Pairwise comparison based interval analysis for group decision aiding with multiple criteria

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ScienceDirect Fuzzy Sets and Systems 274 (2015) 79–96 www.elsevier.com/locate/fss

Pairwise comparison based interval analysis for group decision aiding with multiple criteria Tomoe Entani a , Masahiro Inuiguchi b,∗ a Graduate School of Applied Informatics, University of Hyogo, Kobe, Hyogo 650-0047, Japan b Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

Received 30 September 2013; received in revised form 28 December 2014; accepted 2 March 2015 Available online 10 March 2015

Abstract Interval AHP (Analytic Hierarchy Process) was proposed to obtain interval weights from a given pairwise comparison matrix showing relative importance between criteria. In this paper, Interval AHP is applied to group decision problems. Interval AHP is first revised suitably for comparing alternatives from the viewpoint that the interval weight vector shows the set of agreeable weight vectors for the decision maker. Under individual interval weight vectors obtained from individual pairwise comparison matrices, three approaches to obtaining a consensus interval weight vector are proposed. One is the perfect incorporation approach that obtains consensus interval weight vectors including all individual interval weight vectors. By this approach, we can count out indubitably inferior alternatives. The second is the common ground approach that obtains consensus interval weight vectors included in all individual interval weight vectors. By this approach we can find agreeable group preference between alternatives when all individual opinions are similar. The third is the partial incorporation approach that obtains consensus interval weight vectors intersecting all individual interval weight vectors. By this approach we can find agreeable group preference between alternatives when individual opinions are not similar. The usefulness of the proposed three approaches is demonstrated by simple numerical examples. © 2015 Elsevier B.V. All rights reserved. Keywords: Analytic hierarchy process; Group decision-making; Interval analysis; Linear programming; Dominance relation

1. Introduction AHP (Analytic Hierarchy Process) is an approach to multi-criteria decision making problems and induces the preference of a decision maker from his/her intuitive judgements [1]. In AHP, the decision problem is structured hierarchically as criteria and alternatives. At each node except leaf nodes of the hierarchical tree, a weight vector for criteria or for alternatives is obtained from a pairwise comparison matrix given by a decision maker. The overall priority of an alternative is obtained as the weighted sum of the local weights. As all priorities are estimated by real * Corresponding author.

E-mail addresses: [email protected] (T. Entani), [email protected] (M. Inuiguchi). http://dx.doi.org/10.1016/j.fss.2015.03.001 0165-0114/© 2015 Elsevier B.V. All rights reserved.

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numbers, a weak order of alternatives is obtained. However, the pairwise comparison matrix given by a decision maker is not very accurate and includes inconsistency. In AHP, the consistency index is defined and calculated for each pairwise comparison matrix. The resulting weight vector is accepted when the value of the consistency index is in the acceptable range. To treat the imprecision or vagueness of human judgement, intervals and fuzzy numbers were applied to the expression of the pairwise comparison matrix. van Laarhoven and Pedrycz [2] treated a pairwise comparison matrix with fuzzy components. They applied the logarithmic least squares method to obtain a fuzzy weight vector from the fuzzy pairwise comparison matrix. This original method has some problems in the normalization and some modifications were proposed [3,4]. Buckley [5] also proposed fuzzy hierarchical analysis using a pairwise comparison matrix with fuzzy components. He extended the geometric mean method to fuzzy case to obtain a fuzzy weight vector. Later, Buckley et al. [6] and Csutora and Buckley [7] proposed the fuzzification of the maximum eigenvalue method for obtaining a fuzzy weight vector. Through those approaches, alternatives are evaluated by fuzzy priorities. The ranking of alternatives is done by a method for ranking fuzzy numbers. On the other hand, Saaty and Vargas [8] considered the interval judgement in a pairwise comparison matrix and investigated its effect on the stability of the rank order of alternatives. Arbel [9] treated the interval judgements in pairwise comparison as a range that true relative importance exists. He translated the interval pairwise comparison matrix to linear constraints on weight vectors. From the region expressed by the linear constraints, a consistent normalized weight vector is selected. Weight vector selection methods from the region have been proposed in [10,11]. However, this approach may have a problem when there is no consistent normalized weight vector in the region obtained from the interval pairwise comparison matrix (see [12]). Islam et al. [13] proposed a goal programming approach to obtaining a normalized weight vector from an inconsistent pairwise comparison matrix with interval components. Yu [14] modified this approach to obtain a normalized weight vector by balanced evaluation of deviations from the given interval data using logarithmic transformation of weights. Mikhailov [15,17] and Mikhailov and Tsvetinov [16] introduced tolerances to the linear constraints induced from a pairwise comparison matrix with interval components and proposed a fuzzy programming approach to obtaining a normalized weight vector. Dopazo et al. [18] extended this type of approach to the case of pairwise comparison matrices with fuzzy components. A fuzzy component is transformed to a nested intervals by a finite collection of α-cuts. Then linear constraints for all α-cuts are considered in the logarithmic transformed weight vector space. A goal programming approach is applied to obtaining a normalized weight vector. Those approaches express the uncertain judgement in pairwise comparison explicitly by intervals or by fuzzy numbers but a crisp normalized weight vector suitable for the given uncertain pairwise comparison is obtained. Sugihara and Tanaka [19] proposed an approach to obtaining an interval weight vector from crisp pairwise comparison matrix. This approach is called Interval AHP and based on the idea that the decision maker does not perceive a precise weight vector but a range of weight vectors vaguely in his/her mind. Any weight vector in the range is considered acceptable for the decision maker. Each pairwise comparison by the decision maker is assumed to be done by arbitrarily selected weights from the ranges. This arbitrary selection for each comparison is considered the cause of the inconsistency of the pairwise comparison matrix. An interval weight vector is estimated to include the given pairwise comparison matrix as a possible realization. Later Sugihara et al. [20] extended Interval AHP to treat the pairwise comparison matrix with interval components. Unlike the aforementioned approaches to interval pairwise comparison matrix, interval weight vectors are estimated. In the sense that the interval weight vector is estimated from an interval pairwise comparison matrix, this approach is similar to the fuzzy AHP estimating a fuzzy weight vector from a fuzzy pairwise comparison matrix, but estimation ideas are different. Two interval weight vectors are obtained as the outer and inner estimations in Interval AHP while a fuzzy weight vector is obtained as an approximate estimation in fuzzy AHP. In Interval AHP, the alternatives are ranked by the obtained interval weight vector and the dominance relation between alternatives is often a preorder. Interval AHP approaches can be seen as an AHP counterpart of Tanaka’s fuzzy/interval linear regression analysis [21–23]. This paper extends Interval AHP to group decision making. Namely, we investigate methods for inducing an interval weight vector from m individual crisp pairwise comparison matrices and for finding the consensus interval weight vector. AHP approaches were used to support the interpersonal information exchange [24,25] in group decision making problems. There are two basic approaches for aggregating individual opinions into a group opinion [26]. One approach first aggregates the individual judgements of pairwise comparisons and then provides a consensus weight vector. This approach is based on the assumption that the group is a synergistic unit and called the comparison matrix aggregation

