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A Comparative Study of Ranking Fuzzy Numbers Based on Regular Weighted Function

A Comparative Study of Ranking Fuzzy Numbers Based on Regular Weighted Function Fuzzy Inf. Eng. (2012) 3: 235-248 DOI 10.1007/s12543-012-0113-1 ORIGINAL ARTICLE A Comparative Study of Ranking Fuzzy Numbers Based on Regular Weighted Function R. Saneifard · T. Allahviranloo Received: 1 September 2010/ Revised: 30 March 2012/ Accepted: 10 July 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract Fuzzy systems have gained more and more attention from researchers and practitioners of various fields. In such systems, the output represented by a fuzzy set sometimes needs to be transformed into a scalar value, and this task is known as the defuzzification process. Several analytic methods have been proposed for this prob- lem, but in this paper, the researchers suggest a modified approach to the problem of defuzzification, using the bi-symmetric weighted distance between two fuzzy num- bers. This defuzzification can be used as a crisp approximation with respect to a fuzzy quantity. By considering this and with benchmark between fuzzy numbers, we can present a method for ranking, which can effectively rank various fuzzy numbers and their images and overcome the shortcomings of the previous techniques. After illustrating many numerical examples, following our procedure, the ranking results become valid. Keywords Fuzzy number· Defuzzification· Ranking· Bi-symmetric weighted dis- tance · Decision maker’s strategy. 1. Introduction As most modeling and controlling applications require crisp outputs, when applying fuzzy inference systems, the fuzzy system output A(y) usually has to be converted into a crisp output Y . This operation is called defuzzification. It is essentially a process guided by the output fuzzy subset of the model. One selects a single crisp value as the system output. In [4] and [5], the researchers have introduced a general R. Saneifard () Department of Applied Mathematics, Urmia Branch, Islamic Azad University, Oroumieh, Iran email: srsaneeifard@yahoo.com T. Allahviranloo Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 236 R. Saneifard · T. Allahviranloo (2012) approach to defuzzification based upon the basic defuzzification distributions (BDD) transformator. The output for the fuzzy controller F is a fuzzy subset of the real line; for simplicity the researchers shall assume the support set Y is finite, Y = {y ,··· , y }. 1 n For y ∈ Y, F(y ) = w indicates the degree to which each y is suggested as a good i i i i output value by the rule base under the current input. Two commonly used methods for defuzzification are the center of area (COA) method and the mean of maximum (MOM) method [6, 8]. In the COA method, one calculates the output of the defuzzi- Σ y w i i i fier y as : y = . In the MOM method, one calculates the output of the COA COA Σw controller y = Σ y , where A is the set of elements in Y which provides the MOM y ∈A i m i maximum value of F(y) and m is the cardinality of A. Based upon these observation, one can view the defuzzification process under the COA and MOM methods as first converting the fuzzy subset F of Y into a probability distribution on Y , in the spirit described in the earlier section and then taking the expected value as their output. Keeping with this probabilistic interpretation, the researchers shall in the following use P instead of q to denote the transformation values. Moreover, in [7, 10, 17], i i the researchers have used the concept of the symmetric triangular fuzzy number, and have introduced an approach to defuzzify a fuzzy number based on L -distance and weighted distance. In this paper, the researchers suggest a modified approach to the problem of defuzzification, using the bi-symmetric weighted distance between two fuzzy numbers. In this study, some preliminary results on properties of such defuzzi- fication are to be reported. Having reviewed the previous dictions, the researchers have two objectives in this study. Firstly, they want to introduce a modified defuzzi- fication of a fuzzy quantity. The second objective is applying the proposed method to ranking of fuzzy numbers (i.e., BWDM: bi-symmetric weighted distance method). In addition to its ranking features, this method removes the ambiguities resulted from the comparison of the previous rankings and overcomes the shortcomings. The paper is organized as follows: in Section 2, some fundamental results on fuzzy numbers are recalled. In Section 3, a crisp approximation of a fuzzy number is ob- tained. In this section, some remarks are proposed and illustrated. Also, the proposed method for ranking fuzzy numbers is in this section. Discussion and comparison of this work and other methods are carried out in Section 4. The paper ends with conclusions in Section 5. 2. Preliminaries The basic definitions of a fuzzy number are given in [14, 15, 26, 27] as follows˖ Definition 1 Let U be a universe set. A fuzzy set A of U is defined by a membership function μ (x) → [0, 1], where μ (x), ∀x ∈ U, indicates the degree of x in A. ˜ ˜ A A Definition 2 A fuzzy subset A of universe set U is normal iff supμ (x) = 1, where U x∈U is the universe set. ˜ ˜ Definition 3 A fuzzy set A is a fuzzy number iff A is normal and convex on U. Definition 4 A trapezoidal fuzzy number A is a fuzzy number with a membership Fuzzy Inf. Eng. (2012) 3: 235-248 237 functionμ ˜ defined by : x− a ⎪ 1 , when a ≤ x ≤ a , ⎪ 1 2 a − a ⎪ 2 1 1, when a ≤ x ≤ a , 2 3 μ (x) = (1) ˜ ⎪ ⎪ a − x ⎪ , when a ≤ x ≤ a , 3 4 a − a 4 3 0, otherwise, which can be denoted as a quartet (a , a , a , a ). In these above situations a , a , a 1 2 3 4 1 2 3 and a ,ifa = a , A becomes a triangular fuzzy number. 4 2 3 Definition 5 An extended fuzzy number A is described as any fuzzy subset of the universe set U with membership function μ defined as follows: (a) μ is a continuous mapping from U to the closed interval [0,ω], 0<ω ≤ 1. (b) μ (x) = 0 for all x ∈ (−∞, a ]. ˜ 1 (c) μ is strictly increasing on [a , a ]. 1 2 (d) μ (x) = ω, for all x ∈ [a , a ], as ω is a constant and 0<ω ≤ 1. ˜ 2 3 (e) μ is strictly decreasing on [a , a ]. 3 4 (f) μ (x) = 0 for all x ∈ [a ,+∞). ˜ 4 In these above situations a ,a ,a and a are real numbers. If a = a = a = a , 1 2 3 4 1 2 3 4 A becomes a crisp real number. Definition 6 The membership function μ ˜ of extended fuzzy number A is expressed by ⎪ μ (x), when a ≤ x ≤ a , 1 2 ⎪ A ω, when a ≤ x ≤ a , 2 3 μ (x) = (2) ˜ ⎪ ⎪ R μ (x), when a ≤ x ≤ a , ⎪ 3 4 ⎪ ˜ 0, otherwise, L R where μ :[a , a ] → [0,ω] and μ :[a , a ] → [0,ω]. Based on the basic theories 1 2 3 4 ˜ ˜ A A ˜ ˜ of fuzzy numbers, A is a normal fuzzy number if ω = 1, whereas A is a non-normal fuzzy number if 0 <ω ≤ 1. Therefore, the extended fuzzy number A in Defini- ˜ ˜ tion 6 can be denoted as (a , a , a , a ;ω). The image −Aof A can be expressed by 1 2 3 4 (−a ,−a ,−a ,−a ;ω) [14]. 