**1. Introduction**

48 Recurrent Neural Networks and Soft Computing

[8] J.S. R. Jang "ANFIS: Adaptive Network based Fuzzy Inference Systems", *IEEE Transactions on Systems, Man, and. Cybernetics*, vol. 23, no. 3, (1993) 665–685.

[12] G. Dudek, Michael Jenkin, *Computational Principles of Mobile Robotics*, Cambridge

[13] A. El-Bastawesy , A. El-sayed, M. Abdel-Salam, B. Salah, I. Adel, M. Alaa El-laffy, M.

[10] S. Rao, Optimization, *Theory & Applications 2ed*, July 1984, John Wiley & Sons (Asia). [11] R. Gonzalez, *ANNs for Variational Problems in Engineering,* PhD Thesis, Department of

Department Cairo, EGYPT, November 25-27, 2008.

[9] http://sourceforge.indices-masivos.com/projects/forallahfacon/

University Press; April 15, 2000.

Arab Emirates, 2011

2008.

Faculty of Engineering - Ain Shams University, Computer Engineering & Systems

Computer Languages and Systems, Technical University of Catalonia,21 September

Tariq, B. Magdi, O. Fathi, A New Intelligent Strategy for Optimal Design of High Dimensional Systems, 2011 IEEE GCC Conference & Exhibition, Dubai, United

Fuzzy set theory has been studied extensively over the past 30 years. Most of the early interest in fuzzy set theory pertained to representing uncertainty in human cognitive processes (see for example Zadeh (1965)). Fuzzy set theory is now applied to problems in engineering, business, medical and related health sciences, and the natural sciences. In an effort to gain a better understanding of the use of fuzzy set theory in production management research and to provide a basis for future research, a literature review of fuzzy set theory in production management has been conducted. While similar survey efforts have been undertaken for other topical areas, there is a need in production management for the same. Over the years there have been successful applications and implementations of fuzzy set theory in production management. Fuzzy set theory is being recognized as an important problem modeling and solution technique.

Kaufmann and Gupta (1988) report that over 7,000 research papers, reports, monographs, and books on fuzzy set theory and applications have been published since 1965.

As evidenced by the large number of citations found, fuzzy set theory is an established and growing research discipline. The use of fuzzy set theory as a methodology for modeling and analyzing decision systems is of particular interest to researchers in production management due to fuzzy set theory's ability to quantitatively and qualitatively model problems which involve vagueness and imprecision. Karwowski and Evans (1986) identify the potential applications of fuzzy set theory to the following areas of production management: new product development, facilities location and layout, production scheduling and control, inventory management, quality and cost benefit analysis. Karwowski and Evans identify three key reasons why fuzzy set theory is relevant to production management research. First, imprecision and vagueness are inherent to the decision maker's mental model of the problem under study. Thus, the decision maker's experience and judgment may be used to complement established theories to foster a better understanding of the problem. Second, in the production management environment, the information required to formulate a model's objective, decision variables, constraints and parameters may be vague or not precisely measurable. Third, imprecision and vagueness as a result of personal bias and subjective opinion may further dampen the quality and quantity of available information. Hence, fuzzy set theory can be used to bridge modeling gaps in descriptive and prescriptive decision models in production management research.

Ranking Indices for Fuzzy Numbers 51

We often face difficultly in selecting appropriate defuzzification, which is mainly based on intuition and there is no explicit decision making for these parameters. For more comparison details on most of these methods, in this chapter we review some of ranking

 

*cxb*

*otherwise*

, *R cd <sup>A</sup>* : , 0,

0 ,

*xR dxc*

*xL bxa*

*x* can be defined as

are two strictly

. In other words, both

, then *A* is a

(1)

 . If 1 

.Therefore, triangular fuzzy

*<sup>A</sup>* (2)

First, In general, a generalized fuzzy number *A* is membership ( ) *<sup>A</sup>*

  )(

normal fuzzy number; otherwise, it is a trapezoidal fuzzy number and is usually denoted by

In particular, when *b c* , the trapezoidal fuzzy number is reduced to a triangular fuzzy

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>*

should be integrable on the close interval0,

. <sup>1</sup> *L r a b ar <sup>A</sup>* () ( )/

<sup>1</sup> *R r d d cr <sup>A</sup>* () ( )/

or *A abd* (,,) if 1

be the inverse functions of ( ) *L x <sup>A</sup>* and ( ) *R x <sup>A</sup>* , respectively. Then

should exist. In the case of trapezoidal fuzzy number, the

can be analytically expressed as

0 1

01

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

*SA x X x* ( ) | () 0 

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

*A*

*A*

)(

 

)(

*A*

is a constant, and *L ab <sup>A</sup>* : , 0,

or *A* (,,,) *abcd* if 1

numbers are special cases of trapezoidal fuzzy numbers.

and <sup>1</sup> ( ) *R r <sup>A</sup>*

monotonical and continuous mapping from *R* to closed interval 0,

.

*x*

methods.

**4.1 Definition** 

where 0 1 

*A abcd* (,,,, )

and <sup>1</sup> : , 0, *R ab <sup>A</sup>*

and <sup>1</sup> ( ) *R r <sup>A</sup>*

and <sup>1</sup>

inverse functions <sup>1</sup> ( ) *L r <sup>A</sup>*

support of *A* , i.e.

i.e.

<sup>1</sup> ( ) *L r <sup>A</sup>*

1 <sup>0</sup> ( ) *L r dr <sup>A</sup>* 

number denoted by *A abd* (,,, )

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

(Dubios & Prade, 1978)

**4. Basic notations and definitions** 
