Induced uncertain linguistic OWA operators applied to group decision making
Introduction
The induced aggregation operators are an interesting research topic, which is receiving increasing attention [2], [3], [4], [5], [6]. Yager and Filev [2] introduce a class of induced ordered weighted averaging (IOWA) operators which take as their argument pairs, called OWA pairs, in which one component is used to induce an ordering over the second components which are exact numerical values and then aggregated. Xu and Da [3] introduce a class of induced ordered weighted geometric averaging (IOWGA) operators, which have many desirable properties similar to those of the IOWA operators. Yager [4] develops an induced fuzzy integral aggregation operator, which extends the fuzzy integral aggregation operator by allowing the ordering operation to be based upon values other than those being aggregated. Chen and Chen [5] present some FN-IOWA operators based on fuzzy numbers. Yager [6] suggests a number of applications of the IOWA aggregation operators, and extends the idea of order induced aggregation to the Choquet aggregation resulting in what he calls the induced Choquet ordered averaging (I-COA) operator. All of these operators have been used in situations in which the second components are numerical values. In some situations, however, the input arguments take the form of uncertain linguistic variables rather than numerical ones because of time pressure, lack of knowledge, and the decision maker’s limited attention and information processing capabilities. Therefore, it is necessary to pay attention to this issue. In this paper, we shall develop some induced uncertain linguistic OWA (IULOWA) operators, in which the second components are uncertain linguistic variables, and study some of their desirable properties. Moreover, we shall apply the IULOWA operators to group decision making with uncertain linguistic information.
Section snippets
Preliminaries
The linguistic approach is an approximate technique, which represents qualitative aspects as linguistic values by means of linguistic variables [7], [8], [9], [10], [11], [12], [13], [14]. Suppose that S = {si∣i = −t,…,t} is a finite and totally ordered discrete term set, where si represents a possible value for a linguistic variable. For example, a set of nine terms S could bein
Induced uncertain linguistic OWA operators
In [1], Yager provides a definition of the ordered weighted averaging (OWA) operator as follows: Definition 2 An OWA operator of dimension n is a mapping, OWA:Rn → R, that has an associated weighting vector w = (w1,w2,…,wn)T with the properties wj ∈ [0,1] and , such thatwhere bj is the jth largest of the ai.[1]
An important feature of the OWA operator is the reordering step, which makes this a nonlinear operator. During this step the arguments are ordered by their values. The OWA
An approach based on the IULOWA and the ULWA operators to group decision making with uncertain linguistic information
For a group decision making with uncertain linguistic information, let X = {x1, x2, …, xn} be a discrete set of alternatives, and G = {G1,G2,…,Gm} be the set of attributes. Let ω = (ω1,ω2, …, ωm)T be the weight vector of attributes, where . Let D = {d1,d2,…,dl} be the set of decision makers, and v = (v1,v2, …, vl)T be the weight vector of decision makers. Suppose that is the uncertain linguistic decision matrix, where is a preference value, which takes the
Illustrative example
Let us suppose an investment company, which wants to invest a sum of money in the best option (adapted from [11]). There is a panel with five possible alternatives in which to invest the money:
- (1)
x1 is a car industry;
- (2)
x2 is a food company;
- (3)
x3 is a computer company;
- (4)
x4 is an arms company;
- (5)
x5 is a TV company.
The investment company must make a decision according to the following four attributes (suppose that the weight vector of four attributes is ω = (0.3,0.4,0.2,0.1)T):
- (1)
G1 is the risk analysis;
- (2)
G2 is the
Concluding remarks
The traditional induced aggregation operators are generally suitable for aggregating the information taking the form of numerical values, and yet they will fail in dealing with uncertain linguistic variables. In this paper, we have developed some induced uncertain linguistic OWA (IULOWA) operators, which take as their argument pairs, called ULOWA pairs, in which one component is used to induce an ordering over the second components which are uncertain linguistic values and then aggregated. We
Acknowledgements
The author is very grateful to the editor and the three anonymous referees for their valuable comments and suggestions. The work was supported by China Postdoctoral Science Foundation under Project (2003034366).
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