Ranking fuzzy numbers based on epsilon-deviation degree

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Ranking fuzzy numbers based on epsilon-deviation degree Vincent F. Yu a , Ha Thi Xuan Chi a,∗ , Chien-wen Shen b a b

Department of Industrial Management, National Taiwan University of Science and Technology, 43, Section 4, Keelung Road, Taipei 10607, Taiwan Department of Business Administration, National Central University, No. 300, Jhongda Road, Jhongli City, Taoyuan County 32001, Taiwan

a r t i c l e

i n f o

Article history: Received 15 March 2012 Received in revised form 28 September 2012 Accepted 12 March 2013 Available online xxx Keywords: Ranking fuzzy numbers Centroid Area Decision making

a b s t r a c t Although numerous research studies in recent years have been proposed for comparing and ranking fuzzy numbers, most of the existing approaches suffer from plenty of shortcomings. In particular, they have produced counter-intuitive ranking orders under certain cases, inconsistent ranking orders of the fuzzy numbers’ images, and lack of discrimination power to rank similar and symmetric fuzzy numbers. This study’s goal is to propose a new epsilon-deviation degree approach based on the left and right areas of a fuzzy number and the concept of a centroid point to overcome previous drawbacks. The proposed approach defines an epsilon-transfer coefficient to avoid illogicality when ranking fuzzy numbers with identical centroid points and develops two innovative ranking indices to consistently distinguish similar or symmetric fuzzy numbers by considering the decision maker’s attitude. The advantages of the proposed method are illustrated through several numerical examples and comparisons with the existing approaches. The results demonstrate that this approach is effective for ranking generalized fuzzy numbers and overcomes the shortcomings in recent studies. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Zadeh [1] introduced the fuzzy set theory as a great tool, mathematically representing uncertainty and vagueness in order to efficiently deal with knowledge associated with imprecision. A fuzzy set can greatly reduce uncertainty and has been applied to problems in a variety of fields such as supply chain management, underground mining, weaponry, and so forth [2–9]. One of the most useful applications of the fuzzy set theory is in decision making, a cognitive process. The fuzzy set theory describes the approximate information of values and preferences of the decision maker that involve uncertainty, thus generating a decision by ranking fuzzy numbers representing the imprecise numerical measurement of alternatives. However, selecting an optimal alternative among a set of possibilities under a fuzzy environment is complex and challenging. The literature over the past few decades has proposed numerous methods for ranking fuzzy numbers. Some of them are legendary for ranking fuzzy numbers, such as maximizing sets and minimizing sets, centroid points, and distance minimization. Jain [10] proposed the first ranking method using the maximizing set to order fuzzy numbers for selecting an optimal alternative. Yager [11] presented the centroid-index method, Dubios and Prade [12] used the maximizing set to order fuzzy numbers, and Chen [13]

∗ Corresponding author. Tel.: +886 919042905. E-mail address: [email protected] (H.T.X. Chi).

proposed the maximizing and minimizing set approach. Chu and Tsao [14] ranked fuzzy numbers with an area between the centroid point and original point. Wang and Yang [15] presented the centroid of fuzzy numbers. Abbasbandy and Asady [16] suggested sign distance, while Asady and Zendehnam [17] proposed the distance minimization method. Other methods have offered revisions to achieve more completeness [18,19], but most of them are unable to give satisfied ranking results for all situations due to the complexity. Many researchers have recently employed maximizing set and minimizing set and the concept of a centroid point as the basis for comparing and ranking fuzzy numbers [20–24]. These methods have mainly concern the correlation between L-R areas and centroid point of a fuzzy number. Wang et al. [20] defined the L-R deviation degree of a fuzzy number and came up with the ranking rule, in which the larger the left deviation degree and the smaller the right deviation degree are, the larger the fuzzy number is. Asady [21] and Nejad Mashinchi [22] redefined the L-R deviation degree of a fuzzy number to overcome the shortcomings of Wang et al. [20]. However, most deviation degree approaches still display the same limitations due to the neglected decision maker’s attitude, the incoherent transfer coefficient formula, and the unreliable ranking index computation. To eliminate all these aforementioned drawbacks, this study introduces a new epsilon-deviation degree approach on the basis of maximizing set and minimizing set and centroid point and considers the decision maker’s attitude for ranking fuzzy numbers. The rest of this paper is organized as follows. Section 2 briefly reviews the concept of fuzzy numbers, previous approaches and presents

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the shortcomings through several counter-examples. Section 3 shows the proposed new approach with two innovative indices for ranking fuzzy numbers. The proposed approach is compared with existing approaches. Results and discussions are presented in Section 4. Finally, conclusions are drawn in Section 5, in which the contributions of this research are presented and the major findings are highlighted.

