) D
D A B
A B
(c)
(e)
C
C
(d)
(f)
Fig. 1. Advantages of a sphere model (a) display line segments on a 2D plane (b) display curves on a sphere (c) display four points that are not on the same plane (d) display points that are on the different parts of the sphere (e) display points in a 3D cube (f) display points on the surface of a sphere.
surface of a sphere instead of inside the cube, as shown in Fig. 1e and f. A sphere model has been adopted for displaying nations’ trading data (Cox and Cox, 1991), visualizing even-swap decision processes (Li and Ma, 2008), and selecting portfolios for mutual funds (Ma, 2008), etc. A visualization method employed here which has not been used previously in visualizing preferences in group decisions is the Decision Ball. The Decision Ball model, proposed by Li and Ma (2008) used to assist a decision maker in ranking alternatives and visualizing the decision process, is an extension of MDS which allocates alternatives on the surface of a sphere. This study extends the Decision Ball model to visualize preferences in group decisions. By observing the Decision Balls, each decision maker can see individual ranking and similarities between alternatives, identify the differences between individual judgment and the group’s collective opinion, observe the clusters of alternatives as well as decision makers, and perceive outliers, thereby aiding decision makers in a more objective judgment of a situation. The proposed approach can be extensively applied in many fields. Possible applications are the selection of promotion plans in Marketing, evaluation of suppliers in Supply Chain Management, choice of store locations in Partnership, decision of capital investment in Finance, etc. This paper is organized as follows: Section 2 illustrates group decision making based on multiplicative preference relations; Section 3 develops extended Decision Ball models to allocate individual and group decisions around the surface of spheres, which formulates the main theoretical part of this paper; therefore, those readers only interested in the application of the proposed method can skip Section 3; Section 4 uses a numerical example to demonstrate how to visualize preferences in group decisions on Decision Balls; and Section 5 discusses preference adjustments and sensitivity analyses.
One common format for expressing preferences is the use of pairwise comparisons, an approach which forces one to directly choose one object over another when comparing two objects, rather than requiring the comparison all objects simultaneously (Cook et al., 2005). Pairwise comparison ranking problems can be provided with the magnitude of the degree of preference, intensity ranking or, in terms of ordinal preferences only, a preference ranking. These are sometimes referred to also as cardinal versus ordinal preferences (Hochbaum and Levin, 2006; Li and Ma, 2007). The multiplicative preference relation, as expressed in the Analytic Hierarchy Process (AHP) (Saaty, 1980), is one of the most common forms used to present intensity rankings. There are four common approaches that a group can assume in setting the priorities for multiplicative preference relations: consensus, vote or compromise, geometric mean of the individual judgment, and using separate models (Dyer and Forman, 1992; Lai et al., 2002). Using the first approach, a group of decision makers has to achieve a consensus on each entry in the pairwise comparison matrix. If a consensus cannot be reached, then a vote or compromise can be used to set the entry’s value. If the decision makers are unwilling to vote or to compromise, then a geometric mean of the individuals’ evaluations can be calculated. If the group members have significantly different objectives or a geometric mean cannot be obtained, each group member can make an evaluation