**5. Ranking indices**

#### **a. Methods of centroid point**

In order to determine the centroid points 0 0 (,) *x y* of a fuzzy number *A* , Cheng (Cheng, 1998) provided a formula then Wang et al. (Y. M. Wang et al., 2006) found from the point of view of analytical geometry and showed the corrected centroid points as follows:

$$\begin{aligned} \mathbf{x}\_{0} &= \frac{\int\_{a}^{b} \mathbf{x} L\_{A}(\mathbf{x}) d\mathbf{x} + \int\_{b}^{c} \mathbf{x} d\mathbf{x} + \int\_{c}^{d} \mathbf{x} R\_{A}(\mathbf{x}) d\mathbf{x}}{\int\_{a}^{b} L\_{A}(\mathbf{x}) d\mathbf{x} + \int\_{b}^{c} d\mathbf{x} + \int\_{c}^{d} R\_{A}(\mathbf{x}) d\mathbf{x}} \\\\ \mathbf{y}\_{0} &= \frac{\left[\int\_{0}^{\alpha} y R\_{A}^{-1}(\mathbf{y}) d\mathbf{y} - \int\_{0}^{\alpha} y L\_{A}^{-1}(\mathbf{y}) d\mathbf{y}\right]}{\int\_{0}^{\alpha} R\_{A}^{-1}(\mathbf{y}) d\mathbf{y} - \int\_{0}^{\alpha} L\_{A}^{-1}(\mathbf{y}) d\mathbf{y}}. \end{aligned} \tag{11}$$

For non-normal trapezoidal fuzzy number *A abcd* (,,,, ) formulas (11) lead to following results respectively.

$$\begin{aligned} x\_0 &= \frac{1}{3} \left[ a + b + c + d - \frac{dc - ab}{(d + c) - (a + b)} \right] \\\\ y\_0 &= \frac{ab}{3} \left[ 1 + \frac{c - d}{(d + c) - (a + b)} \right] .\end{aligned} \tag{12}$$

Since non-normal triangular fuzzy numbers are, special cases of normal trapezoidal fuzzy numbers with *b c* , formulas (12) can be simplified as

$$\begin{aligned} x\_0 &= \frac{1}{3}[a+b+d] \\\\ y\_0 &= \frac{a}{3} \end{aligned} \tag{13}$$

6. Let 121 3 *AAA A* , , and *A A* 2 3 be elements of *E* . If 1 2 *A A* , then 1323 *AAAA* .

In order to determine the centroid points 0 0 (,) *x y* of a fuzzy number *A* , Cheng (Cheng, 1998) provided a formula then Wang et al. (Y. M. Wang et al., 2006) found from the point of

1 1

*yR y dy yL y dy*

*A A*

*A A*

3 ( )( )

 

Since non-normal triangular fuzzy numbers are, special cases of normal trapezoidal fuzzy

<sup>0</sup>

*x abd*

. 3

0

*y*

1 3

 

> 1 . 3 ( )( )

*c d*

 

() ()

 

() ()

 

*R y dy L y dy*

.

*dc ab*

( ) ( )

*xL x dx xdx xR x dx*

( ) ( )

*L x dx dx R x dx*

0 0 <sup>0</sup> 1 1 0 0

*dc ab x abcd*

*<sup>y</sup> dc ab*

*b cd A A a bc b cd A A a bc*

view of analytical geometry and showed the corrected centroid points as follows:

12 1 2 (,) , *AA A A* and 2 1 *A A* , we should

123 1 2 (,,) , *AAA A A* and 2 3 *A A* , we

1 2 (,) , *A A* inf{supp( *A*<sup>1</sup> )}>

1 2 (,) , *A A* inf{supp( *A*<sup>1</sup> )}>

(11)

formulas (11) lead to following

(12)

(13)

1. For an arbitrary finite subset of *E* and 1 *A* , 1 1 *A A* . 2. For an arbitrary finite subset of *E* and <sup>2</sup>

3. For an arbitrary finite subset of *E* and <sup>3</sup>

sup{supp( *A*<sup>2</sup> ), we should have 1 2 *A A* .

sup{supp( *A*<sup>2</sup> )}, we should have 1 2 *A A* .

0

*y*

For non-normal trapezoidal fuzzy number *A abcd* (,,,, )

0

numbers with *b c* , formulas (12) can be simplified as

0

1

*x*

4. For an arbitrary finite subset of *E* and <sup>2</sup>

5. For an arbitrary finite subset of *E* and <sup>2</sup>

have *A A* 1 2 .

**5. Ranking indices** 

results respectively.

**a. Methods of centroid point** 

should have 1 3 *A A* .

In this case, normal triangular fuzzy numbers could be compared or ranked directly in terms of their centroid coordinates on horizontal axis.

Cheng (Cheng, 1998) formulated his idea as follows:

$$R(A) = \sqrt{\mathbf{x}\_0(A)^2 + y\_0(A)^2}.\tag{14}$$

To overcome the drawback of Cheng's distance Chu and Tsao's (Chu & Tsao, 2002) computed the area between the centroid and original points to rank fuzzy numbers as:

$$S(A) = \mathfrak{x}\_0(A). \mathfrak{y}\_0(A). \tag{15}$$

Then Wang and Lee (Y. J. Wang, 2008) ranked the fuzzy numbers based on their <sup>0</sup> *x* 's values if they are different. In the case that they are equal, they further compare their <sup>0</sup> *y* 's values to form their ranks.

Further, for two fuzzy numbers *A* and *B* if 0 0 *y* ( ) () *A yB* based on 0 0 *xA xB* ( ) ( ), then *A B* .

By shortcoming of the mentioned methods finally, Abbasbandy and Hajjari (Abbasbandy & Hajjari 2010) improved Cheng's distance centroid as follows:

$$IR(A) = \mathcal{Y}(A)\sqrt{\mathbf{x}\_0(A)^2 + y\_0(A)^2} \tag{16}$$

Where

$$\gamma(A) = \begin{cases} 1 & \int\_0^1 \left( L\_A^{-1}(\mathbf{x}) + R\_A^{-1}(\mathbf{x}) \right) d\mathbf{x} > 0, \\\\ 0 & \int\_0^1 \left( L\_A^{-1}(\mathbf{x}) + R\_A^{-1}(\mathbf{x}) \right) d\mathbf{x} = 0, \\\\ -1 & \int\_0^1 \left( L\_A^{-1}(\mathbf{x}) + R\_A^{-1}(\mathbf{x}) \right) d\mathbf{x} < 0. \end{cases} \tag{17}$$

However, there are some problems on the centroid point methods. In next section, we will present a new index for ranking fuzzy numbers. The proposed index will be constructed by fuzzy distance and centroid point.