2.2. Review of the existing approaches Assume there are n fuzzy numbers A1 , A2 , ... , An , i = 1, 2, . . . , n. Here, Amin and Amax are the minimal reference set and maximal reference set, respectively, and their membership functions are defined as [13]:

xmax − x

xmax − xmin

gAmin (x) =

x − xmax

2.1. Concept of fuzzy numbers This section defines the concept of fuzzy numbers [12]. Definition 1: A real fuzzy number A = (a, b, c, d, ) is described as any fuzzy subset of the real line R with membership function uA which is given by:

uA (x) =

,

uRA (x), ⎪ ⎪ ⎪ ⎩ 0,

b ≤ x ≤ c,

(1)

c ≤ x ≤ d,





interval [0, ]. In other words, 0 gAL (y)dy and 0 gAR (y)dy should both exist. Let E be the set of all fuzzy numbers. A trapezoid fuzzy number A = (a, b, c, d; ) is described as any fuzzy subset of the real line R, with the membership function uA (x) expressed as [13]:

(d − x) ⎪ ⎪ , ⎪ ⎪ d−c ⎪ ⎪ ⎩ 0,

b ≤ x ≤ c,

(2)

c ≤ x ≤ d, otherwise,





x ∈ R : A (x) > 0 .

The image of fuzzy number A = (a, b, c, d) is denoted by −A = (−d, −c, −b, −a) [1,28].

if x ∈ S,

,

(4)

otherwise.

˛i

(uAmin (x) − gAL (x))dx,



(5)

i

xmin xmax

dAR =

(uAmax (x) − gAR (x))dx,

(6)

i

ˇi

where ˛i and ˇi , i = 1, 2, ... , n are the abscissas of the crossover points of gAL , gAR , and uAmin , uAmax , respectively. Assuming a trapei

i

zoid fuzzy number Ai = (ai , bi , ci , di ; i ), the expectation value of the centroid is expressed as [29]:

 di

Mi =

ai

 di ai

xuAi (x)dx

(7)

,

uAi (x)dx

and the transfer coefficient is given by [20]: Ai =

MAi − Mmin Mmax − Mmin

,

(8)





where Mmax = max M1 , M2 , . . . Mn ,





Mmin = min M1 , M2 , . . . Mn , and Mmax = / Mmin . Based on Eqs. (7) and (8), the ranking index value of fuzzy number Ai , i = 1, 2, ..., n is defined as follows [20]:

dAi =

a ≤ x ≤ b,

If  = 1, then A is a normal fuzzy number; otherwise, it is said to be a non-normal fuzzy number. If b = / c, then A is referred to as a fuzzy interval [25] or a flat fuzzy number [26]. If LA (x) and RA (x) are both linear, then A is referred to as a trapezoidal fuzzy number and is usually denoted by A = (a, b, c, d; ) or simply A = (a, b, c, d) if  = 1. In particular, when b = c, the trapezoidal fuzzy number is reduced to a triangular fuzzy number and can be denoted by A = (a, b, d; ) or A = (a, b, d) if  = 1. Thus, triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers. Definition 2: For a fuzzy set A, the support set of A is defined as [27]: S (A) =



i

otherwise,

(3)

where S = ∪ni=1 S(Ai ), xmin = inf S and xmax = sup S. The left and right deviation degrees of Ai , i = 1, 2, ... , n are defined as follows [20]:

i

where 0 ≤  ≤ 1 is a constant. Note that uLA (x) : [a, b] → [0, ] and uRA (x) : [c, d] → [0, ] are strictly monotonic and continuous, mapping from R to closed interval [0, ]. Since uLA (x) and uRA (x) are strictly monotonic and continuous, their inverse functions exist. Let gAL : [0, ] → [a, b] and gAR : [0, ] → [c, d] denote the inverse functions of uLA (x) and uRA (x), respectively. Since uLA (x) : [a, b] → [0, ] is continuous and strictly increasing, gAL : [0, ] → [a, b] is continuous and strictly increasing. Similarly, uRA (x) : [c, d] → [0, ] is continuous and strictly decreasing. Thus, gAR : [0, ] → [c, d] is continuous and strictly decreasing. Both uLA and uRA should be integrable on the close

uA (x) =

0,

dAL =

a ≤ x ≤ b,

⎧ (x − a) ⎪ ⎪ ⎪ b−a , ⎪ ⎪ ⎪ ⎨ ,

xmax − xmin

gAmax (x) =

if x ∈ S, otherwise.