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approach in this paper. In the other approach, individual weight vectors are obtained first and then aggregated to be a consensus weight vector. This approach is based on the idea that the group is simply a collection of individuals and called the weight aggregation approach in this paper. In the both approaches, some aggregation methods such as geometric mean [27,28], clustering [29], and adjusting [30] have been applied to obtain crisp group weight vector. Interval and fuzzy approaches to AHP are also applied to group decision making [18,31,32]. In the same idea as his approach to the interval pairwise comparison matrix [15–17], Mikhailov [32] introduced the membership functions showing the satisfaction degrees of individual pairwise comparison matrices and proposed a fuzzy programming approach to a consensus weight vector by maximizing the satisfaction degrees equally. Dopazo et al. [18] extended their approach with a fuzzy pairwise comparison matrix to group decision making by considering all deviations from linear constraints induced from all individual fuzzy pairwise comparison matrices. In those approaches, individual pairwise comparison judgements are directly translated into goals or fuzzy goals. Therefore, those approaches can be classified into the comparison matrix aggregation approaches. The proposed approach is a weight aggregation approach. Then, individual interval weight vectors are first estimated and then aggregated to obtain a consensus interval weight vector. Under individual interval weight vectors obtained from individually evaluated pairwise comparison matrices, three approaches to obtaining a consensus interval weight vector are proposed. One is the perfect incorporation approach that obtains consensus interval weight vectors including all individual interval weight vectors. By this approach, we can count out indubitably inferior alternatives. The second is the common ground approach that obtains consensus interval weight vectors included in all individual interval weight vectors. By this approach we can find agreeable group preference between alternatives when all individual opinions are similar. The third is the partial incorporation approach that obtains consensus interval weight vectors intersecting all individual interval weight vectors. By this approach we can find agreeable group preference between alternatives when individual opinions are not similar. The proposed three approaches play complementary roles. The usefulness of the proposed three approaches is demonstrated by simple numerical examples. This paper is organized as follows. In next section, Interval AHP for crisp pairwise comparison matrix is briefly reviewed. In Section 3, we revisit Interval AHP. New properties of normalized intervals are shown and new dominance relations between alternatives are proposed. Three approaches to obtaining a consensus interval weight vector from individual pairwise comparison matrices are described in Section 4. Dominance relations under the proposed three approaches are defined and their significance and usage are given. In Section 5, simple numerical examples are given to demonstrate the usefulness of the proposed three approaches. Conclusions and future topics based on the proposed approaches are described in Section 6. 2. Interval AHP 2.1. Estimation of interval weights In AHP, a pairwise comparison matrix is given for obtaining weights on criteria as well as for obtaining evaluations on alternatives in view of each criterion. In this paper, we assume that scores (marginal utilities) of alternatives in view of each criterion are given, and then we concentrate on the elicitation of weights on criteria from a pairwise comparison matrix. We introduce Interval AHP for multiple criteria decision problem with a single decision maker as the basis for treating a group decision problem with m decision makers and n criteria. For simplicity and consistency of notation in subsequent sections, we define M = {1, 2, . . . , m}, N = {1, 2, . . . , n} and N \j = N \{j } = {1, 2, . . . , j − 1, j + 1, . . . , n} for j ∈ N , and describe Interval AHP for the problem of decision maker k ∈ M. Let Ak be the pairwise comparison matrix given by decision maker k. The (i, j )-component akij of Ak shows the relative importance of criterion i over criterion j that decision maker k evaluates. This relative importance is given independently on the other pairs of criteria (i  , j  ) = (i, j ) and the other decision makers k  = k. Namely, we have ⎡ ⎤ 1 · · · ak1n ⎢ .. ⎥ , k = 1, 2, . . . , m. (1) Ak = ⎣ ... a . ⎦ kij

akn1

···

1

In the conventional AHP [1], the (i, j )-component akij takes a value between 1/9 to 9, i.e., 1/9 ≤ akij ≤ 9.

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It is assumed that the comparisons are identical if akii = 1, i ∈ N and reciprocal if akij = 1/akj i , i, j ∈ N , i = j so that s/he has to compare n(n − 1)/2 pairs of the criteria. The larger akij is, the greater importance decision maker k puts on criterion i than on criterion j . If the transitivity, akij = akil aklj , ∀i, j, l ∈ N such that i = j, j = l, l = i,

(2)

is satisfied, the pairwise comparison matrix Ak is consistent. However, Ak is usually inconsistent because decision maker k gives the component of Ak one by one and human evaluation is often inexact, imprecise or vague. Suppose that there exist crisp weights wki and wkj showing the importance of criteria i and j , respectively, that decision maker k perceives, where we assume the normality, i∈N wki = 1. Then we presume that akij is close to wki /wkj , i.e., akij ≈ wki /wkj . In the conventional AHP approaches, wki , i ∈ N are determined so that akij is close to wki /wkj . Especially when Ak is consistent, we obtain weights wki , i ∈ N are uniquely determined so as to satisfy wki /wkj = akij , i, j ∈ N , i < j . However, as we described above, unfortunately, Ak is often inconsistent. In the conventional AHP, a measure of consistency is defined and if it is in the acceptable range, the estimated crisp weights are acceptable and used for the decision analysis. In Interval AHP [19,20], the inconsistency of the pairwise comparison matrix is assumed to come from human vague evaluation. It is assumed that decision maker k perceives vague weight Wki for the i-th criterion.  The vague L , w R ] in Interval AHP. Any weight w ∈ W such that weight is expressed by interval Wki = [wki ki ki i∈N wki = 1 is ki considered acceptable for decision maker k. Each pairwise comparison by the decision maker is assumed to be done by arbitrarily selected weights from the ranges. This arbitrary selection for each comparison is regarded as the cause of the inconsistency of the pairwise comparison matrix in Interval AHP. In this section, we briefly introduce Interval AHP. For the sake of simplicity, we define W k as a vector of interval weights Wki , i ∈ N of decision maker k, i.e., W k = (Wk1 , . . . , Wkn ). Moreover, the sum of widths of interval weights is denoted by R L d(W k ) = (wki − wki ). (3) i

From the assumption of Interval AHP, the relative importance akij is presumed to be in the range of Wki /Wkj = L /w R , w R /w L ], i.e., a [wki kij ∈ Wki /Wkj . If W k satisfies this presumption for all akij , i, j ∈ N , i = j , W k is said to kj ki kj be Ak -feasible. Then, we define Ak -feasible interval weight vector set as 

 L R L R  W(Ak ) = W = ([w1 , w1 ], · · · , [wn , wn ])   wiR wiL R L ≤ akij ≤ L , i, j, ∈ N (i < j ), wi ≥ wi ≥ , i ∈ N , (4) wjR wj where  > 0 is a predetermined very small positive number. In order to apply linear programming later, we use wiL ≥  instead of a positivity constraint wiL > 0. We note that the condition of W in W(Ak ), i.e., wiL /wjR ≤ akij ≤ wiR /wjL , i, j ∈ N (i < j ) is equivalent to wjL /wiR ≤ akj i ≤ wjR /wiL , i, j ∈ N (i < j ) because of the reciprocity, akij = 1/akj i . This is why we wrote the condition only for i, j ∈ N such that i < j . We note that the condition is linear with respect to wiL and wiR , i ∈ N because it is equivalent to wiL ≤ wjR akij , akij wjL ≤ wiR , i, j ∈ N (i < j ) and wiL ≥ wiR ≥ , i ∈ N . The interval weight vector W k should be included in W(Ak ), i.e., W k ∈ W(Ak ). Let us define W 1 ⊆ W 2 by W1i ⊆ W2i , i ∈ N , where W k = (Wk1 , Wk2 , . . . , Wkn ), k = 1, 2. Then we have W 1 ∈ W(Ak ), W 1 ⊆ W 2 imply W 2 ∈ W(Ak ). We define normalized interval weight vector set as ⎧  ⎨  N L R L U  W = W = ([w1 , w1 ], · · · , [wn , wn ])  wjR + wiL ≥ 1, i ∈ N, ⎩  j ∈N\i ⎫ ⎬ wjL + wiR ≤ 1, i ∈ N, wiR ≥ wiL ≥ , i ∈ N . ⎭ j ∈N\i

