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F. Chiclana, F. Herrera, E. Herrera-Viedma (1998)
Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relationsFuzzy Sets Syst., 97
Zeshui Xu, Qingli Da (2005)
A least deviation method to obtain a priority vector of a fuzzy preference relationEur. J. Oper. Res., 164
H. Bustince, P. Burillo (1995)
Correlation of interval-valued intuitionistic fuzzy setsFuzzy Sets Syst., 74
Zeshui Xu (2007)
Intuitionistic preference relations and their application in group decision makingInf. Sci., 177
H. Mitchell (2004)
A correlation coefficient for intuitionistic fuzzy setsInternational Journal of Intelligent Systems, 19
G. Rand (1989)
Non-conventional Preference Relations in Decision MakingJournal of the Operational Research Society
(2004)
On compatibility of interval fuzzy preference matrices, 3
Zeshui Xu (2004)
On Compatibility of Interval Fuzzy Preference RelationsFuzzy Optimization and Decision Making, 3
P. Grzegorzewski (2004)
Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metricFuzzy Sets Syst., 148
Dengfeng Li, Chuntian Cheng (2002)
New similarity measures of intuitionistic fuzzy sets and application to pattern recognitionsPattern Recognition Letters, 23
Dengfeng Li, Chun-tian Cheng (2002)
New similarity measures of intuitionistic fuzzy sets and application to pattern recognitionsPattern Recognit. Lett., 23
T. Tanino (1984)
Fuzzy preference orderings in group decision makingFuzzy Sets and Systems, 12
(2004)
Uninorms in L*-fuzzy set theory
(1989)
Interval-valued intuitionistic fuzzy sets, 31
W. Dudek, B. Davvaz, Y. Jun (2005)
On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroupsInf. Sci., 170
Shu-Jen Chen, C. Hwang (1992)
Fuzzy Multiple Attribute Decision Making - Methods and Applications, 375
Glad Deschrijver, E. Kerre (2005)
Implicators based on binary aggregation operators in interval-valued fuzzy set theoryFuzzy Sets Syst., 153
Yingming Wang, Jianbo Yang, Dongling Xu (2005)
A two-stage logarithmic goal programming method for generating weights from interval comparison matricesFuzzy Sets Syst., 152
K. Atanassov (1986)
Intuitionistic fuzzy setsFuzzy Sets and Systems, 20
L. Zadeh (1996)
Fuzzy sets
S. De, R. Biswas, A. Roy (2001)
An application of intuitionistic fuzzy sets in medical diagnosisFuzzy Sets Syst., 117
E. Szmidt, J. Kacprzyk (2000)
Distances between intuitionistic fuzzy setsFuzzy Sets Syst., 114
W. Hung, Jong-Wuu Wu (2002)
Correlation of intuitionistic fuzzy sets by centroid methodInf. Sci., 144
Jin-Han Park, Jin Park (2004)
Hausdorffness on generalized intuitionistic fuzzy filtersInf. Sci., 168
F. Chiclana, F. Herrera, E. Herrera-Viedma (2001)
Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relationsFuzzy Sets Syst., 122
E. Szmidt, J. Kacprzyk (2002)
Using intuitionistic fuzzy sets in group decision makingControl and Cybernetics, 31
Weiqiong Wang, X. Xin (2005)
Distance measure between intuitionistic fuzzy setsPattern Recognit. Lett., 26
Zeshui Xu, Qingli Da (2002)
The uncertain OWA operatorInternational Journal of Intelligent Systems, 17
Zeshui Xu, R. Yager (2006)
Some geometric aggregation operators based on intuitionistic fuzzy setsInternational Journal of General Systems, 35
F. Herrera, Luis Martínez-López, P. Sánchez (2005)
Managing non-homogeneous information in group decision makingEur. J. Oper. Res., 166
Tadeusz Gerstaenkorn, J. Manko (1991)
Correlation of intuitionistic fuzzy setsFuzzy Sets and Systems, 44
박진한, 권영철, 박종서 (2004)
Intuitionistic Fuzzy Metric Spaces, 14
Javier García, S. Rodabaugh (2005)
Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued "intuitionistic" sets, "intuitionistic" fuzzy sets and topologiesFuzzy Sets Syst., 156
D. Hong, S. Hwang (1995)
Correlation of intuitionistic fuzzy sets in probability spacesFuzzy Sets Syst., 75
Glad Deschrijver, C. Cornelis, E. Kerre (2004)
On the representation of intuitionistic fuzzy t-norms and t-conormsIEEE Transactions on Fuzzy Systems, 12
Zhizhen Liang, P. Shi (2003)
Similarity measures on intuitionistic fuzzy setsPattern Recognit. Lett., 24
Glad Deschrijver, E. Kerre (2003)
On the relationship between some extensions of fuzzy set theoryFuzzy Sets Syst., 133
F. Chiclana, F. Herrera, E. Herrera-Viedma (2002)
A note on the internal consistency of various preference representationsFuzzy Sets Syst., 131
Zeshui Xu (2006)
A C‐OWA operator‐based approach to decision making with interval fuzzy preference relationInternational Journal of Intelligent Systems, 21
S. De, R. Biswas, A. Roy (2000)
Some operations on intuitionistic fuzzy setsFuzzy Sets Syst., 114
S. Orlovsky (1978)
Decision-making with a fuzzy preference relationFuzzy Sets and Systems, 1
(2002)