#### **b. Method of D-distance (Ma et al. 2000)**

Let all of fuzzy numbers are positive or negative. Without less of generality, assume that all of them are positive. The membership function of *a R* is ( ) 1, *<sup>a</sup> x* if *x a* and () 0 *<sup>a</sup> x* if *x a* . Hence if 0 *a* we have the following

$$\mu\_0(0) = \begin{cases} 1 & \text{x = 0} \\ 0 & \text{x \neq 0} \end{cases}$$

Since 0 () , *x E* left fuzziness and right fuzziness are 0, so for each *<sup>A</sup> E*

Ranking Indices for Fuzzy Numbers 57

1 1 1 <sup>0</sup> ( ) [ ( ) ( )] *<sup>A</sup> A A*

1 1 1

1 1 1 ( ( ) ( )) 0

*Lr Rr*

*Lr R r*

0 0 0 0 0 0

(,) (,) (,) (,) (,) (,)

1 ( ( ) ( )) 0 *A A*

*A A*

*sign L r R r dr*

0

() 1 *A* .

is called sign distance.

*i j pi pj i j pi pj i j pi pj*

*A A iff d A d A A A iff d A d A A A iff d A d A*

 

() 1 *A* .

( ) : { 1,1} *A E* be a function that is defined as follows:

( )

() 0 *L r <sup>A</sup>* then

() 0 *R r <sup>A</sup>* then

> 

For *A* and *B E* define the ranking order of *A* and *B* by *<sup>p</sup> d* on *E* . i.e.

1. The function *<sup>p</sup> d* , sign distance has the Wang and Kerre's properties . 2. The function *<sup>p</sup> d* , sign distance for *p* 1 has the following properties

inf {supp ( ), *A* supp ( ), *B* supp ( ), *A C* supp ( ) *B C* } 0

sup {supp ( ), *A* supp ( ), *B* supp( ), *A C* supp ( ) *B C* } 0

b. If *B A* and 11 11 ( ) () () ( ) () () *p p <sup>p</sup> <sup>p</sup> ALr R r BLr R r AA BB*

 

, that for all *<sup>r</sup>* 0,1

3. Suppose *A* and *B E* are arbitrary then

a. If *A B* then *A B* ,

then . *B A* ,

*A*

**5.5 Definition** 

Let 

Where

**5.6 Remark** 

**5.7 Definition** 

**5.8 Definition** 

**5.9 Remark** 

if

or

1. If supp() 0 *A* or inf <sup>1</sup>

2. If supp() 0 *A* or sup <sup>1</sup>

For *A E* **,** 0 0 ( , ) ()( , ) *<sup>p</sup> d A ADA* 

$$D(A\_\prime \mu\_0) = \left[ \bigcup\_{0}^{1} \left( L\_A^{-1}(r)^2 + R\_A^{-1}(r)\_2 \right) dr \right] \tag{18}$$

Thus, we have the following definition

#### **5.1 Definition**

For *A* and *B E* , define the ranking of *A* and *B* by saying

$$\begin{aligned} A &\succ B & \text{iff} & & d(A,\mu\_0) \succ d(B,\mu\_0) \\ A &\prec B & \text{iff} & & d(A,\mu\_0) \prec d(B,\mu\_0) \\ A &\approx B & \text{iff} & & d(A,\mu\_0) = d(B,\mu\_0) \end{aligned}$$

#### **5.2 Property**

Suppose *A* and , *B E* are arbitrary, therefore

If *A B* then . *A B*

If *B A* and 12 12 12 12 () () () () *Lr Rr Lr Rr AABB* for all *r* 0,1 then . *B A*

#### **5.3 Remark**

The distance triangular fuzzy number 0 *A x* ( ,,) of 0 is defined as following**:** 

$$d(A, \mu\_0) = \left[\mathbf{2x}\_0^2 + \sigma^2 \;/\,\mathbf{3} + \boldsymbol{\beta}^2 \;/\,\mathbf{3} + \mathbf{x}\_0(\boldsymbol{\beta} - \sigma)\right] \cdot \mathbf{1}^{1/2} \tag{19}$$

The distance trapezoidal fuzzy number 0 0 *A xy* ( , ,,) of 0 is defined as following

$$d(A, \mu\_0) = \left[\mathbf{2}\mathbf{x}\_0^2 + \sigma^2 \;/\,\mathbf{3} + \boldsymbol{\beta}^2 \;/\,\mathbf{3} - \mathbf{x}\_0 \boldsymbol{\sigma} + y\_0 \boldsymbol{\beta}\right] \cdot \text{l}^{1/2} \tag{20}$$

If *A B* it is not necessary that *A B* .

$$\text{If } A \neq B \text{ and } \left(L\_A^{-1}(r)^2 + R\_A^{-1}(r)^2\right)^{1/2} = \left(L\_B^{-1}(r)^2 + R\_B^{-1}(r)^2\right)^{1/2} \text{ then } A \approx B.$$

#### **c. Method of sign distance (Abbasbandy & Asady 2006)**

#### **5.4 Definition**

For arbitrary fuzzy numbers 1 1 ( ), ( ) *A L rR r A A* and 1 1 ( ), ( ) *B L rR r B B* the function

$$D(A\_\prime A\_0) = \left(\int\_0^1 \left(\left|L\_A^{-1}(\mathbf{x})\right|^p + \left|R\_A^{-1}(\mathbf{x})\right|^p\right) d\mathbf{x}\right)^{\frac{1}{p}}.\tag{21}$$

is the distance between *A* and *B* .

#### **5.5 Definition**

Let ( ) : { 1,1} *A E* be a function that is defined as follows:

$$\varphi(A) = \text{sign}\int\_0^1 [L\_A^{-1}(r) + R\_A^{-1}(r)] dr$$

Where

56 Recurrent Neural Networks and Soft Computing

0 2

( , ) () () . *D A L r R r dr*

*A B iff d A d B A B iff d A d B A B iff d A d B*

 

*A A*

1

0

For *A* and *B E* , define the ranking of *A* and *B* by saying

Thus, we have the following definition

Suppose *A* and , *B E* are arbitrary, therefore

If *B A* and 12 12 12 12

The distance triangular fuzzy number 0 *A x* ( ,,)

The distance trapezoidal fuzzy number 0 0 *A xy* ( , ,,)

If *<sup>A</sup> <sup>B</sup>* and 1 2 1 2 1/2 1 2 1 2 1/2 ( () ()) ( () ()) *Lr Rr Lr Rr AA BB*

For arbitrary fuzzy numbers 1 1

**c. Method of sign distance (Abbasbandy & Asady 2006)** 

If *A B* it is not necessary that *A B* .

is the distance between *A* and *B* .

