**4. Basic notations and definitions**

#### **4.1 Definition**

50 Recurrent Neural Networks and Soft Computing

Ordering fuzzy subsets is an important event in dealing with fuzzy decision problems in many areas. This issue has been of concern for many researchers over the years. Also, in the last several years, there has been a large and energetic upswing in neuroengineering research aimed at synthesizing fuzzy logic with computational neural networks. The two technologies often complement each other: neural networks supply the brute force necessary to accommodate and interpret large amounts of sensor data and fuzzy logic provides a structural framework that utilizes and exploits these low-level results. As a neural network is well known for its ability to represent functions, and the basis of every fuzzy model is the membership function, so the natural application of neural networks in fuzzy models has emerged to provide good approximations to the membership functions that are essential to the success of the fuzzy approach. Many researchers evaluate and analyze the performance of available methods of ranking fuzzy subsets on a set of selected examples that cover possible situations we might encounter as defining fuzzy subsets at each node of a neural network. Along with prosperity of computer and internet technology, more and more pepole used e-learning system to lecture and study. Therefore, how to evaluate the students' proficiency by arranging is the topic that deverses our attention. This chapter focus on fuzzy ranking approaches to evaluate fuzzy numbersas a tool in neural

In many applications, ranking of fuzzy numbers is an important component of the decision process. Since fuzzy numbers do not form a natural linear order, like real numbers, a key issue in operationalzing fuzzy set theory is how to compare fuzzy numbers. Various approaches have been developed for ranking fuzzy numbers. In the existing research, the commonly used technique is to construct proper maps to transform fuzzy numbers into real numbers so called defuzzification. These real numbers are then compared. Herein, in approaches (; Abbasbandy & Asady, 2006; Abbasbandy & Hajjari, 2009, 2011; Asady, 2010; S. J. Chen & S. M. Chen, 2003, 2007, 2009; Deng & Liu, 2005; Deng et al., 2006; Hajjari, 2011a; Hajjari, 2011b; Z.-X. Wang et al. 2009) a fuzzy number is mapped to a real number based on the area measurement. In approaches (L. H. Chen & Lu, 2001, 2002; Liu & Han, 2005),

 cut set and decision-maker's preference are used to construct ranking function. On the other hand, another commonly used technique is the centroid-based fuzzy number ranking approach (Cheng, 1998; Chu, & Tsao, 2002; Y.J. Wang et al. 2008). It should be noted that with the development of intelligent technologies, some adaptive and parameterized defuzzification methods that can include human knowledge have been proposed. Halgamuge et al. (Halgamuge et al. 1996) used neural networks for defuzzification. Song and Leland (Song & Leland, 1996) proposed an adaptive learning defuzzification technique. Yager (1996) proposed knowledge based on defuzzification process, which becomes more intelligent. Similar to methods of Filve and Yager (Filev & Yager, 1991), Jiang and Li (Jiang & Li, 1996) also proposed a parameterized defuzzification method with Gaussian based distribution transformation and polynomial transformation, but in fact, no method gives a right effective defuzzification output. The computational results of these methods are often

**2. Fuzzy ranking and neural network** 

network.

conflict.

**3. Ranking fuzzy numbers** 

First, In general, a generalized fuzzy number *A* is membership ( ) *<sup>A</sup> x* can be defined as (Dubios & Prade, 1978)

$$\mu\_A(\mathbf{x}) = \begin{cases} L\_A(\mathbf{x}) & a \le \mathbf{x} \le b \\ a & b \le \mathbf{x} \le c \\ R\_A(\mathbf{x}) & c \le \mathbf{x} \le d \\ 0 & otherwise, \end{cases} \tag{1}$$

where 0 1 is a constant, and *L ab <sup>A</sup>* : , 0, , *R cd <sup>A</sup>* : , 0, are two strictly monotonical and continuous mapping from *R* to closed interval 0, . If 1 , then *A* is a normal fuzzy number; otherwise, it is a trapezoidal fuzzy number and is usually denoted by *A abcd* (,,,, ) or *A* (,,,) *abcd* if 1 .