Generalized intuitionistic fuzzy sets, 10
C. Cornelis, Glad Deschrijver, E. Kerre (2004)
Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, applicationInt. J. Approx. Reason., 35
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. De SK, Biswas R, Roy AR (2000) Some operations on intuitionistic fuzzy sets. Fuzzy Sets and Systems 114: 477-484 4. De SK, Biswas R, Roy AR (2001) An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets and Systems 117: 209-213 5. Deschrijver G, Kerre EE, On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems, 2003, 133: 227-235 6. Bustince H, Burillo P (1995) Correlation of interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems 74: 237-244 7. Gerstenkorn T, Maʸko J (1991) Correlation of intuitionistic fuzzy sets. Fuzzy Sets and Systems 44: 39-43 8. Hong, D.H., Hwang SY (1995) Correlation of intuitionistic fuzzy sets in probability spaces. Fuzzy Sets and Systems 75: 77-81 9. Hung, W.L., Wu, J.W. (2002) Correlation of intuitionistic fuzzy sets by centroid method. Information Sciences 144: 219-225 10. Mitchell HB (2004) A correlation coeﬃcient for intuitionistic fuzzy sets. International Journal of Intelligent Systems 19: 483-490 11. Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems 114: 505-518 12. Li, D.F., Cheng, C.T. (2002) New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognition Letters 23: 221-225 13. Liang, Z.Z., Shi, P.F. (2003) Similarity measures on intuitionistic fuzzy sets. Pattern Recognition Letters 24: 2687-2693 14. Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorﬀ metric. Fuzzy Sets and Systems 148: 319-328 15. Wang, W.Q., Xin, X.L. (2005) Distance measure between intuitionistic fuzzy sets. Pattern Recogni- tion Letters 26: 2063-2069 16. Deschrijver G, Cornelis C, Kerre EE (2004) On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems 12: 45-61 17. Deschrijver G, Kerre EE (2005) Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets and Systems 153: 229-248 18. Park JH (2004) Intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 22: 1039-1046 19. Park JH, Park JK (2004) Hausdorﬀness on generalized intuitionistic fuzzy ﬁlters. Information Sci- ences 168: 95-110 20. Mondal TK, Samanta SK (2002) Generalized intuitionistic fuzzy sets. Journal of Fuzzy Mathematics 10: 839-861 21. Deschrijver G, Kerre EE (2004) Uninorms in L*-fuzzy set theory. Fuzzy Sets and Systems 148: 243-262 22. Cornelis C, Deschrijver G, Kerre EE (2004) Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classiﬁcation, application. International Journal of Approximate Rea- soning 35: 55-95 23. 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. European Journal of Operational Research 166: 115-132 29. Zadeh LA (1965) Fuzzy Sets. Information and Control 8: 338-353 30. Xu, Z.S. (2004) On compatibility of interval fuzzy preference matrices. Fuzzy Optimization and Decision Making 3: 217-225 31. Xu, Z.S. (2006) A C-OWA operator based approach to decision making with interval fuzzy preference relation. International Journal of Intelligent Systems 21: 1289-1298 32. Orlovsky SA (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets and Systems 1: 155-167 33. Tanino T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems 12: 117-131 34. Kacprzyk J, Roubens M (1988) Non-conventional preference relations in decision-making. Springer, Berlin. 35. Chiclana F, Herrera F, Herrera-Viedma E (1998) Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets and Systems 97: 33- 36. Chiclana F, Herrera F, Herrera-Viedma E (2001) Integrating multiplicative preference relations in a multipurpose decision-making based on fuzzy preference relations. Fuzzy Sets and Systems 122: 277-291 37. Xu, Z.S., Da, Q.L. (2002) The uncertain OWA operator. International Journal of Intelligent Systems 17: 569-575 38. Xu, Z.S., Da, Q.L. (2005) A least deviation method to obtain a priority vector of a fuzzy preference relation. European Journal of Operational Research 164: 206-216 39. Chiclana F, Herrera F, Herrera-Viedma E (2002) A note on the internal consistency of various prefer- ence representations. Fuzzy Sets and Systems 131: 75-78 40. Xu, Z.S. (2004) Uncertain multiple attribute decision making: methods and applications. Tsinghua University Press, Beijing. 41. Wang, Y.M., Yang, J.B., Xu, D.L. (2005) A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets and Systems 152: 475-498
Fuzzy Information and Engineering – Taylor & Francis
Published: Mar 1, 2009
Keywords: Intuitionistic preference relation; Consistent intuitionistic preference relation; Weak transitivity; Priority vector; Linear programming model
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