() () () () *Lr Rr Lr Rr AABB*

**5.1 Definition** 

**5.2 Property** 

**5.3 Remark** 

**5.4 Definition** 

If *A B* then . *A B*

 

12 1 1/2

0 0 0 0 0 0

( , ) ( , ), ( , ) ( , ), ( , ) ( , ).

for all *r* 0,1 then . *B A*

 of 

> 

 

0 0 <sup>0</sup> *dA x* ( , ) 2 /3 /3 ( ) .

0 0 0 0 *dA x* ( , ) 2 /3 /3

then . *A B*

 

 

( ), ( ) *A L rR r A A*

22 2 1/2

 of 

22 2 1/2

and 1 1

1 1 1 <sup>0</sup> <sup>0</sup> ( , ) ( () ()) .

 

*D A A L x R x dx A A*

 *<sup>x</sup>* (19)

 *x y* ) . (20)

( ), ( ) *B L rR r B B*

*p p p*

1

(21)

the function

(18)

0 is defined as following**:** 

0 is defined as following

$$\mathcal{Y}(A) = \begin{cases} 1 & \int\_0^1 (L\_A^{-1}(r) + R\_A^{-1}(r)) > 0 \\ -1 & \int \left(L\_A^{-1}(r) + R\_A^{-1}(r)\right) < 0 \end{cases}$$

#### **5.6 Remark**


#### **5.7 Definition**

For *A E* **,** 0 0 ( , ) ()( , ) *<sup>p</sup> d A ADA* is called sign distance.

#### **5.8 Definition**

For *A* and *B E* define the ranking order of *A* and *B* by *<sup>p</sup> d* on *E* . i.e.

$$\begin{aligned} A\_i &\succ A\_j ⇔& d\_p(A\_i, \mu\_0) \succ d\_p(A\_j, \mu\_0),\\ A\_i &\prec A\_j ⇔& d\_p(A\_i, \mu\_0) \prec d\_p(A\_j, \mu\_0),\\ A\_i &\approx A\_j ⇔& d\_p(A\_i, \mu\_0) = d\_p(A\_j, \mu\_0) \end{aligned}$$

#### **5.9 Remark**


if

inf {supp ( ), *A* supp ( ), *B* supp ( ), *A C* supp ( ) *B C* } 0

or

sup {supp ( ), *A* supp ( ), *B* supp( ), *A C* supp ( ) *B C* } 0

3. Suppose *A* and *B E* are arbitrary then

$$\text{a.} \quad \text{If } \ A = B \text{ then } A \approx B \dots$$

$$\begin{aligned} \text{b.} \quad &\text{If } \; B \subseteq A \text{ and } \; \mathcal{Y}(A) \Big( \left| \mathcal{L}\_A^{-1}(r) \right|^p + \left| \mathcal{R}\_A^{-1}(r)^p \right| \Big) \succ \mathcal{Y}(B) \Big( \left| \mathcal{L}\_B^{-1}(r) \right|^p + \left| \mathcal{R}\_B^{-1}(r) \right|^p \Big), \text{ that for all } \; r \in [0, 1] \\ &\text{then } \; B \prec A. \end{aligned}$$

Ranking Indices for Fuzzy Numbers 59

, we define the magnitude of the trapezoidal fuzzy number *A* as

 <sup>1</sup> 1 1 0 0 <sup>0</sup> <sup>1</sup> ( ) ( () () )() . <sup>2</sup>

where the function *f* ( )*r* is a non-negative and increasing function on 0,1 with

used to rank the fuzzy numbers. The larger *Mag A*( ) , the larger fuzzy number. Therefore for any two fuzzy number *A* and *B E* . We defined the ranking of *A* and *B* by the *Mag A*( )

Then we 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* . In other words, this method is placed in the first class of Kerre's

Let *A abcd* (,,,) be a non-normal trapezoidal fuzzy number with *r* cut

<sup>2</sup> (3 2)( ) (3 2)( ) ( ) . <sup>12</sup> <sup>12</sup> *bc ad Mag A*

5 1 ( ) ( ) ( ). 12 12

In the following, we use an example to illustrate the ranking process of the proposed

1 1 11 1

<sup>1</sup> ( ) ( ( ) ( ) (1) (1)) ( ) . <sup>2</sup>

*<sup>M</sup> A AAA ag A L r dr R r L R f r dr* (26)

 

> 

(24)

*Mag A b c a d* (25)

**g. Method of promoter operator (Hajjari & Abbasbandy 2011)** 

. Consequently, we have

It is clear that for normal trapezoidal fuzzy numbers the formula (24) reduces to

*<sup>M</sup> A A ag A L r R r x y f r dr fr r* ( ) (23)

<sup>1</sup> () . <sup>2</sup> *f r dr* for example, we can use. The resulting scalar value is

 

with parametric

**f. Method of magnitude (Abbasbandy & Hajjari 2009)** 

0

*Mag A Mag B* ( ) () if and only if *A B* . *Mag A Mag B* ( ) () if and only if*A B* . *Mag A Mag B* ( ) () if and only if *A B* **.** 

categories(X. Wang & Kerre 2001).

representation 1 1 ( ), ( ) . *A L rR r A A*

Moreover, for normal fuzzy numbers we have

0

form 1 1 ( ), ( ) *A L rR r A A*

*f f* (0) 0, (1) 1 and <sup>1</sup>

on *E* as follows

method.

For an arbitrary trapezoidal fuzzy number 0 0 *A xy* ( , ,,)

$$\begin{aligned} \text{14.} \quad & \text{If} \quad A \approx B \quad \text{it} \quad \text{is} \quad \text{not} \quad \text{ necessary} \quad \text{that} \quad A = B. \quad \text{Since} \quad \text{if} \quad A \neq B \quad \text{and} \quad \text{25.}\\ \text{?} \quad \chi(A) \left( \left| L\_A^{-1}(r) \right|^p + \left| R\_A^{-1}(r)^p \right| \right) = \chi(B) \left( \left| L\_B^{-1}(r) \right|^p + \left| R\_B^{-1}(r) \right|^p \right) \text{ that for all} \quad r \in [0, 1] \text{ then } B \approx A. \end{aligned}$$

5. If *A B* then . *A B*

Therefore, we can simply rank the fuzzy numbers by the defuzzification of 0 ( , ). *<sup>p</sup> d A* By Remark 3.12 part (5) we can logically infer ranking order of the image of the fuzzy numbers.

