Revision of distance minimization method for ranking of fuzzy numbers

Applied Mathematical Modelling 35 (2011) 1306–1313

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Revision of distance minimization method for ranking of fuzzy numbers B. Asady Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran

a r t i c l e

i n f o

Article history: Received 15 April 2009 Received in revised form 15 August 2010 Accepted 1 September 2010 Available online 24 September 2010 Keywords: Defuzzification Fuzzy number Ranking Distance minimization Epsilon – neighborhood

a b s t r a c t Asady and Zendehnam employed ‘‘distance minimization” to ranking fuzzy numbers in Ref [1]. Then Abbasbandy and Hajjari in [2] found a problem of its. To overcome it problem, they proposed magnitude method to ranking fuzzy numbers. Unfortunately, their method can not to overcome this problem. In this paper, we want to indicate this problem and then propose a revise method of distance minimization method which can avoid problem for ranking fuzzy numbers. Since the revised method is based on the distance minimization method, it is easy to rank fuzzy numbers in a way similar to the original method. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction In many applications, ranking of fuzzy numbers is an important component of the decision making process. In practice, many real-world problems require handling and evaluation of fuzzy data for making decision. To evaluate and compare different alternatives, it is necessary to rank fuzzy numbers. In addition, the concept of optimum or best choice is completely based on ranking or comparison [3]. In addition, Dubois and Prade [4] introduced the relevant concepts of fuzzy numbers, many researches proposed the related methods or applications for ranking fuzzy numbers. For instance, Bortolan and Degani [5] reviewed some methods to rank fuzzy numbers in 1985, Chen and Hwang [6] proposed fuzzy multiple attribute decision making in 1992, Choobineh and Li [7] proposed an index for ordering fuzzy numbers in 1993, Dias [8] ranked alternatives by ordering fuzzy numbers in 1993, Lee et al. [9] ranked fuzzy numbers with a satisfaction function in 1994, Requena et al. [10] utilized artificial neural networks for the automatic ranking of fuzzy numbers in 1994, Fortemps and Roubens [11] presented ranking and defuzzication methods based on area compensation in 1996, and Lee and Li investigated a method for ranking fuzzy numbers based on the uniform and proportional probability distributions. Cheng [12] defined the coefficient variance to improve Lee and Li’s ranking approach. Chu and Tsao found some problems in Cheng’s method and proposed a method to ranking fuzzy number. They employed an area between the centroid and original points to rank fuzzy numbers; however there were some problems with the ranking method [13]. Wang and Lee revised Chu and Tsao’s method which can avoid these problems [14]. Furthermore, Abbasbandy and Asady considered a fuzzy origin for fuzzy numbers and then according to the distance of fuzzy numbers with respect to this origin, they rank them [15]. Asady and Zendehnam proposed a defuzzification method using minimizer of the distance between the two fuzzy numbers [1]. Regarding to several strategies having been reviewed above, these strategies are based on methods including distance between fuzzy sets, weighted mean value, coefficient of variation, centroid point and distance minimization. Since each of these techniques has some problems and they are not completed. For instance, some methods, properly ranking fuzzy numbers, are not able to correctly rank fuzzy number images (see Example 1). In addition, when a fuzzy number is covered

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B. Asady / Applied Mathematical Modelling 35 (2011) 1306–1313