4 3 2 1 Definition 7 The r-cut of a fuzzy number A, where 0 < r ≤ 1 is a set defined as A = {x∈| μ ˜ (x) ≥ r}. According to the definition of a fuzzy number, it is seen once that every r-cut of a fuzzy ˜ ˜ ˜ number is a closed interval. Hence, we have A = [A (r), A (r)], where r L R A (r) = inf{x∈|μ (x) ≥ r}, (3) L A A (r) = sup{x∈|μ (x) ≥ r}. (4) R A 1 1 Definition 8 A function f :[0, 1] → [0, 1] symmetric around , i.e., f ( − r) = 2 2 1 1 1 f ( +r) for all r ∈ [0, ], which reaches its minimum in , is called the bi-symmetrical 2 2 2 weighted function [16]. Moreover, the bi-symmetrical weighted function (BSWF) is called regular if 238 R. Saneifard · T. Allahviranloo (2012) 1) f ( ) = 0, 2) f (0) = f (1) = 1, 3) f (r)dr = . ˜ ˜ ˜ ˜ Definition 9 For two arbitrary fuzzy numbers A and B with r-cuts [A (r), A (r)] and L R ˜ ˜ [B (r), B (r)], respectively, the quantity L R 1 1 2 2 2 ˜ ˜ ˜ ˜ ˜ ˜ d (A, B) = f (r)[A (r)− B (r)] dr+ f (r)[A (r)− B (r)] dr , (5) w L L R R 0 0 where f :[0, 1] → [0, 1] is a bi-symmetrical (regular) weighted function is called the ˜ ˜ bi-symmetrical (regular) weighted distance between A and B based on f . One can, of course, propose many regular BSWF and hence obtain different bi- symmetrical weighted distances. Further on we will consider mainly a following function ⎪ 1 ⎨ 1− 2r, when r ∈ [0, ], f (r) = (6) 2r− 1, when r ∈ [ , 1]. 3. A Novel Fuzzy Ordering Method Ranking of fuzzy numbers was studied by many researchers. Some researchers in- troduced a distance and then compared the fuzzy numbers with it, (Bardossy et al 1992; Bortlan and Degani 2006; Cheng 1998; Tran and Duckstein 2002; Yao and Wu 2000). In recent years, many methods are proposed for ranking different types of fuzzy numbers [1-3, 9, 11, 12], and can be classified into four major classes: prefer- ence relation, fuzzy mean, spread fuzzy scoring, and linguistic expression. But each method appears to have advantages as well as disadvantages. In this section, we pro- pose a novel ranking fuzzy numbers method based on relative weighted distance, and ˜ ˜ ˜ verify this by some examples. In order to rank n fuzzy numbers A , A ,··· , A , let 1 2 n the fuzzy number B be zero in Eq. (5). Then the weighted distance is 1 2 2 2 ˜ ˜ ˜ d (A, 0) = f (r)((A (r)) + (A (r)) )dr . (7) w i i i L R Definition 10 Combining Eqs. (3), (4) and (7) and the concept of metric is the relative distance of p-norm [19]. We can induce the RWD (i.e., relative weighted ˜ ˜ distance of 2-norm) between fuzzy numbers A and B as follows: ˜ ˜ ˜ ˜ |A− B| d (A, B) ˜ ˜ RWD (A, B) = = . (8) 2 2 ˜ ˜ |A| +|B| 2 2 ˜ ˜ d (A, 0)+ d (B, 0) w w For ranking fuzzy numbers, this study firstly defines a minimum crisp valueμ (x) min and maximum crisp valueμ (x) to be the benchmark and its characteristic function max μ (x) andμ (x) is as follows: τ τ min max 1, when x = τ , min μ (x) = (9) τ ⎪ min ⎪ 0, when x  τ min Fuzzy Inf. Eng. (2012) 3: 235-248 239 and 1, when x = τ , ⎨ max μ (x) = (10) max ⎪ 0, when x  τ . max ˜ ˜ ˜ When ranking n fuzzy numbers A , A ,··· , A , the minimum crisp value τ and 1 2 n min maximum crisp valueτ are defined as: max ˜ ˜ ˜ τ = min{x|x ∈ Domain(A , A ,··· , A )}, (11) min 1 2 n ˜ ˜ ˜ τ = max{x|x ∈ Domain(A , A ,··· , A )}. (12) max 1 2 n The advantages of the definition of minimum and maximum crisp value are two- fold: firstly, the minimum and maximum crisp values will be obtained by themselves, besides it is easy to compute. The steps of BWDM algorithm are: Step 1: Computing the left and right inverse functions of each fuzzy number by Eqs. (3) and (4). Step 2: Using Eq. (11) to find minimum crisp value and computing its inverse func- tions to be the benchmark. Step 3: Computing RWD between fuzzy numbers and minimum crisp value (τ ) 2 min by Eq. (8). ˜ ˜ ˜ Example 1 Three fuzzy numbers A, B and C have been illustrated by Chen [24] and their membership functions are shown in Table 1. The inverse functions calculated by Eqs. (3) and (4) are also shown in this table. The fuzzy numbers and the minimum crisp value are illustrated in Fig.1. By Eqs. (3), (4), (11) and (12), this study obtains τ , τ and inverse functions min max as follows: ˜ ˜ ˜ τ = min{x|x ∈ Domain(A, B, C)} min = min{0.01, 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} = 0.01 and ˜ ˜ ˜ τ = max{x|x ∈ Domain(A, B, C)} max = max{0.01, 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} = 0.9 and ⎪ L g (x) = 0.01 min g (x) = min ⎪ R g (x) = 0.01 min and ⎪ L g (x) = 0.9, max g (x) = max ⎪ ⎪ R g (x) = 0.9. max According to Eq. (8), we can get the RWD values between minimum crisp value and ˜ ˜ ˜ fuzzy numbers A, B, and C, which are equal to 0.9836, 0.9818 and 0.9828, respec- tively. Also, the RWD values between maximum crisp value and fuzzy numbers A, ˜ ˜ B and C, which are equal to 0.44, 0.39 and 0.35, respectively. 240 R. Saneifard · T. Allahviranloo (2012) ˜ ˜ ˜ Table 1: Fuzzy numbers A, B and C. FN The membership function The inverse functions 100 1 ( )x+ , 0.01 ≤ x ≤ 0.4, ⎪ ⎪ ⎪ ⎪ ˜ 39 39 A (x) = 0.39x+ 0.01, ⎨ ⎨ L A μ (x) = g (x) = ˜ ⎪ ˜ ⎪ A A ⎪ ⎪ 1, 0.4 ≤ x ≤ 0.7, ⎪ ⎩ A (x) = −0.1x+ 0.8. −10x+ 8, 0.7 ≤ x ≤ 0.8. ⎪ 10 2 ⎪ ⎧ x− , 0.2 ≤ x < 0.5, ⎪ ⎪ ⎪ ⎪ 3 3 B (x) = 0.3x+ 0.2, ⎨ ⎨ L B μ (x) = g (x) = ˜ ˜ ⎪ ⎪ B B ⎪ 1, x = 0.5, ⎪ ⎪ ⎩ ˜ ⎪ B (x) = −0.4x+ 0.9. −2.5x+ 2.25, 0.5 < x ≤ 0.9. 2x− 0.2, 0.1 ≤ x ≤ 0.6, ⎪ ⎪ ⎪ ⎪ ˜ ⎨ ⎨ C (x) = 0.5x+ 0.1, C μ (x) = 1, x = 0.6, g (x) = ˜ ⎪ ˜ ⎪ C C ⎪ ⎪ ⎪ ⎩ C (x) = −0.2x+ 0.8. −5x+ 4, 0.6 ≤ x ≤ 0.8. ˜ ˜ ˜ Fig. 1 Fuzzy numbers A, B, C,τ andτ min max ˜ ˜ Definition 11 Let A and B be two fuzzy numbers characterized by Definition 2 and ˜ ˜ RWD (A, B) is the relative weighted distance of them. Since this article wants to approximate a fuzzy number by a scalar value, thus the researchers have to use an operator RWD : F→ which transforms fuzzy numbers into a family of real line. Operator RWD is a crisp approximation operator. Since ever above defuzzification can be used as a crisp approximation of a fuzzy number, therefore the resultant value is used to rank the fuzzy numbers. Thus, RWD is used to rank fuzzy numbers. The larger RWD , the larger fuzzy number. ˜ ˜ ˜ ˜ Let A,B ∈ F be two arbitrary fuzzy numbers. Define the ranking of A and B by RWD on F as follows: ˜ ˜ ˜ ˜ 1) RWD (A,τ ) > RWD (B,τ ) if only if A B, 2 min 2 min ˜ ˜ ˜ ˜ 2) RWD (A,τ ) < RWD (B,τ ) if only if A ≺ B, 2 min 2 min Fuzzy Inf. Eng. (2012) 3: 235-248 241 ˜ ˜ ˜ ˜ 3) RWD (A,τ ) = RWD (B,τ ) if only if A ∼ B. 2 min 2 min ˜ ˜ ˜ ˜ ˜ ˜ Then, this article formulate the order and as A B if and only if A B or A ∼ B, ˜ ˜ ˜ ˜ ˜ ˜ A  B if and only if A ≺ B or A ∼ B. ˜ ˜ ˜ Remark 1 If inf supp(A) ≥ 0, then RWD (A,τ ) ≥ 0, (RWD (A,τ ) ≥ 0). 2 min 2 max ˜ ˜ ˜ Remark 2 If sup supp(A) ≤ 0, then RWD (A,τ ) ≥ 0, (RWD (A,τ ) ≥ 0). 2 min 2 max Remark 3 Ifτ is equal to zero, then we obtain the same ordering result, for exam- min ˜ ˜ ple, if A = (0, 1, 1, 2), B = (3, 4, 4, 5) are two fuzzy numbers, then we have τ = 0, min ˜ ˜ ˜ ˜ RWD (A,τ ) = 1 and RWD (B,τ ) = 1, i.e., A ∼ B which is unreasonable result. 