separately. A collective model can then be built to aggregate individuals’ judgments. This study assumes that a consensus, through vote or a compromise can not be achieved, and that decision makers may create an incomplete pairwise comparison matrix because, pragmatically, decision makers are sometimes unable to make comparisons among certain criteria. Conventional AHP approaches (Saaty, 1980; Lai et al., 2002) cannot treat incomplete matrices because those methods identify the weights of criteria by computing the eigenvalues of pairwise comparison matrices. It is also difficult to calculate a geometric mean within a group when several pairwise comparison matrices are incomplete. Therefore, instead of adopting a geometric mean approach, this study develops a linear programming model to aggregate each group member’s judgment. Other motivations for building a linear programming model lie in the notion that this type of model is easily extended to a goal programming model which can solve decision problems within a range of preference formats (Yu, 2002; Fan et al., 2006). Given a set of alternatives A ¼ fA1 ; A2 ; . . . ; An g, which are compared with respect to each of the m criteria denoted as C 1 ; . . . ; C m . Let an evaluation group consist of h evaluators E1 ; . . . ; Eh . The pairwise comparison of C f over C g for an evaluator k, denoted as pkf;g , is the preference or importance specified by the evaluator as the ratio of the weights of C f to C g , for all f ; g ¼ 1; . . . ; m. Let pkf;g ¼ ðwkf =wkg Þ akf;g measure the relative dominance of C f over C g in terms of priority weighs wk1 > 0; . . . ; wkm > 0, where akf;g is a multiplicative term accounting for inconsistencies. Following Saaty (1980), the measure of pkf;g is described as a 1/9 to 9 ratio scale and pkf;g ¼ 1=pkg;f is assumed for all k, f, g. The weights of criteria for an evaluator k can be approximated as follows:
Min fwkf g
subject to
m X m wk X f k k pf ;g w g f ¼1 g>f m X
ð1Þ
wkf ¼ 1;
ð2Þ
wkf P 0 8f :
ð3Þ
f ¼1
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The objective here is to minimize the sum of deviations resulting from approximation. This model is a non-linear model because of the absolute term in the objective function. From the work of Li (1996), this model can be transformed into the following linear model: Model 1: Weight approximation model for an evaluator k
Min
Obj1 ¼
fwkf g
m X m X ðwkf pkf;g wkg þ 2qkf;g Þ
ð4Þ
f ¼1 g>f
subject to
wkf pkf;g wkg þ 2qkf;g P 0 8f ; g > f ;
ð5Þ
qkf;g
ð6Þ
P 0 8f ; g > f ;
ð2Þ; ð3Þ Constraints (5) and (6) are added into Model 1 to linearize the objective function. Let wf and wg be the collective weight of criteria k for the group. Considering the degree of importance regarding each evaluator, the degree of importance of an evaluator k, denoted P as the weight bk where hk¼1 bk ¼ 1 and bk P 0 for all k, is combined to form a new collective linear model as follows: Model 2: Weight approximation model for a group
Min
Obj2 ¼
fwf g
h X
bk
k¼1
m X m X ðwf pkf;g wg þ 2qkf;g Þ
! ð7Þ
f ¼1 g>f
subject to wf pkf;g wg þ 2qkf;g P 0 8k; f ; g > f ; qkf;g m X
ð8Þ
P 0 8k; f ; g > f ;
ð9Þ
wf ¼ 1;
ð10Þ
f ¼1
wf P 0 8f :
ð11Þ
The value of bk must be given in advance. However, if the value of bk is unknown, equal weight is assumed commonly. The smaller the value of the Obj2 is, the more consent the group may achieve. After the weights of the criteria are obtained, the individual and group decision can be visualized on Decision Balls. 