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by another fuzzy number, most ranking methods confront several difficulties. Recently, Asady and Zendehnam [1] employed ‘‘Distance Minimization” to rank fuzzy numbers. Then Abbasbandy and Hajjari in [2] found a problem of its. So that, for all triangular fuzzy numbers u(x0, r, b) where x0 ¼ rb and also trapezoidal fuzzy numbers u(x0, y0, r, b) such that x0 þ y0 ¼ rb , 4 2 give the same results. However, it is clear that these fuzzy numbers do not place in an equivalence class. To overcome above problem, they proposed magnitude method to rank fuzzy numbers. Unfortunately, their method cannot overcome this probb lem. Because, for all fuzzy numbers u(x0, r, b) or v(x0, y0, r, b) where x0 ¼ r12 and/or x0 þ y0 ¼ rb , their method give the same 6 results (see set 1 in Example 2). To overcome the above-mention problem, we propose a revise of distance minimization method ranking fuzzy numbers by using the epsilon – neighborhood of fuzzy numbers. Such that a few value of the left and right boundary by an arbitrary epsilon – neighborhood of fuzzy number are deleted. Then, the fuzzy numbers are ranked again by distance Minimization [1]. In Section 2, we recall some fundamental results on fuzzy numbers. Section 3 includes, introduce the ranking approach by the epsilon – neighborhood of the fuzzy numbers and describes some examples and useful properties. Last section is containing conclusion. 2. Preliminaries Definition 1. A fuzzy number is a fuzzy set like A : R ! I ¼ ½0; 1 which satisfies: 1. A is continuous, 2. A(x) = 0 outside some interval [c, d], 3. There are real numbers a, b such that c 6 a 6 b 6 d and 3.1 A(x) is increasing on [c, a], 3.2 A(x) is decreasing on [b, d], 3.3 A(x) = 1, a 6 x 6 b. The set of all these fuzzy numbers is denoted by F(R). An equivalent parametric definitions is also given in [16] as follows: Definition 2. A fuzzy number A in parametric form is a pair ðA; AÞ of functions AðrÞ; AðrÞ; 0 6 r 6 1, which satisfies the following requirements: 1. A(r) is a bounded increasing continuous function, 2. AðrÞ is a bounded decreasing continuous function, 3. AðrÞ 6 AðrÞ; 0 6 r 6 1: A popular fuzzy number is the trapezoidal fuzzy number A(x0, y0, r, b)(L,R) with two defuzzifiers x0, y0 and left fuzziness r and right fuzziness b where the membership function is

8  xx0 þr L > > r > > > > > <1 AðxÞ ¼  R > y0 xþb > > b > > > > : 0

x 0  r 6 x 6 x0 ; ½x0 ; y0 ; y0 6 x 6 y0 þ b; otherwise;

where L, R > 0, is denoted by A = (x0, y0, r, b)(L,R) and if L = R by A = (x0, y0, r, b)L and if L = R = 1 with A = (x0, y0, r, b) . Its parametric form is 1

AðrÞ ¼ x0  r þ rr L ;

1

AðrÞ ¼ y0 þ b  br R :

Set support function is defined as follows:

SðAÞ ¼ fxjAðxÞ > 0g; which that fxjAðxÞ > 0g is closure of set {xjA(x) > 0}. The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows: For arbitrary A ¼ ðA; AÞ; B ¼ ðB; BÞ and k > 0 we define addition (A + B) and multiplication by scaler k as

ðA þ BÞðrÞ ¼ AðrÞ þ BðrÞ; ðkAÞðrÞ ¼ kAðrÞ;

ðA þ BÞðrÞ ¼ AðrÞ þ BðrÞ;

ðkAÞðrÞ ¼ kAðrÞ:

ð1Þ ð2Þ

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B. Asady / Applied Mathematical Modelling 35 (2011) 1306–1313

Definition 3. For arbitrary fuzzy numbers A ¼ ðA; AÞ and B ¼ ðB; BÞ the quantity

DðA; BÞ ¼

Z

1

ðAðrÞ  BðrÞÞ2 dr þ

0

Z

1

 2 1=2 AðrÞ  BðrÞ dr ;

ð3Þ

0

is the distance between A and B. The function D(A, B) is a metric in F(R) and is a particular member of the family of distances dp,q defined as follows:

Z

dp;q ðA; BÞ ¼

1

ð1  qÞjAðrÞ  Bðr Þjp dr þ

0

Z 0

1

 p 1=p   qAðrÞ  BðrÞ dr ;

ð4Þ

where 1 6 p 6 1 and 0 6 q 6 1, for more details see [17,1]. Definition 4. For any the fuzzy number A 2 F(R) and

 2 [0, 1], function

f : FðRÞ  ½; 1 ! FðRÞ; f ðA; Þ ¼ A ¼ ðA ; A Þ ¼ ðA; AÞv½;1

ð5Þ

is called epsilon – neighborhood of the fuzzy number A in which that v is chi function. Hence parametric forms of epsilon – neighborhood for any arbitrary fuzzy number A are as follows

A ðrÞ ¼ Av½;1 ¼

(

A ðrÞ ¼ Av½;1 ¼ Clearly, for any arbitrary

AðrÞ;

if w < r 6 1;

AðÞ; if 0 6 r 6 w; AðrÞ;

if

 < r 6 1;

AðÞ; if 0 6 r 6 :

 2 [0, 1] function A is a fuzzy number.

