# A Method for Estimating Criteria Weights from Intuitionistic Preference Relations

A Method for Estimating Criteria Weights from Intuitionistic Preference Relations Fuzzy Inf. Eng. (2009) 1: 79-89 DOI 10.1007/s12543-009-0006-0 ORIGINAL ARTICLE A Method for Estimating Criteria Weights from Intuitionistic Preference Relations Ze-shui Xu Received: 16 May 2008/ Revised: 27 October 2008/ Accepted: 10 January 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract An intuitionistic preference relation is a powerful means to express de- cision makers’information of intuitionistic preference over criteria in the process of multi-criteria decision making. In this paper, we ﬁrst deﬁne the concept of its con- sistence and give the equivalent interval fuzzy preference relation of it. Then we develop a method for estimating criteria weights from it, and then extend the method to accommodate group decision making based on them And ﬁnally, we use some numerical examples to illustrate the feasibility and validity of the developed method. Keywords Intuitionistic preference relation · Consistent intuitionistic preference relation · Weak transitivity · Priority vector · Linear programming model 1. Introduction In [1], Atanassov introduced the concept of intuitionistic fuzzy set, which emerges from the simultaneous consideration of the degrees of membership and non-members- hip with a degree of hesitancy. The intuitionistic fuzzy set has been studied and ap- plied in a variety of areas. For example, Atanassov and Gargov [2] gave the notion of interval intuitionistic fuzzy set. De et al. [3] deﬁned some operations on intuitionistic fuzzy sets. De et al. [4] applied the intuitionistic fuzzy sets to the ﬁeld of medical di- agnosis. Deschrijver and Kerre [5] established the relationships between intuitionistic fuzzy sets, L- fuzzy sets, interval-valued fuzzy sets and interval valued intuitionistic fuzzy sets. Some authors investigated the correlations [6-10] and similarity measures [11-15] of intuitionistic fuzzy sets. Deschrijver et al. [16] extended the notion of triangular norm and conorm to intuitionistic fuzzy set theory. Deschrijver and Kerre [17] introduced some aggregation operators on the lattice L , and considered some Ze-shui Xu () Institute of Sciences, PLA University of Science and Technology, Nanjing 210007, P.R.China e-mail: xu zeshui@263.net 80 Ze-shui Xu (2009) particular classes of binary aggregation operators based on t-norms on the unit inter- val. They also studied the properties of the implicators generated by these classes. Park [18] deﬁned the notion of intuitionistic fuzzy metric spaces. Park and Park [19] introduced the notion of generalized intuitionistic fuzzy ﬁlters based on the notion of generalized intuitionistic fuzzy sets given by Mondal and Samanta [20], and deﬁned the notion of Hausdorﬀness on generalized intuitionistic fuzzy ﬁlters. Deschrijver and Kerre [21] introduced the notion of uninorm in interval-valued fuzzy set theory. Cor- nelis et al. [22] constructed a representation theorem for Lukasiewicz implicators on the lattice L . which serves as the underlying algebraic structure for both intuitionistic fuzzy and interval-valued fuzzy sets. Gutierrez ´ Garc´ ıa and Rodabaugh [23] demon- strated two meta-mathematical propositions concerning the intuitionistic approaches to fuzzy sets and fuzzy topology, as well as the closely related interval-valued sets and interval-valued intuitionistic ones. Dudek et al. [24] considered the intuition- istic fuzziﬁcation of the concept of sub-hyperquasigroups in a hyperquasigroup and investigated some properties of such sub-hyperquasigroups. Xu and Yager [25] inves- tigated the aggregation of intuitionistic information, and developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric operator, the intuitionistic fuzzy ordered weighted geometric operator, and the intuitionistic fuzzy hybrid geometric operator. They also gave an application of these operators to multiple attribute decision making based on intuitionistic fuzzy sets. Szmidt and Kacprzyk [26], and Xu [27] investigated the group decision making problems with in- tuitionistic preference relations. Herrera et al. [28] developed an aggregation process for combining numerical, interval valued and linguistic information. Furthermore, they proposed diﬀerent extensions of this process to deal with contexts in which can appear other type of information as intuitionistic fuzzy sets or multi-granular linguis- tic information. In the process of multi-criteria decision making under it, intuitionistic preference relation is a powerful tool to express the decision maker’s intuitionistic preference in- formation over criteria, the priority weights derived from the intuitionistic preference relation can be used as the weights of criteria. Thus, how to estimate criteria weights from an intuitionistic preference relation is an interesting and important issue, which no investigation has been devoted to. In this paper, we shall develop a method for es- timating criteria weights from an intuitionistic preference relation. In order to do so, the paper is organized as follows. Section 2 represents some basic concepts. Section 3 introduces the notion of consistent intuitionistic preference relation, and gives the equivalent interval fuzzy preference relation of an intuitionistic preference relation. In Section 4, we develop a method for estimating criteria weights from an intuition- istic preference relation based on linear programming models, and then extend the method to accommodate group decision making based on intuitionistic preference relations. In Section 5, we give some numerical examples, and conclude the paper in Section 6. Fuzzy Inf. Eng. (2009) 1:79-89 81 2. Preliminaries Let X be a universe of discourse. Atanassov [1] introduced the notion of intuitionistic fuzzy set A, which can be shown as follows, A = {< x,μ (x ), v (x ) > |x ∈ X}. (1) j A j A j j Intuitionistic fuzzy set A assigns to each element x ∈ X a membership degree μ (x ) ∈ [0, 1] and a non-membership degree v (x ) ∈ [0, 1], with the condition A j A j 0 ≤ μ (x )+ v (x ) ≤ 1, ∀x ∈ X. (2) A j A j j For each x ∈ X, the value π (x ) = 1−μ (x )− v(x ) A j A j j is called the