3. Decision Ball models for group decision making Given: a set of alternatives A ¼ fA1 ; A2 ; . . . ; An g, which are compared with respect to each of the m criteria C 1 ; . . . ; C m . The values of criteria C 1 ; . . . ; C m for an alternative Ai are expressed as ci;f , for f ¼ 1; . . . ; m. All criterion values are assumed to be continuous data. Denote C f and C f as the lower and upper bounds of the criterion value of C f , respectively. After the weights of the criteria are obtained from Models 1 and 2, the score of alternatives for each evaluator as well as the group can be calculated. The score function in this study is assumed to be in an additive form because this is common practice and thereby more understandable with respect to the decision makers (Belton and Stewart, 2002). Denote Ski and Si as the score value of an alternative Ai for an evaluator k and the group, respectively. An additive score function of an alternative Ai ðci;1 ; ci;2 ; . . . ; ci;m Þ for an evaluator k is defined as below:
Ski ðwk Þ ¼
m X
wkf
f ¼1
Pm
k f ¼1 wf
ci;f C f Cf Cf
ð12Þ
;
wkf
Ski
where ¼ 1 and P 0 for all k. It is obvious that 0 6 6 1, for all k. In order to make sure that all the weights of the criteria and the scores of the alternatives are positive, a criterion value ci;f with cost feature (it should be noted that a criterion is better kept as small as possible) is transferred from ci;f to ðC f ci;f Þ in advance. Following the score function, the dissimilarity function of reflecting the dissimilarities between alternatives Ai and Aj for an evaluator k is defined as
m X
dki;j ðwk Þ ¼
wkf
jci;f cj;f j
f ¼1
Cf Cf
ð13Þ
;
where 0 6 dki;j 6 1 and dki;j ¼ dkj;i for all k. Clearly, if ci;f ¼ cj;f for all f then dki;j ¼ 0 for all k. A Decision Ball model can then be constructed to display all alternatives on the surface of a hemisphere for either a specific evaluator or a group. The arc length between two alternatives is used to represent the dissimilarity between them, i.e., the larger the difference, the longer the arc length. However, because the arc length is monotonically related to the Euclidean distance between two points and both approximation methods make little difference to the resulting configuration (Cox and Cox, 1991), the Euclidean distance is used here for simplification. In addition, the alternative with a higher score is designed to be closer to the North Pole so that alternatives are located on the concentric circles in the order of score from a top view. For the purpose of comparison, an ideal alternative A is defined, where A ¼ A ðC 1 ; C 2 ; . . . ; C m Þ and S ¼ 1: A is designed to be located at the North Pole with coordinate ðx ; y ; z Þ ¼ ð0; 1; 0Þ. k Denote di;j as the Euclidean distance between Ai and Aj for an pffiffiffi k k evaluator k. Let di;j ¼ 2dki;j , such that if dki;j ¼ 0 then di;j ¼ 0 and if pffiffiffi pffiffiffi k k di;j ¼ 1 then di;j ¼ 2, where 2 is used because the distance bepffiffiffi tween the North Pole and equator is 2 when the radius = 1. Denote the coordinates of an alternative Ai on a ball as ðxki ; yki ; zki Þ for an evaluator k. The relationship between dki; and Ski is expressed as:
dki; ¼ 1 Ski
ð14Þ
because:
dki; ¼
m X
wkf
Cf Cf
¼
m X
wkf
ðC f C f Þ ðci;f C f Þ
Cf Cf ! ðC f C f Þ ðci;f C f Þ k k ¼ 1 Ski : wf wf Cf Cf Cf Cf f ¼1 f ¼1
f ¼1
¼
jci;f C f j
f ¼1
m X
m X
The coordinates of Aki and A are ðxki ; yki ; zki Þ and (0, 1, 0), respectively. pffiffiffi k k Since di;j ¼ 2dki;j and dki; ¼ 1 Ski ; ðdi; Þ2 ¼ ðxki 0Þ2 þ ðyki 1Þ2 þ pffiffiffi 2 ðzki 0Þ2 ¼ 2dki; ¼ 2ð1 Ski Þ2 . The relationship between yki and Ski is then expressed as:
yki ¼ 2Ski ðSki Þ2 :
ð15Þ Ski
Ski
yki
yki
It is clear that if ¼ 1 then ¼ 1; if ¼ 0, then ¼ 0. Based on the non-metric multidimensional scaling technique, ^k as a monotonic transformation of dk satisfying following denote d i;j
i;j
^k < d ^k . The coordinates ðxk ; yk ; zk Þ of condition: if dki;j < dku;v , then d u;v i;j i i i all alternatives Ai for an evaluator k can be calculated by the following model: Model 3: (A Decision Ball model for an evaluator k)
Min
fxki ;yki ;zki g
Obj3 ¼
n X n X k ^ k Þ2 ðdi;j d i;j i¼1
j>i
subject to yki ¼ 2Ski ðSki Þ2
8i;
ð16Þ
^k 6 d ^k e 8dk < dk ; d i;j u;v i;j u;v