2 min 2 min In this case, we will use the τ instead τ , then for two arbitrary fuzzy numbers max min ˜ ˜ ˜ ˜ A,B ∈ F, we define the ranking of A and B by RWD on F as follows: ˜ ˜ ˜ ˜ 1) RWD (A,τ ) < RWD (B,τ ) if only if A B, 2 max 2 max ˜ ˜ ˜ ˜ 2) RWD (A,τ ) > RWD (B,τ ) if only if A ≺ B, 2 max 2 max ˜ ˜ ˜ ˜ 3) RWD (A,τ ) = RWD (B,τ ) if only if A ∼ B. 2 max 2 max Here, the following reasonable axioms that Wang and Kerre [9] have proposed for fuzzy quantities ranking are considered. Let RD be an ordering method, S the set of fuzzy quantities for which the method RD can be applied, andA andA finite subsets ˜ ˜ ˜ of S . The statement two elements A and B in A satisfy that A has a higher ranking ˜ ˜ ˜ than B when RD is applied to the fuzzy quantities in A will be written as A B by ˜ ˜ ˜ ˜ RD on A. A ∼ B by RD on A, and A B by RD on A are similarly interpreted. The following axioms show the reasonable properties of the ordering approach RD. ˜ ˜ ˜ A . For A∈A, A  AbyRDonA. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B)∈A , A  B and B  AbyRDon A, we should have A ∼ Bby RD on A. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B, C) ∈A , A  B and B  Cby RD on A, we should have A  Cby RD on A. ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B) ∈A , in f supp(B)>sup supp(A), we should have A  Bby RD on A. ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B) ∈A , in f supp(B)>sup supp(A), we should have A ≺ Bby RD on A. ˜ ˜ ˜ ˜ A . Let (A, B) ∈ (A∩A ) . We obtain the ranking order A  Bby RD on A if and ˜ ˜ only if A  Bby RD onA. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A . Let A, B, AC and BC be elements of S and C ≥ 0.If A  Bby RD on {A, B}, ˜ ˜ ˜ ˜ then AC  BC by RD on{AC, BC}. ˜ ˜ ˜ ˜ Remark 4 If A  B, then−A −B. Hence, this article can infer ranking order of the images of the fuzzy numbers. 4. Examples 242 R. Saneifard · T. Allahviranloo (2012) First of all, this study validates their proposed method with representative examples of [4, 13, 18, 19, 21, 22, 25] with some advantages. Example 2 The RWD values of 12 examples are shown in Fig.2. Table 2 shows the ranking results. From this table, the main findings and BWDM with some advantages show as follows: 1) From Example L, K, some methods use complicated and normalized process to rank and they can’t obtain consistent results. However, their proposed method is more suitable for ranking any kind of fuzzy number without normalization process. 2) Under fuzzy numbers with the same mean (Examples B, I ), Yager [4], Kerre [9], Bass and Kwakernaak [22] have not been able to obtain their orderings. Chang’s method [21] has been able to rank their orderings, but Chang’s results violate the smaller spread, the higher ranking order. In Examples B, I, it obviously shows that the researchers, proposed method can rank instantly and their results comply with intuition of human being (as Table 2). 3) In Examples C, D, L, we can see that the method of Kerre [9], Bass and Kwaker- naak [22] have many limitations on triangle, trapezoid, non-normalized fuzzy num- bers and so on. 4) The proposed method can be used for ranking fuzzy numbers and crisp values. But Yager has not been able to handle the crisp value problem [4]. 5) Kerre’s method would favor a fuzzy number with smaller area measurement, re- gardless of its relative location on the X-axis [9]. The results are against their intuition in Examples C,D. From Table 2, their proposed ranking method can correct the prob- lem. Table 2: The comparison with different ranking approaches. New method Yager [4] Kerre [9] Chang [21] Bass [22] ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A ≺ A A ≺ A A ≺ A A ≺ A A ≺ A 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ B ≺ B B ∼ B B ≺ B B B B ∼ B 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ C ≺ C ≺ C C ≺ C ≺ C C ∼ C ≺ C C ≺ C ≺ C C ∼ C ≺ C 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ D ≺ D ≺ D D ≺ D ≺ D D ≺ D ≺ D D ≺ D ≺ D D ∼ D ≺ D 1 2 3 1 2 3 2 1 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ E E E E E E E E ≺ E 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ F F ≺ F F ≺ F F F F ≺ F 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ G ≺ G ≺ G G ≺ G ≺ G G ≺ G ≺ G G ≺ G ≺ G G ≺ G ≺ G 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ H ≺ H ≺ H H ≺ H ≺ H H ≺ H ≺ H H ≺ H ≺ H H ≺ H ≺ H 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ I I ∼ I I ∼ I I I I ∼ I 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ J ≺ J ≺ J J ≺ J ≺ J J ≺ J ≺ J J ≺ J ≺ J J ≺ J ≺ J 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ K K ≺ K K ≺ K K ≺ K K ≺ K 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ L ≺ L L ≺ L L ≺ L L L L ≺ L 1 2 1 2 1 2 1 2 1 2 Example 3 The other examples in Fig.3 are all positive fuzzy numbers, so they can rank by other methods. In this case, Examples A, I, J, K and L of Tseng and Klein Fuzzy Inf. Eng. (2012) 3: 235-248 243 Fig. 2. The results of Example 2 244 R. Saneifard · T. Allahviranloo (2012) Table 3: The results of comparison using Tseng and Klein’s (1989) Examples. Tseng Kerre Lee Lee and Li Bass [22] Chen and Lu FN’s [9] uniform β = 10.50 A A 0.5 0.95 0.62 0.63 0.57 -0.04 0.005 0.05 A 0.5 0.95 0.6 0.6 0.53 B B 0.87 0.99 0.8 0.8 0.56 0.1 0.3 0.5 B 0.13 0.54 0.5 0.5 0.19 0.3 0.3 0.3 C C 0.87 1.0 0.7 0.7 0.56 0.3 0.3 0.3 C 0.13 0.55 0.4 0.4 0.19 D D 0.47 0.89 0.50 0.50 0.44 0.05 -0.03 -0.1 D 0.53 0.95 0.57 0.53 0.48 E E 0.49 0.45 0.5 0.5 0.36 0.0 0.0 0.0 E 0.51 0.96 0.53 0.50 0.39 F F 0.56 0.93 0.50 0.55 0.40 0.1 0.1 0.1 F 0.44 0.87 0.50 0.45 0.36 G G 0.50 0.90 0.50 0.50 0.38 -0.1 0.0 0.1 G 0.50 0.90 0.50 0.50 0.38 H H 0.52 1.00 0.40 0.40 0.29 0.02 0.02 0.02 H 0.48 0.98 0.39 0.39 0.28 I I 0.56 1.0 0.60 0.60 0.33 0.0 0.025 0.050 I 0.44 0.95 0.57 0.58 0.29 J J 0.64 1.0 0.60 0.60 0.38 0.0 0.075 0.150 J 0.36 0.85 0.53 0.52 0.29 K K 0.58 1.0 0.57 0.58 0.38 0.0 0.05 0.1 K 0.42 0.90 0.53 0.52 0.33 L L 0.52 1.0 0.60 0.60 0.57 0.07 0.05 0.00 L 0.48 0.96 0.60 0.60 0.44 (1989) are chosen to explain the results. They use other methods to explain the results of these methods and Table 3 shows the outcomes. We can easily see that most experimental results are consistent with other methods (Examples B, C, D, E, H, I, J and K). Because of the outcome in Table 3, the results of their method are reconciled with those of other methods except for Example A. In this example, Tseng and Klein (1989) and Kerre (1982) consider the two fuzzy numbers to be the same, but their method and Baldwin and Guild (1979) do not think so, both agree that A is larger than A and their difference is very small. Due to the different β of the Chen and Lu (2001) approach, we may get the results of ranking the reverse. Roughly, there is not much difference in the authors, method and theirs. Example 4 Consider the data used in [1], i.e. the three fuzzy numbers, A = (5, 6, 6, 7), ˜ ˜ B = (5.9, 6, 6, 7), C = (6, 6, 6, 7), as shown in Fig.4. According to Eq. (8), the ˜ ˜ ranking index values are obtained, i.e., RWD (A,τ ) = 0.138, RWD (B,τ ) = 2 min 2 min Fuzzy Inf. Eng. (2012) 3: 235-248 245 Fig. 3. The results of Example 3 246 R. Saneifard · T. Allahviranloo (2012) 0.148 and RWD (C,τ ) = 0.149. Accordingly, the ranking order of fuzzy numbers 2 min ˜ ˜ ˜ is C A. However, by Chu and Tsao’s approach [23], the ranking order is ˜ ˜ ˜ ˜ A. Meanwhile, using CV index proposed [20], the ranking order is A ˜ ˜ C. From Fig.4, it is