Theorem 1. Let A = (x0, y0, r, b)(L,R) be a trapezoidal fuzzy number then parametric form of the epsilon – neighborhood of it is as following:

A ¼ ðA ; A Þ; in which

(

A ðrÞ ¼

1

x0  r þ r r L ;

if

 < r 6 1;

1

( A ðrÞ ¼

x0  r þ rL ; if 0 6 r 6 ; 1

y0 þ b  br R ; 1 R

if

y0 þ b  b ; if

 < r 6 1; 0 6 r 6 :

Proof. From Definitions 2 and 4 the proof is obvious. h

3. Ranking of fuzzy numbers In this section, we will propose a revised of distance minimization method [1] that it is called the best approximation epsilon – neighborhood of a fuzzy number. 3.1. Distance minimization Asady and Zendehnam in [1] proposed a defuzzification using minimizer of the distance between two the fuzzy number. They introduced distance minimization of a fuzzy number A that denoted by M(A) which was defined as follows (see ref [1])

MðAÞ ¼

1 2

Z

1

ðAðrÞ þ AðrÞÞdr:

ð6Þ

0

Consequently, Asady and Zendehnam ranked fuzzy numbers according to the following relations.

MðAÞ > MðBÞ if and only if A  B; MðAÞ ¼ MðBÞ if and only if A  B; MðAÞ < MðBÞ if and only if A  B: They have shown that the crisp approximation M(A) is the best approximation crisp of the fuzzy number A with respect to metric D and is called distance minimization (see Remark 1).

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Remark 1. Let A be a fuzzy number and c a crisp then the function D(A, c) (3) with respect to c is minimum value if c = M(A) and M(A) is unique (see [1]). 3.2. Revision of distance minimization method By Remark 1 in sub Section 3.1, we can say that the best crisp approximation of the epsilon – neighborhood A is as follows

MðA Þ ¼ for all

1 2

Z

1

ðA ÞðrÞ þ A ðrÞÞdr

ð7Þ

0

 2 [0, 1]. Therefore, we get following theorem.

Theorem 2. If A = (x0, y0, r, b)(L,R) be a trapezoidal fuzzy number, then the best approximation epsilon – neighborhood of it is as follows:

MðA Þ ¼

    Rþ1 b r L þ Lþ1 L ðx0 þ y0 Þ þ ðb  rÞ þ 1þL Rþ R  1þR 2

:

ð8Þ

Proof. From Definition 4 and Eq. (7) proof is obvious. Now, we will propose a revision of distance minimization method by following function

MðA Þ ¼

1 2

Z

1

  A ðrÞ þ A ðrÞ dr;

0

in which that A ¼ ðA ðrÞ; A ðrÞÞ is the epsilon – neighborhood of fuzzy number A. As a decision maker can be ranked a pair of the fuzzy numbers, A and B, by using of the following: (1.) If M(A) > M(B), then A  B. (2.) If M(A) < M(B), then A  B. (3.) If M(A) = M(B), then

if MðA Þ > MðB Þ then A  B; else if MðA Þ < MðB Þ; then A  B; else A  B. In short, we rank A and B based on M(A) and M(B) values if they are different. In the case that they are equal, we further compare M(A) and M(B) values to form their ranks. Further, if M(A) P M(B) based on M(A) = M(B), then A B. h Properties 1. Assume that there are n different L  R fuzzy numbers, A1, A2, . . . , An. S is the support set of these fuzzy numbers. Let Ai, Aj, Ak and Ap be any four arbitrary L  R fuzzy numbers, i – j – k – p and 1 6 i, j, k, p 6 n. Then, there are the following properties: (p1) (p2) (p3) (p4) (p5)

If If If If If

Ai  Aj and Aj  Ak then Ai  Ak. inf S(Ai) > sup S(Aj) then Ai  Aj. Ai  Aj then Ai  Aj. Ai  Aj and Ak  Ap then Ai + Ak  Aj + Ap. Ai = (xi, yi, ri, bi) and Aj = (xi, yi, ri + k, bi + k) then Ai  Aj.

