Ranking Fuzzy Numbers with an Area Method Using Circumcenter of Centroids

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1 Fuzzy Inf. Eng. (2013 1: 3-18 DOI /s ORIGINAL ARTICLE Ranking Fuzzy Numbers with an Area Method Using Circumcenter of Centroids P. Phani Bushan Rao N. Ravi Shankar Received: 7 July 2011/ Revised: 25 November 2012/ Accepted: 21 December 2012/ Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2013 Abstract This paper proposes a new method for ranking fuzzy numbers based on the area between circumcenter of centroids of a fuzzy number and the origin. The proposed method not only uses an index of optimism, which reflects the decision maker s optimistic attitude but also makes use of an index of modality which represents the importance of mode and spreads. This method ranks various types of fuzzy numbers which includes normal, generalized trapezoidal and triangular fuzzy numbers along with crisp numbers which are a special case of fuzzy numbers. Some numerical examples are presented to illustrate the validity and advantages of the proposed method. Keywords Fuzzy numbers Centroid points Circumcenter Ranking function Index of optimism Index of modality 1. Introduction Ranking of fuzzy numbers plays a vital role in real time applications, especially in decision making. Hence there is a need for an apt procedure to rank fuzzy numbers in all conditions. Ranking of fuzzy numbers relies on bringing out several characteristics from fuzzy numbers like a centroid, an area bounded by the membership function or several points of intersection between fuzzy numbers. Fuzzy ranking methods bring out a specific characteristic from each fuzzy number and orders them basing on that characteristic feature. Hence, ranking of fuzzy numbers is made more complex because different ranking methods give different ranking orders for the same set of fuzzy numbers. P. Phani Bushan Rao ( Department of Mathematics, GITAM Institute of Technology, GITAM University, Visakhapatnam, Andhra Pradesh , India peddipb@yahoo.com N. Ravi Shankar GITAM Institute of Science, GITAM University, Visakhapatnam, Andhra Pradesh , India

2 4 P. Phani Bushan Rao N. Ravi Shankar (2013 Several researchers came up with different techniques for ranking fuzzy numbers, but still a method to rank all types of fuzzy numbers under different existing conditions is a quest. In fuzzy decision analysis, fuzzy quantities are used to describe the performance of alternatives in modelling a real world problem. Fuzzy numbers being represented by possibility distributions, there are chances of them getting overlapped with each other, hence discriminating them becomes a more complex task than discriminating real numbers where a natural order exists between them. In problems like fuzzy risk analysis, fuzzy optimization, and so on where fuzzy set theory is used, at one or the other stage, fuzzy numbers must be ranked before an action taken by a decision maker. In 1965, Zadeh [1] introduced the theory of fuzzy sets. Ranking fuzzy numbers was first proposed by Jain in the year 1977 for decision making in fuzzy situations by representing the ill-defined quantity as a fuzzy set. Jain [2, 3] proposed a method using the concept of maximizing set to order the fuzzy numbers and the decision maker considered only the right side membership functions. Since then various procedures to rank fuzzy quantities are proposed by various researchers. Chen [7] presented ranking fuzzy numbers with maximizing set and minimizing set. Kim and Park [8] presented a method of ranking fuzzy numbers with index of optimism. Liou and Wang [9] presented ranking fuzzy numbers with integral value. Choobineh and Li [10] presented an index for ordering fuzzy numbers. Since then several methods have been proposed by various researchers including the ranking of fuzzy numbers using area compensation, distance method, decomposition principle and signed distance [11-13]. Wang and Kerre [14, 15] classified the ranking procedures into three classes. The first class consists of ranking procedures based on fuzzy mean and spread and second class includes ranking procedures based on fuzzy scoring whereas the third class contains methods based on preference relations and concluded that the ordering procedures associated with first class are relatively reasonable for the ordering of fuzzy numbers. Later on, ranking fuzzy numbers by left and right dominance [16], area between the centroid point and original point [17], sign distance [18], distance minimization [19] came into existence. Later in 2007, Garcia and Lamata [20] modified the index of Liou and Wang [9] for ranking fuzzy numbers by stating that the index of optimism is not alone sufficient to discriminate fuzzy numbers and proposed an index of modality to rank fuzzy numbers. In 2011, Rao and Shankar [22] proposed a fuzzy ranking method based on circumcenter of centroids and index of modality which allows the participation of decision makers in ranking fuzzy numbers. In this paper, a new method is proposed, based on circumcenter of centroids to rank fuzzy quantities. Assuming a trapezoidal fuzzy number as a trapezoid, a referential triangle is constructed by splitting the trapezoid into three parts which are a triangle, a rectangle and a triangle respectively. Then the centroids of these three parts are calculated followed by the calculation of the circumcenter of these centroids. A ranking function is defined which is the area between the circumcenter point of the given generalized trapezoidal fuzzy numbers and the origin to rank fuzzy numbers. Centroid of trapezoid is used as reference point for most of the ranking procedures proposed in literature, as the centroid is a balancing point of the trapezoid. But the circumcenter of centroids can be considered a better balancing point as this point is