(5)

(6)

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These conditions are obtained in the study of interval probability [33]. Some properties of interval probability are shown later in Section 2.3 and some new results in Section 3.1. The interval weight vector W k should also be normalized, i.e., W k ∈ W N . Unlike W(Ak ), W N does not always satisfy (5). Therefore, neither W(Ak ) ∩ W N does. In order to obtain the interval weight vector as the closest to the given comparison matrix, the sum of widths of d(W k ) is minimized. The optimal interval weight vector set W DMk elicited from Ak is W DMk = {W k | W k ∈ W(Ak ) ∩ W N , d(W k ) ≤ dˆk },

(7)

where dˆk = minimize{d(W k ) | W k ∈ W(Ak ) ∩ W N }. The optimal value dˆk and members of W DMk are obtained as the optimal value and optimal solutions of the following linear programming problem: R L (wki − wki )

minimize

i∈N L ≤ a w R , a w L ≤ w R , i, j, ∈ N (i < j ), subject to wkj kij ki kij ki kj R L wkj + wki ≥ 1, i ∈ N, j ∈N\i



(8)

L R wkj + wki ≤ 1, i ∈ N,

j ∈N\i R ≥ w L ≥ , i ∈ N. wki ki L = w R = w , i ∈ N . This happens if and only if given compariIf dˆk = 0, we obtain a unique optimal solution wki ki ki ∗ son matrix Ak is consistent. The widest solution Wki = [, 1 − (n − 1)]i ∈ N is a feasible solution to Problem (8).

2.2. Dominance relation between alternatives L , w R ], i ∈ N are obtained, each Once linear programming problem (8) is solved and interval weights Wki = [wki ki alternative op is evaluated by overall interval priority Ykp , p ∈ M. Sugihara and Tanaka [19] defined Ykp by

L R Ykp = [ykp , ykp ]=

yp =

i∈N

   k wi ui (op )  wi ∈ Wi , i ∈ N , 

(9)

L and y R are defined by where ykp kp L ykp =



L wki ui (op ),

i∈N

R ykp =



R wki ui (op ).

(10)

i∈N

Then they proposed the following dominance relation k between alternatives op and oq : L L R R op k oq if and only if ykp ≥ ykq and ykp ≥ ykq .

(11)

This dominance relation is a preorder. L and y R , the normality condition of W k , i ∈ N is discarded. Guo and Tanaka [34] However, in the definitions of ykp i kp L and y R as follows: modified ykp kp

L ykp

= min



  L R wi ui (op )  wi = 1, wki ≤ wi ≤ wki , i ∈ N ,

i∈N

R ykp

= max

i∈N

(12)

i∈N

  L R wi ui (op )  wi = 1, wki ≤ wi ≤ wki , i ∈ N . i∈N

(13)

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2.3. Properties of normalized interval weights In this subsection, we briefly review the previously known properties of normalized interval weights. The following two propositions were proven by Tanaka et al. [33]. L , w R ], . . . , [w L , w R ]) ∈ W N . There exist w ∈ W , i ∈ N such that Proposition 1. Let W k = ([wk1 ki ki kn kn k1

 i∈N

wki = 1.

L , w R ], . . . , [w L , w R ]) ∈ W N , l ∈ Q = {1, 2, . . . , q}. Then interval weight vector W = Proposition 2. Let W l = ([wl1 ln ln l1 ([w1L , w1R ], . . . , [wnL , wnR ] defined by

wiL = min wliL , l∈Q

wiR = max wliR

(14)

l∈Q

satisfies W ∈ W N , i.e., W is also a normalized interval weight vector. Tanaka et al. [33] proved this proposition when n = 2 (i.e., |N | = 2). Proposition 2 is a direct extension of their result. Moreover, Proposition 2 holds even when Q is generalized to any compact set. Given normalized interval weight vectors W l , l ∈ Q, we define an interval closure of W l , l ∈ Q by the smallest ˆ such that W l ⊆ W ˆ , l ∈ Q. Obviously, the i-th component interval [wˆ L , wˆ R ] of W ˆ is defined by (14). interval vector W i i Proposition 2 implies that the interval closure of normalized interval weight vectors is also a normalized interval weight vector. 3. Interval AHP revisited 3.1. More properties Before extending Interval AHP to a Group Interval AHP, we revisit Interval AHP. We first describe a few new useful properties of normalized interval weight vectors. We obtain the following proposition stronger than Proposition 1. L , w R ], . . . , [w L , w R ]) ∈ W N . For any w ◦ ∈ W = [w L , w R ], there exist w ∈ Proposition 3. Let W k = ([wk1 ki kj kn kn ki ki ki k1 L R ◦ = 1. Wkj = [wkj , wkj ], j ∈ N \i such that j ∈N \i wkj + wki

  ◦ ∈ W , where w L ≤ w ◦ ≤ w R . Because of W ∈ W N , R ◦ R Proof. Suppose wki ki k j ∈N\i wkj + wki ≥ j ∈N\i wkj + kj ki ki     L ≥ 1 and L ◦ L R L ◦ R ◦ wki j ∈N\i wkj + wki ≤ j ∈N \i wkj + wki ≤ 1. Then, j ∈N\i wkj + wki ≤ 1 ≤ j ∈N\i wkj + wki implies  L R ◦ that there exist wkj ∈ Wkj , j ∈ N\i such that wkj ≤ wkj ≤ wkj and j ∈N\i wkj + wki = 1. Hence, we obtain the proposition. 2 Consider any interval weight vector V = (V1 , V2 , . . . , Vn ) with Vi = [viL , viR ] ⊆ [0, 1], i ∈ N which is not necessarily normalized. We define a set of normalized weight vector w = (w1 , w2 , . . . , wn ) belonging to V by 

  η(V ) = w = (w1 , w2 , . . . , wn )  wi ∈ Vi , i ∈ N, wi = 1 . (15)  i∈N

We easily obtain that η(V ) is a compact set or an empty set because Vi , i ∈ N are compact sets. When η(V ) = ∅, we moreover define interval weight vector H (V ) = (H1 (V ), . . . , Hn (V )) with Hi (V ) = [νiL (V ), νiR (V )], i ∈ N by νiL (V ) =

min

(w1 ,w2 ,...,wn )∈η(V )

wi ,

νiR (V ) =

max

(w1 ,w2 ,...,wn )∈η(V )

wi .

(16)

H (V ) is the interval closure of all normalized weight vectors w belonging to V . From Proposition 2, we know that H (V ) is a normalized interval weight vector. The following proposition shows that H (V ) is the unique maximal normalized interval weight vector included in a given interval weight vector V and called the normalization of V .

