2010 International Conference on Artificial Intelligence and Computational Intelligence
A Mean Deviation Based Method for Intuitionistic Fuzzy Multiple Attribute Decision Making Yejun Xu Business School HoHai University Nanjing, Jiangsu 210098, P R China
[email protected] geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric(IFOWG) operator, the intuitionistic fuzzy hybrid geometric (IFHG) operator to multiple attribute group decision making with intuitionistic fuzzy information. Xu[14] developed the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid averaging (IFHA) operator. However, when using these operators, the associated weighting vector is more or less determined subjectively and the decision making information itself is not taken into consideration sufficiently. All of the above methods will be unsuitable for dealing with such situations. Therefore, it is necessary to develop a method for determining the weights objectively of the multiple attribute decision making problems under intuitionistic fuzzy environment. In this paper, we focus our attention on developing a method objectively named mean deviation method to determine the attribute weights under the condition that the attribute weights are completely unknown, and the attribute values are taking the form of intuitionistic fuzzy numbers, to overcome the above limitations. To do so, the rest of the paper is organized as follows. In Section 2, we introduce some basic concepts of intuitionistic fuzzy sets. In Section 3, we establish an optimization model based on the mean deviation method. By solving this model, a simple and exact formula is derived to determine the attribute weights. We utilize the intuitionistic fuzzy weighted averaging (IFWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. In Section 4, a practical example is used to illustrate the developed models. In Section 5, we conclude the paper and give some remarks.
Abstract—The aim of this paper is to develop a method to determine the weights of attributes objectively under intuitionistic fuzzy environment. Based on the mean deviation, we establish an optimization model in which the information about attribute weights is completely unknown. By solving the model, we get a simple and exact formula which can be used to determine the attribute weights. After that, we utilize the intuitionistic fuzzy weighted average (IFWA) operator to aggregate the given intuitionistic fuzzy information corresponding to each alternative, and then select the most desirable alternative according to the score function and accuracy function. Finally, a practical example is given to verify the developed method and to demonstrate its practicality and effectiveness. Keywords-Intuitionistic fuzzy set; multiple attribute decision making; mean deviation;
I.
INTRODUCTION
Intuitionistic fuzzy sets(IFS) introduced by Atanassov[1, 2] have been found to be well suited to dealing with vagueness. IFS characterized by a membership function and a non-membership function, is an extension of Zadeh’s fuzzy set[3] whose basic component is only a membership function. Since its appearance, the IFS have received more and more attention and applied it to the field of decision making. Gau and Buehrer[4] presented the concept of vague sets. Burillo and Bustince[5] showed that the notion of vague sets coincides with that of intuitionistic fuzzy sets. Based on vague sets, Chen and Tan[6], and Hong and Choi [7] utilized the minimum and maximum operations to develop some approximate technique for handling multiattribute decision making problems under fuzzy environment. Szmidt and Kacprzyk [8] proposed some solution concepts such as the intuitionistic fuzzy core and consensus winner in group decision making with intuitionistic (individual and social) fuzzy preference relations, and proposed a method to aggregate the individual intuitionistic fuzzy preference relations into a social fuzzy preference relation on the basis of fuzzy majority equated with a fuzzy linguistic quantifier. Li and Cheng[9], Liang and Shi[10], Huang and Yang[11], and Wang and Xin[12] introduced some similarity measures of intuitionistic fuzzy sets and applied them to pattern recognition. Xu and Yager[13] developed some aggregation operators, such as the intuitionistic fuzzy weighted 978-0-7695-4225-6/10 $26.00 © 2010 IEEE DOI 10.1109/AICI.2010.244
II.