0,

2. Literature review

⎧ L uA (x), ⎪ ⎪ ⎪ ⎨

,

⎧ ⎪ ⎪ ⎪ ⎨

dAL Ai i

1 + dAR (1 − Ai )

,

Mmax = / Mmin , i = 1, 2, . . . , n. (9)

i

dAL ⎪ ⎪ i ⎪ ⎩ 1 + dR ,

Mmax = / Mmin , i = 1, 2, . . . , n.

Ai

The ranking order is defined as follows [20]: 1) Ai  Aj , if and only if dAi > dAj , 2) Ai ≺ Aj , if and only if dAi < dAj , 3) Ai ∼Aj , if and only if dAi = dAj . 2.3. Shortcomings of the existing approaches Despite many efforts, most of the recent approaches still display the weaknesses analyzed and discussed in detail through the following three counter-examples. Example 1 first shows limitations of the computation of the transfer coefficient, constructed by an expectation value of the centroid in Wang et al.’s method [20]. Example 1: Consider the following sets of fuzzy numbers adopted from [12], as shown in Figs. 1 and 2. Set 1: A1 = (2, 4, 6), A2 = (1, 5, 6), A3 = (3, 5, 6). Set 2: A1 = (2, 3, 8), A2 = (2, 3, 7, 8), A3 = (2, 3, 10). Nejad and Mashinchi [22] explained and pointed out that Wang et al.’s method [20] failed in ranking fuzzy numbers with centroid

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y

y A1

1

0.5

A2

0.5

0

1

2

A3

3

4

5

x

6

0

points that are equal to a maximal or minimal centroid point, thus the transfer coefficient is zero. Set 1 considers three triangular fuzzy numbers, as shown in Fig. 1, where MA1 = MA2 = MAmin . According to Wang et al.’s method, the transfer coefficients obtained are 1 = 2 = 0, leading to d1 = d2 = 0. It is hence concluded that if transfer coefficients 1 , 2 are equal to zero, the left deviation degrees d1L , d2L are worthless. Set 2 considers two triangular fuzzy numbers and one trapezoidal fuzzy number, as shown in Fig. 2, where MA2 = MA3 = MAmax leading to 2 = 3 = 1 ⇒ 1 − 2 = 1 − 3 = 0 ⇒ d2 = d3 = 0. The transfer coefficients 2 , 3 are equal to one, and thus the right deviation degrees d2R , d3R become worthless. Obviously, when the centroid point is identical to a minimal or maximal one, the ranking indices of Wang et al. [20] exhibit inadequate discrimination. Example 2 analyses the illogical ranking orders of the fuzzy numbers’ images existing in the methods proposed in [20–22]. Example 2: Consider the following sets of fuzzy numbers adopted from [30]. Set 1: A1 = (1, 2, 6) , A2 = (2.5, 2.75, 3) and A3 = (2, 3, 4) .

⎧ ⎨ x − 1, if x ∈ [1, 2], ⎩

3 − x,

if x ∈ [2, 3],

0,

otherwise.

⎧ 1/2 ⎪ 1 − (x − 2)2 , if x ∈ [2, 4], ⎪ ⎪ ⎪ ⎪ ⎨

1/2 1 gA2 (x) = 1 − (x − 2)2 , if x ∈ [2, 4], ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎩ 0,

1

2

3

4

5

x

6

Fig. 3. Fuzzy numbers A1 , A2 , and A3 in Set 1 of Example 2.

Fig. 1. Fuzzy numbers A1 , A2 , and A3 in Set 1 of Example 1.

Set 2 : gA1 (x) =

A3

A2

A1

1

otherwise.