ð17Þ
k
ðdi;j Þ2 ¼ ðxki xkj Þ2 þ ðyki ykj Þ2 þ ðzki zkj Þ2 ðxki Þ2
þ
ðyki Þ2
16
xki ; zki
þ
ðzki Þ2
6 1;
¼ 1 8i; 06
yki
e is a tolerable error:
8i; j;
ð18Þ ð19Þ
6 1 8i; ð20Þ
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The objective of Model 3 is to minimize the sum of squared difk ^k . Expression (16) is from Eq. (15), ferences between di;j and d i;j where the alternative with a higher score is designed to be closer to the North Pole. Expression (17) is the monotonic transformation ^k . Expression (18) is from the definition of Euclidean from dki;j to d i;j distance. All alternatives are graphed on the surface of the northern hemisphere as restricted by (19) and (20). The Decision Ball model for a group decision can be similarly formulated. Model 3 is a non-linear model, which can be solved using some commercial optimization software, such as Global Solver of Lingo 9.0, to obtain an optimum solution. This model performs well when the number of alternatives is less than 10. However, in cases of alternatives number more than 10, some classification techniques, such as K-means (MacQueen, 1967) for instance, can be used to reduce the solving time by dividing the alternatives into several groups. The coordinates of group centers are calculated first; then, those group centers are treated as anchor points. The coordinates of alternatives can be obtained by calculating dissimilarities between the alternatives and the anchor points. Thus, all alternatives can be displayed on the Decision Ball within a reasonable amount of time. The sphere tool, developed by Li and Ma (2008), is adopted here to present spatial locations of alternatives. Decision makers can change viewpoints easily by rotating the sphere interactively.
Similarly, the weights of criteria and scores of alternatives for partners E2 and E3 can be obtained, with ðw21 ; w22 ; w23 ; w24 Þ ¼ ð0:35; 0:35; 0:18; 0:12Þ; ðw31 ; w32 ; w33 ; w34 Þ ¼ð0:09; 0:26;0:13; 0:52Þ;ðS21 ; S24 ; S25 ; S26 ; S27 ; S28 Þ ¼ ð0:41; 0:62; 0:44; 0:46; 0:18; 0:25;0:47; S22 ; S23 ; 0:55Þ, and ðS31 ; S32 ; S33 ; S34 ; S35 ; S36 ; S37 ; S38 Þ ¼ ð0:69; 0:48; 0:61; 0:27; 0:10; 0:33; 0:52; 0:66Þ. The Decision Balls for partners E2 and E3 are depicted in Figs. 3 and 4, with the ordering of eight alternatives A2 A8 A7 A4 A3 A1 A6 A5 and A1 A8 A3 A7 A2 A6 A4 A5 , respectively. Suppose that the importance degrees relative to the three partners are the same. That is, b1 ¼ b2 ¼ b3 ¼ 1=3. Applying Model 2 to
Table 1 Criteria values for 8 candidate locations.
A1 A2 A3 A4 A5 A6 A7 A8 Maximum Minimum
C1 Store size
C2 Population
C3 Traffic
C4 Rental fee
850 1600 600 1000 900 1000 1500 800 1600 600
960 960 1140 950 840 750 840 1260 1260 750
510 520 550 660 450 500 440 500 660 440
3200 4500 4000 5600 5500 4400 3800 4000 5600 3200
4. A numerical example The choice of a store location has a profound effect on the entire business life of a retail operation. Suppose there are three partners E1 ; E2 , and E3 , who would like to run a convenience store. After filtering out clearly undesired locations, eight alternative spots A ¼ fA1 ; . . . ; A8 g are proposed. The partners set four criteria to fulfill: ðC 1 Þ sufficient space, ðC 2 Þ high population density, ðC 3 Þ convenient accessibility, and ðC 4 Þ low cost. Store size is measured in square feet. The number of people who live within a one-mile radius is used to calculate population density. The average number of vehicle traffic passing the location per hour is used to evaluate the accessibility. Cost, measured by monthly rent, is the only criterion in this example which the partners would like to keep as small as possible. The criteria values of eight candidate locations are listed in Table 1. The last two rows of Table 1 are the maximum and minimum values of criteria. These three partners use pairwise