PRELIMINARIES
In the following, we introduce some basic concepts related to intuitionistic fuzzy sets. In[1], Atanassov introduced a generalized fuzzy set called intuitionistic fuzzy set, shown as follows. Definition 1. An IFS in X is given by A = {< x, μ A ( x), v A ( x) >| x ∈ X } (1)
which is characterized by a membership function μ A : X → [0,1] and a non-membership function v A : X → [0,1] , with the condition 12
0 ≤ μ A ( x) + v A ( x) ≤ 1 , ∀x ∈ X where the numbers μ A ( x) and v A ( x) represent, respectively, the degree of membership and the degree of non-membership of the element x to the set A . Definition 2. For each IFS A in X , if π A ( x) = 1 − μ A ( x) − v A ( x) , ∀x ∈ X (2) is called the indeterminacy degree or hesitation degree of x to A . Especially, if π A ( x ) = 1 − μ A ( x) − v A ( x) = 0 , ∀x ∈ X (3) Then, the intuitionistic fuzzy set A is reduced to a common fuzzy set[3]. For convenience, we call α = ( μα , vα ) an intuitionistic fuzzy number(IFN)([15]), where μα ∈ [0,1] , vα ∈ [0,1] , and
Definition 6[16]. Let α = ( μα , vα ) and β = ( μ β , vβ ) be two
intuitionistic fuzzy numbers, then (1) α + β = ( μα + μ β − μα ⋅ μ β , vα ⋅ vβ ) ; (2) α ⋅ β = ( μα ⋅ μ β , vα + vβ − vα ⋅ vβ ) ; (3) λα = (1 − (1 − μα )λ , vαλ ) , λ > 0 ; (4) α λ = ( μαλ ,1 − (1 − vα )λ ) , λ > 0 . Definition 7[15]. Let α = ( μα , vα ) , β = ( μ β , vβ ) be two
intuitionistic fuzzy numbers, then we call 1 d (α , β ) = α − β = (| μα − μ β | + | vα − vβ |) 2 the deviation between α and β . III.
μα + vα ≤ 1 .
(6)
MEAN DEVIATION METHOD
The multiple-attribute decision-making problems under study can be described in detail as follows. Let X = {x1 , x2 ,..., xm } ( m ≥ 2 ) be a discrete set of m
Definition 3[13]. Let α = ( μα , vα ) be an intuitionistic fuzzy number, a score function S of an intuitionistic fuzzy number can be represented as follows: S (α ) = μα − vα (4)
feasible alternatives, U = {u1 , u2 ,..., un } be a finite set of attributes. For each alternative xi ∈ X , the decision maker
where S (α ) ∈ [−1,1] . For an IFN α = ( μα , vα ) , it is clear that if the deviation between μα and vα gets greater, which means the value μα
gives his/her preference value rij with respect to attribute u j ∈ U , where rij takes the form of intuitionistic fuzzy numbers, that is rij = ( μ ij , vij ) , μij ∈ [0,1] , vij ∈ [0,1] , and
gets bigger and the value vα gets smaller, then the IFN α gets greater. Definition 4[13]. Let α = ( μα , vα ) be an intuitionistic fuzzy number, an accuracy function H to evaluate the degree of accuracy of the intuitionistic fuzzy number can be represented as follows: H (α ) = μα + vα (5) where H (α ) ∈ [0,1] . The larger the value of H (α ) , the higher the degree of accuracy of the degree of membership of the IFN α . Xu[13] introduced an order relation between two intuitionistic fuzzy numbers in the following. Definition 5. Let α = ( μα , vα ) and β = ( μ β , vβ ) be two
μij + vij ≤ 1 , i = 1, 2,..., m , j = 1, 2, ..., n , then all the preference values of the alternatives consists the decision matrix R = (rij ) m× n . Definition 8[14]. Let R = (rij ) m× n be the intuitionistic fuzzy
decision matrix, ri = (ri1 , ri 2 ,..., rin ) be the vector of attribute values corresponding to the alternative xi , i = 1, 2,..., m , then we call zi ( w) = IF WA w (ri1 , ri 2 ,..., rin ) = w1ri1 + w2 ri 2 + ... + wn rin n n ⎛ w w = ⎜ 1 − ∏ (1 − μij ) j , ∏ (vij ) j j =1 j =1 ⎝ the overall value of the alternative
intuitionistic fuzzy numbers, S (α ) = μα − vα and S ( β ) = μ β
⎞ (7) ⎟ ⎠ xi , where w = ( w1 , w2 ,
..., wn )T is the weighting vector of attributes. In the situation where the information about attribute weights is completely known, i.e., each attribute weight can be provided by the expert with crisp numerical value, we can aggregate all the weighted attribute values corresponding to each alternative into an overall one by using (7). Based on the overall attribute values zi ( w) of the
−vβ be the scores of α and β , respectively, and let H (α ) = μα + vα and H ( β ) = μ β + vβ be the accuracy degrees of
α and β , then • If S (α ) < S ( β ) , then α is smaller than β , denoted by α