Proof. It is easy to verifies of P1 until P5. For instance, we proof part P5. P5 : Let Ai = (xi, yi, ri, bi) and Aj = (xi, yi, ri + k, bi + k) be two trapezoidal fuzzy numbers therefore, L = R = 1 and by using Eq. (8) we have

ðxi þ yi Þ þ ðbi þ k  ðri þ kÞÞ þ ri2þk ð1 þ 2 Þ  bi2þk ð1 þ 2 Þ 2 ðxi þ yi Þ þ ðbi  ri Þ þ r2i ð1 þ 2 Þ  b2i ð1 þ 2 Þ ¼ ¼ MðAi Þ 2

MðAj Þ ¼

for all

 2 [0, 1]. Or equality, we get Ai  Aj.

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B. Asady / Applied Mathematical Modelling 35 (2011) 1306–1313

To present the rationality and necessity of this revision of Asady and Zendehnams method, some examples are proposed to illustrate these methods and compared with others methods in [1,2,14,18–20]. h Example 1. Consider the three triangular fuzzy number A = (2, 1, 4), B = (2.75, 0.25, 0.25), and C = (3, 1, 1). By Wang et al. method [20] we have d1 = 0.4167, d2 = 0, and d3 = 1.667 where the ranking order is B  A  C. From this, we can logically infer the ranking order of the images of these fuzzy numbers as B  A  C. However, by the Wang et al. [20] method the ranking order is A  C  B. Therefore, Wang et al. method has shortcoming. Since their method is not able to correctly rank fuzzy numbers images. On the other hand, we rank these fuzzy numbers with the revised method.

MðAÞ ¼ MðBÞ ¼ 2:75 and MðCÞ ¼ 3: Since M(A) = M(B) also, we have to compare M(A) with M(B), where

MðA Þ ¼ 2:75  0:752 ;

MðB Þ ¼ 2:75:

Consequently, we have M(A) < M(B) for all

 2 (0, 1]. Therefore, the ranking order is

A  B  C: Oppositely,

MðAÞ ¼ MðBÞ ¼ 2:75 and MðCÞ ¼ 3: Hence

MðA Þ ¼ 2:75 þ 0:752 ;

MðB Þ ¼ 2:75:

So, we have

A  B  C: Clearly, our method can overcome the shortcomings of the inconsistency of Wang’s method in ranking fuzzy numbers and their images. Example 2. Consider four sets follows: Set Set Set Set

1: 2: 3: 4:

A = (1, 13, 1), B(1/12, 2, 1) and C = (0, 1, 6, 0). A = (1, 5, 1), B(1/4, 2, 1) and C = (2, 9,1) in Ref. [2].  pffiffi A = (1/3, 2, 1), B(0, 0.1, 0.1) and C ¼ 0; 1=3; 33 ; 0 . A = (2, 7, 1), B(1, 0, 2) and C = (0, 1, 1).

In sets 1, 3 and 4 ranking order by our method is same as distance minimization in Table 1. But for set 2 by our method

MðAÞ ¼ MðBÞ ¼ MðCÞ ¼ 0: On the other hand defuzzification of epsilon – neighborhood of the fuzzy numbers A, B and C is as follows:

MðA Þ ¼ 2 ;

MðB Þ ¼

1 2  2

and MðC  Þ ¼ 22 :

Therefore, the ranking order is

B  A  C: The results of other methods are given in Table 1. In set 1, by Abbasbandy and Hajjari method [2], the ranking order is A  B  C and ranking order for Nasseri and Sohrabi [19] is B  C  A. It is the shortcoming of Abbasbandy and Nasseri methods, because are not consistent with human intuition. But our method has the same result as other four techniques, which has not the shortcomings mentioned above and its A  C  B (see Fig. 1). In set 2, A  B  C is the result of Distance, minimization [1] which is unreasonable. The result of proposed method is B  A  C which is similar to the results of Abbasbandy [2] and Nasseri and Sohrabi methods [12], but result of Wang et al. [20] and Asady [18] is C  A  B. Finally, the result of Wang and Lee [14] method is C  B  A. We conclude from Fig. 2 that A  C  B is better than A  B  C. As to set 3, Wang and Lee’s method [14] cannot rank the general fuzzy numbers B,C. As, the result of Asady and Zendehnam [1] is B  C  A which is similar to the our method. However, the results for other methods is C  B  A. From Fig. 3, it can be seen that the result obtained by our approach is consistent with human intuition, because result is base on distance minimization or the best approximation (see Theorem 2) . Also, by the Abbasbandy and Hajjari method is C  B  A. Finally, in set 4, we have B  C  A which is similar to the result of Abbasbandy and Hajjari and Distance minimization method, but for Wang et al. [20] method is B  A  C and C  B  A for Nasseri and Sohrabi

