Arithmetic operations using sigmoidal function under fuzzy environment and Its application to decision making

Abstract

Decision making problems are the important parts of modern decision theory due to the rapid development of economic and social uncertainties. Today's, decision maker wants to attain more than one goal in selecting the course of action while satisfying the constraints dictated by environmental processes and resources. But the increasing complexity of the society has brought new problems that involve very large number of variables. The decision maker may be subjective and uncertain about his level of preference due to incomplete information or knowledge, inherent complexity and uncertainty within the decision environment, lack of an appropriate measure or scale. Thus, if the assessment values are known to have various types of vagueness/imprecision or subjectiveness, then the classical decision making techniques are not useful for such problems. To cope with such situation, a fuzzy set theory is one of the successful and widely used theory for dealing with the uncertainties and hence play an important role in the areas of engineering and management disciplines. As compared to other research domains, the fuzzy arithmetic gained great interest in scientific areas such as decision problems, reliability analysis, optimization etc. In order to perform operations on fuzzy observations, fuzzy numbers came into existence.

The objective of this work is divided into two folds. In the first fold, generalized sigmoidal fuzzy number has been introduced and based on it various arithmetic operations are executed for analyzing the system performance. For this, data related to the system are handled with the help of nonlinear sigmoidal type membership function, instead of linear or parabolic fuzzy numbers. Based on it, an arithmetic operations have been performed in the form of fuzzy membership functions by using the concept of distribution and complementary distribution functions. The advantage of the proposed approach is that it gives compressed range of prediction and they do not need the computation of $\alpha-$ cut of the fuzzy number. Thus, results obtained by using fuzzy numbers are practically much better than those obtained by classical methods. During the second fold, an analysis has been conducted under fuzzy environment for depicting the best type of fuzzy number, namely linear, parabolic and sigmoidal fuzzy numbers, under the different set of criteria. Finally, based on the collected information in the form of decision matrix all the individual decision makers' opinion for rating the candidate are aggregated using the fuzzy positive and negative ideal solutions.

The present thesis is organized into five chapters which are briefly summarized as follows:

A brief account of the related work of various authors in the evaluation of arithmetic operations, membership functions, decision makers etc., by using conventional, fuzzy and optimization techniques is presented in the first chapter. In Chapter 2, the basic and preliminaries related to the arithmetic operation and to be used in the subsequent chapters are given. Chapter 3 presents a concept of generalized sigmoidal fuzzy number. Also, the definition of trapezoidal sigmoidal fuzzy number and their arithmetic operations between two generalized sigmoidal fuzzy numbers are introduced. Various arithmetic operations, such as addition, subtraction, multiplication, inverse, division etc are studied by using the concept of the distribution and complementary distribution functions. The operations have been validated through some elementary applications and results are compared with that of $\alpha-$ cut method and shows the supremacy of the result. In Chapter 4, the decision making technique has been introduced for computing the best type of the fuzzy number out of linear, parabolic and sigmoidal fuzzy numbers under the different set of criteria. Since handling the uncertainty in the data is the big task for the decision maker, so results computed in Chapter 3 has been used here in the form of fuzzy decision matrix and then aggregated by using fuzzy positive and negative ideal solutions. Finally, a closeness coefficient is defined for each alternative, to determine the rankings of all alternatives. Chapter 5 deals with the overall concluding observations of this study and a brief discussion on the scope for future work.