#### **d. Method of H-distance**

#### **5.10 Definition**

A continuous function *s*: 0,1 0,1 with the following properties is a source function *<sup>s</sup>*(0) 0, *<sup>s</sup>*(1) 1, *s r*( ) is increasing, and <sup>1</sup> 0 <sup>1</sup> () . <sup>2</sup> *s r dr*

In fact, a reducing has the reflection of weighting the influence of the different r-cuts and diminishes the contribution of the lower r-levels. This is reasonable since these levels arise from values of membership function for which there is a considerable amount of uncertainty. For example, we can use *sr r* () .

#### **5.11 Definition**

For *A* and *B E* we define H-distance of *A* and *B* by

$$D\_H^\*(A,B) = \frac{1}{2} \left| \left| Val(A) - Val(B) \right| + \left| Amb(A) - Amb(B) \right| + d\_H \left( \left| A \right|^1 + \left| B \right|^1 \right) \right| \tag{22}$$

where *dH* is the Housdorf metric between intervals and <sup>1</sup> . is the 1-cut representation of a fuzzy number.

#### **e. Method of source distance (Ma et al., 2000)**

#### **5.12 Definition**

For *A* and *B E* we define source distance of *A* and *B* by

$$D\_s(A, B) = \frac{1}{2} \left( \left| \operatorname{Val}\_s(A) - \operatorname{Val}\_s(B) \right| + \left| \operatorname{Amb}\_s(A) - \operatorname{Amb}\_s(B) \right| + \max\{ |\operatorname{t}\_A - \operatorname{t}\_B|, |\operatorname{t}\_\ell| \le m\_A - m\_B \} \right),$$

where [ ,] *m t <sup>A</sup> <sup>A</sup>* and [ ,] *m t B B* are the cores of fuzzy numbers *A* and *B* respectively.

#### **5.13 Property**

The source distance metric *Ds* is a metric on *ETR* and a pseudo-metric on . *E*

4. If *A B* it is not necessary that *A B* . Since if *A B* and

Therefore, we can simply rank the fuzzy numbers by the defuzzification of 0 ( , ). *<sup>p</sup> d A*

Remark 3.12 part (5) we can logically infer ranking order of the image of the fuzzy numbers.

A continuous function *s*: 0,1 0,1 with the following properties is a source function

In fact, a reducing has the reflection of weighting the influence of the different r-cuts and diminishes the contribution of the lower r-levels. This is reasonable since these levels arise from values of membership function for which there is a considerable amount of

\* <sup>1</sup> 1 1 ( ,) ( ) () ( ) () <sup>2</sup>

 <sup>1</sup> ( , ) ( ) ( ) ( ) ( ) max{| |,| |} , <sup>2</sup> *D A B Val A Val B Amb A Amb B t t m m s ss s s A <sup>B</sup> <sup>A</sup> <sup>B</sup>*

where [ ,] *m t <sup>A</sup> <sup>A</sup>* and [ ,] *m t B B* are the cores of fuzzy numbers *A* and *B* respectively.

The source distance metric *Ds* is a metric on *ETR* and a pseudo-metric on . *E*

*D A B Val A Val B Amb A Amb B d A B H H* (22)

<sup>1</sup> () . <sup>2</sup> *s r dr*

0

that for all *<sup>r</sup>* 0,1 then . *B A* ,

By

. is the 1-cut representation of a

11 11 ( ) () () ( ) () () *p p <sup>p</sup> <sup>p</sup> ALr R r BLr Rr AA BB*

 

5. If *A B* then . *A B*

**d. Method of H-distance** 

*<sup>s</sup>*(0) 0, *<sup>s</sup>*(1) 1, *s r*( ) is increasing, and <sup>1</sup>

uncertainty. For example, we can use *sr r* () .

For *A* and *B E* we define H-distance of *A* and *B* by

where *dH* is the Housdorf metric between intervals and <sup>1</sup>

For *A* and *B E* we define source distance of *A* and *B* by

**e. Method of source distance (Ma et al., 2000)** 

**5.10 Definition** 

**5.11 Definition** 

fuzzy number.

**5.12 Definition** 

**5.13 Property** 

#### **f. Method of magnitude (Abbasbandy & Hajjari 2009)**

For an arbitrary trapezoidal fuzzy number 0 0 *A xy* ( , ,,) with parametric form 1 1 ( ), ( ) *A L rR r A A* , we define the magnitude of the trapezoidal fuzzy number *A* as

$$\text{Mag}(A) = \frac{1}{2} \left( \int\_0^1 (L\_A^{-1}(r) + R\_A^{-1}(r) + \mathbf{x}\_0 + y\_0) f(r) dr \right). \quad f(r) = r \tag{23}$$

where the function *f* ( )*r* is a non-negative and increasing function on 0,1 with *f f* (0) 0, (1) 1 and <sup>1</sup> 0 <sup>1</sup> () . <sup>2</sup> *f r dr* for example, we can use. The resulting scalar value is used to rank the fuzzy numbers. The larger *Mag A*( ) , the larger fuzzy number. Therefore for any two fuzzy number *A* and *B E* . We defined the ranking of *A* and *B* by the *Mag A*( ) on *E* as follows

*Mag A Mag B* ( ) () if and only if *A B* .

*Mag A Mag B* ( ) () if and only if*A B* .

*Mag A Mag B* ( ) () if and only if *A B* **.** 

Then we 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* . In other words, this method is placed in the first class of Kerre's categories(X. Wang & Kerre 2001).

#### **g. Method of promoter operator (Hajjari & Abbasbandy 2011)**

Let *A abcd* (,,,) be a non-normal trapezoidal fuzzy number with *r* cut representation 1 1 ( ), ( ) . *A L rR r A A* . Consequently, we have

$$\text{Mag}(A) = \frac{(\text{G}\,o^2 + \text{2})(b+c)}{12o} + \frac{(\text{G}\,o - \text{2})(a+d)}{12o}.\tag{24}$$

It is clear that for normal trapezoidal fuzzy numbers the formula (24) reduces to

$$\text{Mag}(A) = \frac{5}{12}(b+c) + \frac{1}{12}(a+d). \tag{25}$$

In the following, we use an example to illustrate the ranking process of the proposed method.

Moreover, for normal fuzzy numbers we have

$$\text{Mag}(A) = \frac{1}{2} \left[ \int\_0^1 (L\_A^{-1}(r) dr + R\_A^{-1}(r) + L\_A^{-1}(1) + R\_A^{-1}(1)) f(r) dr \right]. \tag{26}$$

Ranking Indices for Fuzzy Numbers 61

Based on mentioned formulae, the ranking index value of fuzzy numbers

Now, by using (15), for any two fuzzy numbers *Ai* and *Aj* the ranking order is based on the

The revised method of ranking L-R fizzy number based on deviation degree (Asady,

Asady (Asady, 2010) revised Wang et al. (Z.X. Wang et al. 2009) method and suggested

Consider two fuzzy numbers *A* and *B* the ranking order is based on the following

( ) <sup>1</sup>

( ) <sup>1</sup>

*<sup>D</sup> D A*

*<sup>D</sup> D A*

*L A R A*

*L A R A*

*D* 

*<sup>D</sup>* (33)

(34)

max min

*<sup>d</sup> <sup>M</sup> Mi n*

, , 1,2,..., , 1 (1 )

max min

*<sup>d</sup> <sup>M</sup> Mi n*

, 1,2,..., .