easy to see that the ranking results obtained by the existing approaches [20], [23] are unreasonable and are not consistent with human intuition. ˜ ˜ ˜ On the other hand, in [1], the ranking result is C A, which is the same as the one obtained by the writers approach. However, their approach is simpler in the computation procedure. Based on the analysis results from [1], the ranking results using their approach and other approaches are listed in Table 4. ˜ ˜ ˜ Fig. 4 Fuzzy numbers A, B and C of Example 4 Table 4: Comparative results of Example 4. Fuzzy New Sign distance Chu-Tsao Cheng CV number approach p=2 distance index A 0.138 8.52 3 6.021 0.028 B 0.148 8.82 3.126 6.349 0.0098 C 0.149 8.85 3.085 6.3519 0.0089 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Results A ≺ B ≺ C A ≺ B ≺ C A ≺ C ≺ B A ≺ B ≺ C A ≺ C ≺ B Example 5 Consider the following set: ˜ ˜ ˜ A = (1, 2, 2, 5), B = (0, 3, 3, 4) and C = (2, 2.5, 2.5, 3) ˜ ˜ (see Fig.5). By using new approach, RWD (A,τ ) = 0.9962, RWD (B,τ ) = 2 min 2 min ˜ ˜ ˜ ˜ 0.9965 and RWD (C,τ ) = 0.9960. Hence, the ranking order is B C too. It 2 min seems that the result obtained by “Distance Minimization” method is unreasonable. To compare with some of the other methods in [23], the readers can refer to Table 5. ˜ ˜ Furthermore, in the mentioned example, RWD (−A,τ ) = 0.44, RWD (−B,τ ) = 2 min 2 min ˜ ˜ ˜ ˜ 0.43 and RWD (−C,τ ) = 0.50, consequently the ranking order is−C −A −B. 2 min Clearly, this proposed method has consistency in ranking fuzzy numbers and their images, which could not be guaranteed by CV-index method. Through Fig.5, it is easy to see that neither of them is consistent with human intuition. Fuzzy Inf. Eng. (2012) 3: 235-248 247 ˜ ˜ ˜ Fig. 5 Fuzzy numbers A, B and C of Example 5 Table 5: Comparative results of Example 5. New Sign distance Distance Chu-Tsao CV FN approach p=2 minimization (revisited) index A 0.9962 3.9157 2.5 0.74 0.32 B 0.9965 3.9157 2.5 0.74 0.36 C 0.9960 3.5590 2.5 0.75 0.08 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Results C ≺ A ≺ B C ≺ A ∼ B C ∼ A ∼ B A ∼ B ≺ C B ≺ A ≺ C 5. Conclusion In this paper, the researchers propose a modified defuzzification using weighted dis- tance between two fuzzy numbers and by using this, they have proposed a method for ranking fuzzy numbers. The method can effectively rank various fuzzy numbers and their images. From experimental results, the new method with some advantages: (a) without normalizing process, (b) fit all kind of ranking fuzzy number (without limitations), and (c) correct Kerre’s concept (regardless of it relative location on the X-axis). Therefore, we can apply the BWDM in practical examples. Acknowledgments Authors would like to thank referees for their helpful comments. References 1. Abbasbandy S, Asady B (2006) Ranking of fuzzy numbers by sign distance. Information Science 176: 2405-2416 2. Abbasbandy S, Hajjari T (2009) A new approach for ranking of trapezoidal fuzzy numbers. Computer and Mathematics with Appl. 57: 413-419 3. Asady B, Zendehnam A (2007) Ranking fuzzy numbers by distance minimization. Appl. Math. Model. 31: 2589-2598 4. Yager R R, Filev D P (1993) On the issue of defuzzification and selection based on a fuzzy set. Fuzzy Sets and Systems 55: 255-272 5. Filev D P, Yager R R (1991) A generalized defuzzification method under BAD distribution. Internat. J. Intelligent Systems 6: 687-697 6. Larkin L I (1985) A fuzzy logic controller for aircraft flight control. Industrial Applications of Fuzzy Control: 87-104 7. Ming M, Kandel A, Friedman M (2000) A new approach for defuzzification. Fuzzy Sets and Systems 111: 351-356 248 R. Saneifard · T. Allahviranloo (2012) 8. Liu X (2001) Measuring the satisfaction of constraints in fuzzy linear programing. Fuzzy Sets and Systems 122: 263-275 9. Wang X, Kerre E E (2001) Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems 118: 378-405 10. Saneifard R (2009) A method for defuzzification by weighted distance. International Journal of Industrial Mathematics 3: 209-217 11. Saneifard R (2009) Ranking L-R fuzzy numbers with weighted averaging based on levels. Interna- tional Journal of Industrial Mathematics 2: 163-173 12. Saneifard R, Allahviranloo T, Hosseinzadeh F, Mikaeilvand N (2007) Euclidean ranking DMUs with fuzzy data in DEA. Applied Mathematical Sciences 60: 2989-2998 13. Ezzati R, Saneifard R (2010) A new approach for ranking of fuzzy numbers with continuous weighted quasi-arithmetic means. Mathematical Sciences 4: 143-158 14. Kauffman A, Gupta M M (1991) Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York 15. Zimmermann H J (1991) Fuzzy sets theory and its applications. Kluwer Academic Press, Dordrecht 16. Grzegorzewski P, Winiarska K (2009) Weighted trapezoidal approximations of fuzzy numbers. Proc. IFSA-Eusflat 1531-1534 17. Chang J R, Cheng C H, Teng K H, Kuo C Y (2007) Selecting weapon system using relative distance metric method. Soft Computing 11: 573-584 18. Chang J R, Cheng C H, Kuo C Y (2006) Conceptual procedure for ranking fuzzy numbers based on adaptive two-dimensions dominance. Soft Computing 10: 74-103 19. Lee E C, Li R L (1988) Comparison of fuzzy numbers based on the probability measure of fuzzy events. Comput Math Appl. 105: 887-896 20. Cheng C H (1999) Ranking alternatives with fuzzy weights using maximizing set and minimizing set. Fuzzy Sets and System 105: 365-375 21. Chang W (1981) Ranking of fuzzy utilities with triangular membership function. Proceeding of the International Conference on Policy Analysis Information System 105: 263-272 22. Bass S M, Kwakernaak H (1977) Rating and ranking of multiple aspect alternatives using fuzzy sets. Automatica 13: 47-58 23. Chu T, Tsao C (2002) Ranking fuzzy numbers with an area between the centroid point and original point. Comput. Math. Appl. 43: 112-117 24. Chen S H (1985) Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets and Systems 17: 113-129 25. Deng Y, Zhu Z F, Liu Q (2006) Ranking fuzzy numbers with an area method using radius of gyration. Computers and Mathematics with Applications 51: 1127-1136 26. Wang Y J, Lee H S (2008) The revised method of ranking fuzzy numbers with an area between the centroid and original points. Computers and Mathematics with Applications 55: 2033-2042 27. Nasseri S H, Sohrabi M (2010) Ranking fuzzy numbers by using radius of gyration. Australian Journal of Basic and Applied Sciences 4: 658-664 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

A Comparative Study of Ranking Fuzzy Numbers Based on Regular Weighted Function

A Comparative Study of Ranking Fuzzy Numbers Based on Regular Weighted Function

Abstract

AbstractFuzzy systems have gained more and more attention from researchers and practitioners of various fields. In such systems, the output represented by a fuzzy set sometimes needs to be transformed into a scalar value, and this task is known as the defuzzification process. Several analytic methods have been proposed for this problem, but in this paper, the researchers suggest a modified approach to the problem of defuzzification, using the bi-symmetric weighted distance between two fuzzy...