comparisons to express preferences among pairs of criteria, as listed in Tables 2–5. For example, the relative dominance of C 1 over C 4 for the E1 partner is 2 (expressed as p11;4 = 2) in Table 2. Because E2 and E3 partners are unable to make comparisons among some criteria, the relationships p22;3 and p31;4 are left blank in Tables 3 and 4, respectively. Applying Model 1 to the preference values of partner E1 yields ðw11 ; w12 ; w13 ; w14 Þ ¼ ð0:13; 0:51; 0:26; 0:10Þ with an objective value of 0.17. From Eq. (12), the scores of alternatives are calculated as ðS11 ; S12 ; S13 ; S14 ; S15 ; S16 ; S17 ;S18 Þ ¼ ð0:43; 0:48; 0:59; 0:51; 0:14; 0:17; 0:28; 0:68Þ. The dissimilarities between alternatives can also be obtained from Eq. (13). Applying Model 3 to these scores and dissimilarities yields coordinates of alternatives for partner E1 . The resulting Decision Ball for E1 is displayed in Fig. 2. Because the alternative with a higher score is designed to be closer to the North Pole, the order of alternatives can be read along the latitudes of the alternative: the higher the latitude, the higher the score. The order of eight alternatives for the partner E1 is A8 A3 A4 A2 A1 A7 A6 A5 . Also, recall that the distance between two alternatives represents the dissimilarity between them: the longer the distance, the larger the dissimilarity. For instance, the dissimilarity between A8 and A2 ðd18;2 ¼ 0:45Þ is larger than that between A8 and A3 ðd18;3 ¼ 0:20Þ.
Table 2 Preference matrix for the partner E1 . E1
C1
C1 C2 C3 C4
C2
C3
C4
1/4
1/3 2
2 5 2
C2
C3
C4
1
2
3 2 3
C2
C3
C4
1/3
1/2 2
Table 3 Preference matrix for the partner E2 . E2
C1
C1 C2 C3 C4
Table 4 Preference matrix for the partner E3 . E3
C1
C1 C2 C3 C4
1/2 1/4
Table 5 Weight vectors for three partners and the group.
C1 C2 C3 C4
E1
E2
E3
Group
0.13 0.51 0.26 0.10
0.35 0.35 0.18 0.12
0.09 0.26 0.13 0.52
0.19 0.48 0.24 0.10
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Fig. 2. A Decision Ball of E1 ðA8 A3 A4 A2 A1 A7 A6 A5 Þ.
the preference values listed in Tables 2–4 yields Obj2 ¼ 0:66; ðw1 ; w2 ; w3 ; w4 Þ ¼ ð0:19; 0:48; 0:24; 0:09Þ, and ðS1 ; S2 ; S3 ; S4 ; S5 ; S6 ; S7 ; S8 Þ ¼ ð0:41; 0:52; 0:55; 0:50;0:16; 0:19; 0:33; 0:64Þ. The corresponding Group Decision Ball is shown in Fig. 5, with A8 A3 A2 A4 A1 A7 A6 A5 . In this example, the order of alternatives for partner E1 is exactly the same as the final group decision. It is also important to note that the coordinates of alternatives will be shifted once the sphere is rotated. In order to compare the decisions among different partners and the group, the x coordinate of one arbitrary alternative could be fixed. In this example, the x coordinate of A8 is fixed, that is, a constraint x8 ¼ 0 is added into Model 3. Combined with certain classification techniques, the Decision Balls can also display the grouping results. Take a well known classification technique, the K-means method (MacQueen, 1967), for instance. Suppose the number of groups required is 3, the grouping results of each partner and the final group decision are displayed in Figs. 2–5, where each group is marked with a dash circle. The alternatives in the final group decision (Fig. 5) are classified as the following three groups: fA8 ; A3 g; fA2 ; A4 g; fA1 ; A7 ; A6 ; A5 g. The classification results of alternatives for partner E1 (Fig. 2) is the same as that of the final group decision (Fig. 5). The grouping of (Fig. 3) is fA8 ; A3 ; A1 g; alternatives for partner E2 fA2 ; A7 ; A4 g; fA6 ; A5 g, and for partner E3 (Fig. 4) is fA8 ; A3 ; A2 g; fA1 ; A7 g; fA6 ; A4 ; A5 g. All partners classify fA8 ; A3 g and fA6 ; A5 g in the same group. In addition to rankings of alternatives and dissimilarities among alternatives, partners can observe differences among partners and
Fig. 3. A Decision Ball of E2 ðA2 A8 A7 A4 A3 A1 A6 A5 Þ.