Set 1 considers three fuzzy numbers, as shown in Fig. 3, where MA1 = MA3 = MAmax . Nejad and Mashinchi [22] demonstrated that their method can overcome the shortcoming of illogically ranking images by preventing the transfer coefficients from being zero or one. However, the ranking index has inadequate discrimination as it obtains an inconsistent ranking outcome as A1 ≺ A2 ≺ A3 , and −A3 ≺ −A1 ≺ −A2 .

y A2

1

0.5 A1

1

0

2

3

4

x

Fig. 4. Fuzzy numbers A1 , A2 in Set 2 of Example 2.

Applying Wang et al.’s method [20], the ranking order of the three fuzzy numbers is A2 ≺ A1 ≺ A3 , and the ranking order of their images is −A1 ∼ − A3 ≺ −A2 , which are illogical. The same shortcoming also occurs in Nejad and Mashinchi’s method, whereby the ranking order of fuzzy numbers and their images obtained are A1 ≺ A2 ≺ A3 , and −A3 ≺ −A1 ≺ −A2 , respectively. Applying Asady’s method [21] to Set 2 shown in Fig. 4., the ranking order obtained is A1 ≺ A2 . However, the ranking order of their images is −A1 ≺ −A2 , which is illogical. Because the approaches of Wang et al. [20] and Nejad and Mashinchi [22] also cannot produce the correct ranking order under symmetric fuzzy number circumstances, their methods lead to the same ranking orders as shown in Example 3. Example 3: Consider symmetric triangular fuzzy numbers A1 = (0.2, 0.5, 0.8) , A2 = (0.4, 0.5, 0.6) adopted from [30], as shown in Fig. 5, where MA1 = MA2 = MAmax = MAmin , dAL = dAR , and dAL = dAR . 1 1 2 2 The ranking orders reported by Wang et al. [20] and Nejad and Mashinchi [22] are A1 ≺ A2 and −A1 ≺ −A2 , which is illogical.

y y

1

A2

1

0

1

2

3

4

5

6

A1

0.5

A3

A1

0.5

7

8

9

Fig. 2. Fuzzy numbers A1 , A2 , and A3 in Set 2 of Example 1.

10

x

0

0.1

0.2

A2

0.3

0.4

0.5

0.6

0.7

0.8

x

Fig. 5. Fuzzy numbers A1 , A2 in Example 3.

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3. The proposed epsilon-deviation degree method To overcome the weaknesses of the existing approaches based on deviation degree, this study introduces an epsilondeviation degree method for ranking fuzzy numbers Ai = (ai , bi , ci , di ; i ), i = 1, 2, . . . , n. The proposed ranking index integrates an epsilon-transfer coefficient, L-R areas of a fuzzy number and decision maker’s attitude. Definition 3: The L-R areas of a generalized fuzzy number are defined by the following equations:





bi

diL =

bi

i uLi (x)dx

i dx − amin





di

i uRi (x)dx

i dx − ci

(11)

MAi − Mmin +

ε1 2

Mmax − Mmin + ε1

Di =

ε2 + diL i ε2 + diR (1 − i )

 ˛

Di =

,

(12)



ε2 + (1 − ˛) diL i ε2 + ˛diR (1 − i )

,

(13)

 ,

(14)

where i = 1, 2, . . ., n,  > 0,0 < ε2 < 1, ˛ ∈ [0,1]. Definition 6: The ranking order is defined on the basis of the following properties: 1) Ai  Aj , if and only if DAi > DAj , 2) Ai ≺ Aj , if and only if DAi < DAj , 3) DAi = DAj , If DA˛ > DA˛ , then Ai  Aj , i

j

else Ai ∼Aj . The index of optimism ˛ ∈ [0, 1] reflects the degree of optimism by a decision maker in decision making [31]. The larger the index of optimism is, the higher the degree of optimism will be for the decision maker. In particular, ˛ = 1, Di˛ = Di1 reflects an optimistic decision maker by considering the right side of the fuzzy number, whereas ˛ = 0, Di˛ = Di0 represents a pessimistic decision maker and considers the left side of the fuzzy number. Theorem 1: Assume a group of fuzzy numbers Ai = and their images −Ai = (ai , bi , ci , di ; i ), i = 1, 2, ... , n (−di , −ci , −bi , −ai ; i ), i = 1, 2, ... , n. The proposed ranking index satisfies that DAi > DAj if and only if D−Ai < D−Aj . Proof: As can be seen in Fig. 6, the properties of a fuzzy number express the following equations: R L dAL = d−A , dAR = d−A ,

(15)

Ai = (1 − −Ai ), 1 − Ai = −Ai .