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Table 1 Utility values of three alternatives on each criterion.

o1 o2 o3

cr1

cr2

cr3

0.34 0.33 0.33

0.33 0.34 0.33

0.33 0.33 0.34

Proposition 4. Assume η(V ) = ∅. H (V ) is the unique maximal normalized interval weight vector included in a given interval weight vector V . Proof. Suppose that there exists a normalized interval weight vector W = (W1 , W2 , . . . , Wn ) with Wi = [wiL , wiR ], i ∈ N such that W ⊆ V and W  H (V ). W  H (V ) implies that there exists j ∈ N such that Wj  Hj (V ). Let wj◦ ∈  Wj \ Hj (V ). By Proposition 3, there exist wi ∈ Wi , i ∈ N \j such that i∈N \j wi + wj◦ = 1. Because W ⊆ V , we have w◦ = (w1 , . . . , wj −1 , wj◦ , wj +1 , . . . , wn ) ∈ V . This implies w◦ ∈ η(V ). By (16), we obtain wj◦ ∈ [νjL (V ), νjR (V )] = Hj (V ). This contradicts to wj◦ ∈ Wj \ Hj (V ). Therefore, we obtain the theorem. 2 Proposition 4 ensures the uniqueness of the maximal normalized interval weight vector W satisfying W ⊆ V . Based on Proposition 4, we may find the maximal normalized interval weight vector belonging to a given interval weight vector V by solving the following linear programming problems for each i ∈ N : wiL = min{wi | j ∈N wj = 1,  wj ∈ Vj , j ∈ N } and wiR = max{wi | j ∈N wj = 1, wj ∈ Vj , j ∈ N }. 3.2. The proposed dominance relation between alternatives In this paper, we interpret the interval weight vector obtained by solving problem (8) as a set of weight vectors that the decision maker accepts as his/her opinion. From this point of view, we do not totally agree with the comparison of alternatives using the dominance relation defined by (11). Let us consider three alternatives o1 , o2 and o3 shown in Table 1. There are three criteria, cr1 , cr2 and cr3 and their interval weights of decision maker k are given as W1 = [0.28, 0.4], W2 = [0.29, 0.41] and W3 = [0.3, 0.42], respectively. We note that interval weight vector W = (W1 , W2 , W3 ) satisfy the normality condition, i.e., W ∈ W N . Applying (12) and (13), we obtain overall interval priorities Y1 = [0.3328, 0.334], Y2 = [0.3329, 0.3341] and Y3 = [0.333, 0.3342]. Based on (11), we obtain o3 k o2 k o1 . However, in our interpretation of the interval weight vector, the decision maker accepts weight vector (0.4, 0.3, 0.3) yielding order o1  o2 ∼ o3 , weight vector (0.3, 0.4, 0.3) yielding order o2  o1 ∼ o3 and others yielding different orders. In such a case, it is natural to conclude that decision maker thinks three alternatives o1 , o2 and o3 indifferent one another or hard to tell. The comparison of alternatives using the dominance relation defined by (11) sometimes leads to counterintuitive results. Considering the variety of weight vectors the decision maker accepts, we propose two dominance relations: a possible dominance relation πk showing an alternative dominates another for at least one weight vector in the given interval weight vectors and a necessary dominance relation νk showing an alternative dominates another for all weight vectors in the given interval weight vectors. Given an interval weight vector W k , those dominance relations are defined by op πk oq if and only if ∃w = (w1 , . . . , wn ) ∈ W k such that wi = 1, wi (ui (op ) − ui (oq )) ≥ 0, i∈N

op νk oq if and only if ∀w = (w1 , . . . , wn ) ∈ W k such that wi = 1, wi (ui (op ) − ui (oq )) ≥ 0. i∈N

(17)

i∈N

(18)

i∈N

Namely, op πk oq implies that decision maker k accepts that op is not worse than oq in some way, while op νk oq implies that decision maker k accepts that op is not worse than oq by all means.

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Those dominance relations op πk oq and op νk oq are verified by solving the following continuous knapsack problems, respectively: maximize dkπ (op , oq ) = (ui (op ) − ui (oq ))wi , i∈N (19) wi = 1, subject to i∈N

wki ≤ wi ≤ wki , i ∈ N, minimize dkν (op , oq ) = (ui (op ) − ui (oq ))wi , i∈N wi = 1, subject to

(20)

i∈N

wki ≤ wi ≤ wki , i ∈ N, and we obtain dˆkπ (o1 , o2 ) ≥ 0 if and only if op πk oq , dˆkν (o1 , o2 ) ≥ 0 if and only if op νk oq ,

(21) (22)

where dˆkπ (o1 , o2 ) and dˆkν (o1 , o2 ) are the optimal values of Problems (19) and (20), respectively. In Interval AHP, W DMk may have multiple members. In such a case, we assume that decision maker k accepts any of optimal interval weight vectors as an interval weight vector reflecting his/her opinion. Then, to deal with this multiplicity, we modify the definitions of π and ν as op πk oq if and only if ∃W k ∈ W DMk , ∃w = (w1 , . . . , wn ) ∈ W k such that wi = 1, wi (ui (op ) − ui (oq )) ≥ 0, i∈N

(23)

i∈N DM k

, ∀w = (w1 , . . . , wn ) ∈ W k op νk oq if and only if ∃W k ∈ W such that wi = 1, wi (ui (op ) − ui (oq )) ≥ 0.

(24)

i∈N i∈N of Table 1, πk holds for all possible pairs o3 πk o2 while νk does not hold for any

In the case of alternatives, i.e., o1 πk o2 , o2 πk o1 , o1 πk o3 , o3 πk o1 , o2 πk o3 and pair of alternatives. Thus, the proposed dominance relations πk and νk suggest the three alternatives are indifferent and non-comparable, respectively. Those results are close to our intuition. Relation πk satisfies the completeness or comparability (for every pair (op , oq ) of objects, op πk oq or oq πk op holds) and negative transitivity (for any objects op , oq , or , op πk oq and oq πk or implies op πk or ). On the other hand, relation νk satisfies reflexivity (for every object op , op νk op holds) and transitivity (for any objects op , oq , or , op νk oq and oq νk or implies op νk or ), i.e., it is a preorder. Therefore, we cannot always find the best alternative. However, relation νk is useful to narrow down the candidates by erasing dominated alternatives without any further information about individual preferences. 4. Interval AHP approaches to group decision aiding 4.1. Outline of the proposed approaches Interval AHP is extended to group decision making problems. We assume that n individuals in a group gave their own comparison matrices Ak , k ∈ N . From those matrices, we calculate interval weights of criteria, Wi , i ∈ M representing a group consensus. There are basically two ways to aggregate multiple opinions given by Ak , k ∈ N . One is to aggregate Ak , k ∈ N first, and then to calculate Wi , i ∈ M using the aggregated pairwise comparison matrix. The other is to calculate individual interval weight vectors first, and then to aggregate them. Since each component akij L /w R , w R /w L ] under our assumption that each decision maker perceives of Ak is considered arbitrary from range [wki ki ki ki vague weights, the aggregation of Ak , k ∈ N may lead to improper results. Thus, we take the latter approach.

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As described in the previous sections, decision maker k’s opinion can be represented by an interval weight vector W k ∈ W DMk using Ak . We investigate the aggregation of W k ∈ W DMk , k ∈ M. 4.2. The perfect incorporation of all individual opinions The easiest solution for the group consensus is to incorporate all individual opinions. Namely we consider an interval weight vector W = (W1 , W2 , . . . , Wn ) satisfying W ⊇ W k , W k ∈ W DMk , ∀k ∈ M. Because we are seeking a suitable set of normalized weight vectors, we render weight vector W normalized, i.e., W ∈ W N . By adopting the minimization of the sum of widths, the interval weight vector W representing the group consensus is obtained by solving the following linear programming problem: min{d(W ) | W ⊇ W k , W k ∈ W DMk , k ∈ M, W ∈ W N }.