Fig. 4. A Decision Ball of E3 ðA1 A8 A3 A7 A2 A6 A4 A5 Þ.
Fig. 5. A group Decision Ball ðA8 A3 A2 A4 A1 A7 A6 A5 Þ.
the final group decision regarding a specific alternative. Take A8 for instance, the evaluation results of three partners (marked with solid triangles) on a group Decision Ball are displayed in Fig. 6. As can be seen in Fig. 6, partner E3 ’s evaluation is the closest to the final group decision for A8 ; whereas, E2 ’ evaluation is the most different in relation to the others. Fig. 7 shows another example for evaluations of A2 , where E2 ’s evaluation is also the most different in relation to the others’. By observing the Decision Ball, the clusters of decision makers, as well as outliers, can be detected. The differences between an individual partner’s evaluation and the final group decision can also be easily seen using Decision Balls. For instance, partner E1 ’s evaluation is depicted on the group Decision Ball in Fig. 8, where the triangle indicates E1 ’s evaluation and the circle represents the final group opinion. In addition, partner E2 ’s evaluation is displayed on the group Decision Ball in Fig. 9. Comparing Figs. 8 and 9, the overall evaluation results of E1 is much closer to the final group decision than those of E2 . This problem was solved by Global Solver of Lingo 9.0 on a Pentium 4 personal computer. The coordinates of alternatives for E1 ; E2 ; E3 , and the group are listed in Table 6. It is noteworthy that the points corresponding to alternatives may be placed on the back part of the ball since the value of xki and zki is between 1 and 1. Take E3 for instance, the points corresponding to A1 and A4 should be placed on the back part of the ball because z31 and z34 are negative. However, in order to
L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
Fig. 6. Comparisons of A8 on a group Decision Ball.
Fig. 7. Comparisons of A2 on a group Decision Ball.
clearly display spatial relationships among alternatives, all alternatives in this example are presented in the front of the ball by rotating the sphere tool developed by Li and Ma (2008).
5. Preference adjustments and sensitivity analyses The proposed method can provide visual aids in helping decision making teams to observe the background context and to perceive outliers; thus it can reduce the possibility of decision makers’ bias in their judgments, in order to obtain a preferred result. The proposed method can also assist decision makers in adjusting their judgments to find a consensus. Take partner E2 in the store selection problem for example. As shown in Fig. 9, the scores of A7 and A2 are much higher than those of the final group opinion (since the latitudes of A2 and A7 marked with a ‘‘triangle” are much higher than those marked with a ‘‘circle”). That is, the scores of A7 and A2 evaluated by E2 are too high. As listed in Table 1, A2 and A7 have the highest values in store size ðC 1 Þ, which indicates E2 over weighted criterion C 1 . Suppose E2 adjusts judgments regarding criterion C 1 , from fp21;2 ¼ 1; p21;3 ¼ 2; p21;4 ¼ 3g to fp21;2 ¼ 1=3; p21;3 ¼ 1=2; p21;4 ¼ 2g, as listed in Table 7. Denote E2a as E2 after adjustment. The weights of criteria and 2a scores of alternatives for E2a can be obtained, with ðw2a 1 ; w2 ; 2a 2a 2a 2a 2a 2a 2a 2a 2a ; w Þ ¼ ð0:20; 0:27; 0:40; 0:13Þ; ðS ; S ; S ; S ; S ; S ; S ; S Þ ¼ w2a 3 4 1 2 3 4 5 6 7 8 ð0:42; 0:52; 0:49; 0:58; 0:13; 0:26; 0:33; 0:50Þ. The Decision Ball for
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Fig. 8. Comparisons of E1 (triangle) and the group decision (circle).
Fig. 9. Comparisons of E2 (triangle) and the group decision (circle).