(16)

i

⇔

i

ε2 + dAR (1 − Ai )

i

i

 >

i

i

ε2 + dAL Aj j

ε2 + dAR (1 − Aj )

 ,

(17)

j

Let gAi = dAL Ai , hAi = dAR (1 − Ai ), then gAi = h−Ai , hAi = g−Ai . i

i

The inequality (17) can be written as:



ε2 + gAi



ε2 + hAi

 >

ε2 + gAj



ε2 + hAj



⇔ ε2 2 + ε2 hAj + ε2 gAi + gAi hAj

 

> ε2 2 + ε2 hAi + ε2 gAj + gAj hAi



,

⇔ ε2 2 + ε2 g−Aj + ε2 h−Ai + h−Ai g−Aj



2

 

> ε2 + ε2 g−Ai + ε2 h−Aj + h−Aj g−Ai

 ⇔

ε2 + g−Ai ε2 + h−Ai



 <

ε2 + g−Aj ε2 + h−Aj

(18) ,

 ,

⇔ −D−Ai < −D−Aj . The order of the images is therefore −Ai ≺ −Aj . The algorithm for ranking fuzzy numbers based on the epsilondeviation degree can be summarized as follows. Step 1. Calculate diL and diR of each fuzzy number, according to Eqs. (10) and (11). Step 2. Determine the epsilon-transfer coefficient Ai through Eq. (12). Step 3. Obtain the ranking index values Di and Di˛ , using Eqs. (13) and (14). Step 4. Rank fuzzy numbers A1 , A2 , ..., An based on their ranking index values, using Definition 6. 4. Results and discussions

j

else if DA˛ < DA˛ , then Ai ≺ Aj , i



ε2 + dAL Ai



where Mmin = min{MA1 , MA2 , ..., MAn } and Mmax = max{MA1 , MA2 , ..., MAn }, 0 < ε1 < 1. ε1 is defined as an extremely small value to avoid the new epsilon-transfer coefficient from being zero. Definition 5: The innovative ranking index value is defined as:





ci

where uLi (x) and uRi (x) are L-R membership functions of a fuzzy number defined by Eq. (1). Moreover, amin = min{a1 , a2 , ..., an } and dmax = max{d1 , d2 , ..., dn }. Definition 4: Assume a group of fuzzy numbers Ai = (ai , bi , ci , di ; i ), an epsilon-transfer coefficient of Ai , i = 1, 2, . . . , n, is given by: Ai =

Assume Ai  Aj , then DAi > DAj

(10)

ai

dmax

diR =

Fig. 6. The image of a fuzzy number.

4.1. Results In this section, the proposed method is used to solve the three counter-examples discussed in Section 2.3 to demonstrate its reliability. Two additional numerical examples are also presented to show the superiority of the proposed approach. Let ε1 = 0.01, ε2 = 0.01,  = 1 in all the examples. Example 1 Set 1: Examine the three fuzzy numbers A1 , A2 and A3 shown in Fig. 1. By using the new ranking index value, the ranking orders of the fuzzy numbers and their images are A1 ≺ A2 ≺ A3 and −A1  −A2  −A3 , respectively. Set 2: Consider the three fuzzy numbers shown in Fig. 2. Using the proposed method, the ranking orders of the fuzzy numbers and their images are A2  A3  A1 and −A1  −A3  −A2 , respectively. Table 1 shows a comparison of the ranking results of Example 1. It can be seen that the proposed ranking results are consistent with the existing approaches [18,22,32,33] and intuition. As a result, the

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Table 1 Comparing the ranking results of the previous studies of Example 1. Methods Abbasbandy and Hajari [16]

Wang et al. [20]

Asady [21]

Nejad and Mashinchi [22]

Asady [18]

Chen and Sanguansat [33]