(32)

where *M MM M* max max , ,..., 1 2 *<sup>n</sup>* and *Mmin = min{M1, M2, ..., Mn}*

*L i i R i i*

*i R i*

*d*

1

*d*

*i L*

*D*(.) operator for ranking of fuzzy numbers as follows:

*d*

*Ai* , 1, . . ., , *i n* is given by

1. *Ai Aj* if and only if , *<sup>i</sup> <sup>j</sup> d d* 2. *A A <sup>i</sup> <sup>j</sup>* if and only if , *<sup>i</sup> <sup>j</sup> d d* 3. *Ai Aj* if and only if . *<sup>i</sup> <sup>j</sup> d d*

1. If *DA DB* ( ) ( ), then . *A B* 2. If *DA DB* ( ) ( ), then . *A B*

*A B* , ( ) () *DA DB* then , *A B*

*A B* , ( ) () *DA DB* then , *A B*

\*

3. If *DA DB* ( ) ( ), then

if \* \*

if \* \*

following rules.

2010)

situations:

 

 

where

where

else . *A B*

#### **h. Methods of deviation degree**

Ranking L-R fuzzy numbers based on deviation degree (Z.X. Wang et al., 2009)

#### **5.14 Definition**

For any groups of fuzzy numbers 1 2 , ,..., *AA An* in *E* with support sets *SA i n* ( ), 1, . . ., . *<sup>i</sup>* Let 1 ( ) *<sup>n</sup> <sup>i</sup> <sup>i</sup> S SA* and min *x S* inf and max *x S* sup . Then minimal and maximal reference sets *A*min and *A*max are defined as

$$\begin{aligned} \, \, \mu\_{A\_{\min}} \left( \mathbf{x} \right) = \begin{cases} \frac{\mathbf{x}\_{\max} - \mathbf{x}}{\mathbf{x}\_{\max} - \mathbf{x}\_{\min}}, & \quad \text{if} \quad \mathbf{x} \in \mathcal{S} \\\ 0, & \quad \text{otherwise} \end{cases} \end{aligned} \tag{27}$$

$$\mu\_{A\_{\text{max}}} \left( \mathbf{x} \right) = \begin{cases} \frac{\mathbf{x} - \mathbf{x\_{min}}}{\mathbf{x\_{max}} - \mathbf{x\_{min}}}, & \text{if} \quad \mathbf{x} \in \mathcal{S} \\ \mathbf{x\_{max}} - \mathbf{x\_{min}} \\ \mathbf{0}, & \text{otherwise}. \end{cases} \tag{28}$$

#### **5.15 Definition**

For any groups of fuzzy numbers 1 2 , ,..., *A A An* in *E* , let *A*min and *A*max be minimal and maximal reference sets of these fuzzy numbers, respectively. Then left and right deviation degree of *Ai* , 1, . . ., , *i n* are defined as follows:

$$d\_i^L = \int\_{x\_{\min}}^{t\_i} \left(\mu\_{A\_{\min}}\left(\mathbf{x}\right) - L\_A^{-1}\left(\mathbf{x}\right)\right) d\mathbf{x}$$

$$d\_i^R = \int\_{u\_i}^{x\_{\max}} \left(\mu\_{A\_{\max}}\left(\mathbf{x}\right) - R\_A^{-1}\left(\mathbf{x}\right)\right) d\mathbf{x} \tag{29}$$

where *it* and *ui n <sup>i</sup>* , 1,2,..., are the abscissas of the crossover points of *Ai L* and min , *A* and *Ai R* and max , *<sup>A</sup>* respectively.

#### **5.16 Definition**

For any groups of fuzzy numbers *Ai = (ai, bi, ci, di, )* in *E* , its expectation value of centroid is defined as follows:

$$\mathcal{M}\_i = \frac{\int\_{a\_i}^{d\_i} \mathbf{x} \,\mu\_{A\_i}(\mathbf{x}) d\mathbf{x}}{\int\_{a\_i}^{d\_i} \mu\_A(\mathbf{x}) d\mathbf{x}} \tag{30}$$

$$\mathcal{A}\_{i} = \frac{M\_{i} - M\_{\text{min}}}{M\_{\text{max}} - M\_{\text{min}}} \tag{31}$$

For any groups of fuzzy numbers 1 2 , ,..., *AA An* in *E* with support sets *SA i n* ( ), 1, . . ., . *<sup>i</sup>*

max max min , ( )

 

> 

*i x R*

*M*

*i*

*i AA <sup>u</sup> d x* 

*x x x*

*x x x*

*<sup>i</sup> <sup>i</sup> S SA* and min *x S* inf and max *x S* sup . Then minimal and maximal reference

min max min , ( )

For any groups of fuzzy numbers 1 2 , ,..., *A A An* in *E* , let *A*min and *A*max be minimal and maximal reference sets of these fuzzy numbers, respectively. Then left and right deviation

> min min <sup>1</sup> () () *it <sup>L</sup> i AA <sup>x</sup> d x*

> > max max

where *it* and *ui n <sup>i</sup>* , 1,2,..., are the abscissas of the crossover points of *Ai L* and min ,

*i <sup>i</sup> <sup>i</sup> i i*

*<sup>A</sup> <sup>a</sup>*

*i*

*M M M M*

*d <sup>A</sup> <sup>a</sup> i d*

<sup>1</sup> () ()

( )

*x x dx*

( )

*x dx*

min max min

*<sup>R</sup> <sup>x</sup> dx* (29)

*)* in *E* , its expectation value of centroid is

(30)

(31)

*<sup>L</sup> <sup>x</sup> dx*

*x x if x S*

*x x if x S*

0, ,

0, .

*otherwise*

*otherwise*

(27)

(28)

*A*

Ranking L-R fuzzy numbers based on deviation degree (Z.X. Wang et al., 2009)

min

max

*A*

degree of *Ai* , 1, . . ., , *i n* are defined as follows:

*<sup>A</sup>* respectively.

