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Int. J. Pure Appl. Sci. Technol., 6(1) (2011), pp. 54-61 International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in Research Paper Weighted Average Rating (WAR) Method for Solving Group Decision Making Problem Using an Intuitionistic Trapezoidal Fuzzy Hybrid Aggregation (ITFHA) Operator A. Nagoor Gani 1,*, N. Sritharan 2 and C. Arun Kumar 3 1,2,3 PG & Research Department of Mathematics, Jamal Mohamed College, Trichy, Tamilnadu, India * Corresponding author, e-mail: ([email protected]) (Received: 22-7-11; Accepted: 30-8-11) Abstract: Intuitionistic Fuzzy numbers each of which is characterized by the degree of membership and the degree of non-membership of an element are a very useful means to depict the decision information in the process of decision making. The aim of this article is to investigate the approach to multiple attribute group decision making with intuitionistic trapezoidal fuzzy numbers, some operational laws of intuitionistic trapezoidal fuzzy numbers are applied. We investigate the group decision making problems in which all the information provided by the decision makers is expressed as decision matrices where each of the elements are characterized by intuitionistic trapezoidal fuzzy numbers and the information about attribute weights are known. We first use the intuitionistic trapezoidal fuzzy hybrid aggregation (ITFHA) operator to aggregate all individual fuzzy decision matrices provided by the decision makers into the collective intuitionistic fuzzy decision matrix. Furthermore, we utilize weighted average rating method and score function to give an approach to ranking the given alternatives and selecting the most desirable one. Finally we give an illustrative example. Keywords: Intuitionistic Fuzzy Set, Multiple Attribute Group Decision Making(MAGDM), Intuitionistic Trapezoidal Fuzzy Hybrid Aggregation Operator, Weighted Average Rating, Score Function. Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61. 55 1. Introduction: Atanassov[1] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set[2] whose basic component is only a membership function. Li[5] investigated MADM with intuitionistic fuzzy information and Lin[6] presented a new method for handling multiple attribute fuzzy decision making problems, where the characteristics of the alternatives are represented by intuitionistic fuzzy sets. Furthermore, the proposed method allows the decision maker to assign the degree of membership and the degree of non-membership of the attribute to the fuzzy concept ‘importance’. Wang[8] gave the definition of intuitionistic trapezoidal fuzzy number and interval valued intuitionistic trapezoidal fuzzy number. Wang and Zhang[9] gave the definition of expected values of intuitionistic trapezoidal fuzzy number and proposed the programming method of multi-criteria decision making based on intuitionistic trapezoidal fuzzy number incomplete certain information. In this paper, we focus our attention on the issue of multi attribute decision making under intuitionistic fuzzy environment where all the information provided by the decision makers is characterized by intuitionistic trapezoidal fuzzy numbers, and the information about the attribute weights are known. We first use the intuitionistic trapezoidal fuzzy hybrid aggregation (ITFHA) [4] operator to aggregate all individual fuzzy decision matrices provided by the decision makers into the collective intuitionistic fuzzy decision matrix. Next we calculate the weighted average rating [7] by using the aggregated matrix and the given criteria weights. Finally we find the best alternative by using the score function. This Paper is organized as follows: In section 2, we give a review of basic concepts and operator related with intuitionistic trapezoidal fuzzy numbers. Section 3, presents an algorithm for weighted average rating method for solving Group decision making problem using an Intuitionistic trapezoidal fuzzy Hybrid aggregation operator. Section 4, provides a practical example to illustrate the developed approach and finally, we conclude the paper in Section 5. 2. Basic Concepts: 2.1. Definition: Let a set X={x1, x2, x3, …, xn} be fixed, an IFS A in X is an object of the following form, Where the functions µ A : X → [0,1], x ∈ X , µ A ( x) ∈ [0,1] and υ A : X → [0,1], x ∈ X ,υ A ∈ [0,1] with the condition , the numbers denote the to the set degree of membership and the degree of non-membership of the element A respectively. 2.2. Definition: Let its membership function as be an intuitionistic trapezoidal fuzzy number, Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61. 