Proposed method

A3 4.91

−A1 −4

−A2 −4.75

−A3 −4.91

−3.333

−5

−3.5

Set 1

A1 4

A2 4.75

2

3.33

5

3.5

1

0

0

1.857

0.714

0.417

0

2

0

0.444

0.444

2.615

0

0

1

1.2

1.444

2.142

0.571

0.466

0.294

2

0.307

0.545

0.416

2.4

1.4285

1.833

1

1.35

1.5

2.630

0.2703

0.2162

0.106

2

0.145

0.259

0.230

2.979

1.756

2.027

1

4

4.25

4.75

−4

−4.25

−4.75

2

4

5

4.5

−4

−5

−4.5

1

2.994

5.974

2

0.046

0.12

1

2.95

5.76

2

0.049

0.126

16.38

0.334

0.086

21.71

15.4

0.34

0.09

proposed method herein can overcome the identical shortcomings in Wang et al.’s method [20]. Example 2 Set 1: Consider three triangle fuzzy numbers, as shown in Fig. 3, in which the same centroid point problems are again emphasized with MA1 = MA3 . Although Nejad and Mashinchi [22] attempted to overcome this shortcoming in Wang et al.’s method, they failed to consider the ranking order of images of fuzzy numbers in the ranking index computation, resulting in the unreasonable ranking orders as A1 ≺ A2 ≺ A3 , −A2  −A1  −A3 . By using the proposed method, the ranking order of fuzzy numbers is consistent with that of their images as A1 ≺ A2 ≺ A3 , −A1  −A2  −A3 . These results agree with human intuition and the results of Abbasbandy Hajjari [16] and Asady [18], as can be seen in Table 2. Therefore, the new approach not only can deal with identical centroid points, but also the ranking order of images of fuzzy numbers. The following examples clearly show more advantages of the proposed method by employing the decision maker’s attitude for ranking symmetric fuzzy numbers.

20.49

0.167 8.2938 0.17

Ranking results A1 ≺ A3 ≺ A2 −A1  −A2  −A3 A1 ≺ A3 ≺ A2 −A1  −A3  −A2 A1 ∼A2 ≺ A3 −A1  −A2  −A3 A1 ≺ A2 ∼A3 −A1  −A2 ∼ − A3

A1 ≺ A2 ≺ A3 −A1  −A2  −A3 A1 ≺ A3 ≺ A2 −A1  −A3  −A2 A1 ≺ A2 ≺ A3 −A1  −A2  −A3 A1 ≺ A3 ≺ A2 −A1  −A3  −A2

0.061 11.61 0.065

7.957

A1 ≺ A2 ≺ A3 −A1  −A2  −A3 A1 ≺ A3 ≺ A2 −A1  −A3  −A2

11.12

A1 ≺ A2 ≺ A3 −A1  −A2  −A3 A1 ≺ A3 ≺ A2 −A1  −A3  −A2 A1 ≺ A2 ≺ A3 −A1  −A2  −A3 A1 ≺ A3 ≺ A2 −A1  −A3  −A2

Set 2: Consider the triangle fuzzy number A1 with a linear membership function and the fuzzy number A2 with a non-linear membership function, as shown in Fig. 4. Asady’s method [21] shows the limitation by coming up with the counter-intuitive ranking order A1 ≺ A2 , whereas the proposed method herein obtains the reasonable and intuitive result as A1  A2 , −A2  −A1 with the ranking indices DA = 0.612, DA2 = 0.072, D−A1 = 1.631, and D−A2 = 1.397. Thus, the proposed method can deal with a non-linear membership function of a fuzzy number. Example 3: Consider symmetric triangle fuzzy numbers A1 = (0.2, 0.5, 0.8) and A2 = (0.4,0.5,0.6) as one of the typical problems for comparing fuzzy numbers, adopted from [30]. As shown in Fig. 5, where the fuzzy numbers have the same spread and the same mode that make some approaches incapable of discrimination [20]. While the approaches of Asady and Zendehnam [17], Chu and Tsao [14], Nejad and Mashinchi [22], Asady [21], Wang and Lee [34], and Chen [35] produce the same ranking order A1 ∼A2 , the ranking order obtained by Wang et al.’s [20] and Wang et al.’s approach [36] is A1  A2 . In fact, most of the existing approaches cannot give a satisfactory result due to ignoring the decision maker’s prefer-

Table 2 Comparing the ranking results of the previous studies of Set 1 in Example 2. Method

A1

A2

A3

−A1

−A2

−A2

Abbasbandy and Hajari [32]