indeterminacy degree or hesitation degree of x to A. Especially, if π (x ) = 1−μ (x )− v (x ) = 0, for each x ∈ X, A j A j A j j then, the intuitionistic fuzzy set A is reduced to a common fuzzy set [29]. Consider a multi-criteria decision making problem with a ﬁnite set of n criteria, and let X = {x , x ,··· , x } be the set of criteria. In [27], we introduced the notion of 1 2 n intuitionistic preference relation as follows: Deﬁnition 1[27] An intuitionistic preference relation B on X is represented by a matrix B = (b ) ⊂ X × X with b =< (x, x ),μ(x, x ), v(x, x ) >, for all i, j = ij ij i j i j i j 1, 2,··· , n. For convenience, we let b = (μ , v ), for all i, j = 1, 2,··· , n, where b ij ij ij ij is an intuitionistic fuzzy value, composed by the certainty degree μ to which x is ij i preferred to x and the certainty degree v to which x is non-preferred to x , and j ij i j 1 − μ − v is interpreted as the hesitation degree to which x is preferred to x . ij ij i j Furthermore,μ and v satisfy the following characteristics: ij ij 0 ≤ μ + v ≤ 1,μ = v , v = μ ,μ = v = 0.5, for all i, j = 1, 2,··· , n. (3) ij ij ji ij ji ij ii ii 3. Consistent Intuitionistic Preference Relation By Deﬁnition 1, we know that each element b in the intuitionistic preference rela- ij tion B consists of the pair (μ , v ). Consider that each pair (μ , v ) must satisfy the ij ij ij ij conditionμ + v ≤ 1, i.e.,μ ≤ 1− v . This condition is exactly the condition under ij ij ij ij which two real numbers form an interval [28]. As a result, we can transform the pair b = (μ , v ) into the interval number b = [μ , 1− v ], and thus, the intuitionistic ij ij ij ij ij ij preference relation B = (b ) is equivalent to an interval fuzzy preference relation ij n×n − + ˙ ˙ ˙ ˙ ˙ [30,31] B = (b ) , where b = [b , b ] = [μ , 1− v ], for all i, j = 1, 2,··· , n, and ij n×n ij ij ij ij ij − + + − + − + − ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ b + b = b + b = 1, b  b  0, b = b = 0.5, for all i, j = 1, 2,··· , n. ij ji ij ji ij ij ii ii 82 Ze-shui Xu (2009) For convenience, we denote by the intuitionistic preference relation B = (b ) , ij n×n where b = [μ , 1− v ], for all i, j = 1, 2,··· , n,. Especially, if ij ij ij μ + v = 1, for all i, j = 1, 2,··· , n, ij ij then the intuitionistic preference relation B = (b ) is reduced to a fuzzy preference ij n×n relation [32-38] R = (r ) , where ij n×n 0 ≤ r ≤ 1, r + r = 1, r = 0.5, for all i, j = 1, 2,··· , n. ij ij ji ii Let w = (w , w ,··· , w ) be the vector of priority weights, where w reﬂects the 1 2 n i importance degree of the criterion x , and w  0, i = 1, 2,··· , n, w = 1, (4) i i i=1 then, a fuzzy preference relation R = (r ) is called a consistent fuzzy preference ij n×n relation, if the following additive transitivity [33] is satisﬁed r = r − r + 0.5, for all i, j, k = 1, 2,··· , n ij ik jk and such a fuzzy preference relation is given by [39,40]: r = 0.5(w − w + 1), for all i, j = 1, 2,··· , n. (5) ij i j By (5), in the following, we deﬁne the concept of consistent intuitionistic prefer- ence relation: Deﬁnition 2 Let B = (b ) be an intuitionistic preference relation, where b = ij n×n ij [μ , 1 − v ], for all i, j = 1, 2,··· , n, if there exists a vector w = (w , w ,··· , w ) , ij ij 1 2 n such that μ ≤ 0.5(w − w + 1) ≤ 1− v , (6) ij i j ij for all i = 1, 2,··· , n − 1; j = i + 1,··· , n, where w satisﬁes the condition (4), then we call B a consistent intuitionistic preference relation; otherwise, we call B an inconsistent intuitionistic preference relation. In the next section, we shall develop a method for estimating criteria weights from an intuitionistic preference relation. 4. A Method for Estimating Criteria Weights If B = (b ) is a consistent intuitionistic preference relation, then the priority vector ij n×n w = (w , w ,··· , w ) of B should satisfy (4) and (6). Thus, motivated by the idea 1 2 n [41], we utilize (4) and (6) to establish the following linear programming model: − + (M-1) w = Min w and w = Max w i i i i s.t. 0.5(w − w + 1)  μ , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij 0.5(w − w + 1) ≤ 1− v , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij w  0, i = 1, 2,··· , n, w = 1. i i i=1 Fuzzy Inf. Eng. (2009) 1:79-89 83 − + Solving the model (M-1), we can get the weight intervals [w , w ], i = 1, 2,··· , n. i i − + Especially, if w = w , for all i, then we get a unique priority vector w = (w , w ,··· , 1 2 i i w ) from the intuitionistic preference relation B. If B = (b ) is an inconsistent intuitionistic preference relation, then (6) does ij n×n not always hold. In this case, we relax (6) by introducing the deviation variables d ij and d , i = 1, 2,··· , n− 1; j = i+ 1,··· , n: ij − + μ − d ≤ 0.5(w − w + 1) ≤ 1− v + d , (7) ij i j ij ij ij − + for all i = 1, 2,··· , n− 1; j = i+ 1,··· , n, where d and d are both nonnegative real ij ij − + numbers. Obviously, the smaller the deviation variables d and d , the closer B to an ij ij inconsistent intuitionistic preference relation. As a result, we establish the following optimization model. n−1 n ∗ − + (M-2) J = Min (d + d ) ij ij i=1 j=i+1 s.t. 0.5(w − w + 1)+ d  μ , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij ij 0.5(w − w + 1)− d ≤ 1− v , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij ij w  0, i = 1, 2,··· , n, w = 1, i i i=1 − + d , d  0, i = 1, 2,··· , n− 1; j = i+ 1,··· , n. ij ij − + ˙ ˙ Solving this model, we can get the optimal deviation values d , and d , i = 1, 2,··· , n− ij ij 1; j = i + 1,··· , n, and the optimal priority vector w = (w , w ,··· , w ) from the 1 2 n intuitionistic preference relation B. In what follows, we further extend the above results to group decision making based on intuitionistic preference relations: For a multi-criteria group decision making problem, let E = {e , e ,··· , e } be the 1 2 m set of decision makers. Each decision maker e compares each pair of the criteria (k) x and x , and then constructs an intuitionistic preference relation B = (b ) , i j k n×n ij (k) (k) (k) (k) (k) where b = (μ , v ), for all i, j = 1, 2,··· , n. Furthermore, μ and v satisfy the ij ij ij ij ij characteristics (6). Let w = (w , w ,··· , w ) be the weight vector of the criteria x (i = 1, 2,··· , n), 