3 Fuzzy Inf. Eng. (2013 1: equidistant from all the vertices which are centroids. Further, the method, an index of optimism, is used to reflect the decision maker s optimistic attitude and also an index of modality is used that represents the importance of using mode and spreads of fuzzy numbers. The work is organized as follows. Section 2 briefly introduces the basic concepts and definitions of fuzzy numbers. Section 3 presents the proposed new method. In Section 4, a procedure to rank fuzzy numbers is presented. In Section 5, by considering some numerical examples, a comparative study is made with few methods of literature. In Section 6, the method is validated and finally the conclusions of the work are presented in Section Preliminaries In this section, some basic concepts of fuzzy numbers, arithmetic operations and few existing methods for ranking fuzzy numbers are reviewed Fuzzy Sets-basic Definitions In this subsection, some basic definitions of fuzzy numbers are reviewed from [27]. Definition 2.1 Let U be a universe of discourse. A fuzzy set à of U is defined by a membership function f à : U [0, 1], where fã(x is the degree of x in à and is represented as à = {(x, fã(x/xɛu}. Definition 2.2 A fuzzy set à defined on the universe set U happens to be normal iff Sup xɛu fã(x = 1. Definition 2.3 A fuzzy set à of universe ( set U is said to be convex iff fã(λx + (1 λy min fã(x, fã(y x, yɛu and λɛ[0, 1]. Definition 2.4 A fuzzy set à of universe set U is a fuzzy number iff à is normal and convex on U. Definition 2.5 A real fuzzy number à is described as any fuzzy subset of the real line R with membership function f à (x possessing the following properties: 1 fã is a continuous mapping from R to the closed interval [0, wã], 0 wã 1. 2 fã(x = 0 for all x ɛ(, a (d,. 3 fã(x is strictly increasing on [a, b]. 4 fã(x = 1 for all x ɛ [b, c]. 5 fã(x is strictly decreasing on [c, d], where a, b, c, d are real numbers. Definition 2.6 The membership function of the real fuzzy number à is given by f L (x, a x b, à wã, b x c, fã(x = f R (x, c x d, à 0, otherwise, where 0 wã 1 is a constant, a, b, c, d are real numbers and f L : [a, b] à [0, wã], f R :[c, d] [0, à wã ] are two strictly monotonic and continuous functions

4 6 P. Phani Bushan Rao N. Ravi Shankar (2013 from R to the closed interval [0, wã]. It is customary to write a fuzzy number as à = (a, b, c, d; wã. Ifwà = 1, then à = (a, b, c, d;1 is a normalized fuzzy number, otherwise à is said to be a generalized or non-normal fuzzy number. If the membership function is fã(x piecewise linear, then à is said to be a trapezoidal fuzzy number. Definition 2.7 The membership function of a trapezoidal fuzzy number is given by w à (x a, a x b, b a wã, b x c, fã(x = wã(x d, c x d, c d 0, otherwise. If wã = 1, then à = (a, b, c, d;1is a normalized trapezoidal fuzzy number and Ãisa generalized trapezoidal fuzzy number if 0 wã 1 as shown in Fig.1. Fig. 1 Trapezoidal fuzzy number Definition 2.8 The image of à = (a, b, c, d : wã is à = ( d, c, b, a; wã. If a = b, c = d and wã = 1, then à is called a crisp interval. If a = b = c = d and wã = 1, then à is called a crisp value. As a particular case, if b = c, the trapezoidal fuzzy number reduces to a triangular fuzzy number given by à = (a, b, d; wã as shown in Fig.2. Fig. 2 Triangular fuzzy number