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Fuzzy Inf. Eng. (2012) 3: 235-248 DOI 10.1007/s12543-012-0113-1 ORIGINAL ARTICLE A Comparative Study of Ranking Fuzzy Numbers Based on Regular Weighted Function R. Saneifard · T. Allahviranloo Received: 1 September 2010/ Revised: 30 March 2012/ Accepted: 10 July 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract Fuzzy systems have gained more and more attention from researchers and practitioners of various fields. In such systems, the output represented by a fuzzy set sometimes needs to be transformed into a scalar value, and this task is known as the defuzzification process. Several analytic methods have been proposed for this prob- lem, but in this paper, the researchers suggest a modified approach to the problem of defuzzification, using the bi-symmetric weighted distance between two fuzzy num- bers. This defuzzification can be used as a crisp approximation with respect to a fuzzy quantity. By considering this and with benchmark between fuzzy numbers, we can present a method for ranking, which can effectively rank various fuzzy numbers and their images and overcome the shortcomings of the previous techniques. After illustrating many numerical examples, following our procedure, the ranking results become valid. Keywords Fuzzy number· Defuzzification· Ranking· Bi-symmetric weighted dis- tance · Decision maker’s strategy. 1. Introduction As most modeling and controlling applications require crisp outputs, when applying fuzzy inference systems, the fuzzy system output A(y) usually has to be converted into a crisp output Y . This operation is called defuzzification. It is essentially a process guided by the output fuzzy subset of the model. One selects a single crisp value as the system output. In [4] and [5], the researchers have introduced a general R. Saneifard () Department of Applied Mathematics, Urmia Branch, Islamic Azad University, Oroumieh, Iran email: srsaneeifard@yahoo.com T. Allahviranloo Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 236 R. Saneifard · T. Allahviranloo (2012) approach to defuzzification based upon the basic defuzzification distributions (BDD) transformator. The output for the fuzzy controller F is a fuzzy subset of the real line; for simplicity the researchers shall assume the support set Y is finite, Y = {y ,··· , y }. 1 n For y ∈ Y, F(y ) = w indicates the degree to which each y is suggested as a good i i i i output value by the rule base under the current input. Two commonly used methods for defuzzification are the center of area (COA) method and the mean of maximum (MOM) method [6, 8]. In the COA method, one calculates the output of the defuzzi- Σ y w i i i fier y as : y = . In the MOM method, one calculates the output of the COA COA Σw controller y = Σ y , where A is the set of elements in Y which provides the MOM y ∈A i m i maximum value of F(y) and m is the cardinality of A. Based upon these observation, one can view the defuzzification process under the COA and MOM methods as first converting the fuzzy subset F of Y into a probability distribution on Y , in the spirit described in the earlier section and then taking the expected value as their output. Keeping with this probabilistic interpretation, the researchers shall in the following use P instead of q to denote the transformation values. Moreover, in [7, 10, 17], i i the researchers have used the concept of the symmetric triangular fuzzy number, and have introduced an approach to defuzzify a fuzzy number based on L -distance and weighted distance. In this paper, the researchers suggest a modified approach to the problem of defuzzification, using the bi-symmetric weighted distance between two fuzzy numbers. In this study, some preliminary results on properties of such defuzzi- fication are to be reported. Having reviewed the previous dictions, the researchers have two objectives in this study. Firstly, they want to introduce a modified defuzzi- fication of a fuzzy quantity. The second objective is applying the proposed method to ranking of fuzzy numbers (i.e., BWDM: bi-symmetric weighted distance method). In addition to its ranking features, this method removes the ambiguities resulted from the comparison of the previous rankings and overcomes the shortcomings. The paper is organized as follows: in Section 2, some fundamental results on fuzzy numbers are recalled. In Section 3, a crisp approximation of a fuzzy number is ob- tained. In this section, some remarks are proposed and illustrated. Also, the proposed method for ranking fuzzy numbers is in this section. Discussion and comparison of this work and other methods are carried out in Section 4. The paper ends with conclusions in Section 5. 2. Preliminaries The basic definitions of a fuzzy number are given in [14, 15, 26, 27] as follows˖ Definition 1 Let U be a universe set. A fuzzy set A of U is defined by a membership function μ (x) → [0, 1], where μ (x), ∀x ∈ U, indicates the degree of x in A. ˜ ˜ A A Definition 2 A fuzzy subset A of universe set U is normal iff supμ (x) = 1, where U x∈U is the universe set. ˜ ˜ Definition 3 A fuzzy set A is a fuzzy number iff A is normal and convex on U. Definition 4 A trapezoidal fuzzy number A is a fuzzy number with a membership Fuzzy Inf. Eng. (2012) 3: 235-248 237 functionμ ˜ defined by : x− a ⎪ 1 , when a ≤ x ≤ a , ⎪ 1 2 a − a ⎪ 2 1 1, when a ≤ x ≤ a , 2 3 μ (x) = (1) ˜ ⎪ ⎪ a − x ⎪ , when a ≤ x ≤ a , 3 4 a − a 4 3 0, otherwise, which can be denoted as a quartet (a , a , a , a ). In these above situations a , a , a 1 2 3 4 1 2 3 and a ,ifa = a , A becomes a triangular fuzzy number. 4 2 3 Definition 5 An extended fuzzy number A is described as any fuzzy subset of the universe set U with membership function μ defined as follows: (a) μ is a continuous mapping from U to the closed interval [0,ω], 0<ω ≤ 1. (b) μ (x) = 0 for all x ∈ (−∞, a ]. ˜ 1 (c) μ is strictly increasing on [a , a ]. 1 2 (d) μ (x) = ω, for all x ∈ [a , a ], as ω is a constant and 0<ω ≤ 1. ˜ 2 3 (e) μ is strictly decreasing on [a , a ]. 3 4 (f) μ (x) = 0 for all x ∈ [a ,+∞). ˜ 4 In these above situations a ,a ,a and a are real numbers. If a = a = a = a , 1 2 3 4 1 2 3 4 A becomes a crisp real number. Definition 6 The membership function μ ˜ of extended fuzzy number A is expressed by ⎪ μ (x), when a ≤ x ≤ a , 1 2 ⎪ A ω, when a ≤ x ≤ a , 2 3 μ (x) = (2) ˜ ⎪ ⎪ R μ (x), when a ≤ x ≤ a , ⎪ 3 4 ⎪ ˜ 0, otherwise, L R where μ :[a , a ] → [0,ω] and μ :[a , a ] → [0,ω]. Based on the basic theories 1 2 3 4 ˜ ˜ A A ˜ ˜ of fuzzy numbers, A is a normal fuzzy number if ω = 1, whereas A is a non-normal fuzzy number if 0 <ω ≤ 1. Therefore, the extended fuzzy number A in Defini- ˜ ˜ tion 6 can be denoted as (a , a , a , a ;ω). The image −Aof A can be expressed by 1 2 3 4 (−a ,−a ,−a ,−a ;ω) [14]. 