1 2 n i where w reﬂects the importance degree of the criterion x . If the weight vector w i i satisﬁes (4) and (k) (k) μ ≤ 0.5(w − w + 1) ≤ 1− v , (8) i j ij ij (k) for all i = 1, 2,··· , n− 1; j = i+ 1,··· , n, then B = b ) is a consistent intuition- k n×n ij istic preference relation. If (8) holds for all k = 1, 2,··· , m, i.e., all the intuitionistic (k) preference relations B = b ) are consistent, then we call the group is of consen- k n×n ij sus. In this case, similar to the model (M-1), we can utilize (4) and (8) to establish the following linear programming model: 84 Ze-shui Xu (2009) − + (M-3) w = Min w and w = Max w i i i i s.t. 0.5(w − w + 1)  μ , i = 1, 2,··· , n− 1; i j ij j = i+ 1,··· , n; k = 1, 2,··· , m, 0.5(w − w + 1) ≤ 1− v , i = 1, 2,··· , n− 1; i j ij j = i+ 1,··· , n; k = 1, 2,··· , m, w  0, i = 1, 2,··· , n, w = 1. i i i=1 − + Solving the model (M-3), we can get the weight intervals [w , w ], i = 1, 2,··· , n. i i − + T Especially, if w = w , for all i, then we get a unique weight vector w = (w , w ,··· , w ) 1 2 n i i for the criteria x (i = 1, 2,··· , n). If the group is not of consensus, then (8) does not always hold for all k = 1, 2,··· , m. −(k) +(k) In this case, we relax (8) by introducing the deviation variables d and d , ij ij i = 1, 2,··· , n− 1; j = i+ 1,··· , n; k = 1, 2,··· , m : (k) −(k) (k) +(k) μ − d ≤ 0.5(w − w + 1) ≤ 1− v + d , i j ij ij ij ij (9) for all i = 1, 2,··· , n− 1; j = i+ 1,··· , n; k = 1, 2,··· , m, −(k) +(k) where d and d are both nonnegative real numbers. ij ij Based on (9), we establish the following optimization model: m n−1 n −(k) +(k) (M-4) J = Min (d + d ) 2 ij ij k=1 i=1 j=i+1 s.t. 0.5(w − w + 1)+ d  μ , i = 1, 2,··· , n− 1; i j ij ij j = i+ 1,··· , n; k = 1, 2,··· , m, 0.5(w − w + 1)− d ≤ 1− v , i = 1, 2,··· , n− 1; i j ij ij j = i+ 1,··· , n; k = 1, 2,··· , m, w  0, i = 1, 2,··· , n, w = 1, i i i=1 −(k) +(k) d , d  0, i = 1, 2,··· , n− 1; j = i+ 1,··· , n; ij ij k = 1, 2,··· , m. − + ˙ ˙ Solving this model, we can get the optimal deviation values d and d , i = 1, 2,··· , n− ij ij 1; j = i+ 1,··· , n; k = 1, 2,··· , m, and the optimal priority vector w = (w , w ,··· , 1 2 w ) of the criteria x (i = 1, 2,··· , n). n i 5. Illustrative Examples Example 1 For a multi-criteria decision making problem, there are ﬁve criteria x (i = 1, 2,··· , 5). A decision maker compares each pair of criteria x and x , and i i j provides his/her intuitionistic fuzzy preference value a = (μ , v ), composed by the ij ij ij certainty degreeμ to which x is preferred to x and the certainty degree v to which ij i j ij x is non-preferred to x , and then constructs the following intuitionistic preference i j Fuzzy Inf. Eng. (2009) 1:79-89 85 relation: ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.6, 0.3) (0.4, 0.2) (0.7, 0.2) (0.4, 0.5) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (0.3, 0.6) (0.5, 0.5) (0.5, 0.3) (0.6, 0.1) (0.3, 0.6) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = (a ) = . ij 5×5 ⎢ (0.2, 0.4) (0.3, 0.5) (0.5, 0.5) (0.6, 0.2) (0.4, 0.5)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.7) (0.1, 0.6) (0.2, 0.6) (0.5, 0.5) (0.3, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.5, 0.4) (0.6, 0.3) (0.5, 0.4) (0.6, 0.3) (0.5, 0.5) We ﬁrst transform the intuitionistic preference relation A into its equivalent interval fuzzy preference relation B = (b ) (here, b = [μ , 1− v ], i, j = 1, 2,··· , 5): ij n×n ij ij ij ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.6, 0.7] [0.4, 0.8] [0.7, 0.8] [0.4, 0.5] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.3, 0.4] [0.5, 0.5] [0.5, 0.7] [0.6, 0.9] [0.3, 0.4]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B = (b ) = , [0.2, 0.6] [0.3, 0.5] [0.5, 0.5] [0.6, 0.8] [0.4, 0.5] ij 5×5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.2, 0.3] [0.1, 0.4] [0.2, 0.4] [0.5, 0.5] [0.3, 0.4]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.5, 0.6] [0.6, 0.7] [0.5, 0.6] [0.6, 0.7] [0.5, 0.5] then by solving the model (M-2), we get J = 0.075, the optimal deviation values: − − − + − + − + − + ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = d = 0, d = d = 0, d = 0.025, d = 0, d = d = 0, d = d = 0, 12 12 13 13 14 14 15 15 23 23 − + − + − + − + − + ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0.025, d = 0, d = d = 0, d = 0.025, d = 0, d = d = 0, d = d = 0 24 24 25 25 34 34 35 35 45 45 and the optimal priority vector w = (0.35, 0.15, 0.15, 0, 0.35) , i.e., the weights of the criteria x (i = 1, 2,··· , 5) are w = 0.35, w = 0.15, w = 0.15, w = 0 and i 1 2 3 4 w = 0.35, respectively. Example 2 Suppose that a decision maker provides his/her preference information over a collection of criteria x , x , x , x with the following intuitionistic preference 1 2 3 4 relation: ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.6, 0.2) (0.5, 0.4) (0.7, 0.1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.6) (0.5, 0.5) (0.4, 0.3) (0.6, 0) ⎥ ⎢ ⎥ ⎢ ⎥ A = (a ) = ⎢ ⎥ . ij 4×4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (0.4, 0.5) (0.3, 0.4) (0.5, 0.5) (0.7, 0.1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.1, 0.7) (0, 0.6) (0.1, 0.7) (0.5, 0.5) We ﬁrst transform the intuitionistic preference relation A into its equivalent interval fuzzy preference relation ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.6, 0.8] [0.5, 0.6] [0.7, 0.9] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.2, 0.4] [0.5, 0.5] [0.4, 0.7] [0.6, 1] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B = (b ) = , ⎢ ⎥ ij 4×4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.4, 0.5] [0.3, 0.6] [0.5, 0.5] [0.7, 0.9] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.1, 0.3] [0, 0.4] [0.1, 0.3] [0.5, 0.5] then by solving the model (M-2), we get J = 0, and all the optimal deviation values − + ˙ ˙ d and d (i = 1, 2, 3; j = i+ 1,··· , 4) are equal to zero. Thus, by Deﬁnition 2, A is ij ij 86 Ze-shui Xu (2009) a consistent intuitionistic preference relation, and then we solve the model (M-1) and get a unique priority vector w = (0.4, 0.2, 0.4, 0) , i.e., the weights of the criteria x (i = 1, 2,··· , 4) are w = 0.4, w = 0.2, w = 0.4 and w = 0, respectively. i 1 2 3 4 Example 3 For a multi-criteria group decision making problem, there