**6.2 Example**

Given two triangular fuzzy number *A* (0.2,0.5,0.8) and *B* (0.4,0.5,0.6) (Nejad & Mashinchi, 2011), which are indicated in Fig. 3.

The ranking order by Nejad and Mashinchi is *A B* . The images of two numbers *A* and *B* are *A=(-0.8, -0.5, -0.2), B=(-0.6, -0.5, -0.4)* respectively, then the ranking order is *-B-A*.

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

This example could be indicated that all methods are reasonable. However, we will show that functions of all three methods are not the same in different conditions.

#### **6.3 Example**

Consider the three triangular fuzzy numbers A=(1, 2, 6), B = (2.5, 2.75,3) and C = (2, 3, 4), which are taken from Asady's revised (Asady, 2010) (See Fig. 4).

Ranking Indices for Fuzzy Numbers 69

Let the triangular fuzzy number *A* (1,2,3) and the fuzzy number *B* (1,2,4) with the

2/1 2

Using Asady's method the ranking order is obtained . *A B* However, the ranking order of

From mentioned examples, we can theorize that ranking fuzzy numbers based on deviation

2

)2(1 ,21

*x x*

*x x*

*otherwise*

0 . )2( ,42 <sup>4</sup>

   

which is unreasonable.

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

)( 2/1

 

*x B*

degree cannot rank fuzzy numbers in all situations.

Fig. 4.

**6.4 Example** 

membership function (See Fig. 5)

their images is *BA* ,

Utilizing Nejad and Mashinchi's method the ranking order is *ABC* and the ranking order of their images will be obtained *-C-A-B*, which is illogical.

By using Wang et al.'s method the ranking order is *B A C* and for their images is *-A -C -B,* which is unreasonable too.

From point of revised deviation degree method (Asady, 2010) the ranking orders are *B AC, -C-A-B*, respectively.

From this example, it seems the revised method can rank correctly.

In the next example, we will indicate that none of the methods based on deviation degree can rank correctly in all situations.

Utilizing Nejad and Mashinchi's method the ranking order is *ABC* and the ranking

By using Wang et al.'s method the ranking order is *B A C* and for their images is

From point of revised deviation degree method (Asady, 2010) the ranking orders are

In the next example, we will indicate that none of the methods based on deviation degree

order of their images will be obtained *-C-A-B*, which is illogical.

From this example, it seems the revised method can rank correctly.

 *-C -B,* which is unreasonable too.

can rank correctly in all situations.

*-B*, respectively.

*-A* 

*B AC, -C-A*

Fig. 3.

#### Fig. 4.

#### **6.4 Example**

Let the triangular fuzzy number *A* (1,2,3) and the fuzzy number *B* (1,2,4) with the membership function (See Fig. 5)

$$\mu\_{\mathfrak{g}}(\mathbf{x}) = \begin{cases} \left[1 - (\mathbf{x} - \mathbf{2})^2\right]^{1/2} & 1 \le \mathbf{x} \le \mathbf{2}, \\\\ \left[1 - \frac{1}{4}(\mathbf{x} - \mathbf{2})^2\right]^{1/2} & 2 \le \mathbf{x} \le \mathbf{4}, \\\\ 0 & otherwise. \end{cases}$$

Using Asady's method the ranking order is obtained . *A B* However, the ranking order of their images is *BA* , which is unreasonable.

From mentioned examples, we can theorize that ranking fuzzy numbers based on deviation degree cannot rank fuzzy numbers in all situations.

Ranking Indices for Fuzzy Numbers 71

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