(25)

This approach to obtaining a consensus interval weight vector is called the perfect incorporation approach. The group consensus based on this approach is called the perfect incorporation of all individual opinions. We have the following proposition which shows that we can drop constraint W ∈ W N from Problem (25). Proposition 5. (25) is equivalent to the following simpler linear programming problem: min{d(W ) | W ⊇ W k , W k ∈ W DMk , k ∈ M}, or equivalently, minimize

(26)

(wiR − wiL ), i∈N

subject to



R L (wki − wki ) ≤ dˆk , k ∈ M,

i∈N L wki

R , a w L ≤ w R , i, j ∈ N, i < j, k ∈ M, ≤ akij wkj kij kj ki R L wki + wkj ≥ 1, k ∈ M, j ∈ N,

i∈N\j

(27)

L R wki + wkj ≤ 1, k ∈ M, j ∈ N,

i∈N\j L ≤ w R ≤ w R , k ∈ M, i ∈ N,  ≤ wiL ≤ wki ki i L , w R ], . . . , [w L , w R ]). where W = ([w1L , w1R ], . . . , [wnL , wnR ]) and W k = ([wk1 kn kn k1

¯ = ([w¯ L , w¯ R ], . . . , [w¯ nL , w¯ nR ]) obtained by solving Problem (26) satisfies W ¯ ∈ W N . Let W ¯k= Proof. It suffices that W 1 1 L R L R DM k ¯ by solving Problem (26). From the first ([w¯ k1 , w¯ k1 ], . . . , [w¯ kn , w¯ kn ]) ∈ W , k ∈ M be obtained together with W ¯ obviously satisfy ¯ ⊇W ¯ k , k ∈ M. By minimizing the sum of widths of W , W constraint of Problem (26), we have W L w¯ iL = min w¯ ki , k∈M

R w¯ iR = max w¯ ki .

(28)

k∈M

¯ k ∈ W DMk ⊆ W N , from Proposition 2, we obtain W ¯ ∈ W N. Since W

2

Let dˆP be the optimal value of Problem (26), i.e., dˆP = min{d(W ) | W ⊇ W k , W k ∈ W DMk , k ∈ M}. Then the set of optimal interval weight vectors W representing the perfect incorporation consensus is defined by WP = {W | d(W ) ≤ dˆP , W ⊇ W k , W k ∈ W DMk , k ∈ M}.

(29)

The dominance relation P between alternatives by the perfect incorporation consensus is defined by op P oq if and only if ∀W ∈ WP , ∀w = (w1 , . . . , wn ) ∈ W such that wi = 1, wi (ui (op ) − ui (oq )) ≥ 0. i∈N

i∈N

(30)

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For convenience, this dominance relation is called perfect dominance relation.  The perfect dominance relation op P oq is verified by solvingthe following linear programming problem: min{ i∈N wi (ui (op ) − ui (oq )) | w = (w1 , . . . , wn ) ∈ W , W ∈ WP , i∈N wi = 1}, or equivalently, minimize wi (ui (op ) − ui (oq )), i∈N

R L (wki − wki ) ≤ dˆk , k ∈ M, subject to i∈N L ≤ a w R , a w L ≤ w R , i, j ∈ N, i < j, k ∈ M, wki kij kj kij kj ki R L wki + wkj ≥ 1, k ∈ M, j ∈ N, i∈N\j



L R wki + wkj ≤ 1, k ∈ M, j ∈ N,

(31)

i∈N\j



(wiR − wiL ) ≤ dˆP ,

i∈N



wi = 1,

i∈N L ≤ w R ≤ w R , k ∈ M, i ∈ N,  ≤ wiL ≤ wki ki i

wiL ≤ wi ≤ wiR , i ∈ N. Let dP (op , oq ) be the optimal value of Problem (31). If dP (op , oq ) is nonnegative, op P oq holds. When WP is a singleton, problem (31) is reduced to a continuous knapsack problem similar to Problem (20) which is solved much more easily. We have the following proposition. Proposition 6. The following assertion is valid: op P oq implies ∀k ∈ M, op νk oq .

(32)

Proof. Because W ∈ WP implies ∀k ∈ M, ∃W k ∈ W DMk , W ⊇ W k , the proposition is easily obtained. 2 From Proposition 6, if op P oq , we know that all decision makers agree to opinion that op is not worse than oq by all means, i.e., the opinion that op is not worse than oq is very strongly supported. However, when there is a wide range of diverse opinions, W ∈ WP tends to be very wide so that almost no dominance is obtained. In such cases, this approach does not work well. On the other hand, if decision makers have similar opinions, this approach works well to narrow down the candidates by erasing dominated alternatives. The inverse of (32) does not always hold. Indeed, when n = 2, m = 3 and W DMk , k = 1, 2 are singletons ¯ = ([0.35, 0.45], [0, 0.2], with members w1 = (0.45, 0, 0.55) and w2 = (0.35, 0.2, 0.45), respectively, we obtain W [0.45, 0.55]). If u1 (op ) − u1 (oq ) = 0.385,u2 (op ) − u2 (oq ) = 0.035 and u3 (op ) − u3 (oq ) = −0.315, we obtain op νk oq , k = 1, 2 while op P oq because i∈N wi (ui (op ) −ui (oq )) ≤ 0 for w = (w1 , w2 , w3 ) = (0.35, 0.1, 0.55) ∈  ¯ satisfying i∈N wi = 1. W WP includes weight vectors interpolated among W k ∈ W DMk , k ∈ M. Such interpolated weight vectors are regarded as possible opinions after decision makers’ revision in consideration of other opinions. 4.3. Taking common ground with all individual opinions The other easy solution for the group consensus is to find the common ground with all individual opinions, i.e., to find normalized weight vectors included in W k ∈ W DMk , ∀k ∈ M. In our problem setting, the common ground exists only when there exist W k ∈ W DMk , k ∈ M such that η( k∈M Wk ) = ∅, where η isthe operation defined by (15). If the common ground exists, it is defined by the normalization of k∈M W k , i.e., H ( k∈M W k ).

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 The common ground H ( k∈M W k ) is obtained as W by solving the following linear programming problem:   max d(w) | W ⊆ W k , W k ∈ W DMk , k ∈ M, W ∈ W N ,

(33)

or equivalently, maximize

(wiR − wiL ), i∈N

subject to



R L (wki − wki ) ≤ dˆk , k ∈ M,

i∈N L wki

R L R ≤ akij wkj , akij wkj ≤ wki , i, j ∈ N, i < j, k ∈ M, R L wki + wkj ≥ 1, k ∈ M, j ∈ N,

i∈N \j



L R wki + wkj ≤ 1, k ∈ M, j ∈ N,

(34)

i∈N \j



wiR + wjL ≥ 1, j ∈ N,

i∈N \j



wiL + wjR ≤ 1, j ∈ N,

i∈N \j L ≤ w L ≤ w R ≤ w R , k ∈ M, i ∈ N,  ≤ wki i i ki L , w R ], . . . , [w L , w R ]). If no feasible solution exists, no comwhere W = ([w1L , w1R ], . . . , [wnL , wnR ]) and W k = ([wk1 kn kn k1 mon ground can be found. This approach obtaining a consensus interval weight vector is called common ground approach and the obtained group consensus is called the common ground consensus. Let dˆC be the optimal value of Problem (33), i.e., dˆC = max{d(w) | W ⊆ W k , W k ∈ W DMk , k ∈ M, W ∈ W N } if the common ground consensus exists; −∞ otherwise. Then the set of optimal interval weight vectors W representing the common ground consensus is defined by

WC = {W | d(W ) ≥ dˆC , W ⊆ W k , W k ∈ W DMk , k ∈ M, W ∈ W N }.