5 Fuzzy Inf. Eng. (2013 1: If wã = 1, then à = (a, b, d is a normalized triangular fuzzy number and à = (a, b, d, wã is a generalized or non-normal triangular fuzzy number if 0 wã Arithmetic Operations between Fuzzy Numbers In this subsection, arithmetic operations between two generalized fuzzy numbers are reviewed from Chen and Chen [25]. Assume that à = (a 1, b 1, c 1, d 1 ; wã and B = (a 2, b 2, c 2, d 2 ; w B are two generalized trapezoidal fuzzy numbers. 1 Fuzzy numbers ( addition : à B = a 1 + a 2, b 1 + b 2, c 1 + c 2, d 1 + d 2 ; min(wã, w B. 2 Fuzzy numbers ( subtraction : à B = a 1 d 2, b 1 c 2, c 1 b 2, d 1 a 2 ; min(wã, w B. 3 Fuzzy numbers ( multiplication : à B = a, b, c, d; min(wã, w B, where a = min(a 1 a 2, a 1 d 2, d 1 a 2, d 1 d 2, b = min(b 1 b 2, b 1 c 2, c 1 b 2, c 1 c 2, c = max(b 1 b 2, b 1 c 2, c 1 b 2, c 1 c 2 and d = max(a 1 a 2, a 1 d 2, d 1 a 2, d 1 d 2. 4 Fuzzy numbers ( division : à B = a 1 /d 2, b 1 /c 2, c 1 /b 2, d 1 /a 2 ; min(wã, w B. 5 Multiplication by a scalar k: (ka 1, kb 1, kc 1, kd 1 ; wã, k > 0, kã = (kd 1, kc 1, kb 1, ka 1 ; wã, k < A Review of Some Existing Methods for Ranking Fuzzy Numbers In this subsection, few existing methods for ranking of fuzzy numbers are reviewed such as Cheng [12], Chu and Tsao [17], Liou and Wang [9] and Garcia and Lamata [20] Cheng s Fuzzy Ranking Method Cheng [12] proposed a centroid-index ranking method for calculating the centroid point ( x 0, y 0 of each fuzzy number à = (a, b, c, d; w. The centroid point ( x 0, y 0 is given by ( x 0, y 0 = [( w(d 2 2c 2 + 2b 2 a 2 + dc ab + 3(c 2 b 2 3w(d c + b a + 6(c b w ( 2(b + c a d + 3(a + dw ]. 3 (b + c a d + 3(a + dw Cheng proposed a ranking function R(à = x y 2 0, which is the distance from centroid point and the origin. The larger the value of R(Ã, the better is the ranking of Ã. Cheng s method cannot rank crisp fuzzy numbers because the denominator becomes zero for crisp fuzzy numbers, hence his centroid formulae is undefined for crisp numbers Chu and Tsao s Fuzzy Ranking Method,

6 8 P. Phani Bushan Rao N. Ravi Shankar (2013 Chu and Tsao [17] presented a method for ranking fuzzy numbers based on a ranking function R(à = x 0 y 0 which is the area between the centroid point ( x 0, y 0 of each fuzzy number à = (a, b, c, d; w and original point. Chu and Tsao s centroid formula is given by [( w(d 2 2c 2 + 2b 2 a 2 + dc ab + 3(c 2 b 2 ( x 0, y 0 =, w ( (b + c ] 1+. 3w(d c + b a + 6(c b 3 a + b + c + d The larger the value of R(Ã, the better is the ranking of Ã. Chu and Tsao s centroid formulae cannot rank crisp numbers which are a special case of fuzzy numbers, as can be seen from the above formulae that the denominator in the first coordinate of their centroid formulae is zero, hence his centroid formulae is undefined for crisp numbers Liou and Wang s Fuzzy Ranking Method Liou and Wang [9] ranked fuzzy numbers with total integral value. For a fuzzy number à = (a, b, c, d; w, the total integral value is defined as I α T (à = αi R (à + (1 αi L (Ã, where I R (à = 1 0 gr(ydy and I à L(à = 1 0 gl (ydy are the right and left integral values à of à respectively and αɛ[0, 1] is the index of optimism which represents the degree of optimism from a decision maker. If α = 0, the total integral value represents a pessimistic decision maker s view point and is equal to left integral value. If α = 1, the total integral value represents an optimistic decision maker s view point and is equal to the right integral value and when α = 0.5, the total integral value represents a moderate decision maker s view point and is equal to the mean of right and left integral values. For a decision maker, the larger the value of α, the higher is the degree of optimism Garcia and Lamata s Fuzzy Ranking Method Garcia and Lamata [20] modified the index of Liou and Wang [9] for ranking fuzzy numbers. This method, an index of optimism, is used to reflect the decision maker s optimistic attitude, and also an index of modality is used to represent the neutrality of the decision maker. For a fuzzy number à = (a, b, d; w, Garcia and Lamata [20] proposed an index associated with the ranking as the convex combination I β,α (à = βs M (à + (1 βi α T (Ã, where S M (à is the area of the core of the fuzzy number which is equal to b for a triangular fuzzy number defined by à = (a, b, c, d; w and the average value of the plateau for a trapezoidal fuzzy number given by à = (a, b, c, d; w, βɛ[0, 1] is the index of modality that represents the importance of central value against the extreme values, αɛ[0, 1] is the degree of optimism of the decision maker and I α T (à = αi R (à + (1 αi L (Ã. 3. Propose Method for Ranking Fuzzy Numbers Based on Circumcenter of Centroids In this section, a new method for ranking fuzzy numbers is proposed. This method unifies the concepts such as centroids, circumcenter of centroids, index of optimism