are ﬁve crite- ria x (i = 1, 2,··· , 5), and three decision makers e (k = 1, 2, 3). The decision maker i k e (k = 1, 2, 3) compare each pair of criteria x and x , and construct the following k i j (k) intuitionistic preference relations A = (a ) : k n×n ij ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.5, 0.2) (0.6, 0.2) (0.7, 0.1) (0.3, 0.5) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.5) (0.5, 0.5) (0.3, 0.1) (0.5, 0.2) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ A = (a ) = ⎢ ⎥ , (0.2, 0.6) (0.1, 0.3) (0.5, 0.5) (0.7, 0.3) (0.5, 0.3) 1 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.1, 0.7) (0.2, 0.5) (0.3, 0.7) (0.5, 0.5) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.5, 0.3) (0.6, 0.4) (0.3, 0.5) (0.6, 0.4) (0.5, 0.5) ⎡ ⎤ ⎢ ⎥ ⎢ (0.5, 0.5) (0.6, 0.2) (0.7, 0.3) (0.7, 0.2) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.6) (0.5, 0.5) (0.4, 0.2) (0.5, 0.3) (0.3, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (2) ⎢ ⎥ ⎢ ⎥ A = (a ) = (0.3, 0.7) (0.2, 0.4) (0.5, 0.5) (0.6, 0.2) (0.5, 0.4) , 2 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.7) (0.3, 0.5) (0.2, 0.6) (0.5, 0.5) (0.3, 0.5)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.4, 0.6) (0.6, 0.3) (0.4, 0.5) (0.5, 0.3) (0.5, 0.5) ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.4, 0.3) (0.5, 0.4) (0.6, 0.3) (0.3, 0.7) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.3, 0.4) (0.5, 0.5) (0.4, 0.3) (0.6, 0.2) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = (a ) = . (0.4, 0.5) (0.3, 0.4) (0.5, 0.5) (0.5, 0.3) (0.7, 0.2) 3 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.3, 0.6) (0.2, 0.6) (0.3, 0.5) (0.5, 0.5) (0.4, 0.5)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.7, 0.3) (0.6, 0.4) (0.2, 0.7) (0.5, 0.4) (0.5, 0.5) We ﬁrst transform the intuitionistic preference relations A (k = 1, 2, 3) into their (k) (k) equivalent interval fuzzy preference relations B = (b ) (k = 1, 2, 3) (here, b = k n×n ij ij (k) (k) [μ , 1− v ], i, j = 1, 2,··· , 5; k = 1, 2, 3): ij ij ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.5, 0.8] [0.6, 0.8] [0.7, 0.9] [0.3, 0.5] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.2, 0.5] [0.5, 0.5] [0.3, 0.9] [0.5, 0.8] [0.4, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ ⎢ ⎥ B = (b ) = [0.2, 0.4] [0.1, 0.7] [0.5, 0.5] [0.7, 0.7] [0.5, 0.7] , 1 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.1, 0.3] [0.2, 0.5] [0.3, 0.3] [0.5, 0.5] [0.4, 0.4]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.5, 0.7] [0.6, 0.6] [0.3, 0.5] [0.6, 0.6] [0.5, 0.5] ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.6, 0.8] [0.7, 0.7] [0.7, 0.8] [0.4, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.2, 0.4] [0.5, 0.5] [0.4, 0.8] [0.5, 0.7] [0.3, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (2) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B = (b ) = , 2 5×5 ⎢ [0.3, 0.3] [0.2, 0.6] (0.5, 0.5) [0.6, 0.8] [0.5, 0.6]⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.2, 0.3] [0.3, 0.5] [0.2, 0.4] [0.5, 0.5] [0.3, 0.5]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.4, 0.4] [0.6, 0.7] [0.4, 0.5] [0.5, 0.7] [0.5, 0.5] Fuzzy Inf. Eng. (2009) 1:79-89 87 ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.4, 0.7] [0.5, 0.6] [0.6, 0.7] [0.3, 0.3] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.3, 0.6] [0.5, 0.5] [0.4, 0.7] [0.6, 0.8] [0.4, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3) ⎢ ⎥ ⎢ ⎥ B = (b ) = [0.4, 0.5] [0.3, 0.6] [0.5, 0.5] [0.5, 0.7] [0.7, 0.8] , 3 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.3, 0.4] [0.2, 0.4] [0.3, 0.5] [0.5, 0.5] [0.4, 0.5]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.7, 0.7] [0.6, 0.6] [0.2, 0.3] [0.5, 0.6] [0.5, 0.5] then by solving the model (M-4), we get J = 1.067, the optimal deviation values: −(1) +(1) −(1) +(1) −(1) +(1) −(1) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.033, d = 0, d = 0, d = 0, d = 0, 12 12 13 13 14 14 15 +(1) −(1) +(1) −(1) +(1) −(1) +(1) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0.067, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, 15 23 23 24 24 25 25 −(1) +(1) −(1) +(1) −(1) +(1) ˙ ˙ ˙ ˙ ˙ ˙ d = 0.067, d = 0, d = 0, d = 0, d = 0.033, d = 0, 34 34 35 35 45 45 −(2) +(2) −(2) +(2) −(2) +(2) −(2) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.133, d = 0, d = 0, d = 0, d = 0, 12 12 13 13 14 14 15 +(2) −(2) +(2) −(2) +(2) −(2) +(2) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0.167, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, 15 23 23 24 24 25 25 −(2) +(2) −(2) +(2) −(2) +(2) −(3) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, 34 34 35 35 45 45 12 +(3) −(3) +(3) −(3) +(3) −(3) +(3) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0.267, 12 13 13 14 14 15 15 −(3) +(3) −(3) +(3) −(3) +(3) ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.067, d = 0, d = 0, d = 0, 23 23 24 24 25 25 −(3) +(3) −(3) +(3) −(3) +(3) ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.2, d = 0, d = 0.033, d = 0 34 34 35 35 45 45 and the optimal priority vector: w = (0.4, 0.06, 0.27, 0, 0.27) , i.e., the weights of the criteria x (i = 1, 2,··· , 5) are w = 0.4, w = 0.06, w = i 1 2 3 0.27, w = 0.27 and w = 0.35, respectively. 4 5 6. Conclusions We have introduced the notion of consistent intuitionistic preference relation and es- tablished some simple linear programming models to develop a method for estimating criteria weights from intuitionistic preference relations. The method can be applica- ble to multi-criteria decision making problems in many ﬁelds, such as the high tech- nology project investment of venture capital ﬁrms, supply chain management, and medical diagnosis, etc. In the future, we shall study the approach to improve the consistency of inconsistent intuitionistic preference relations. Acknowledgments The work was supported by the National Science Fund for Distinguished Young Scholars of China (No.70625005). References 1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20: 87-96 88 Ze-shui Xu (2009) 2. Atanassov K, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems 31: 343-349 3. 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Gutierrez ´ Garc´ ıa J, Rodabaugh SE (2005) Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued ”intuitionistic” sets, ”intuitionistic” fuzzy sets and topologies. Fuzzy Sets and Systems, 156: 445-484 24. Dudek WA, Davvaz B, Jun YB (2005) On intuitionistic fuzzy sub-hyperquasigroups of hyperquasi- groups. Information Sciences 170: 251-262 25. Xu, Z.S., Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems 35: 417-433 26. Szmidt E, Kacprzyk J (2002) Using intuitionistic fuzzy sets in group decision making. Control and Cybernetics 31: 1037-1053 27. Xu, Z.S. (2007) Intuitionistic preference relations and their application in group decision making. Information Sciences 177: 2363-2379 28. Herrrera F, Mart´ ınez L, Sanchez ´ PJ (2005) Managing non-homogeneous information in group deci- Fuzzy Inf. Eng. (2009) 1:79-89 89 sion making. 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# A Method for Estimating Criteria Weights from Intuitionistic Preference Relations