4 3 2 1 Definition 7 The r-cut of a fuzzy number A, where 0 < r ≤ 1 is a set defined as A = {x∈| μ ˜ (x) ≥ r}. According to the definition of a fuzzy number, it is seen once that every r-cut of a fuzzy ˜ ˜ ˜ number is a closed interval. Hence, we have A = [A (r), A (r)], where r L R A (r) = inf{x∈|μ (x) ≥ r}, (3) L A A (r) = sup{x∈|μ (x) ≥ r}. (4) R A 1 1 Definition 8 A function f :[0, 1] → [0, 1] symmetric around , i.e., f ( − r) = 2 2 1 1 1 f ( +r) for all r ∈ [0, ], which reaches its minimum in , is called the bi-symmetrical 2 2 2 weighted function [16]. Moreover, the bi-symmetrical weighted function (BSWF) is called regular if 238 R. Saneifard · T. Allahviranloo (2012) 1) f ( ) = 0, 2) f (0) = f (1) = 1, 3) f (r)dr = . ˜ ˜ ˜ ˜ Definition 9 For two arbitrary fuzzy numbers A and B with r-cuts [A (r), A (r)] and L R ˜ ˜ [B (r), B (r)], respectively, the quantity L R 1 1 2 2 2 ˜ ˜ ˜ ˜ ˜ ˜ d (A, B) = f (r)[A (r)− B (r)] dr+ f (r)[A (r)− B (r)] dr , (5) w L L R R 0 0 where f :[0, 1] → [0, 1] is a bi-symmetrical (regular) weighted function is called the ˜ ˜ bi-symmetrical (regular) weighted distance between A and B based on f . One can, of course, propose many regular BSWF and hence obtain different bi- symmetrical weighted distances. Further on we will consider mainly a following function ⎪ 1 ⎨ 1− 2r, when r ∈ [0, ], f (r) = (6) 2r− 1, when r ∈ [ , 1]. 3. A Novel Fuzzy Ordering Method Ranking of fuzzy numbers was studied by many researchers. Some researchers in- troduced a distance and then compared the fuzzy numbers with it, (Bardossy et al 1992; Bortlan and Degani 2006; Cheng 1998; Tran and Duckstein 2002; Yao and Wu 2000). In recent years, many methods are proposed for ranking different types of fuzzy numbers [1-3, 9, 11, 12], and can be classified into four major classes: prefer- ence relation, fuzzy mean, spread fuzzy scoring, and linguistic expression. But each method appears to have advantages as well as disadvantages. In this section, we pro- pose a novel ranking fuzzy numbers method based on relative weighted distance, and ˜ ˜ ˜ verify this by some examples. In order to rank n fuzzy numbers A , A ,··· , A , let 1 2 n the fuzzy number B be zero in Eq. (5). Then the weighted distance is 1 2 2 2 ˜ ˜ ˜ d (A, 0) = f (r)((A (r)) + (A (r)) )dr . (7) w i i i L R Definition 10 Combining Eqs. (3), (4) and (7) and the concept of metric is the relative distance of p-norm [19]. We can induce the RWD (i.e., relative weighted ˜ ˜ distance of 2-norm) between fuzzy numbers A and B as follows: ˜ ˜ ˜ ˜ |A− B| d (A, B) ˜ ˜ RWD (A, B) = = . (8) 2 2 ˜ ˜ |A| +|B| 2 2 ˜ ˜ d (A, 0)+ d (B, 0) w w For ranking fuzzy numbers, this study firstly defines a minimum crisp valueμ (x) min and maximum crisp valueμ (x) to be the benchmark and its characteristic function max μ (x) andμ (x) is as follows: τ τ min max 1, when x = τ , min μ (x) = (9) τ ⎪ min ⎪ 0, when x  τ min Fuzzy Inf. Eng. (2012) 3: 235-248 239 and 1, when x = τ , ⎨ max μ (x) = (10) max ⎪ 0, when x  τ . max ˜ ˜ ˜ When ranking n fuzzy numbers A , A ,··· , A , the minimum crisp value τ and 1 2 n min maximum crisp valueτ are defined as: max ˜ ˜ ˜ τ = min{x|x ∈ Domain(A , A ,··· , A )}, (11) min 1 2 n ˜ ˜ ˜ τ = max{x|x ∈ Domain(A , A ,··· , A )}. (12) max 1 2 n The advantages of the definition of minimum and maximum crisp value are two- fold: firstly, the minimum and maximum crisp values will be obtained by themselves, besides it is easy to compute. The steps of BWDM algorithm are: Step 1: Computing the left and right inverse functions of each fuzzy number by Eqs. (3) and (4). Step 2: Using Eq. (11) to find minimum crisp value and computing its inverse func- tions to be the benchmark. Step 3: Computing RWD between fuzzy numbers and minimum crisp value (τ ) 2 min by Eq. (8). ˜ ˜ ˜ Example 1 Three fuzzy numbers A, B and C have been illustrated by Chen [24] and their membership functions are shown in Table 1. The inverse functions calculated by Eqs. (3) and (4) are also shown in this table. The fuzzy numbers and the minimum crisp value are illustrated in Fig.1. By Eqs. (3), (4), (11) and (12), this study obtains τ , τ and inverse functions min max as follows: ˜ ˜ ˜ τ = min{x|x ∈ Domain(A, B, C)} min = min{0.01, 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} = 0.01 and ˜ ˜ ˜ τ = max{x|x ∈ Domain(A, B, C)} max = max{0.01, 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} = 0.9 and ⎪ L g (x) = 0.01 min g (x) = min ⎪ R g (x) = 0.01 min and ⎪ L g (x) = 0.9, max g (x) = max ⎪ ⎪ R g (x) = 0.9. max According to Eq. (8), we can get the RWD values between minimum crisp value and ˜ ˜ ˜ fuzzy numbers A, B, and C, which are equal to 0.9836, 0.9818 and 0.9828, respec- tively. Also, the RWD values between maximum crisp value and fuzzy numbers A, ˜ ˜ B and C, which are equal to 0.44, 0.39 and 0.35, respectively. 240 R. Saneifard · T. Allahviranloo (2012) ˜ ˜ ˜ Table 1: Fuzzy numbers A, B and C. FN The membership function The inverse functions 100 1 ( )x+ , 0.01 ≤ x ≤ 0.4, ⎪ ⎪ ⎪ ⎪ ˜ 39 39 A (x) = 0.39x+ 0.01, ⎨ ⎨ L A μ (x) = g (x) = ˜ ⎪ ˜ ⎪ A A ⎪ ⎪ 1, 0.4 ≤ x ≤ 0.7, ⎪ ⎩ A (x) = −0.1x+ 0.8. −10x+ 8, 0.7 ≤ x ≤ 0.8. ⎪ 10 2 ⎪ ⎧ x− , 0.2 ≤ x < 0.5, ⎪ ⎪ ⎪ ⎪ 3 3 B (x) = 0.3x+ 0.2, ⎨ ⎨ L B μ (x) = g (x) = ˜ ˜ ⎪ ⎪ B B ⎪ 1, x = 0.5, ⎪ ⎪ ⎩ ˜ ⎪ B (x) = −0.4x+ 0.9. −2.5x+ 2.25, 0.5 < x ≤ 0.9. 2x− 0.2, 0.1 ≤ x ≤ 0.6, ⎪ ⎪ ⎪ ⎪ ˜ ⎨ ⎨ C (x) = 0.5x+ 0.1, C μ (x) = 1, x = 0.6, g (x) = ˜ ⎪ ˜ ⎪ C C ⎪ ⎪ ⎪ ⎩ C (x) = −0.2x+ 0.8. −5x+ 4, 0.6 ≤ x ≤ 0.8. ˜ ˜ ˜ Fig. 1 Fuzzy numbers A, B, C,τ andτ min max ˜ ˜ Definition 11 Let A and B be two fuzzy numbers characterized by Definition 2 and ˜ ˜ RWD (A, B) is the relative weighted distance of them. Since this article wants to approximate a fuzzy number by a scalar value, thus the researchers have to use an operator RWD : F→ which transforms fuzzy numbers into a family of real line. Operator RWD is a crisp approximation operator. Since ever above defuzzification can be used as a crisp approximation of a fuzzy number, therefore the resultant value is used to rank the fuzzy numbers. Thus, RWD is used to rank fuzzy numbers. The larger RWD , the larger fuzzy number. ˜ ˜ ˜ ˜ Let A,B ∈ F be two arbitrary fuzzy numbers. Define the ranking of A and B by RWD on F as follows: ˜ ˜ ˜ ˜ 1) RWD (A,τ ) > RWD (B,τ ) if only if A B, 2 min 2 min ˜ ˜ ˜ ˜ 2) RWD (A,τ ) < RWD (B,τ ) if only if A ≺ B, 2 min 2 min Fuzzy Inf. Eng. (2012) 3: 235-248 241 ˜ ˜ ˜ ˜ 3) RWD (A,τ ) = RWD (B,τ ) if only if A ∼ B. 2 min 2 min ˜ ˜ ˜ ˜ ˜ ˜ Then, this article formulate the order and as A B if and only if A B or A ∼ B, ˜ ˜ ˜ ˜ ˜ ˜ A  B if and only if A ≺ B or A ∼ B. ˜ ˜ ˜ Remark 1 If inf supp(A) ≥ 0, then RWD (A,τ ) ≥ 0, (RWD (A,τ ) ≥ 0). 2 min 2 max ˜ ˜ ˜ Remark 2 If sup supp(A) ≤ 0, then RWD (A,τ ) ≥ 0, (RWD (A,τ ) ≥ 0). 2 min 2 max Remark 3 Ifτ is equal to zero, then we obtain the same ordering result, for exam- min ˜ ˜ ple, if A = (0, 1, 1, 2), B = (3, 4, 4, 5) are two fuzzy numbers, then we have τ = 0, min ˜ ˜ ˜ ˜ RWD (A,τ ) = 1 and RWD (B,τ ) = 1, i.e., A ∼ B which is unreasonable result. 