7 Fuzzy Inf. Eng. (2013 1: (Liou and Wang [9] and index of modality (Garcia and Lamata [20]. The centroid of a trapezoid is considered the balanceing point of the trapezoid (Fig.3. Fig. 3 Circumcenter of centroids Assuming a trapezoidal fuzzy number as a trapezoid, divide the trapezoid into three plane figures. These three plane figures are a triangle (APB, a rectangle (BPQC and again a triangle (CQD respectively. The circumcenter of the centroids of these three plane figures is taken as the point of reference to define the ranking of generalized fuzzy numbers. The reason for selecting this point as a point of reference is that each centroid point (G 1 of triangle APB, G 2 of rectangle BPQC and G 3 of triangle CQD is a balancing point of each individual plane figure and the circumcenter of these centroid points is equidistant from each vertex (which are centroids. Therefore, this point would be a better reference point than a centroid point of the trapezoid. Consider a generalized trapezoidal fuzzy number à = (a, b, c, d; w, the centroids of the three plane figures are G 1 = ( a+2b 3, w 3, G 2 = ( b+c 2, w 2 and G 3 = ( 2c+d w 3 3, respectively. Equation of the line G 1 G 3 is y = w 3 and G 2 does not lie on the line G 1 G 3. Therefore, G 1, G 2 and G 3 are non-collinear and they form a triangle. We define the circumcenter SÃ( x 0, y 0 of the triangle with vertices G 1, G 2 and G 3 of the generalized trapezoidal fuzzy number à = (a, b, c, d; w as S Ã( x 0, y 0 = ( a + 2b + 2c + d, 6 (2a + b 3c(2d + c 3b + 5w2. (1 12w As a special case, for triangular fuzzy number à = (a, b, d; w, i.e., c = b, the circumcenter of centroids is given by 4. Ranking Procedure S Ã( x 0, y 0 = ( a + 4b + d, 6 4(a b(d b + 5w2. (2 12w An efficient way to rank fuzzy numbers is defuzzification. In this section, the procedure involved in ranking fuzzy quantities using the process of defuzzification is presented.