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Fuzzy Inf. Eng. (2009) 1: 79-89 DOI 10.1007/s12543-009-0006-0 ORIGINAL ARTICLE A Method for Estimating Criteria Weights from Intuitionistic Preference Relations Ze-shui Xu Received: 16 May 2008/ Revised: 27 October 2008/ Accepted: 10 January 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract An intuitionistic preference relation is a powerful means to express de- cision makers’information of intuitionistic preference over criteria in the process of multi-criteria decision making. In this paper, we ﬁrst deﬁne the concept of its con- sistence and give the equivalent interval fuzzy preference relation of it. Then we develop a method for estimating criteria weights from it, and then extend the method to accommodate group decision making based on them And ﬁnally, we use some numerical examples to illustrate the feasibility and validity of the developed method. Keywords Intuitionistic preference relation · Consistent intuitionistic preference relation · Weak transitivity · Priority vector · Linear programming model 1. Introduction In [1], Atanassov introduced the concept of intuitionistic fuzzy set, which emerges from the simultaneous consideration of the degrees of membership and non-members- hip with a degree of hesitancy. The intuitionistic fuzzy set has been studied and ap- plied in a variety of areas. For example, Atanassov and Gargov [2] gave the notion of interval intuitionistic fuzzy set. De et al. [3] deﬁned some operations on intuitionistic fuzzy sets. De et al. [4] applied the intuitionistic fuzzy sets to the ﬁeld of medical di- agnosis. Deschrijver and Kerre [5] established the relationships between intuitionistic fuzzy sets, L- fuzzy sets, interval-valued fuzzy sets and interval valued intuitionistic fuzzy sets. Some authors investigated the correlations [6-10] and similarity measures [11-15] of intuitionistic fuzzy sets. Deschrijver et al. [16] extended the notion of triangular norm and conorm to intuitionistic fuzzy set theory. Deschrijver and Kerre [17] introduced some aggregation operators on the lattice L , and considered some Ze-shui Xu () Institute of Sciences, PLA University of Science and Technology, Nanjing 210007, P.R.China e-mail: xu zeshui@263.net 80 Ze-shui Xu (2009) particular classes of binary aggregation operators based on t-norms on the unit inter- val. They also studied the properties of the implicators generated by these classes. Park [18] deﬁned the notion of intuitionistic fuzzy metric spaces. Park and Park [19] introduced the notion of generalized intuitionistic fuzzy ﬁlters based on the notion of generalized intuitionistic fuzzy sets given by Mondal and Samanta [20], and deﬁned the notion of Hausdorﬀness on generalized intuitionistic fuzzy ﬁlters. Deschrijver and Kerre [21] introduced the notion of uninorm in interval-valued fuzzy set theory. Cor- nelis et al. [22] constructed a representation theorem for Lukasiewicz implicators on the lattice L . which serves as the underlying algebraic structure for both intuitionistic fuzzy and interval-valued fuzzy sets. Gutierrez ´ Garc´ ıa and Rodabaugh [23] demon- strated two meta-mathematical propositions concerning the intuitionistic approaches to fuzzy sets and fuzzy topology, as well as the closely related interval-valued sets and interval-valued intuitionistic ones. Dudek et al. [24] considered the intuition- istic fuzziﬁcation of the concept of sub-hyperquasigroups in a hyperquasigroup and investigated some properties of such sub-hyperquasigroups. Xu and Yager [25] inves- tigated the aggregation of intuitionistic information, and developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric operator, the intuitionistic fuzzy ordered weighted geometric operator, and the intuitionistic fuzzy hybrid geometric operator. They also gave an application of these operators to multiple attribute decision making based on intuitionistic fuzzy sets. Szmidt and Kacprzyk [26], and Xu [27] investigated the group decision making problems with in- tuitionistic preference relations. Herrera et al. [28] developed an aggregation process for combining numerical, interval valued and linguistic information. Furthermore, they proposed diﬀerent extensions of this process to deal with contexts in which can appear other type of information as intuitionistic fuzzy sets or multi-granular linguis- tic information. In the process of multi-criteria decision making under it, intuitionistic preference relation is a powerful tool to express the decision maker’s intuitionistic preference in- formation over criteria, the priority weights derived from the intuitionistic preference relation can be used as the weights of criteria. Thus, how to estimate criteria weights from an intuitionistic preference relation is an interesting and important issue, which no investigation has been devoted to. In this paper, we shall develop a method for es- timating criteria weights from an intuitionistic preference relation. In order to do so, the paper is organized as follows. Section 2 represents some basic concepts. Section 3 introduces the notion of consistent intuitionistic preference relation, and gives the equivalent interval fuzzy preference relation of an intuitionistic preference relation. In Section 4, we develop a method for estimating criteria weights from an intuition- istic