E2a is depicted in Fig. 10, with the ordering of eight alternatives: A4 A2 A8 A3 A1 A7 A6 A5 . Applying Model 2 to the preference values of E1 ; E2a , and E3 yields Obj2 = 0.44, ðw1 ; w2 ; w3 ; w4 Þ ¼ ð0:16; 0:49; 0:25; 0:10Þ, and ðS1 ; S2 ; S3 ; S4 ; S5 ; S6 ; S7 ; S8 Þ ¼ ð0:42; 0:50; 0:56; 0:50; 0:15; 0:18; 0:31; 0:66Þ. The corresponding group Decision Ball is shown in Fig. 11, with A8 A3 A4 A2 A1 A7 A6 A5 . The value of Obj2 is reduced from 0.66 to 0.44. Since the smaller the value of the Obj2 is, the more consent the group may achieve, it means the group achieves greater consent after E2 adjusts its preferences. Fig. 12 shows the evaluations of E1 ; E2a ; E3 , and the group for a specific alternative A2 . In comparison with Fig. 7, E2a ’s score (latitude) in Fig. 12 is much closer to the group than that of E2 in Fig. 7. E2a ’s evaluation is depicted on the adjusted group Decision Ball in Fig. 13, where the triangle indicates E2a ’s evaluation and the circle represents the adjusted final group opinion. The sensitivity of the proposed approach for adding/removing an evaluator or an alternative is analyzed as follows. Suppose a partner E4 joins the group decision with preference matrix as listed in Table 8. The weights of criteria and scores of alternatives for E4 can be obtained, with ðw41 ; w42 ; w43 ; w44 Þ ¼ ð0:32; 0:42; 0:21; 0:05Þ; ðS41 ; S42 ; S43 ; S44 ; S45 ; S46 ; S47 ; S48 Þ ¼ð0:37; 0:60; 0:46; 0:50; 0:18; 0:21; 0:40; 0:57Þ. The Decision Ball for E4 is depicted in Fig. 14, with the ordering of eight alternatives A2 A8 A4 A3 A7 A1 A6 A5 . Applying Model 3 to the preferences matrices of E1 ; E2a ;E3 , and E4 yields Obj2 = 0.63, ðw1 ;w2 ; w3 ;w4 Þ ¼ ð0:17;0:51;0:26;0:06Þ, and ðS1 ;S2 ;
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Table 6 Coordinates of alternatives for E1 ; E2 ; E3 , and the group. E1
A1 A2 A3 A4 A5 A6 A7 A8
E2
E3
Group
x
y
z
x
y
z
x
y
z
x
y
z
0.50 0.58 0.13 0.65 0.74 0.61 0.72 0.00
0.67 0.73 0.83 0.76 0.27 0.32 0.49 0.90
0.55 0.37 0.54 0.08 0.62 0.72 0.50 0.45
0.31 0.47 -0.01 0.47 0.56 0.64 0.68 0.00
0.65 0.85 0.68 0.70 0.33 0.43 0.72 0.80
0.69 0.23 0.73 0.53 0.76 0.63 0.17 0.60
0.41 0.53 0.31 0.88 0.95 0.65 0.61 0.00
0.91 0.73 0.85 0.46 0.19 0.55 0.77 0.89
-0.09 0.43 0.43 -0.07 0.24 0.53 0.21 0.46
0.45 0.54 0.08 0.65 0.63 0.75 0.70 0.00
0.66 0.77 0.79 0.75 0.29 0.34 0.55 0.87
0.60 0.35 0.60 0.10 0.72 0.57 0.45 0.49
Table 7 Adjusted preference matrix for the partner E2 .
S3 ;S4 ;S5 ;S6 ;S7 ;S8 Þ ¼ ð0:39;0:50;0:56;0:53; 0:16;0:17;0:29; 0:66Þ. The corresponding group Decision Ball and comparisons of A2 for E1 ;E2a ;E3 , and E4 are shown in Fig. 15. Because the preference matrices of the original partners are not affected by adding a new evaluator, the Decision Balls for E1 ; E2a , and E3 are not changed. However, since the weights of criteria for the final group decision are affected by adding a new evaluator, the final group Decision Ball is altered. Comparisons of group Decision Ball with three partners (marked with triangle) and 4 partners (marked with circle) are shown in Fig. 16. The sensitivity analysis for removing an existing evaluator is similar to that of adding one.