For any groups of fuzzy numbers *Ai = (ai, bi, ci, di,* 

*A*

**h. Methods of deviation degree** 

sets *A*min and *A*max are defined as

**5.14 Definition** 

Let 1 ( ) *<sup>n</sup>*

**5.15 Definition** 

and *Ai R* and max ,

**5.16 Definition** 

defined as follows:

where *M MM M* max max , ,..., 1 2 *<sup>n</sup>* and *Mmin = min{M1, M2, ..., Mn}*

Based on mentioned formulae, the ranking index value of fuzzy numbers *Ai* , 1, . . ., , *i n* is given by

$$d\_i = \begin{cases} \frac{d\_i^L \mathcal{A}\_i}{1 + d\_i^R (1 - \mathcal{A}\_i)}, & M\_{\text{max}} \neq M\_{\text{min}}, \ i = 1, 2, \dots, n, \\\frac{d\_i^L}{1 + d\_i^R} & M\_{\text{max}} = M\_{\text{min}}, \ i = 1, 2, \dots, n. \end{cases} \tag{32}$$

Now, by using (15), for any two fuzzy numbers *Ai* and *Aj* the ranking order is based on the following rules.


Asady (Asady, 2010) revised Wang et al. (Z.X. Wang et al. 2009) method and suggested *D*(.) operator for ranking of fuzzy numbers as follows:

Consider two fuzzy numbers *A* and *B* the ranking order is based on the following situations:


where

$$D(A) = \frac{D\_A^L}{1 + D\_A^R} \tag{33}$$

$$D^\*(A) = \frac{D\_A^L \mathcal{Y}}{1 + D\_A^R \mathcal{Y}} \tag{34}$$

where

Ranking Indices for Fuzzy Numbers 63

Then they ranked fuzzy numbers 1 2 , ,..., *AA An* based on the ranking area values 1 2 , ,..., *<sup>n</sup> ss s* .

In the next section, we discuss on those methods that based on deviation degree by a

Let two fuzzy numbers *A= (3,6,9)* and *B= (5,6,7)* from (Z.-X. Wang et al., 2009) as shown in

Through the approaches in this paper, the ranking index can be obtained as *Mag (A)=Mag(B)=12* and *EV(A) = EV(B) = 6*. Then the ranking order of fuzzy numbers is

 *B*. Because fuzzy numbers *A* and *B* have the same mode and symmetric spread, most of existing approaches have the identical results. For instance, by Abbasbandy and Asady's approach (Abbasbandy & Asady, 2006), different ranking orders are obtained when different index values *p* are taken. When *p = 1* and *p = 2* the ranking order is the same, i.e.,

 *B* Nevertheless, the same results produced when distance index, *CV* index of Cheng's approach and Chu and Tsao's area are respectively used, i.e., xA = xB = 6 and <sup>1</sup>

then from Cheng's distance and Chau and Tsao's area we get that *R(A) = R(B) = 2.2608*,

*B=(5, 6, 7)*. Now we review the ranking approaches by promoter operator. Since *A* and *B* have the same ranking order and the same centroid points we then compute their

ambiguities. Hence, from (Deng et al., 2006) it will be obtained *amb(A)* = 1 and <sup>1</sup> ( ) <sup>3</sup>

 *B*, for two triangular fuzzy numbers *A=(3, 6, 9)* and

<sup>3</sup> *A B y y*

*amb B* .

Nevertheless, the new ranking method has drawback.

number numerical counter examples.

**6.1 Example** 

Fig. 1.

*A* 

*A* 

**6. Discussion and counter examples** 

Fig. 1. Fuzzy numbers *A=(3,6,9)* and *B=(5,6,7)*

*S(A) = S(B) = 1.4142* respectively.

From the obtained results we have *A* 

Consequently, by using promoter operator we have

$$D\_A^L = \int\_0^1 \left( R\_A^{-1}(\mathbf{x}) + L\_A^{-1}(\mathbf{x}) - 2\mathbf{x}\_{\text{min}} \right) d\mathbf{x} \tag{35}$$

$$D\_A^R = \int\_0^1 \left( 2\mathbf{x}\_{\text{max}} - R\_A^{-1}(\mathbf{x}) - L\_A^{-1}(\mathbf{x}) \right) d\mathbf{x} \tag{36}$$

**Ranking fuzzy numbers based on the left and the right sides of fuzzy numbers** (Nejad & Mashinchi, 2011)

Recently Nejad and Mashinchi (Nejad & Mashinchi, 2011) pointed out the drawback of Wang et al. (Z.X. Wang et al. , 2009) hen they presented a novel ranking method as follows.

#### **5.17 Definition**

Let *Ai iii i* ( , , , , ), 1,2,..., , *abcd i n* are fuzzy numbers in *E* , *a aa a* min min , ,..., 1 2 *<sup>n</sup>* and *d dd d* max max , ,..., . 1 2 *<sup>n</sup>* The areas *<sup>L</sup> is* and *<sup>R</sup> is* of the left and right sides of the fuzzy number *Ai* are defined as

$$s\_i^L = \int\_0^\alpha (L\_A^{-1}(r) - a\_{\min}) dr \tag{37}$$

$$\left. \right|\_{\ast} \mathbf{s}\_{\ast}^{\ast} = \int\_{0}^{\upsilon} \left( d\_{\max} - \boldsymbol{R}\_{\mathcal{A}}^{-1}(r) \right) dr. \tag{38}$$

Based on above definitions, the proposed ranking index is

$$s\_{\iota} = \frac{s\_{\iota}^{L}\mathcal{A}\_{\iota}}{1 + s\_{\iota}^{R}(1 - \mathcal{A}\_{\iota})}, \quad i = 1, 2, \ldots, n. \tag{39}$$

Then the ranking order follows next rules.


To obtain the reasonable they added two triangular fuzzy numbers *A*0 and *An*<sup>1</sup> , where

$$\begin{aligned} A\_0 &= (a\_0, b\_0, d\_0), \\ a\_0 &= 2b\_0 - d\_0, \quad b\_0 = \min\{a\_i, \ i = 1, 2, \dots, n\}, \\ d\_0 &= (d + b\_0) / 2, \quad d = \min\{d\_i, \ i = 1, 2, \dots, n\} \end{aligned} \tag{40}$$

and

$$\begin{aligned} A\_{n+1} &= (a\_{n+1}, b\_{n+1}, d\_{n+1}), \\ a\_{n+1} &= (b\_{n+1} + a) / 2, \quad b\_{n+1} = \max\{d\_i, \quad i = 1, 2, \dots, n\}, \\ d\_{n+1} &= 2b\_{n+1} - a\_{n+1}), \quad a = \max\{a\_i, \quad i = 1, 2, \dots, n\}. \end{aligned} \tag{41}$$

Then they ranked fuzzy numbers 1 2 , ,..., *AA An* based on the ranking area values 1 2 , ,..., *<sup>n</sup> ss s* .

Nevertheless, the new ranking method has drawback.