2 min 2 min In this case, we will use the τ instead τ , then for two arbitrary fuzzy numbers max min ˜ ˜ ˜ ˜ A,B ∈ F, we define the ranking of A and B by RWD on F as follows: ˜ ˜ ˜ ˜ 1) RWD (A,τ ) < RWD (B,τ ) if only if A B, 2 max 2 max ˜ ˜ ˜ ˜ 2) RWD (A,τ ) > RWD (B,τ ) if only if A ≺ B, 2 max 2 max ˜ ˜ ˜ ˜ 3) RWD (A,τ ) = RWD (B,τ ) if only if A ∼ B. 2 max 2 max Here, the following reasonable axioms that Wang and Kerre [9] have proposed for fuzzy quantities ranking are considered. Let RD be an ordering method, S the set of fuzzy quantities for which the method RD can be applied, andA andA finite subsets ˜ ˜ ˜ of S . The statement two elements A and B in A satisfy that A has a higher ranking ˜ ˜ ˜ than B when RD is applied to the fuzzy quantities in A will be written as A B by ˜ ˜ ˜ ˜ RD on A. A ∼ B by RD on A, and A B by RD on A are similarly interpreted. The following axioms show the reasonable properties of the ordering approach RD. ˜ ˜ ˜ A . For A∈A, A  AbyRDonA. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B)∈A , A  B and B  AbyRDon A, we should have A ∼ Bby RD on A. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B, C) ∈A , A  B and B  Cby RD on A, we should have A  Cby RD on A. ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B) ∈A , in f supp(B)>sup supp(A), we should have A  Bby RD on A. ˜ ˜ ˜ ˜ ˜ ˜ A . For (A, B) ∈A , in f supp(B)>sup supp(A), we should have A ≺ Bby RD on A. ˜ ˜ ˜ ˜ A . Let (A, B) ∈ (A∩A ) . We obtain the ranking order A  Bby RD on A if and ˜ ˜ only if A  Bby RD onA. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A . Let A, B, AC and BC be elements of S and C ≥ 0.If A  Bby RD on {A, B}, ˜ ˜ ˜ ˜ then AC  BC by RD on{AC, BC}. ˜ ˜ ˜ ˜ Remark 4 If A  B, then−A −B. Hence, this article can infer ranking order of the images of the fuzzy numbers. 4. Examples 242 R. Saneifard · T. Allahviranloo (2012) First of all, this study validates their proposed method with representative examples of [4, 13, 18, 19, 21, 22, 25] with some advantages. Example 2 The RWD values of 12 examples are shown in Fig.2. Table 2 shows the ranking results. From this table, the main findings and BWDM with some advantages show as follows: 1) From Example L, K, some methods use complicated and normalized process to rank and they can’t obtain consistent results. However, their proposed method is more suitable for ranking any kind of fuzzy number without normalization process. 2) Under fuzzy numbers with the same mean (Examples B, I ), Yager [4], Kerre [9], Bass and Kwakernaak [22] have not been able to obtain their orderings. Chang’s method [21] has been able to rank their orderings, but Chang’s results violate the smaller spread, the higher ranking order. In Examples B, I, it obviously shows that the researchers, proposed method can rank instantly and their results comply with intuition of human being (as Table 2). 3) In Examples C, D, L, we can see that the method of Kerre [9], Bass and Kwaker- naak [22] have many limitations on triangle, trapezoid, non-normalized fuzzy num- bers and so on. 4) The proposed method can be used for ranking fuzzy numbers and crisp values. But Yager has not been able to handle the crisp value problem [4]. 5) Kerre’s method would favor a fuzzy number with smaller area measurement, re- gardless of its relative location on the X-axis [9]. The results are against their intuition in Examples C,D. From Table 2, their proposed ranking method can correct the prob- lem. Table 2: The comparison with different ranking approaches. New method Yager [4] Kerre [9] Chang [21] Bass [22] ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A ≺ A A ≺ A A ≺ A A ≺ A A ≺ A 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ B ≺ B B ∼ B B ≺ B B B B ∼ B 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ C ≺ C ≺ C C ≺ C ≺ C C ∼ C ≺ C C ≺ C ≺ C C ∼ C ≺ C 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ D ≺ D ≺ D D ≺ D ≺ D D ≺ D ≺ D D ≺ D ≺ D D ∼ D ≺ D 1 2 3 1 2 3 2 1 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ E E E E E E E E ≺ E 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ F F ≺ F F ≺ F F F F ≺ F 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ G ≺ G ≺ G G ≺ G ≺ G G ≺ G ≺ G G ≺ G ≺ G G ≺ G ≺ G 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ H ≺ H ≺ H H ≺ H ≺ H H ≺ H ≺ H H ≺ H ≺ H H ≺ H ≺ H 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ I I ∼ I I ∼ I I I I ∼ I 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ J ≺ J ≺ J J ≺ J ≺ J J ≺ J ≺ J J ≺ J ≺ J J ≺ J ≺ J 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ K K ≺ K K ≺ K K ≺ K K ≺ K 1 2 1 2 1 2 1 2 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ L ≺ L L ≺ L L ≺ L L L L ≺ L 1 2 1 2 1 2 1 2 1 2 Example 3 The other examples in Fig.3 are all positive fuzzy numbers, so they can rank by other methods. In this case, Examples A, I, J, K and L of Tseng and Klein Fuzzy Inf. Eng. (2012) 3: 235-248 243 Fig. 2. The results of Example 2 244 R. Saneifard · T. Allahviranloo (2012) Table 3: The results of comparison using Tseng and Klein’s (1989) Examples. Tseng Kerre Lee Lee and Li Bass [22] Chen and Lu FN’s [9] uniform β = 10.50 A A 0.5 0.95 0.62 0.63 0.57 -0.04 0.005 0.05 A 0.5 0.95 0.6 0.6 0.53 B B 0.87 0.99 0.8 0.8 0.56 0.1 0.3 0.5 B 0.13 0.54 0.5 0.5 0.19 0.3 0.3 0.3 C C 0.87 1.0 0.7 0.7 0.56 0.3 0.3 0.3 C 0.13 0.55 0.4 0.4 0.19 D D 0.47 0.89 0.50 0.50 0.44 0.05 -0.03 -0.1 D 0.53 0.95 0.57 0.53 0.48 E E 0.49 0.45 0.5 0.5 0.36 0.0 0.0 0.0 E 0.51 0.96 0.53 0.50 0.39 F F 0.56 0.93 0.50 0.55 0.40 0.1 0.1 0.1 F 0.44 0.87 0.50 0.45 0.36 G G 0.50 0.90 0.50 0.50 0.38 -0.1 0.0 0.1 G 0.50 0.90 0.50 0.50 0.38 H H 0.52 1.00 0.40 0.40 0.29 0.02 0.02 0.02 H 0.48 0.98 0.39 0.39 0.28 I I 0.56 1.0 0.60 0.60 0.33 0.0 0.025 0.050 I 0.44 0.95 0.57 0.58 0.29 J J 0.64 1.0 0.60 0.60 0.38 0.0 0.075 0.150 J 0.36 0.85 0.53 0.52 0.29 K K 0.58 1.0 0.57 0.58 0.38 0.0 0.05 0.1 K 0.42 0.90 0.53 0.52 0.33 L L 0.52 1.0 0.60 0.60 0.57 0.07 0.05 0.00 L 0.48 0.96 0.60 0.60 0.44 (1989) are chosen to explain the results. They use other methods to explain the results of these methods and Table 3 shows the outcomes. We can easily see that most experimental results are consistent with other methods (Examples B, C, D, E, H, I, J and K). Because of the outcome in Table 3, the results of their method are reconciled with those of other methods except for Example A. In this example, Tseng and Klein (1989) and Kerre (1982) consider the two fuzzy numbers to be the same, but their method and Baldwin and Guild (1979) do not think so, both agree that A is larger than A and their difference is very small. Due to the different β of the Chen and Lu (2001) approach, we may get the results of ranking the reverse. Roughly, there is not much difference in the authors, method and theirs. Example 4 Consider the data used in [1], i.e. the three fuzzy numbers, A = (5, 6, 6, 7), ˜ ˜ B = (5.9, 6, 6, 7), C = (6, 6, 6, 7), as shown in Fig.4. According to Eq. (8), the ˜ ˜ ranking index values are obtained, i.e., RWD (A,τ ) = 0.138, RWD (B,τ ) = 2 min 2 min Fuzzy