Fig. 10. A Decision Ball of E2a ðA4 A2 A8 A3 A1 A7 A6 A5 Þ.
Fig. 12. Comparisons of A2 on an adjusted group Decision Ball.
Fig. 11. An adjusted group Decision Ball ðA8 A3 A4 A2 A1 A7 A6 A5 Þ.
Fig. 13. Comparisons of E2a (triangle) and the adjusted group decision (circle).
L.-C. Ma / European Journal of Operational Research 203 (2010) 176–184
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Table 8 Preference matrix for a new partner E4 . E4 C1 C2 C3 C4
C1
C2
C3
C4
1/3
2 2
7 9 5
Next, the sensitivity of the proposed approach for adding an alternative is analyzed. Suppose an alternative A9 is added as a candidate spot with c9;1 ¼ 950; c9;2 ¼ 1000; c9;3 ¼ 480, and c9;4 ¼ 4800. According to Model 1 and Model 2, adding an alternative does not change the weights of the criteria for the partners and the group. If the c9;f ; f ¼ 1; . . . ; 4, do not alter the original upper and lower bounds of C f , the scores of, and dissimilarities among, alternatives will remain the same for A1 ; . . . ; A8 . Therefore, the latitudes of alternatives will stay the same. Because relationships among A9 and the original eight alternatives are included into the Decision Ball model, the relative distances among alternatives may be slightly changed. Take E1 for instance, the corresponding Decision Ball after adding A9 is shown in Fig. 17. The Decision Ball before (marked with circle) and after adding A9 (marked with triangle) are shown in Fig. 18.
Fig. 16. Comparisons of group Decision Ball with three partners (triangle) and 4 partners (circle).
Fig. 17. A Decision Ball of E1 after adding A9 .
Fig. 14. A Decision Ball of E4 ðA2 A8 A4 A3 A7 A1 A6 A5 Þ.
The analyses of other partners and the group for adding A9 are similar to that of partner E1 . Sensitivity of the proposed approach for removing an alternative is also similar to that of adding an alternative. 6. Conclusions
Fig. 15. Comparisons of A2 for E1 ; E2a ; E3 , and E4 on a group Decision Ball.
This study extends the Decision Ball model to visualize group decision making. The arc length between two alternatives is used to represent the dissimilarity between them, e.g., the larger the difference, the longer the arc length. The alternative with a higher score is designed to be closer to the North Pole. By observing the Decision Balls, each evaluator can see individual ranking and similarities among alternatives, identify the differences between individual judgment and the group’s collective opinion, observe the clusters of alternatives as well as decision makers, and perceive outliers. Because outliers can be found easily, it can reduce the possibility of decision makers’ bias in their judgments to get a preferred result, and thereby help decision makers to achieve a more objective judgment. The proposed method can also assist decision makers in adjusting their judgments to find a consensus. One restriction of this approach is the running time that may considerably increase when the number of alternatives becomes large. In future studies, the question of how to linearize this nonlinear model in order to deal with larger problems can be
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Fig. 18. Comparisons of E1 before (circle) and after (triangle) adding A9 .
addressed. Nevertheless, because the pairwise comparison preference relationships are good for smaller problems or the final stage of decision making, the proposed approach has been shown helpful in the cases where the number of alternatives is fewer than 10. Acknowledgement This research is supported by the National Science Council of the Republic of China under contract NSC 96-2416-H-239-002. References Andrews, R.L., Manrai, A.K., 1999. MDS maps for product attributes and market response: an application to scanner panel data. Marketing Science 18 (4), 584– 604. Belton, V., Stewart, T.J., 2002. Multiple criteria decision analysis. An Integrated Approach. Kluwer Academic Publishers, Norwell, MA. Borg, I., Groenen, P., 1997. Modern Multidimensional Scaling. Springer, New York. Chintagunta, P.K., 1992. Estimating a multinomial probit model of brand choice using the method of simulated moments. Marketing Science 11 (4), 386–407.
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