8 10 P. Phani Bushan Rao N. Ravi Shankar (2013 Consider a set of n generalized trapezoidal fuzzy numbers à i = (a i, b i, c i, d i ; w i, i = 1, 2,, n. For any decision maker whether pessimistic (α = 0, optimistic (α = 1 or neutral (α = 0.5, the ranking value of the trapezoidal fuzzy number is defined by a ranking function which associates a real number to each trapezoidal fuzzy number à i with circumcenter of centroids SÃi ( x 0, y 0 defined by (1. This is given by R(à i = ( x 0 Ãi.( y 0 Ãi. (3 This represents the area between the circumcenter of centroids for the generalized trapezoidal fuzzy number à i and the origin. The larger is the value of R(à i, 1, 2,, n, the bigger is the fuzzy number à i. As this area is not sufficient to discriminate fuzzy numbers in all situations, the following areas are also used to discriminate fuzzy numbers. Areas associated with mode, total spread, left spread, right spread For any generalized trapezoidal fuzzy number à i = (a i, b i, c i, d i ; w i, i = 1, 2,, n, the areas associated with mode and spreads are defined as follows: Area associated with mode or central value is defined as M A (à i = w( b + c 2. Total spread area is defined as T SA (à i = w(d a. Left spread area is defined as L SA (à i = w(b a. Right spread area is defined as R SA (à i = w(d c. Remark 1 For generalized trapezoidal fuzzy number à i = (a i, b i, c i, d i ; w i, i = 1, 2,, n, with circumcenter of centroids SÃi ( x 0, y 0 defined by (1, we define the index associated with the ranking as I α (à i = α( y 0 Ãi + (1 α( x 0 Ãi, (4 where αɛ[0, 1] is the index of optimism which represents the degree of optimism of a decision maker. The larger the value of α is, the higher the degree of optimism of the decision maker. The index of optimism alone is not sufficient to discriminate fuzzy numbers as this uses only the coordinates of the circumcenter of centroids. Hence, we upgrade this by using an index of modality which represents the importance of areas associated with central value and spreads along with index of optimism. Remark 2 For a generalized trapezoidal fuzzy number à i = (a i, b i, c i, d i ; w i, i = 1, 2,, n, with circumcenter of centroids SÃi ( x 0, y 0 defined by (1, we define the index associated with the ranking as I 1 β,α (à i = βm A (à i + (1 βi α (à i, (5 I 2 β,α (à i = βt SA (à i + (1 βi α (à i, (6 I 3 β,α (à i = βl SA (à i + (1 βi α (à i, (7

9 Fuzzy Inf. Eng. (2013 1: I 4 β,α (à i = βr SA (à i + (1 βi α (à i, (8 where βɛ[0, 1] is the index of modality which represents the importance of area associated with central value and the area associated with spreads against the coordinates of circumcenter x 0 and y Working Rule to Rank Fuzzy Numbers Suppose that given a set of n fuzzy numbers S = ( A 1, A 2,, A n which are to be ranked. The mapping from the set à into the real line is defined as: R : à i R(à i = ( x 0 Ã.( y 0 Ãi, i = 1, 2,, n. (9 Using (3 calculate the ranking indexes R( A 1, R( A 2,, R( A n, of the fuzzy numbers A 1, A 2,, A n. The larger the value of R(à i where i = 1, 2,, n, the better the ranking of à i is. Hence for two fuzzy numbers à i and à j, If R(à i > R(à j, then à i > à j. If R(à i < R(à j, then à i < à j. If R(à i = R(à j, then discrimination of fuzzy numbers is not possible. In such cases, use (5 to rank fuzzy numbers. The larger the value of Iβ,α 1 (à i, i = 1, 2,, n, the better the ranking of à i is. If Iβ,α 1 (à i = Iβ,α 1 (à j ; i j, then use (6 to rank fuzzy numbers. The larger the value of Iβ,α 2 (à i, i = 1, 2,, n, the better the ranking of à i is. If Iβ,α 2 (à i = Iβ,α 2 (à j ; i j, then use (7 to rank fuzzy numbers. The larger the value of Iβ,α 3 (à i, i = 1, 2,, n, the better the ranking of à i is. If Iβ,α 3 (à i = Iβ,α 3 (à j ; i j, then use (8 to rank fuzzy numbers. The larger the value of Iβ,α 4 (à i, i = 1, 2,, n, the better the ranking of à i is. If Iβ,α 4 (à i = Iβ,α 4 (à j ; i j, then the fuzzy number with larger value of wãi is ranked higher. 5. Comparative Study In this section, the ranking results of various existing methods Yager s method [6], Murakami et al s method [23], Liou and Wang s method [9], Fortemps and Roubens s method [11], Cheng s method [12], Chen and Lu s method [16], Chu and Tsao s method [17], Asady and Zendehnam s method [19], Wang et al s method [21], Chen and Chen s method [25], Deng and Liu s method [24] and Chen and Chen s method [26] are compared with proposed method. These are studied in three cases. Comparative study-case I In this case, four fuzzy numbers are considered and the proposed method is compared with the ranking results of Yager [6], Fortemps and Roubens [11], Liou and Wang [9] and Chen and Lu [16] and the results are shown in Table 1. Consider the four normal fuzzy numbers A 1 = (0.1, 0.2, 0.3; A 2 = (0.2, 0.5, 0.8; A 3 = (0.3, 0.4, 0.9; A 4 = (0.6, 0.7, 0.8 as shown in Fig.4.