preference relation based on linear programming models, and then extend the method to accommodate group decision making based on intuitionistic preference relations. In Section 5, we give some numerical examples, and conclude the paper in Section 6. Fuzzy Inf. Eng. (2009) 1:79-89 81 2. Preliminaries Let X be a universe of discourse. Atanassov [1] introduced the notion of intuitionistic fuzzy set A, which can be shown as follows, A = {< x,μ (x ), v (x ) > |x ∈ X}. (1) j A j A j j Intuitionistic fuzzy set A assigns to each element x ∈ X a membership degree μ (x ) ∈ [0, 1] and a non-membership degree v (x ) ∈ [0, 1], with the condition A j A j 0 ≤ μ (x )+ v (x ) ≤ 1, ∀x ∈ X. (2) A j A j j For each x ∈ X, the value π (x ) = 1−μ (x )− v(x ) A j A j j is called the indeterminacy degree or hesitation degree of x to A. Especially, if π (x ) = 1−μ (x )− v (x ) = 0, for each x ∈ X, A j A j A j j then, the intuitionistic fuzzy set A is reduced to a common fuzzy set [29]. Consider a multi-criteria decision making problem with a ﬁnite set of n criteria, and let X = {x , x ,··· , x } be the set of criteria. In [27], we introduced the notion of 1 2 n intuitionistic preference relation as follows: Deﬁnition 1[27] An intuitionistic preference relation B on X is represented by a matrix B = (b ) ⊂ X × X with b =< (x, x ),μ(x, x ), v(x, x ) >, for all i, j = ij ij i j i j i j 1, 2,··· , n. For convenience, we let b = (μ , v ), for all i, j = 1, 2,··· , n, where b ij ij ij ij is an intuitionistic fuzzy value, composed by the certainty degree μ to which x is ij i preferred to x and the certainty degree v to which x is non-preferred to x , and j ij i j 1 − μ − v is interpreted as the hesitation degree to which x is preferred to x . ij ij i j Furthermore,μ and v satisfy the following characteristics: ij ij 0 ≤ μ + v ≤ 1,μ = v , v = μ ,μ = v = 0.5, for all i, j = 1, 2,··· , n. (3) ij ij ji ij ji ij ii ii 3. Consistent Intuitionistic Preference Relation By Deﬁnition 1, we know that each element b in the intuitionistic preference rela- ij tion B consists of the pair (μ , v ). Consider that each pair (μ , v ) must satisfy the ij ij ij ij conditionμ + v ≤ 1, i.e.,μ ≤ 1− v . This condition is exactly the condition under ij ij ij ij which two real numbers form an interval [28]. As a result, we can transform the pair b = (μ , v ) into the interval number b = [μ , 1− v ], and thus, the intuitionistic ij ij ij ij ij ij preference relation B = (b ) is equivalent to an interval fuzzy preference relation ij n×n − + ˙ ˙ ˙ ˙ ˙ [30,31] B = (b ) , where b = [b , b ] = [μ , 1− v ], for all i, j = 1, 2,··· , n, and ij n×n ij ij ij ij ij − + + − + − + − ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ b + b = b + b = 1, b  b  0, b = b = 0.5, for all i, j = 1, 2,··· , n. ij ji ij ji ij ij ii ii 82 Ze-shui Xu (2009) For convenience, we denote by the intuitionistic preference relation B = (b ) , ij n×n where b = [μ , 1− v ], for all i, j = 1, 2,··· , n,. Especially, if ij ij ij μ + v = 1, for all i, j = 1, 2,··· , n, ij ij then the intuitionistic preference relation B = (b ) is reduced to a fuzzy preference ij n×n relation [32-38] R = (r ) , where ij n×n 0 ≤ r ≤ 1, r + r = 1, r = 0.5, for all i, j = 1, 2,··· , n. ij ij ji ii Let w = (w , w ,··· , w ) be the vector of priority weights, where w reﬂects the 1 2 n i importance degree of the criterion x , and w  0, i = 1, 2,··· , n, w = 1, (4) i i i=1 then, a fuzzy preference relation R = (r ) is called a consistent fuzzy preference ij n×n relation, if the following additive transitivity [33] is satisﬁed r = r − r + 0.5, for all i, j, k = 1, 2,··· , n ij ik jk and such a fuzzy preference relation is given by [39,40]: r = 0.5(w − w + 1), for all i, j = 1, 2,··· , n. (5) ij i j By (5), in the following, we deﬁne the concept of consistent intuitionistic prefer- ence relation: Deﬁnition 2 Let B = (b ) be an intuitionistic preference relation, where b = ij n×n ij [μ , 1 − v ], for all i, j = 1, 2,··· , n, if there exists a vector w = (w , w ,··· , w ) , ij ij 1 2 n such that μ ≤ 0.5(w − w + 1) ≤ 1− v , (6) ij i j ij for all i = 1, 2,··· , n − 1; j = i + 1,··· , n, where w satisﬁes the condition (4), then we call B a consistent intuitionistic preference relation; otherwise, we call B an inconsistent intuitionistic preference relation. In the next section, we shall develop a method for estimating criteria weights from an intuitionistic preference relation. 4. A Method for Estimating Criteria Weights If B = (b ) is a consistent intuitionistic preference relation, then the priority vector ij n×n w = (w , w ,··· , w ) of B should satisfy (4) and (6). Thus, motivated by the idea 1 2 n [41], we utilize (4) and (6) to establish the following linear programming model: − + (M-1) w = Min w and w = Max w i i i i s.t. 0.5(w − w + 1)  μ , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij 0.5(w − w + 1) ≤ 1− v , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij w  0, i = 1, 2,··· , n, w = 1. i i i=1 Fuzzy Inf. Eng. (2009) 1:79-89 83 − + Solving the model (M-1), we can get the weight intervals [w , w ], i = 1, 2,··· , n. i i − + Especially, if w = w , for all i, then we get a unique priority vector w = (w , w ,··· , 1 2 i i w ) from the intuitionistic preference relation B. If B = (b ) is an inconsistent intuitionistic preference relation, then (6) does ij n×n not always hold. In this case, we relax (6) by introducing the deviation variables d ij and d , i = 1, 2,··· , n− 1; j = i+ 1,··· , n: ij − + μ − d ≤ 0.5(w − w + 1) ≤ 1− v + d , (7) ij i j ij ij ij − + for all i = 1, 2,··· , n− 1; j = i+ 1,··· , n, where d and d are both nonnegative real ij ij − + numbers. Obviously, the smaller the deviation variables d and d , the closer B to an ij ij inconsistent intuitionistic preference relation. As a result, we establish the following optimization model. n−1 n ∗ − + (M-2) J = Min (d + d ) ij ij i=1 j=i+1 s.t. 0.5(w − w + 1)+ d  μ , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij ij 0.5(w − w + 1)− d ≤ 1− v , i = 1, 2,··· , n− 1; j = i+ 1,··· , n, i j ij ij w  0, i = 1, 2,··· , n, w = 1, i i i=1 − + d , d  0, i = 1, 2,··· , n− 1; j = i+ 1,··· , n. ij ij − + ˙ ˙ Solving this model, we can get the optimal deviation values d , and d , i = 