(35)

S Two dominance relations W C and C between alternatives by the common ground consensus are defined by

op W C oq if and only if ∃W ∈ WC , ∃w = (w1 , . . . , wn ) ∈ W wi = 1, wi (ui (op ) − ui (oq )) ≥ 0, such that i∈N

op SC oq if and only if ∀W ∈ WC , ∀w = (w1 , . . . , wn ) ∈ W such that wi = 1, wi (ui (op ) − ui (oq )) ≥ 0. i∈N

(36)

i∈N

(37)

i∈N

Those relations are called weak common dominance relation and strong common dominance relation, respectively. Similar to perfect dominance relation op P oq , weak and strong common dominance relations op W C oq and  op SC oq are verified by solving linear programming problems, respectively: dCW(op , oq ) = max{ i∈N wi (ui (op ) −   ui (oq )) | w = (w1 , . . . , wn ) ∈ W , W ∈ WC , wi = 1}, dCS (op , oq ) = min{ i∈N wi (ui (op ) − ui (oq )) | w = i∈N  W W S (w1 , . . . , wn ) ∈ W , W ∈ WC , i∈N wi = 1}. We have op C oq if and only if dC (op , oq ) ≥ 0 and op C S oq if and only if dC (op , oq ) ≥ 0. Moreover, when WC is a singleton, the problems are reduced to continuous knapsack problems similar to Problems (19) and (20). We have the following proposition.

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Proposition 7. The following assertion is valid: op SC oq implies op W C oq , op W C op W C

oq implies ∀k ∈ M, oq implies

op πk

(38) (39)

oq ,

oq SC op .

(40)

k , W ⊆ W . (40) is Proof. (38) is obvious. (39) is obtained from this fact that W ∈ WC implies ∀k ∈ M,∃W k ∈ W DM k o implies ∀W ∈ W , ∀w = (w , . . . , w ) ∈ W such that w = 1, proved as follows. op W C 1 n i∈N i i∈N wi (ui (op ) − C q ui (oq )) < 0. This implies oq SC op . 2

From Proposition 7, if op W C oq , we know that all decision makers agree to the opinion that op is not worse than oq in some way, i.e., the opinion that op is not worse than oq can be supported somehow by all decision makers. If op SC oq , we also have op W C oq , and thus all decision makers can somehow support the opinion that op is not worse S than oq . However op SC oq is usually stronger than op W op is not worse than oq for C oq because op C oq implies   W all w ∈ WC while op C oq implies op is not worse than oq for at least one w ∈ WC . op SC oq is equivalent to  op W WC is a singleton, i.e., a singleton {w} is a unique member of WC . C oq when  When there is a wide range of diverse opinions, η k∈M W k = ∅ is hard to be satisfied. In this case, the common ground approach does not work well. On the other hand, if the decision makers have similar or vague opinions, this approach works well to narrow down the candidates by erasing dominated alternatives.   ˆ = { k∈M W k | η( k∈M W k ) = ∅, W k ∈ Remark 1. Instead of W ∈ WC , we can define the common ground by W ˆ = ([wˆ L , wˆ R ], . . . , [wˆ nL , wˆ nR ]) is obtained by solving the following linear programming probW DMk , k ∈ N }. This W 1 1 lems:

   L DM k wˆ i = min wi  (w1 , w2 , . . . , wn ) ∈ W k , W k ∈ W , k ∈ M, wi = 1 , (41)  i∈N

   R DM k wˆ i = max wi  (w1 , w2 , . . . , wn ) ∈ W k , W k ∈ W , k ∈ M, wi = 1 . (42)  i∈N

ˆ By and draw similar results to those with W ∈ WC . Especially when  using W , wecan define dominance relations { k∈M W k | η( k∈M W k ) = ∅, Wk ∈ W DMk , k ∈ M} has a unique member, the common ground approach in this remark is reduced to the common ground approach described earlier.   Remark 2. The existence of W k ∈ W DMk , k ∈ M such that η k∈M W k = ∅ is confirmed by the consistency of the following system of linear inequalities: w = (w1 , w2 , . . . , wn ) ∈ W k , W k ∈ W(Ak ), k ∈ M, wj = 1. (43) j ∈N

4.4. The partial incorporation of all individual opinions Both perfect incorporation and common ground approaches do not work well when there is a wide range of diverse opinions. The solution for the group consensus proposed in this subsection may work in such cases. The proposed approach incorporates all individual opinions not perfectly but only partially. Then we consider relaxation of constraints W ⊇ W k , W k ∈ W DMk , ∀k ∈ M on W representing the group consensus to constraints W k = ∅ does not imply the existence of normalized like W ∩ W k = ∅, W k ∈ W DMk , ∀k ∈ M. However, W ∩  vector w = (w1 , w2 , . . . , wn ) such that w ∈ W ∩ W k and i∈N wi = 1 even when W ∈ W N . Indeed, consider W = ([0.2, 0.2], [0.2, 0.2], [0.2, 0.3], [0.3, 0.4]) ∈ W N and W k = ([0.2, 0.3], [0.2, 0.3], [0.2, 0.2], [0.3, 0.3]) ∈ W N , we find a unique vector w = (0.2, 0.2, 0.2, 0.3) ∈ W ∩ W k . This  vector is not normalized although W ∩ W k = ∅. If there is no normalized vector such that w ∈ W ∩ W k and i∈N wi = 1, the opinion of decision maker k is not reflected in the weight vector W representing the group consensus.

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such that wk ∈ W ∩ W k , W k ∈ W DMk , ∀k ∈ M and  Then we use the constraints ∃wk = (wk1 , wk2 , . . . , wkn ) DM k , ∀k ∈ M. Under this relaxed constraints with the w = 1 relaxed from constraints W ⊇ W , W ∈ W k k i∈N ki normality of W (W ∈ W N ), we minimize the sum of widths so that we may find a dominance relation more effectively narrowing down the candidates from alternatives. Accordingly, the interval weight vector W representing the group consensus is obtained by solving the following linear programming problem: 

  DM k N min d(W )  wk ∈ W ∩ W k , wki = 1, W k ∈ W , k ∈ M, W ∈ W . (44)  i∈N  We note that ∃wk ∈ W ∩ W k such that i∈N wki = 1 can be rewritten as η(W ∩ W k ) = ∅. This approach to obtaining a consensus interval weight vector is called the partial incorporation approach. The group consensus based on this approach is called the partial incorporation of all individual opinions. We have the following proposition which shows that we can drop constraint W ∈ W N from Problem (44). Proposition 8. (44) is equivalent to the following simpler linear programming problem: 

  DM k min d(W )  wk ∈ W ∩ W k , wki = 1, W k ∈ W , k∈M , 

(45)

i∈N

or equivalently, minimize

(wiR − wiL ), i∈N

subject to



R L (wki − wki ) ≤ dˆk , k ∈ M,

i∈N L ≤ a w R , a w L ≤ w R , i, j ∈ N, i < j, k ∈ M, wki kij kj kij kj ki R L wki + wkj ≥ 1, k ∈ M, j ∈ N,