In particular, when *b c* , the trapezoidal fuzzy number is reduced to a triangular fuzzy number denoted by *A abd* (,,, ) or *A abd* (,,) if 1 .Therefore, triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers.

Since *LA* and *RA* are both strictly monotonical and continuous functions, their inverse functions exist and should be continuous and strictly monotonical. Let <sup>1</sup> : , 0, *L ab <sup>A</sup>* and <sup>1</sup> : , 0, *R ab <sup>A</sup>* be the inverse functions of ( ) *L x <sup>A</sup>* and ( ) *R x <sup>A</sup>* , respectively. Then <sup>1</sup> ( ) *L r <sup>A</sup>* and <sup>1</sup> ( ) *R r <sup>A</sup>* should be integrable on the close interval0, . In other words, both 1 <sup>0</sup> ( ) *L r dr <sup>A</sup>* and <sup>1</sup> <sup>0</sup> ( ) *R r dr <sup>A</sup>* should exist. In the case of trapezoidal fuzzy number, the inverse functions <sup>1</sup> ( ) *L r <sup>A</sup>* and <sup>1</sup> ( ) *R r <sup>A</sup>* can be analytically expressed as

$$\begin{aligned} \cdot L\_A^{-1}(r) &= a + (b - a)r \;/\; \alpha \; \; \; 0 \le \alpha \le 1 \\\\ R\_A^{-1}(r) &= d - (d - c)r \;/\; \; \; 0 \le \alpha \le 1 \end{aligned}$$

The set of all elements that have a nonzero degree of membership in *A* , it is called the support of *A* , i.e.

$$S(A) = \{ \mathbf{x} \in X \mid \mu\_A(\mathbf{x}) \succ \mathbf{0} \}\tag{2}$$

The set of elements having the largest degree of membership in *A* , it is called the core of *A* , i.e.

Ranking Indices for Fuzzy Numbers 53

Another parameter is utilized for representing the typical value of the fuzzy number is the middle of the expected interval of a fuzzy number and it is called the expected value of a

> 1 1 1 1 0 0 <sup>1</sup> ( ) () () . <sup>2</sup> *EV A L r dr R r dr A A*

and 1 1 ( ), ( ) *B L rR r B B*

 1/2 <sup>2</sup> 1 1 <sup>2</sup> 11 1 1 0 0 ( , ) () () () () *D A B L r L r dr R r R r dr AB A B* (10)

is the distance between A and B. The function *DAB* (,)is a metric in E and (, ) *E D* is a

The ordering indices are organized into three categories by Wang and Kerre (Wang & Kerre,

 **Defuzzification method**: Each index is associated with a mapping from the set of fuzzy quantities to the real line. In this case, fuzzy quantities are compared according to the

**Reference set method**: in this case, a fuzzy set as a reference set is set up and all the

**Fuzzy relation method**: In this case, a fuzzy relation is constructed to make pair wise

Let M be an ordering method on E. The statement two elements *A*1 and *A*2 in E satisfy that *A*1 has a higher ranking than *A*2 when M is applied will be written as *A*1 1 *A* by M. *A*1 1 *A* and *A*1 1 *A* are similarly interpreted. The following reasonable properties for the

fuzzy quantities to be ranked are compared with the reference set.

ordering approaches are introduced by Wang and Kerre (Wang & Kerre 2001).

comparisons between the fuzzy quantities involved.

fuzzy number *A* i.e. number *A* is given by (Bodjanova, 2005)

The first of maxima (FOM) is the smallest element of *core A*( ).i.e.

The last of maxima (LOM) is the greatest element of *core A*( ).i.e.

For arbitrary fuzzy numbers 1 1 ( ), ( ) *A L rR r A A*

**4.6 Definition** 

**4.7 Definition** 

**4.8 Definition** 

complete metric space.

corresponding real numbers.