1, 2,··· , n− ij ij 1; j = i + 1,··· , n, and the optimal priority vector w = (w , w ,··· , w ) from the 1 2 n intuitionistic preference relation B. In what follows, we further extend the above results to group decision making based on intuitionistic preference relations: For a multi-criteria group decision making problem, let E = {e , e ,··· , e } be the 1 2 m set of decision makers. Each decision maker e compares each pair of the criteria (k) x and x , and then constructs an intuitionistic preference relation B = (b ) , i j k n×n ij (k) (k) (k) (k) (k) where b = (μ , v ), for all i, j = 1, 2,··· , n. Furthermore, μ and v satisfy the ij ij ij ij ij characteristics (6). Let w = (w , w ,··· , w ) be the weight vector of the criteria x (i = 1, 2,··· , n), 1 2 n i where w reﬂects the importance degree of the criterion x . If the weight vector w i i satisﬁes (4) and (k) (k) μ ≤ 0.5(w − w + 1) ≤ 1− v , (8) i j ij ij (k) for all i = 1, 2,··· , n− 1; j = i+ 1,··· , n, then B = b ) is a consistent intuition- k n×n ij istic preference relation. If (8) holds for all k = 1, 2,··· , m, i.e., all the intuitionistic (k) preference relations B = b ) are consistent, then we call the group is of consen- k n×n ij sus. In this case, similar to the model (M-1), we can utilize (4) and (8) to establish the following linear programming model: 84 Ze-shui Xu (2009) − + (M-3) w = Min w and w = Max w i i i i s.t. 0.5(w − w + 1)  μ , i = 1, 2,··· , n− 1; i j ij j = i+ 1,··· , n; k = 1, 2,··· , m, 0.5(w − w + 1) ≤ 1− v , i = 1, 2,··· , n− 1; i j ij j = i+ 1,··· , n; k = 1, 2,··· , m, w  0, i = 1, 2,··· , n, w = 1. i i i=1 − + Solving the model (M-3), we can get the weight intervals [w , w ], i = 1, 2,··· , n. i i − + T Especially, if w = w , for all i, then we get a unique weight vector w = (w , w ,··· , w ) 1 2 n i i for the criteria x (i = 1, 2,··· , n). If the group is not of consensus, then (8) does not always hold for all k = 1, 2,··· , m. −(k) +(k) In this case, we relax (8) by introducing the deviation variables d and d , ij ij i = 1, 2,··· , n− 1; j = i+ 1,··· , n; k = 1, 2,··· , m : (k) −(k) (k) +(k) μ − d ≤ 0.5(w − w + 1) ≤ 1− v + d , i j ij ij ij ij (9) for all i = 1, 2,··· , n− 1; j = i+ 1,··· , n; k = 1, 2,··· , m, −(k) +(k) where d and d are both nonnegative real numbers. ij ij Based on (9), we establish the following optimization model: m n−1 n −(k) +(k) (M-4) J = Min (d + d ) 2 ij ij k=1 i=1 j=i+1 s.t. 0.5(w − w + 1)+ d  μ , i = 1, 2,··· , n− 1; i j ij ij j = i+ 1,··· , n; k = 1, 2,··· , m, 0.5(w − w + 1)− d ≤ 1− v , i = 1, 2,··· , n− 1; i j ij ij j = i+ 1,··· , n; k = 1, 2,··· , m, w  0, i = 1, 2,··· , n, w = 1, i i i=1 −(k) +(k) d , d  0, i = 1, 2,··· , n− 1; j = i+ 1,··· , n; ij ij k = 1, 2,··· , m. − + ˙ ˙ Solving this model, we can get the optimal deviation values d and d , i = 1, 2,··· , n− ij ij 1; j = i+ 1,··· , n; k = 1, 2,··· , m, and the optimal priority vector w = (w , w ,··· , 1 2 w ) of the criteria x (i = 1, 2,··· , n). n i 5. Illustrative Examples Example 1 For a multi-criteria decision making problem, there are ﬁve criteria x (i = 1, 2,··· , 5). A decision maker compares each pair of criteria x and x , and i i j provides his/her intuitionistic fuzzy preference value a = (μ , v ), composed by the ij ij ij certainty degreeμ to which x is preferred to x and the certainty degree v to which ij i j ij x is non-preferred to x , and then constructs the following intuitionistic preference i j Fuzzy Inf. Eng. (2009) 1:79-89 85 relation: ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.6, 0.3) (0.4, 0.2) (0.7, 0.2) (0.4, 0.5) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (0.3, 0.6) (0.5, 0.5) (0.5, 0.3) (0.6, 0.1) (0.3, 0.6) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = (a ) = . ij 5×5 ⎢ (0.2, 0.4) (0.3, 0.5) (0.5, 0.5) (0.6, 0.2) (0.4, 0.5)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.7) (0.1, 0.6) (0.2, 0.6) (0.5, 0.5) (0.3, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.5, 0.4) (0.6, 0.3) (0.5, 0.4) (0.6, 0.3) (0.5, 0.5) We ﬁrst transform the intuitionistic preference relation A into its equivalent interval fuzzy preference relation B = (b ) (here, b = [μ , 1− v ], i, j = 1, 2,··· , 5): ij n×n ij ij ij ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.6, 0.7] [0.4, 0.8] [0.7, 0.8] [0.4, 0.5] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.3, 0.4] [0.5, 0.5] [0.5, 0.7] [0.6, 0.9] [0.3, 0.4]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B = (b ) = , [0.2, 0.6] [0.3, 0.5] [0.5, 0.5] [0.6, 0.8] [0.4, 0.5] ij 5×5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.2, 0.3] [0.1, 0.4] [0.2, 0.4] [0.5, 0.5] [0.3, 0.4]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.5, 0.6] [0.6, 0.7] [0.5, 0.6] [0.6, 0.7] [0.5, 0.5] then by solving the model (M-2), we get J = 0.075, the optimal deviation values: − − − + − + − + − + ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = d = 0, d = d = 0, d = 0.025, d = 0, d = d = 0, d = d = 0, 12 12 13 13 14 14 15 15 23 23 − + − + − + − + − + ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0.025, d = 0, d = d = 0, d = 0.025, d = 0, d = d = 0, d = d = 0 24 24 25 25 34 34 35 35 45 45 and the optimal priority vector w = (0.35, 0.15, 0.15, 0, 0.35) , i.e., the weights of the criteria x (i = 1, 2,··· , 5) are w = 0.35, w = 0.15, w = 0.15, w = 0 and i 1 2 3 4 w = 0.35, respectively. Example 2 Suppose that a decision maker provides his/her preference information over a collection of criteria x , x , x , x with the following intuitionistic preference 1 2 3 4 relation: ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.6, 0.2) (0.5, 0.4) (0.7, 0.1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.6) (0.5, 0.5) (0.4, 0.3) (0.6, 0) ⎥ ⎢ ⎥ ⎢ ⎥ A = (a ) = ⎢ ⎥ . ij 4×4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (0.4, 0.5) (0.3, 0.4) (0.5, 0.5) (0.7, 0.1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.1, 0.7) (0, 0.6) (0.1, 0.7) (0.5, 0.5) We ﬁrst transform the intuitionistic preference relation A into its equivalent interval fuzzy preference relation ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.6, 0.8] [0.5, 0.6] [0.7, 0.9] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.2, 0.4] [0.5, 0.5] [0.4, 0.7] [0.6, 1] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B = (b ) = , ⎢ ⎥ ij 4×4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.4, 0.5] [0.3, 0.6] [0.5, 0.5] [0.7, 0.9] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.1, 0.3] [0, 0.4] [0.1, 0.3] [0.5, 0.5] then by solving the model (M-2), we get J = 0, and all the optimal deviation values − + ˙ ˙ d and d (i = 1, 2, 3; j = i+ 1,··· , 4) are equal to zero. Thus, by Deﬁnition 2, A is ij ij 86 Ze-shui Xu (2009) a consistent intuitionistic preference relation, and then we solve the model (M-1) and get a unique priority vector w = (0.4, 0.2, 0.4, 0) , i.e., the weights of the criteria x (i = 1, 2,··· , 4) are w = 0.4, w = 0.2, w = 0.4 and w = 0, respectively. i 1 2 3 4 Example 3 For a multi-criteria group decision making problem, there are ﬁve crite- ria x (i = 1, 2,··· , 5), and three decision makers e (k = 1, 2, 3). The decision maker i k e (k = 1, 2, 3) compare each pair of criteria x and x , and construct the following k i j (k) intuitionistic preference relations A = (a ) : k n×n ij ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.5, 0.2) (0.6, 0.2) (0.7, 0.1) (0.3, 0.5) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.5) (0.5, 0.5) (0.3, 0.1) (0.5, 0.2) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ A = (a ) = ⎢ ⎥ , (0.2, 0.6) (0.1, 0.3) (0.5, 0.5) (0.7, 0.3) (0.5, 0.3) 1 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.1, 0.7) (0.2, 0.5) (0.3, 0.7) (0.5, 0.5) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.5, 0.3) (0.6, 0.4) (0.3, 0.5) (0.6, 0.4) (0.5, 0.5) ⎡ ⎤ ⎢ ⎥ ⎢ (0.5, 0.5) (0.6, 0.2) (0.7, 0.3) (0.7, 0.2) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.6) (0.5, 0.5) (0.4, 0.2) (0.5, 0.3) (0.3, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (2) ⎢ ⎥ ⎢ ⎥ A = (a ) = (0.3, 0.7) (0.2, 0.4) (0.5, 0.5) (0.6, 0.2) (0.5, 0.4) , 2 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.2, 0.7) (0.3, 0.5) (0.2, 0.6) (0.5, 0.5) (0.3, 0.5)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.4, 0.6) (0.6, 0.3) (0.4, 0.5) (0.5, 0.3) (0.5, 0.5) ⎡ ⎤ ⎢ ⎥ (0.5, 0.5) (0.4, 0.3) (0.5, 0.4) (0.6, 0.3) (0.3, 0.7) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.3, 0.4) (0.5, 0.5) (0.4, 0.3) (0.6, 0.2) (0.4, 0.6)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = (a ) = . (0.4, 0.5) (0.3, 0.4) (0.5, 0.5) (0.5, 0.3) (0.7, 0.2) 3 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (0.3, 0.6) (0.2, 0.6) (0.3, 0.5) (0.5, 0.5) (0.4, 0.5)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (0.7, 0.3) (0.6, 0.4) (0.2, 0.7) (0.5, 0.4) (0.5, 0.5) We ﬁrst transform the intuitionistic preference relations A (k = 1, 2, 3) into their (k) (k) equivalent interval fuzzy preference relations B = (b ) (k = 1, 2, 3) (here, b = k n×n ij ij (k) (k) [μ , 1− v ], i, j = 1, 2,··· , 5; k = 1, 2, 3): ij ij ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.5, 0.8] [0.6, 0.8] [0.7, 0.9] [0.3, 0.5] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.2, 0.5] [0.5, 0.5] [0.3, 0.9] [0.5, 0.8] [0.4, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) ⎢ ⎥ ⎢ ⎥ B = (b ) = [0.2, 0.4] [0.1, 0.7] [0.5, 0.5] [0.7, 0.7] [0.5, 0.7] , 1 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.1, 0.3] [0.2, 0.5] [0.3, 0.3] [0.5, 0.5] [0.4, 0.4]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.5, 0.7] [0.6, 0.6] [0.3, 0.5] [0.6, 0.6] [0.5, 0.5] ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.6, 0.8] [0.7, 0.7] [0.7, 0.8] [0.4, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.2, 0.4] [0.5, 0.5] [0.4, 0.8] [0.5, 0.7] [0.3, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (2) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B = (b ) = , 2 5×5 ⎢ [0.3, 0.3] [0.2, 0.6] (0.5, 0.5) [0.6, 0.8] [0.5, 0.6]⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.2, 0.3] [0.3, 0.5] [0.2, 0.4] [0.5, 0.5] [0.3, 0.5]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.4, 0.4] [0.6, 0.7] [0.4, 0.5] [0.5, 0.7] [0.5, 0.5] Fuzzy Inf. Eng. (2009) 1:79-89 87 ⎡ ⎤ ⎢ ⎥ [0.5, 0.5] [0.4, 0.7] [0.5, 0.6] [0.6, 0.7] [0.3, 0.3] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [0.3, 0.6] [0.5, 0.5] [0.4, 0.7] [0.6, 0.8] [0.4, 0.4] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3) ⎢ ⎥ ⎢ ⎥ B = (b ) = [0.4, 0.5] [0.3, 0.6] [0.5, 0.5] [0.5, 0.7] [0.7, 0.8] , 3 5×5 ⎢ ⎥ ij ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ [0.3, 0.4] [0.2, 0.4] [0.3, 0.5] [0.5, 0.5] [0.4, 0.5]⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ [0.7, 0.7] [0.6, 0.6] [0.2, 0.3] [0.5, 0.6] [0.5, 0.5] then by solving the model (M-4), we get J = 1.067, the optimal deviation values: −(1) +(1) −(1) +(1) −(1) +(1) −(1) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.033, d = 0, d = 0, d = 0, d = 0, 12 12 13 13 14 14 15 +(1) −(1) +(1) −(1) +(1) −(1) +(1) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0.067, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, 15 23 23 24 24 25 25 −(1) +(1) −(1) +(1) −(1) +(1) ˙ ˙ ˙ ˙ ˙ ˙ d = 0.067, d = 0, d = 0, d = 0, d = 0.033, d = 0, 34 34 35 35 45 45 −(2) +(2) −(2) +(2) −(2) +(2) −(2) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.133, d = 0, d = 0, d = 0, d = 0, 12 12 13 13 14 14 15 +(2) −(2) +(2) −(2) +(2) −(2) +(2) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0.167, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, 15 23 23 24 24 25 25 −(2) +(2) −(2) +(2) −(2) +(2) −(3) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, 34 34 35 35 45 45 12 +(3) −(3) +(3) −(3) +(3) −(3) +(3) ˙ ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0, d = 0, d = 0, d = 0, d = 0.267, 12 13 13 14 14 15 15 −(3) +(3) −(3) +(3) −(3) +(3) ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.067, d = 0, d = 0, d = 0, 23 23 24 24 25 25 −(3) +(3) −(3) +(3) −(3) +(3) ˙ ˙ ˙ ˙ ˙ ˙ d = 0, d = 0, d = 0.2, d = 0, d = 0.033, d = 0 34 34 35 35 45 45 and the optimal priority vector: w = (0.4, 0.06, 0.27, 0, 0.27) , i.e., the weights of the criteria x (i = 1, 2,··· , 5) are w = 0.4, w = 0.06, w = i 1 2 3 0.27, w = 0.27 and w = 0.35, respectively. 4 5 6. 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### Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Mar 1, 2009

Keywords: Intuitionistic preference relation; Consistent intuitionistic preference relation; Weak transitivity; Priority vector; Linear programming model