(46)

i∈N\j



L R wki + wkj ≤ 1, k ∈ M, j ∈ N,

i∈N\j



wki = 1, k ∈ M,

i∈N R ,  ≤ w L ≤ w ≤ w R , k ∈ M, i ∈ N,  ≤ wiL ≤ wki ≤ wki ki ki i L , w R ], . . . , [w L , w R ]). where W = ([w1L , w1R ], . . . , [wnL , wnR ]) and W k = ([wk1 kn kn k1

Proof. It can be proved in the same way as Proposition 5 from the fact that wiL = mink∈M wki and wiR = maxk∈M wki at an optimal solution to Problem (46). 2 Let dˆQ be the optimal value of Problem (45), i.e., 

  DM k ˆ wki = 1, W k ∈ W , k∈M . dQ = min d(W )  w k ∈ W ∩ W k ,  i∈N

Then the set of optimal interval weight vectors W representing the partial incorporation consensus is defined by

   DM WQ = W  d(W ) ≤ dˆQ , w k ∈ W ∩ W k , wki = 1, W k ∈ W k , k ∈ M . (47)  i∈N

 We note that dˆQ = 0 implies η( k∈M W k ) = ∅, i.e., there exists a common ground with all individual opinions. Then, in this case, it is better to apply the common ground approach than the partial incorporation approach. The dominance relation Q between alternatives by the partial incorporation consensus is defined by

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op Q oq if and only if ∀W ∈ WQ , ∀w = (w1 , . . . , wn ) ∈ W such that wi = 1, wi (ui (op ) − ui (oq )) ≥ 0. i∈N

(48)

i∈N

This dominance relation is called partial dominance relation and satisfies the reflexivity and transitivity. The partial dominance relation op Q oq is verified by solving the following  linear programming problem: dQ (op , oq ) =  min{ i∈N wi (ui (op ) − ui (oq )) | w = (w1 , . . . , wn ) ∈ W , W ∈ WQ , i∈N wi = 1}. We have op Q oq if and only if dQ (op , oq ) ≥ 0. Moreover, when WQ is a singleton, the linear programming problem is reduced to a continuous knapsack problem solved much more easily. We have the following proposition. Proposition 9. The following assertion is valid: op Q oq implies ∀k ∈ M, op πk oq .

(49)

 Proof. Because W ∈ WQ implies ∀k ∈ M, ∃w k ∈ W ∩ W k , such that i∈N wki = 1, W k ∈ W DMk and wk satisfies  i∈N wi (ui (op ) − ui (oq )) ≥ 0. Then the proposition is immediately obtained. 2 In the same way as the perfect incorporation approach, WQ includes weight vectors not included in any of W k ∈ W DMk , k ∈ M. Such weight vectors are regarded as possible opinions after decision makers’ revision in consideration of other opinions. 4.5. Three approaches and fuzzy dominance relation We have proposed three solutions of group consensus under individual comparison matrices of m decision makers. The perfect incorporation approach yields the most reliable dominance relation all decision makers support by all means. However, there will not be so many pairs of alternatives that the dominance relation holds. This perfect dominance relation is useful to count out indubitably inferior alternatives. DM k , k ∈ M The common  ground and partial incorporation approaches are complementary. If there exist W k ∈ W such that η( k∈M W k ) = ∅, the common ground approach is useful, otherwise the partial incorporation approach is. The weak common dominance relation and the partial dominance relation are similar in the sense that both imply ∀k ∈ M, op πk oq . W W However, the weak common dominance relation W C is complete (comparable). Namely, op C oq or oq C op W holds for all alternative pairs (op , oq ). Moreover both of op W C oq and oq C op hold for a number of alternative pairs W (op , oq ). Thus the significance of the satisfaction of op C oq is weak. On the other hand, as shown in Proposition 7, W S op W C oq implies not only oq C op but also oq C op . Therefore, the negative weak common dominance relation is useful and similar to the strong common dominance relation. The negative weak common dominance relation is stronger than the strong common dominance relation because the dissatisfaction of weak common dominance relation implies the satisfaction of strong common dominance relation. The strong common dominance relation and the partial dominance relation are not like the weak common dominance relation but effective for the comparison of alternatives. Using negative weak common dominance relation and strong common dominance relation as well as the partial dominance relation, we can suggest the agreeable group preferences with respect to pairs of alternatives to the group of decision makers. The proposed approaches are useful in complementary purposes for narrowing down the candidates from alternatives as described above. When W DMk is a singleton {W k } for all k ∈ M, we have the following implications:  if η( W k ) = ∅, op P oq implies op SC oq , (50) k∈M

otherwise, op P oq implies op Q oq .

(51)

From those inclusion relations, we may build a fuzzy dominance relation  with the following discrete membership function:

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Table 2 Marginal utilities of three alternatives. Alternative o1 o2 o3

⎧ 1 ⎪ ⎪ ⎪ ⎨ 0.5 μ (op , oq ) = ⎪ 0.5 ⎪ ⎪ ⎩ 0

Criterion cr1

cr2

cr3

cr4

0.7 0.75 1

1 0.1 0.3

0.3 1 0.45

0.2 1 0.8

if op P oq ,   if op P oq , η W k = ∅ and op SC oq , k∈M  if op P oq , η( k∈M W k ) = ∅ and op Q oq , otherwise.

(52)

The membership grade shows the strength of group preference. When we introduce the compromises of decision makers into the proposed approaches, we may refine the fuzzy dominance relation representing group preference. However, this is beyond the scope of this paper. 5. Numerical examples 5.1. Outline We give numerical examples to illustrate the proposed three approaches. Two examples with three decision makers, three alternatives and four criteria are given. Both examples share the same alternatives whose marginal utilities in view of each criterion are given in Table 2. On the other hand, the sets of individual pairwise comparison matrices are different: one without common ground and the other with common ground. 5.2. Case where no common ground exists We consider a group of DM 1 , DM 2 and DM 3 who give individual pairwise comparison matrices shown in the first five column of Table 3. For each decision maker, one of interval weight vectors obtained from pairwise comparison matrix Ak , W k is shown at the rightmost column of Table 3. We note that those are not always unique. Indeed, W 3 = (0.3750, [0.2188, 0.3750], [0.1250, 0.1875], [0.0625, 0.2188]) is also a member of W DM3 . From those interval weight vectors of three decision makers, we know that all decision makers evaluate cr1 the Checking the consistency of (43) for this most important criterion while the weight of cr1 varies by the individual.   =  ∅. Then, we apply the perfect and W example, we find that there are no W k ∈ WDMk , k ∈ M such that η k k∈M partial incorporation approaches. Applying the perfect incorporation approach, we obtain one of consensus interval weight vectors, W P = (W1P , W2P , W3P , W4P ) with W1P = [0.3750, 0.5714], W2P = [0.1905, 0.3750], W3P = [0.0714, 0.1905] and W4P = [0.0476, 0.2188]. Solving Problem (31) for all ordered pairs of alternatives, we obtain dP (o1 , o2 ) = −0.1569, dP (o2 , o1 ) = −0.2241, dP (o1 , o3 ) = −0.1644, dP (o3 , o1 ) = −0.0893, dP (o2 , o3 ) = −0.1560 and dP (o3 , o2 ) = −0.0116. Thus, no perfect dominance holds between alternatives. Now we apply the partial incorporation approach. We obtain one of consensus interval weight vectors, W Q = (W1Q , W2Q , W3Q , W4Q ) with W1Q = [0.3750, 0.5714], W2Q = [0.1905, 0.2500], W3Q = [0.1131, 0.1250] and W4Q = [0.1250, 0.2500]. For verifying the partial dominance relation, we calculate dQ (o1 , o2 ) = −0.1378, dQ (o2 , o1 ) = −0.1622, dQ (o1 , o3 ) = −0.1676, dQ (o3 , o1 ) = −0.0857, dQ (o2 , o3 ) = −0.116 and dQ (o3 , o2 ) = 0.0031. We only obtain an agreeable dominance of o3 over o2 . 5.3. Case where common ground exists We consider a group of DM 4 , DM 5 and DM 6 . The given individual pairwise comparison matrices and the obtained interval weight vectors are shown in Table 4. We note that the pairwise matrix of DM 4 is the same as that of DM 3 in the previous example.