10 12 P. Phani Bushan Rao N. Ravi Shankar (2013 Fig. 4 Set of four fuzzy numbers By using the proposed method, we have the following: S A1 ( x 0, y 0 = (0.2, , S A2 ( x 0, y 0 = (0.5, , S A3 ( x 0, y 0 = (0.4666, 0.4, S A4 ( x 0, y 0 = (0.7, 0.413, R(A 1 = , R(A 2 = , R(A 3 = , R(A 4 = A 4 > A 2 > A 3 > A 1. Table 1: Comparison results of various ranking methods. Method \ Fuzzy number A 1 A 2 A 3 A 4 Yager [6] Fortemps and Roubens [11] Liou and Wang [9] α = α = α = Chen and Lu [6] α = α = α = Proposed method From Table 1, we can see that Yager s method [6], Fortemps and Roubens method [11], Liou and Wang s method [9], Chen and Lu s [16] method failed to discriminate the fuzzy numbers A 2, A 3 and Chen and Lu s method [16] even failed to discriminate fuzzy numbers A 1, A 4, whereas, the proposed method properly discriminated fuzzy numbers appropriately. Comparative study-case II In this case, three sets of fuzzy numbers are adopted from (Deng and Liu [24] as shown in Fig.5. The proposed method is compared with the ranking results of Yager

11 Fuzzy Inf. Eng. (2013 1: [6], Murakami et al [23] and Cheng [12], Chen and Chen [25], Deng and Liu [24] and Chen and Chen [26] and the results are shown in Table 2. Fig. 5 Three sets of fuzzy numbers (Deng and Liu [24] Table 2: Comparison results of various ranking methods. Set 1 Set 2 Set 3 Method \ Fuzzy number à B C à B à B Yager [6] Murakami et al [23] Cheng [12] Chen and Chen [25] Deng and Liu [24] Chen and Chen [26] Proposed method From Table 2, the observations for the fuzzy numbers are Set 1: Yager s method [6], Murakami et al s method [23] and Cheng s method [12] fail to discriminate fuzzy numbers à and B and fail to calculate the ranking value of the fuzzy number C, whereas, the ranking order of the proposed method and other methods are the same, i.e., à < B < C. Set 2: Cheng s method [12] failed to discriminate the fuzzy numbers à and B, whereas, the ranking order of the proposed method and other methods are the same,

12 14 P. Phani Bushan Rao N. Ravi Shankar (2013 i.e., à > B. Set 3: Yager s method [6], Murakami et al s method [23], Cheng s method [12] failed to calculate the ranking value of the fuzzy numbers à and B, the ranking orders of Chen and Chen [25], Chen and Chen [26] are à < B and Deng and Liu [24] and the proposed method are à > B. Comparative study-case III In this case, four sets of fuzzy numbers are adopted from (Asady and Zendehnam [19] as shown in Fig.6. The proposed method is compared with the ranking results of Yager [6], Chu and Tsao [17], Asady and Zendehnam [19], Chen [7], Wang et al [28] and the results are shown in Table 3. Fig. 6 Four sets of fuzzy numbers (Asady and Zendehnam [19] From Table 3, the observations for the fuzzy numbers are Set 1: The ranking order of the proposed method is in coincidence with other methods, i.e., A 1 < A 2 < A 3. Set 2: The ranking order of Wang et al [28] is A 2 < A 3 < A 1, whereas the ranking order of the proposed method coincides with other methods, i.e., A 1 < A 2 < A 3. Set 3: Asady and Zendehnam s method [19] failed to discriminate fuzzy numbers A 1 and A 2, whereas the ranking order of the proposed method coincides with other methods, i.e., A 1 < A 2. Set 4: Yager s method [6], Chu and Tsao s method [17], Asady and Zendehnam s

13 Fuzzy Inf. Eng. (2013 1: Table 3: Comparison results of various ranking methods. Set 1 Set 2 Set 3 Set 4 Methods A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 1 A 2 Yager [6] Chen [7] Chu et al [17] Asady et al [19] Wang et al [28] Proposed method method [19] and Wang et al s method [28] failed to discriminate the fuzzy numbers A 1 and A 2, whereas the proposed method discriminates the two given fuzzy numbers and gives the correct ranking order, i.e., A 1 < A 2. Comparative study-case IV In this case, six sets of fuzzy numbers are adopted from (Chen and Chen [25] as shown in Fig.7. The proposed method is compared with the ranking results of Yager [6], Murakami et al [23], Cheng [12], Chu and Tsao [17] and Chen and Chen [25], and the results are shown in Table 4. Fig. 7 Six sets of fuzzy numbers (Chen and Chen [25]