Inf. Eng. (2012) 3: 235-248 245 Fig. 3. The results of Example 3 246 R. Saneifard · T. Allahviranloo (2012) 0.148 and RWD (C,τ ) = 0.149. Accordingly, the ranking order of fuzzy numbers 2 min ˜ ˜ ˜ is C A. However, by Chu and Tsao’s approach [23], the ranking order is ˜ ˜ ˜ ˜ A. Meanwhile, using CV index proposed [20], the ranking order is A ˜ ˜ C. From Fig.4, it is easy to see that the ranking results obtained by the existing approaches [20], [23] are unreasonable and are not consistent with human intuition. ˜ ˜ ˜ On the other hand, in [1], the ranking result is C A, which is the same as the one obtained by the writers approach. However, their approach is simpler in the computation procedure. Based on the analysis results from [1], the ranking results using their approach and other approaches are listed in Table 4. ˜ ˜ ˜ Fig. 4 Fuzzy numbers A, B and C of Example 4 Table 4: Comparative results of Example 4. Fuzzy New Sign distance Chu-Tsao Cheng CV number approach p=2 distance index A 0.138 8.52 3 6.021 0.028 B 0.148 8.82 3.126 6.349 0.0098 C 0.149 8.85 3.085 6.3519 0.0089 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Results A ≺ B ≺ C A ≺ B ≺ C A ≺ C ≺ B A ≺ B ≺ C A ≺ C ≺ B Example 5 Consider the following set: ˜ ˜ ˜ A = (1, 2, 2, 5), B = (0, 3, 3, 4) and C = (2, 2.5, 2.5, 3) ˜ ˜ (see Fig.5). By using new approach, RWD (A,τ ) = 0.9962, RWD (B,τ ) = 2 min 2 min ˜ ˜ ˜ ˜ 0.9965 and RWD (C,τ ) = 0.9960. Hence, the ranking order is B C too. It 2 min seems that the result obtained by “Distance Minimization” method is unreasonable. To compare with some of the other methods in [23], the readers can refer to Table 5. ˜ ˜ Furthermore, in the mentioned example, RWD (−A,τ ) = 0.44, RWD (−B,τ ) = 2 min 2 min ˜ ˜ ˜ ˜ 0.43 and RWD (−C,τ ) = 0.50, consequently the ranking order is−C −A −B. 2 min Clearly, this proposed method has consistency in ranking fuzzy numbers and their images, which could not be guaranteed by CV-index method. Through Fig.5, it is easy to see that neither of them is consistent with human intuition. Fuzzy Inf. Eng. (2012) 3: 235-248 247 ˜ ˜ ˜ Fig. 5 Fuzzy numbers A, B and C of Example 5 Table 5: Comparative results of Example 5. New Sign distance Distance Chu-Tsao CV FN approach p=2 minimization (revisited) index A 0.9962 3.9157 2.5 0.74 0.32 B 0.9965 3.9157 2.5 0.74 0.36 C 0.9960 3.5590 2.5 0.75 0.08 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Results C ≺ A ≺ B C ≺ A ∼ B C ∼ A ∼ B A ∼ B ≺ C B ≺ A ≺ C 5. Conclusion In this paper, the researchers propose a modified defuzzification using weighted dis- tance between two fuzzy numbers and by using this, they have proposed a method for ranking fuzzy numbers. The method can effectively rank various fuzzy numbers and their images. From experimental results, the new method with some advantages: (a) without normalizing process, (b) fit all kind of ranking fuzzy number (without limitations), and (c) correct Kerre’s concept (regardless of it relative location on the X-axis). Therefore, we can apply the BWDM in practical examples. Acknowledgments Authors would like to thank referees for their helpful comments. References 1. Abbasbandy S, Asady B (2006) Ranking of fuzzy numbers by sign distance. Information Science 176: 2405-2416 2. Abbasbandy S, Hajjari T (2009) A new approach for ranking of trapezoidal fuzzy numbers. Computer and Mathematics with Appl. 57: 413-419 3. Asady B, Zendehnam A (2007) Ranking fuzzy numbers by distance minimization. Appl. Math. Model. 31: 2589-2598 4. Yager R R, Filev D P (1993) On the issue of defuzzification and selection based on a fuzzy set. Fuzzy Sets and Systems 55: 255-272 5. Filev D P, Yager R R (1991) A generalized defuzzification method under BAD distribution. Internat. J. Intelligent Systems 6: 687-697 6. Larkin L I (1985) A fuzzy logic controller for aircraft flight control. Industrial Applications of Fuzzy Control: 87-104 7. Ming M, Kandel A, Friedman M (2000) A new approach for defuzzification. Fuzzy Sets and Systems 111: 351-356 248 R. Saneifard · T. Allahviranloo (2012) 8. Liu X (2001) Measuring the satisfaction of constraints in fuzzy linear programing. Fuzzy Sets and Systems 122: 263-275 9. Wang X, Kerre E E (2001) Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems 118: 378-405 10. Saneifard R (2009) A method for defuzzification by weighted distance. International Journal of Industrial Mathematics 3: 209-217 11. Saneifard R (2009) Ranking L-R fuzzy numbers with weighted averaging based on levels. Interna- tional Journal of Industrial Mathematics 2: 163-173 12. Saneifard R, Allahviranloo T, Hosseinzadeh F, Mikaeilvand N (2007) Euclidean ranking DMUs with fuzzy data in DEA. Applied Mathematical Sciences 60: 2989-2998 13. Ezzati R, Saneifard R (2010) A new approach for ranking of fuzzy numbers with continuous weighted quasi-arithmetic means. Mathematical Sciences 4: 143-158 14. Kauffman A, Gupta M M (1991) Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York 15. Zimmermann H J (1991) Fuzzy sets theory and its applications. Kluwer Academic Press, Dordrecht 16. Grzegorzewski P, Winiarska K (2009) Weighted trapezoidal approximations of fuzzy numbers. Proc. IFSA-Eusflat 1531-1534 17. Chang J R, Cheng C H, Teng K H, Kuo C Y (2007) Selecting weapon system using relative distance metric method. Soft Computing 11: 573-584 18. Chang J R, Cheng C H, Kuo C Y (2006) Conceptual procedure for ranking fuzzy numbers based on adaptive two-dimensions dominance. Soft Computing 10: 74-103 19. Lee E C, Li R L (1988) Comparison of fuzzy numbers based on the probability measure of fuzzy events. Comput Math Appl. 105: 887-896 20. Cheng C H (1999) Ranking alternatives with fuzzy weights using maximizing set and minimizing set. Fuzzy Sets and System 105: 365-375 21. Chang W (1981) Ranking of fuzzy utilities with triangular membership function. Proceeding of the International Conference on Policy Analysis Information System 105: 263-272 22. Bass S M, Kwakernaak H (1977) Rating and ranking of multiple aspect alternatives using fuzzy sets. Automatica 13: 47-58 23. Chu T, Tsao C (2002) Ranking fuzzy numbers with an area between the centroid point and original point. Comput. Math. Appl. 43: 112-117 24. Chen S H (1985) Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets and Systems 17: 113-129 25. Deng Y, Zhu Z F, Liu Q (2006) Ranking fuzzy numbers with an area method using radius of gyration. Computers and Mathematics with Applications 51: 1127-1136 26. Wang Y J, Lee H S (2008) The revised method of ranking fuzzy numbers with an area between the centroid and original points. Computers and Mathematics with Applications 55: 2033-2042 27. Nasseri S H, Sohrabi M (2010) Ranking fuzzy numbers by using radius of gyration. Australian Journal of Basic and Applied Sciences 4: 658-664

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Sep 1, 2012

Keywords: Fuzzy number; Defuzzification; Ranking; Bi-symmetric weighted distance; Decision maker's strategy

References