2.25 A1 ∼A2 ≺ A3 0.417 A2 ≺ A3 ≺ A1 3.5 A2 ≺ A1 ≺ A3 0.1846 A1 ≺ A2 ≺ A3 2.743 A1 ≺ A2 ≺ A3 0.174 A1 ≺ A2 ≺ A3

2.75

3

−2.75

−3

0.000

1.167

0

4

0.2204

0.2817

2.75

3

0.284

0.404

−2.25 −A1 ∼ − A2  −A3 0.000 −A2  −A1 ∼ − A3 0 −A2  −A1 ∼ − A3 1.4063 −A2  −A1  −A3 −2.743 −A1  −A2  −A3 5.757 −A1  −A2  −A3

Wang et al. [20] Asady [21] Nejad and Mashinchi[22] Asady [18] Proposed method

Evaluation Inconsistent

2.119

0.000

6.5

0

1.6515

1.2750

Inconsistent Inconsistent Inconsistent −2.75

−3 Consistent

3.522

2.473 Consistent

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Table 3 Different ranking results corresponding with degrees of optimism of Example 3. −A1

Degree of optimism

A1

A2

˛=0

751

1251

˛ = 0.5

1

1

˛=1

0.00133

0.0008

y

y

1

1

−A2

Ranking results

0.00133

0.0008

1

1

751

1251

A1 ≺ A2 −A1  −A2 A1 = A2 −A1 = −A2 A1  A2 −A1 ≺ −A2

A2 A1

0.5

0.5 A1

A2

A3

-2 0 2 4 6 8 10 12 14 16

5

x

5.9 6

A3

x

7

Fig. 8. Fuzzy numbers A1 , A2 , and A3 in Example 5. Fig. 7. Fuzzy numbers A1 , A2 , and A3 in Example 4.

ence, which significantly influences the final decision, leading to inconsistent outcomes. To deal with this situation, the proposed method considers the decision maker’s attitude ˛ as one of the significant factors for calculating the overall ranking index when the ranking indices have the same value: DA1 = DA2 = 1. In this case, decision makers are allowed to choose which alternative they prefer. As a result, the ranking order depends on the decision maker’s preference. As can be seen from Table 3, the different degrees of optimisms correspond with different ranking results. If decision makers select A2 , then they show a small degree of optimism. In contrast, if decision makers would rather have A1 than A2 , then they show a high degree of optimism. Finally, decision makers are neutral if they believe that A1 is the same as A2 , and then they can choose one of them. Example 4: Consider three fuzzy numbers, as shown in Fig. 7, where both A2 , a triangle fuzzy number, and A3 , a trapezoidal fuzzy number, are symmetric. The ranking indices obtained by the proposed method are DA1 = 0.489, DA2 = DA3 = 0.341. Therefore, the decision maker’s attitude is considered to calculate DA˛ and 2 DA˛ for fuzzy numbers A2 , A3 More specifically, for a pessimistic 3

decision maker, ˛ = 0: DA0 = DA0 = 106.9, or for an optimistic deci2

3

sion maker, ˛ = 1: DA1 = DA1 = 0.0019. The ranking order is thus 2

3

A1  A2 ∼A3 , which is consistent with human intuition. While the magnitude concept of Abbasdandy and Hajjari [32] produces an ranking order of A1 ≺ A2 ∼A3 , Ezzati et al.’s method [37] results in

a different ranking order of A1 ≺ A3 ≤ A2 . However, both of their results are counter-intuitive. Example 5: Consider the three triangular fuzzy numbers A1 = (5, 6, 7) A2 = (5.9, 6, 7), and A3 = (6, 6, 6, 7) as shown in Fig. 8. Using the proposed method, the ranking order of the fuzzy numbers is A1 ≺ A2 ≺ A3 and the ranking order of their images is −A1  −A2  −A3 . Obviously, the proposed method can give a logical ranking result. As can be seen in Table 4, the ranking order of the proposed method is consistent with that of the earlier methods [14,18,20,22,32,35,38]. Based on the explanation from Wang et al. [20], the area between the centroid point and the original point method of Chu and Tsao [14] and the CV index of Cheng [35] produce A1 ≺ A3 ≺ A2 and A3 ≺ A2 ≺ A1 respectively, which are inconsistent with human intuition [20]. 4.2. Discussions In the proposed method, the computational transfer coefficient and the index value fully consider all factors, which affect the ranking order, result in adequate discrimination for distinguishing a symmetric, a non-linear membership function and identical centroid points of fuzzy numbers, and produce an illogical ranking order of fuzzy numbers’ images. Overall, the ranking results make sense and are consistent with earlier approaches that consider the optimum index in ranking fuzzy numbers [19,31,39]. This study indicates that a decision maker’s degree of optimism, one of the most significant factors in making a consistent