14 16 P. Phani Bushan Rao N. Ravi Shankar (2013 Table 4: Comparison results of various ranking methods. Sets Method \ Fuzzy number à B à B à B Yager [6] Murakami et al [23] Cheng [12] Chu and Tsao [17] Chen and Chen [25] Proposed method α = β 0.6 α = β β α = β β Sets Method \ Fuzzy number à B à B à B C Yager [6] Murakami et al [23] Cheng [12] Chu and Tsao [17] Chen and Chen [25] Proposed method From Table 4, the observations for the fuzzy numbers are Set 1: The ranking order of the proposed method is in coincidence with other methods, i.e., à < B. Set 2: Yager s method [6], Cheng s method [12] and Chu and Tsao s method [17] failed to discriminate the fuzzy numbers à and B. The ranking order of other methods and the proposed method are the same, i.e., à < B. Set 3: The ranking order of all the methods is à > B, whereas the proposed method initially failed to discriminate the fuzzy numbers à and B, but by taking into consideration the decision maker s choice and index of modality, we see that ranking order of the proposed method is à < B, where βɛ[0, 1]. Set 4: Yager s method [6] failed to discriminate the fuzzy numbers à and B, whereas the ranking order of the proposed method is in coincidence with other methods, i.e., à < B. Set 5: Yager s method [6], Murakami et al s method [23], Cheng s method [12] and Chu and Tsao s method [17] failed to calculate the ranking value of the fuzzy num-

15 Fuzzy Inf. Eng. (2013 1: bers, whereas the ranking order of Chen and Chen s method [25] and the proposed method are the same, i.e., à < B. Set 6: Yager s method [6], Murakami et al s method [23], Chen and Chen s method [25] get the ranking order which is à < C < B, whereas the ranking order of the proposed method and other methods are the same, i.e., à < B < C. 6. Validation of the Proposed Ranking Function The proposed ranking function is valid as it satisfies the reasonable properties for ordering fuzzy quantities proposed by Wang et al [14]. 7. Conclusion This paper proposes a method that ranks fuzzy numbers which is simple and concrete. This method ranks trapezoidal as well as triangular fuzzy numbers and their images. It also ranks crisp numbers which are special case of fuzzy numbers. Besides the decision maker s degree of optimism, an index of modality representing the importance of mode and spreads against the coordinates of centroid, is used in ranking fuzzy numbers. This method which is simple in calculation not only gives satisfactory results to well defined problems, but also gives a correct ranking order and agrees with human intuition. Acknowledgments Authors would like to thank referees for their helpful comments. References 1. Zadeh L A (1965 Fuzzy sets. Information and Control 8: Jain R (1976 Decision making in the presence of fuzzy variables. IEEE Transactions on Systems, Man and Cybernetics 6: Jain R (1978 A procedure for multi-aspect decision making using fuzzy sets. International Journal of Systems Science 8: Baldwin J F, Guild N C F(1979 Comparison of fuzzy sets on the same decision space. Fuzzy Sets and Systems 2: Yager R R (1978 Ranking fuzzy subsets over the unit interval. Proc. IEEE Conference on Decision and Control, San Diego, California 17: Yager R R (1981 A procedure for ordering fuzzy subsets of the unit interval. Information Sciences 24: Chen S H (1985 Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets and Systems 17: Kim K, Park K S (1990 Ranking fuzzy numbers with index of optimism. Fuzzy Sets and Systems 35: Liou T S, Wang M J (1992 Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems 50: Choobineh F, Li H (1993 An index for ordering fuzzy numbers. Fuzzy Sets and Systems 54: Fortemps P, Roubens M (1996 Ranking and defuzzification methods based on area compensation. Fuzzy Sets and Systems 82: Cheng C H (1998 A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems 95(3: Yao J S, Wu K (2000 Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems 116: Wang X, Kerre E E (2001 Reasonable properties for the ordering of fuzzy quantities (I. Fuzzy Sets and Systems 118:

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