2001) as follows:

1 1 1 1 0 0 ( ) () , () *EI A L r dr R r dr A A*

. (6)

(7)

*FOM core A* min ( ). (8)

*LOM core A* max ( ). (9)

the equality

$$\mathcal{C}(A) = \left\{ \mathbf{x} \in X \mid \mu\_A(\mathbf{x}) = \sup\_{\mathbf{x} \in X} L\_A(\mathbf{x}) \right\} \tag{3}$$

In the following, we will always assume that *A* is continuous and bounded support *S A*( ) . The strong support of *A* should be *SA ad* () , .

#### **4.2 Definition**

The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalent represented in (Zadeh, 1965; Ma et al., 1999; Dubois & Prade, 1980) as follows.

For arbitrary 1 1 ( ), ( ) *A L rR r A A* and we define addition 1 1 ( ), ( ) *B L rR r B B A B* and multiplication by scalar 0 *k* as

$$\begin{aligned} \left(\underline{A+B}\right)(r) &= \underline{A}(r) + \underline{B}(r) \\ \left(\overline{A+B}(r)\right) &= \overline{A}(r) + \overline{B}(r) \\ \left(\underline{k}\underline{A}\right)(r) &= k\underline{A}(r), \left(\overline{k}\overline{A}\right)(r) = k\overline{A}(r) .\end{aligned}$$

To emphasis, the collection of all fuzzy numbers with addition and multiplication as defined by (8) is denoted by E, which is a convex cone. The image (opposite) of *A* (,,,) *abcd* is *A dcba* (,,,) (Zadeh, L.A, 1965; Dubois, D. and H. Prade, 1980).

#### **4.3 Definition**

A function *f* : 0,1 0,1 is a reducing function if is *s* increasing and *f* (0) 0 and *f* (1) 1 . We say that s is a regular function if *f r dr* ( ) 1/2 .

#### **4.4 Definition**

If *A* is a fuzzy number with r-cut representation, 1 1 ( ), ( ) *L rR r A A* and s is a reducing function, then the value of *A* (with respect to s); it is defined by

$$Val(A) = \int\_0^1 f(r)[L\_A^{-1}(r) + R\_A^{-1}(r)]dr\tag{4}$$

#### **4.5 Definition**

If *A* is a fuzzy number with r-cut representation 1 1 ( ), ( ) *L rR r A A* , and s is a reducing function then the ambiguity of *A* (with respect to s) is defined by

$$Amb(A) = \int\_0^1 f(r)[R\_A^{-1}(r) - L\_A^{-1}(r)] dr\tag{5}$$

Let also recall that the expected interval *EI A*( ) of a fuzzy number *A* is given by

$$EI(A) = \left[ \int\_0^1 L\_A^{-1}(r) dr, \int\_0^1 R\_A^{-1}(r) dr \right]. \tag{6}$$

Another parameter is utilized for representing the typical value of the fuzzy number is the middle of the expected interval of a fuzzy number and it is called the expected value of a fuzzy number *A* i.e. number *A* is given by (Bodjanova, 2005)

$$EV(A) = \frac{1}{2} \left[ \int\_0^1 L\_A^{-1}(r) dr + \int\_0^1 R\_A^{-1}(r) dr \right]. \tag{7}$$

#### **4.6 Definition**

52 Recurrent Neural Networks and Soft Computing

( ) | ( ) sup ( ) *A A*

In the following, we will always assume that *A* is continuous and bounded support *S A*( ) .

The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalent represented in (Zadeh, 1965; Ma et al., 1999; Dubois &

> () () () ( () ()

*kA r kA r kA r kA r*

To emphasis, the collection of all fuzzy numbers with addition and multiplication as defined by (8) is denoted by E, which is a convex cone. The image (opposite) of *A* (,,,) *abcd* is

A function *f* : 0,1 0,1 is a reducing function if is *s* increasing and *f* (0) 0

1 1 1

and we define addition 1 1

( ) ( ), ( ) ( ).