In the next section, we discuss on those methods that based on deviation degree by a number numerical counter examples.

### **6. Discussion and counter examples**

#### **6.1 Example**

62 Recurrent Neural Networks and Soft Computing

 <sup>1</sup> 1 1 min <sup>0</sup> () () 2 *<sup>L</sup> D R x L x x dx A AA*

 <sup>1</sup> 1 1 max <sup>0</sup> 2 () () *<sup>R</sup> D x R x L x dx <sup>A</sup> A A*

**Ranking fuzzy numbers based on the left and the right sides of fuzzy numbers** (Nejad &

Recently Nejad and Mashinchi (Nejad & Mashinchi, 2011) pointed out the drawback of Wang et al. (Z.X. Wang et al. , 2009) hen they presented a novel ranking method as

*is* and *<sup>R</sup>*

*R*

*s s*

*R i*

Based on above definitions, the proposed ranking index is

*s*

0 000 0 0 00 0 0

*A abd*

( , , ),

),,,(

*n nn i n n n i*

1 11 1 1 1 1 111

 

*dbaA*

*n nnn*

 <sup>1</sup> min <sup>0</sup> ( ) *<sup>L</sup> i A s L r a dr*

> .)( <sup>0</sup> 1

.,...,2,1, )1(1

*i*

*ni*

max *drrRds <sup>A</sup>*

*<sup>i</sup>* 

*i L i <sup>i</sup>* 

To obtain the reasonable they added two triangular fuzzy numbers *A*0 and *An*<sup>1</sup> , where

*a b d b ai n d db d d i n*

 

2 , min{ , 1,2,..., }, ( ) / 2, min{ , 1,2,..., }

2 ), }.,...,2,1,max{ ,2/)( },,...,2,1,max{

*niaaabd baba nid*

 

*i i*

Mashinchi, 2011)

**5.17 Definition** 

Let *Ai iii i* ( , , , , ), 1,2,..., , *abcd i n* 

number *Ai* are defined as

*d dd d* max max , ,..., . 1 2 *<sup>n</sup>* The areas *<sup>L</sup>*

Then the ranking order follows next rules.

1. *A A <sup>i</sup> <sup>j</sup>* if and only if , *<sup>i</sup> <sup>j</sup> s s* 2. *Ai Aj* if and only if , *<sup>i</sup> <sup>j</sup> s s* 3. *Ai Aj* if and only if . *<sup>i</sup> <sup>j</sup> s s*

follows.

and

(35)

(36)

*is* of the left and right sides of the fuzzy

(38)

(39)

(40)

(41)

(37)

are fuzzy numbers in *E* , *a aa a* min min , ,..., 1 2 *<sup>n</sup>* and

Let two fuzzy numbers *A= (3,6,9)* and *B= (5,6,7)* from (Z.-X. Wang et al., 2009) as shown in Fig. 1.

Fig. 1. Fuzzy numbers *A=(3,6,9)* and *B=(5,6,7)*

Through the approaches in this paper, the ranking index can be obtained as *Mag (A)=Mag(B)=12* and *EV(A) = EV(B) = 6*. Then the ranking order of fuzzy numbers is *A B*. Because fuzzy numbers *A* and *B* have the same mode and symmetric spread, most of existing approaches have the identical results. For instance, by Abbasbandy and Asady's approach (Abbasbandy & Asady, 2006), different ranking orders are obtained when different index values *p* are taken. When *p = 1* and *p = 2* the ranking order is the same, i.e., *A B* Nevertheless, the same results produced when distance index, *CV* index of Cheng's

approach and Chu and Tsao's area are respectively used, i.e., xA = xB = 6 and <sup>1</sup> <sup>3</sup> *A B y y*

then from Cheng's distance and Chau and Tsao's area we get that *R(A) = R(B) = 2.2608*, *S(A) = S(B) = 1.4142* respectively.

From the obtained results we have *A B*, for two triangular fuzzy numbers *A=(3, 6, 9)* and *B=(5, 6, 7)*. Now we review the ranking approaches by promoter operator. Since *A* and *B* have the same ranking order and the same centroid points we then compute their ambiguities. Hence, from (Deng et al., 2006) it will be obtained *amb(A)* = 1 and <sup>1</sup> ( ) <sup>3</sup> *amb B* .

Consequently, by using promoter operator we have

Ranking Indices for Fuzzy Numbers 65

Fig. 2.

$$P(A) = \left(M \text{ag}(A), \frac{1}{1 + amb(A)}\right) = (12, \frac{1}{2}) \quad P(B) = \left(M \text{ag}(B), \frac{1}{1 + amb(B)}\right) = (12, \frac{3}{4})$$

$$P(A) = \left(EV(A), \frac{1}{1 + amb(A)}\right) = (6, \frac{1}{2}) \quad P(B) = \left(EV(B), \frac{1}{1 + amb(B)}\right) = (6, \frac{3}{4})$$

$$P(A) = \left(R(A), \frac{1}{1 + amb(A)}\right) = (2.2608, \frac{1}{2}) \quad P(B) = \left(R(B), \frac{1}{1 + amb(B)}\right) = (2.2608, \frac{3}{4})$$

$$P(A) = \left(S(A), \frac{1}{1 + amb(A)}\right) = (12, \frac{1}{2}) \quad P(B) = \left(S(B), \frac{1}{1 + amb(B)}\right) = (12, \frac{3}{4})$$

The ranking order is *A B* Through the proposed approach by Wang et al., the ranking index values can be obtained as *d1* = 0.1429 and *d2* = 0.1567. Then the ranking order of fuzzy numbers is also *A B*.

In the following, we use the data sets shown in Chen and Chen (S. J. Chen et al. 2009) to compare the ranking results of the proposed approaches with Cheng method (Cheng, 1998), Chu and Tsao's method (Chu & Tsao 2002) and Chen and Chen (S. J. Chen et al. 2009). The comparing of ranking results for different methods will be explained in the following.

For the fuzzy numbers *A* and *B* shown in Set 1 of Fig. 4, Cheng's method (Cheng, C. H., 1998), Chu's method (Chu, T. and Tsao, C., 2002), Chen and Chen's method (Chen, S. J. and Chen, S. M., 2007; Chen, S.-M. and Chen, J.-H., 2009) and *Mag-* method (Abbasbandy, S. and Hajjari, T., 2009) get the same ranking order *A B***.**


) 2 1

) 2 1

) 2 1

) 2 1

,

The ranking order is *A B* Through the proposed approach by Wang et al., the ranking index values can be obtained as *d1* = 0.1429 and *d2* = 0.1567. Then the ranking order of fuzzy

In the following, we use the data sets shown in Chen and Chen (S. J. Chen et al. 2009) to compare the ranking results of the proposed approaches with Cheng method (Cheng, 1998), Chu and Tsao's method (Chu & Tsao 2002) and Chen and Chen (S. J. Chen et al. 2009). The comparing of ranking results for different methods will be explained in the following.