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B. Asady / Applied Mathematical Modelling 35 (2011) 1306–1313 Table 1 Comparative results of Example 2. Authors

Fuzzy number

Set 1

Set 2

Set 3

Set 4

Wang method

A B C

0 6.5206 2.5193

1.2293 4.0599 0

0.4364 0.5853 0.5417

1.8667 0 3.1111

A B C

ACB 0 0 0

CAB 0.6666 0.1666 1.3334

ACB 0.5 0.3333 .23710

BAC 3 1.6667 0

ABC 3 .25 1.375

BA C 0.03333 .062499 0.6666

CBA

BCA

A B C

-

-

A B C

ACB 2 0.1667 1.3333

CBA 0 0 0

0.0833 0 0.0223

0.5 0.5 0

A B C

ACB 11.6425 1.9353 3.3608

ABC 3.8944 1.8385 6.8337

BCA 2.4669 0.9605 0.7223

BCA 4.9666 1.7248 1.2248

A B C

BCA 2.2222 4.4375 3.1429

BA C (2, 2.24) (2, 14) (2, 0)

CBA 1.2128 1.3636 1.3928

CBA 1.8333 1.1250 1.4286

ACB

CAB

CBA

BCA

Results Abbasbandy and Hajjari

Results Wang and Lee

Results Asady and Zendehnam

Results Nasseri and Sohrabi

Results Asady

Results (–) : The ranking method cannot calculate the ranking value.

1 0.8 0.6

A

0.4

B

C

0.2 0 −12

−10

−8

−6

−4

−2

0

2

Fig. 1. Set 1 of Example 2.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

A

C

B

0.1 0 −7

−6

−5

−4

−3

−2

−1

Fig. 2. Set 2 of Example 2.

0

1

2

3

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B. Asady / Applied Mathematical Modelling 35 (2011) 1306–1313

1 0.8 0.6 0.4

A

B

C

0.2 0 −2

−1.5

−1

−0.5

0

0.5

1

1.5

Fig. 3. Set 3 of Example 2.

1 0.8 B

0.6

C

0.4 A

0.2 0 −5

−4

−3

−2

−1

0

1

2

3

Fig. 4. Set 4 of Example 2.

method . But again , Wang and Lee’s method [14] cannot rank the general fuzzy number B. From Fig. 4, it is easy to see that the ranking results obtained by the existing approaches [14,19,20] are unreasonable and are not consistent with human intuition. 4. Conclusion In this paper, we have been suggested an interesting approach to crisp function approximation of fuzzy numbers and define the epsilon – neighborhood of the fuzzy number. The proposed method leads to the crisp function which is the best related to a certain measure of distance between the fuzzy number and a crisp function of set support function. Finally, we have been revised distance minimization [1] for ranking of the fuzzy numbers which very simple work with its . It has been some new point which are not mentioned anywhere [1–30]. For example (a) We define the following two variable function in space fuzzy numbers

f : FðRÞ  ½0; 1 ! R; f ðA; Þ ¼ ðAv½;1 ; Av½;1 Þ: (b) We define epsilon – neighborhood from a fuzzy number (see Definition 4). (c) We obtain the function M(A) : F(R)  [0, 1] ? R, which is a distance minimization with respect to epsilon – neighborhood from the fuzzy number A (see Theorem 1). (d) The continues function M(A) is the best approximation of epsilon – neighborhood A and applied for ranking the fuzzy numbers (see Theorem 2). (e) In this ranking method, A B if M(A) > M(B) or M(A) P M(B) based on M(A) = M(B).

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