*CA x X x L x* 

The strong support of *A* should be *SA ad* () , .

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

 

*A dcba* (,,,) (Zadeh, L.A, 1965; Dubois, D. and H. Prade, 1980).

and *f* (1) 1 . We say that s is a regular function if *f r dr* ( ) 1/2 .

function, then the value of *A* (with respect to s); it is defined by

If *A* is a fuzzy number with r-cut representation 1 1

then the ambiguity of *A* (with respect to s) is defined by

<sup>1</sup> 1 1

Let also recall that the expected interval *EI A*( ) of a fuzzy number *A* is given by

If *A* is a fuzzy number with r-cut representation, 1 1

*A B r Ar Br A Br Ar Br*

 

**4.2 Definition** 

**4.3 Definition** 

**4.4 Definition** 

**4.5 Definition** 

Prade, 1980) as follows.

For arbitrary 1 1

multiplication by scalar 0 *k* as

*x X*

(3)

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

<sup>0</sup> ( ) ( )[ ( ) ( )] *Val A A A <sup>f</sup> r L r R r dr* (4)

<sup>0</sup> ( ) ( )[ ( ) ( )] *Amb A A A <sup>f</sup> r R r L r dr* (5)

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

and s is a reducing

, and s is a reducing function

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

*A B* and

The first of maxima (FOM) is the smallest element of *core A*( ).i.e.

$$FOM = \min \operatorname{core}(A). \tag{8}$$

#### **4.7 Definition**

The last of maxima (LOM) is the greatest element of *core A*( ).i.e.

$$LOM = \max core(A). \tag{9}$$

#### **4.8 Definition**

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

$$D(A,B) = \left[\int\_0^1 \left(L\_A^{-1}(r) - L\_B^{-1}(r)\right)^2 dr + \int\_0^1 \left(R\_A^{-1}(r) - R\_B^{-1}(r)\right)^2 dr\right]^{1/2} \tag{10}$$

is the distance between A and B. The function *DAB* (,)is a metric in E and (, ) *E D* is a complete metric space.

The ordering indices are organized into three categories by Wang and Kerre (Wang & Kerre, 2001) as follows:


Let M be an ordering method on E. The statement two elements *A*1 and *A*2 in E satisfy that *A*1 has a higher ranking than *A*2 when M is applied will be written as *A*1 1 *A* by M. *A*1 1 *A* and *A*1 1 *A* are similarly interpreted. The following reasonable properties for the ordering approaches are introduced by Wang and Kerre (Wang & Kerre 2001).

Ranking Indices for Fuzzy Numbers 55

In this case, normal triangular fuzzy numbers could be compared or ranked directly in

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:

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

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

By shortcoming of the mentioned methods finally, Abbasbandy and Hajjari (Abbasbandy &

0 0 *IR A A x A y A* () () () ()

0

0

0

2 2

*L x R x dx*

1 () () 0,

*A A*

1 1 1

1 1 1

( ) 0 () () 0,

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

Let all of fuzzy numbers are positive or negative. Without less of generality, assume that all

1 0 (0) 0 0. *x x*

*A L x R x dx*

1 1 1

*A A*

*L x R x dx*

1 ( ) ( ) 0.

*A A*

2 2

0 0 *RA x A y A* ( ) ( ) ( ). (14)

0 0 *SA x A y A* ( ) ( ). ( ). (15)

(16)

(17)

*x* if *x a* and () 0 *<sup>a</sup>*

*<sup>A</sup> E* *x* if

terms of their centroid coordinates on horizontal axis. Cheng (Cheng, 1998) formulated his idea as follows:

Hajjari 2010) improved Cheng's distance centroid as follows:

 

 

of them are positive. The membership function of *a R* is ( ) 1, *<sup>a</sup>*

0

() , *x E* left fuzziness and right fuzziness are 0, so for each

fuzzy distance and centroid point.

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

*x a* . Hence if 0 *a* we have the following

form their ranks.

*A B* .

Where

Since 0 