56 Its non-membership function is Where and . 2.3. Definition: Let two intuitionistic trapezoidal fuzzy number and λ ≥ 0, then 1. 2. 3. 4. be 2.4. Definition: An intuitionistic trapezoidal fuzzy hybrid aggregation (ITFHA) operator of dimension n is a mapping , that has an associated vector such that wj> 0 and Furthermore, Where is the jth largest of the weighted intuitionistic trapezoidal fuzzy numbers be the weight vector of , and n is the balancing coefficient. Where is a permutation of (1,2,…,n), such that for all j=2,…,n. 2.5. Definition: We defined a method to compare two intuitionistic trapezoidal fuzzy numbers which is based on the score function and the accuracy function. Let and be two intuitionistic fuzzy values, and be the scores of and , respectively, and let, and be the accuracy degrees of and , respectively, then (i) if S( ) < S( ), then is smaller than , denoted by < ; (ii) if S( ) = S( ), then, (a) if H( ) = H( ), then and are the same, denoted by = ; (b) if H( ) < H( ), then is smaller than , denoted by < . 3. Weighted Average Rating (WAR) Algorithm: Step 1: Form a intuitionistic fuzzy decision matrix Step 2: Utilize the ITFHA operator of k decision makers. Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61. 57 to derive the of collective overall preference intuitionistic trapezoidal fuzzy values the alternative Ai, where V=(v1, v2, …, vn) be the weighting vector of decision makers with vk in [0,1], is the associated weighting vector of the ITFHA operator, with Step 3: Calculate the weighted aggregated decision matrix , using the multiplication formula, where is the intuitionistic trapezoidal fuzzy number. Step 4: calculate the weighted average rating by using the aggregated matrix and the given criteria weights with help of the formula Step 5: To find the best alternative by using the score function. 5. Numerical Example: Let us suppose there is a risk investment company which wants to invest a sum of money in the best option. There is a panel with four possible alternatives to invest the money. The risk investment company must take a decision according to the following four attributes. 1. G1 is the risk analysis, 2. G2 is the growth analysis, 3. G3 is the social political impact analysis, 4. G4 is the Environmental impact analysis. The four possible alternatives Ai (i=1,2,3,4) are to be evaluated using the intuitionistic trapezoidal fuzzy numbers by the three decision makers with their weighting vector v=(0.35,0.40,0.25)T under the above four attributes weighting vector w=(0.2,0.1,0.3,0.4)T and construct the decision matrices as follows, Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61. 58 By step 2 using the ITFHA Operator to aggregate all the three decision matrices into single collective decision matrix with Intuitionistic trapezoidal fuzzy ratings. Consider, Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61. The Aggregated Matrix is, 59 Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61. 60 Next we calculate the weighted aggregated decision matrix by using step 3, we get Calculate the Weighted Average Rating for each alternative by using step 4, we get D(A1) = ([0.4233,0.5270,0.6623,0.7654];0.2810,0.5300) D(A2) = ([1.2734,1.3860,1.5248,1.6730];0.5980,0.2880) D(A3) = ([0.9573,1.1650,1.3870,1.5420];0.5060,0.2990) D(A4) = ([1.0770,1.2310,1.3850,1.5370];0.3841,0.3520) To find the ranking order of the alternatives, use the score function, S(A1) = 0.2810 - 0.5300 = -0.249 S(A2) = 0.5980 – 0.2880 = 0.3100 S(A3) = 0.5060 – 0.2990 = 0.2070 S(A4) = 0.3841 – 0.3520 = 0.0321 S(A2) > S(A3) > S(A4) > S(A1) i.e. A2 > A3 > A4 > A1 The Best option is A2. 4. Conclusions: We have investigated the multiattribute decision making problems under intuitionistic fuzzy environment and developed an approach to handling the situations where the attribute values are characterized by intuitionistic trapezoidal fuzzy numbers and the information about attribute weights are known. The approach first fuses all individual intuitionistic fuzzy decision matrices into the collective intuitionistic fuzzy decision matrix by using the intuitionistic trapezoidal fuzzy hybrid aggregation operator, then based on the collective intuitionistic fuzzy decision matrix, we can utilize the weighted average rating method and the score function to get the best alternative. References [1] [2] [3] [4] [5] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. K. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33(1989), 37-46. S. 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