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Table 3 Individual evaluations of DM 1 , DM 2 and DM 3 . DM 1 ’s A1

cr1

cr2

cr3

cr4

W1

cr1 cr2 cr3 cr4

1 – – –

2 1 – –

3 2 1 –

4 3 2 1

0.5000 0.2500 [0.1250, 0.1667] [0.0833, 0.1250] dˆ1 = 0.0834

DM 2 ’s A2

cr1

cr2

cr3

cr4

W2

cr1 cr2 cr3 cr4

1 – – –

3 1 – –

3 3 1 –

4 3 4 1

0.5714 [0.1905, 0.2143] [0.0714, 0.1905] [0.0476, 0.1429] dˆ2 = 0.2382

DM 3 ’s A3

cr1

cr2

cr3

cr4

W3

cr1 cr2 cr3 cr4

1 – – –

1 1 – –

2 3 1 –

2 1 3 1

0.3750 [0.2500, 0.3750] [0.1250, 0.1875] [0.0625, 0.2500] dˆ3 = 0.3750

Table 4 Individual evaluations of DM 4 , DM 5 and DM 6 . DM 4 ’s A4

cr1

cr2

cr3

cr4

W4

cr1 cr2 cr3 cr4

1 – – –

1 1 – –

2 3 1 –

2 1 3 1

0.3750 [0.2500, 0.3750] [0.1250, 0.1875] [0.0625, 0.2500] dˆ4 = 0.3750

DM 5 ’s A5

cr1

cr2

cr3

cr4

W5

cr1 cr2 cr3 cr4

1 – – –

1 1 – –

3 2 1 –

3 5 2 1

0.3750 [0.3667, 0.3750] [0.1250, 0.1833] [0.0750, 0.1250] dˆ5 = 0.1167

DM 6 ’s A6

cr1

cr2

cr3

cr4

W6

cr1 cr2 cr3 cr4

1 – – –

1 1 – –

2 3 1 –

6 2 3 1

0.3750 [0.3333, 0.3750] [0.1250, 0.1875] [0.0625, 0.1667] dˆ6 = 0.2084

From those interval weight vectors of three decision makers, we know that all decision makers’ evaluations are similar. the consistency of (43) for this example, we find that there are W k ∈ WDMk , k ∈ M such that   Checking η k∈M W k = ∅. Then, we apply the perfect incorporation and common ground approaches. Applying the perfect incorporation approach, we obtain one of consensus interval weight vectors, W P = (W1P , W2P , W3P , W4P ) with W1P = 0.3750, W2P = [0.2188, 0.3750], W3P = [0.1250, 0.1875] and W4P = [0.0625, 0.2188]. Solving Problem (31) for all ordered pairs of alternatives, we obtain dP (o1 , o2 ) = −0.1281, dP (o2 , o1 ) = −0.1375,

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dP (o1 , o3 ) = −0.1188, dP (o3 , o1 ) = −0.0844, dP (o2 , o3 ) = −0.0750 and dP (o3 , o2 ) = −0.0094. Thus, no perfect dominance relation holds between alternatives. Now we apply the common ground approach. We obtain one of consensus interval weight vectors, W C = (W1C , W2C , W3C , W4C ) with W1C = 0.3750, W2C = [0.3667, 0.3750], W3C = [0.1250, 0.1833] and W4C = [0.0750, 0.1250]. For verifying the strong common dominance relation, we calculate dCS (o1 , o2 ) = 0.1179, dCS (o2 , o1 ) = −0.1363, dCS (o1 , o3 ) = 0.0492, dCS (o3 , o1 ) = −0.0788, dCS (o2 , o3 ) = −0.0750 and dCS (o3 , o2 ) = 0.0513. We obtain agreeable dominances for ordered pairs (o1 , o2 ), (o1 , o3 ) and (o3 , o2 ). From this result, we may have a group preference, o1 SC o3 SC o2 . This example shows a case where opinions of decision makers are similar so that we may expect a total order among alternatives, i.e., o1 SC o3 SC o2 . 6. Concluding remarks Three approaches based on Interval AHP to group decision making have been proposed. They are perfect incorporation, common ground and partial incorporation approaches. In the perfect incorporation approach, consensus interval weight vectors including all individual interval weight vectors are calculated and used for counting out indubitably inferior alternatives. The common ground approach is useful when all individual opinions are similar while the partial incorporation approach is useful when individual opinions are not similar. Group interval weight vectors included in all individual interval weight vectors are calculated in the common ground approach, while consensus interval weight vectors intersecting all individual interval weight vectors are calculated in the partial incorporation approach. Those consensus interval weight vectors are used for finding agreeable group preference between alternatives. The roles of obtained consensus interval weight vectors are different between the perfect incorporation approach and the others. The applicable scopes are different between the common ground and the partial incorporation approaches. Therefore, the proposed three approaches are complementary. All approaches proposed in this paper work simply by solving linear programming problems. In the conventional approaches, a crisp weight vector is calculated and the consistency of the given pairwise comparison matrix is evaluated. Once the consistency is in the acceptable range, the crisp weight vector is usually used to rank the alternatives without reconsideration of the original data inconsistency. On the other hand, in the proposed approach, the data inconsistency is translated into the interval, and the calculated interval weight vector is used for the analysis. Because of the interval weights, the obtained dominance relations are not always weak orders, but they are useful for narrowing down the candidates by erasing inferior alternatives. For the alternative pairs whose dominance relations are unclear, we may ask the decision makers for further consideration and negotiation toward the consensus. The proposed approaches can be extended to fuzzy/interval pairwise comparison matrices. On the other hand, we have not yet introduced compromise concepts to the proposed approaches in this paper. By the compromise, we will obtain more effective dominance relations guiding the decision makers toward the consensus. Moreover, we will be able to associate the proposed three approaches with the construction of a fuzzy group dominance relation with multiple membership grades. These will be our future studies. Acknowledgement This work was partially supported by JSPS KAKENHI Grant Number 26350423. References [1] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [2] P.J.M. van Laarhoven, W. Pedrycz, A fuzzy extension of Saaty’s priority theory, Fuzzy Sets Syst. 11 (1983) 199–227. [3] Y.-M. Wang, T.M.S. Elhag, Z. Hua, A modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process, Fuzzy Sets Syst. 157 (2006) 3055–3071. [4] J. Ramík, R. Perzina, A method for solving fuzzy multicriteria decision problems with dependent criteria, Fuzzy Optim. Decis. Mak. 9 (2) (2010) 123–141. [5] J.J. Buckley, Fuzzy hierarchical analysis, Fuzzy Sets Syst. 17 (1985) 233–247. [6] J.J. Buckley, T. Feuring, Y. Hayashi, Fuzzy hierarchical analysis revisited, Eur. J. Oper. Res. 129 (2001) 48–64.

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