Table 4 Comparing the ranking results of the previous studies of Example 5. Method

A1

A2

A3

Ranking

Cheng’s CV index [35] Chu and Tsao [14] Abbasbandy and Hajari [32] Wang et al. [21] Asady [21] Nejad and Mashinchi [22] Asady [18] Kumar [38] Proposed method

0.028 3 6 0.000 0.666 0.5 6 6 1

0.0098 3.126 6.075 0.620 0.960 0.938 6.225 6.225 3.381

0.0089 3.085 6.083 0.750 1 1 6.25 6.25 3.817

A3 A1 A1 A1 A1 A1 A1 A1 A1

≺ A2 ≺ A3 ≺ A2 ≺ A2 ≺ A2 ≺ A2 ≺ A2 ≺ A2 ≺ A2

≺ A1 ≺ A2 ≺ A3 ≺ A3 ≺ A3 ≺ A3 ≺ A3 ≺ A3 ≺ A3

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ranking outcome, is efficient enough to distinguish symmetric fuzzy numbers, which have the same mode and the same spread. Decision makers evaluate alternatives and make a decision based on their optimism or preference, resulting in the decision maker’s optimistic attitude having a significant impact on ranking fuzzy numbers. The larger the index of optimism is, the higher the degree of optimism will be for the decision maker [19,31,39,40]. In order to achieve ranking fuzzy numbers, the ranking index should include the physical situations of a fuzzy number and the decision maker’s preference, because a decision maker evaluates alternatives and makes a decision based on feeling or preference. 5. Conclusion This study proposes a new epsilon-deviation degree approach for ranking fuzzy numbers. The proposed method not only considers the left and right side areas of a fuzzy number and the expectation value of centroid, but also the index of optimism. The new epsilon-transfer coefficient shows strong discrimination to overcome the identical centroid points of fuzzy numbers, which are the shortcomings in earlier studies [20,21]. In comparison to other approaches, what is especially noteworthy is that our proposed ranking index gives flexible options for a decision maker by considering the decision maker’s degree of optimism, which significantly impacts on how to distinguish symmetric fuzzy numbers. Through the counter-examples and numerical examples demonstrated herein, the innovative ranking index is capable of discrimination to rank non-linear membership function fuzzy numbers and to produce a logical ranking order of fuzzy numbers’ images, which may not be obtained under recent existing approaches [20–22]. One more advantage of the proposed method is in ranking generalized fuzzy numbers that have different weights. In conclusion, our proposed method possesses high-quality characteristics that make up a valuable tool for comparing and ranking fuzzy numbers. Thus, in the future, the proposed method can be improved for ranking generalized fuzzy numbers with different left and right heights to handle fuzzy risk analysis problems. References [1] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [2] T.C. Chu, Y.C. Lin, A fuzzy TOPSIS method for robot selection, International Journal of Advanced Manufacturing Technology 21 (2003) 284–290. [3] S.-Y. Chou, Y.-H. Chang, A decision support system for supplier selection based on a strategy-aligned fuzzy SMART approach, Expert Systems with Applications 34 (2008) 2241–2253. [4] S.-Y. Chou, Y.-H. Chang, C.-Y. Shen, A fuzzy simple additive weighting system under group decision-making for facility location selection with objective/subjective attributes, European Journal of Operational Research 189 (2008) 132–145. [5] M.-L. Tseng, An assessment of cause and effect decision-making model for firm environmental knowledge management capacities in uncertainty, Environmental Monitoring and Assessment 161 (2010) 549–564. [6] M.-L. Tseng, Using a hybrid MCDM model to evaluate firm environmental knowledge management in uncertainty, Applied Soft Computing 11 (2011) 1340–1352. [7] M.-L. Tseng, Green supply chain management with linguistic preferences and incomplete information, Applied Soft Computing 11 (2011) 4894–4903. [8] A. Azadeh, M. Osanloo, M. Ataei, A new approach to mining method selection based on modifying the Nicholas technique, Applied Soft Computing 10 (2010) 1040–1061. [9] M. Dagdeviren, S. Yavuz, N. KilInc¸, Weapon selection using the AHP and TOPSIS methods under fuzzy environment, Expert Systems with Applications 36 (2009) 8143–8151.

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