For the fuzzy numbers *A* and *B* shown in Set 1 of Fig. 4, Cheng's method (Cheng, C. H., 1998), Chu's method (Chu, T. and Tsao, C., 2002), Chen and Chen's method (Chen, S. J. and Chen, S. M., 2007; Chen, S.-M. and Chen, J.-H., 2009) and *Mag-* method (Abbasbandy, S. and

4. For the fuzzy numbers *A* and *B* shown in Set 2 of Fig. 4, Cheng's method (Cheng, C. H., 1998), Chu 's method (Chu, T. and Tsao, C., 2002) and *Mag-* method (Abbasbandy, S.

(Abbasbandy, S. and Hajjari, T., 2009) get an inaccurate ranking order *A*

(Abbasbandy, S. and Hajjari, T., 2009) get the same ranking order: *A B***.**

by applying the promoter operator the ranking order is the same as Chen and Chen's method (Chen, S. J. and Chen, S. M., 2007; Chen, S.-M. and Chen, J.-H., 2009), i.e. *A B.* 5. For the fuzzy numbers *A* and *B* shown in Set 3 of Fig. 4, Cheng's method (Cheng, C. H., 1998), Chu and Tsao's method (Chu, T. and Tsao, C., 2002) and *Mag-* method

applying the promoter operator the ranking order is the same as Chen and Chen's method (Chen, S. J. and Chen, S. M., 2007; Chen, S.-M. and Chen, J.-H., 2009) i.e. *A B*. 6. For the fuzzy numbers *A* and *B* shown in Set 4 of Fig. 4, Cheng's method (Cheng, C. H., 1998), Chu and Tsao's method (Chu, T. and Tsao, C., 2002), Chen and Chen's method (Chen, S. J. and Chen, S. M., 2007; Chen, S.-M. and Chen, J.-H., 2009) and *Mag-* method

7. For the fuzzy numbers *A* and *B* shown in Set 5 of Fig. 2, Cheng's method (Cheng, C. H., 1998), Chu and Tsao's method (Chu, T. and Tsao, C., 2002) cannot calculate the crispvalue fuzzy number, whereas Chen and Chen's method (S. J. Chen, 2009) and *Mag*

*Mag-* method (Abbasbandy & Hajjari, 2009) get the same ranking order: *A B.*

,

,

,

) 4 3

) 4 3

> ) 4 3

) 4 3

*B*, which is unreasonable. Whereas

*B* whereas by

.

,12( )(1

,6( )(1

,2608.2( )(1

,12( )(1

 

   

> 

<sup>1</sup> ),()(

<sup>1</sup> ),()(

*Bamb*

<sup>1</sup> ),()(

*Bamb*

<sup>1</sup> ),()(

*Bamb*

*Bamb BMagBP*

 

> 

*BEVBP*

 

> 

*BSBP*

*BRBP*

,12( )(1

,6( )(1

,2608.2( )(1

 

,12( )(1

   

 

<sup>1</sup> ),()(

<sup>1</sup> ),()(

*Aamb*

*Aamb AMagAP*

<sup>1</sup> ),()(

*Aamb*

<sup>1</sup> ),()(

*Aamb*

Hajjari, T., 2009) get the same ranking order *A B***.**

and Hajjari, T., 2009) get the same ranking order *A*

 

> 

*AEVAP*

 

> 

*ASAP*

numbers is also *A B*.

*ARAP*

Ranking Indices for Fuzzy Numbers 67

8. For the fuzzy numbers *A* and *B* shown in Set 6 of Fig. 2, Cheng's method (Cheng, C. H., 1998), Chu and Tsao's method (Chu & Tsao, 2002), Chen and Chen's method (S. J. Chen & S. M. Chen, 2007; S.-M. Chen & J.-H. Chen, 2009) and *Mag-* method (Abbasbandy &

9. For the fuzzy numbers *A* and *B* shown in Set 7 of Fig. 2, Cheng's method (Cheng, 1998), Chu and Tsao's method (Chu & Tsao, 2002), Chen and Chen's method (S. J. Chen et al. 2009) get the same ranking order: *B A* , whereas the ranking order by *Mag-* method (Abbasbandy & Hajjari, 2009) is *A B* . By comparing the ranking result of *Mag*method with other methods with respect to Set 7 of Fig. 2, we can see that *Mag-* method considers the fact that defuzzified value of a fuzzy number is more important than the

10. For the fuzzy numbers *A* and *B* shown in Set 8 of Fig. 2, Cheng's method (Cheng, 1998), Chu and Tsao's method (Chu & Tsao, 2002), Chen and Chen's method (S. J. Chen et al., 2009) and *Mag-* method (Abbasbandy & Hajjari, 2009) get the same ranking order: *ABC* , whereas the ranking order by Chen and Chen's method is *AC B* . By comparing the ranking result of mentioned method with other methods with respect to Set 8 of Fig. 4, we can see that Chen's method considers the fact that the spread of a

The idea of ranking fuzzy numbers by deviation degree is useful, but a significant

Now we give some numerical example to show the drawback of the aforementioned

Given two triangular fuzzy number *A* (0.2,0.5,0.8) and *B* (0.4,0.5,0.6) (Nejad &

The ranking order by Nejad and Mashinchi is *A B* . The images of two numbers *A* and *B*

On the other hand, ranking order for *A* and *B* and their images by Wang et al.'s method

This example could be indicated that all methods are reasonable. However, we will show

Consider the three triangular fuzzy numbers A=(1, 2, 6), B = (2.5, 2.75,3) and C = (2, 3, 4),

are *A=(-0.8, -0.5, -0.2), B=(-0.6, -0.5, -0.4)* respectively, then the ranking order is *-B-A*.

that functions of all three methods are not the same in different conditions.

which are taken from Asady's revised (Asady, 2010) (See Fig. 4).

fuzzy number is more important than defuzzified value of a fuzzy number.

approaches should be reserved the important properties such that

Mashinchi, 2011), which are indicated in Fig. 3.

and Asady's revised are *AB A B* , respectively.

Hajjari, 2009) get the same ranking order: *A B.*

spread of a fuzzy number.

 *A B BA A B AC BC AB BC AC*

methods.

**6.2 Example** 

**6.3 Example** 


#### Table 1.

Table 1.


The idea of ranking fuzzy numbers by deviation degree is useful, but a significant approaches should be reserved the important properties such that


Now we give some numerical example to show the drawback of the aforementioned methods.
