Generalized Parabolic Fuzzy Numbers and it's Applications

Abstract

Probability theory is being used extensively in engineering and management for various analyses. As the conventional theory is based on the probabilities and binary state structure for analyzing the performance of the structure and therefore result based on it do not always provide useful results to the practitioners due to its limitation of being able to handle only quantitative information. So the result obtained are therefore not of much practical value. This is primarily due to the fact that there is significant impact on subjective information in relation to the available quantitative information. One way to handle the subjective information is the use of fuzzy set theory. Research in last two decades, however, has shown that probability theory is not the only possible way to represent imprecision and uncertainty. Fuzzy set theory provides a significant alternative to the probabilistic approach to find the various arithmetic operations for evaluating the performance of the system. In recent times, the use of fuzzy sets has been gaining popularity and is playing 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 to carry out the analyze of the various arithmetic operations using generalized parabolic fuzzy numbers. For this various arithmetic operations has been studied by taking parabolic fuzzy numbers. As most of the data collected from the various resources are generally imprecise, vague and uncertain. So to handle these types of data, fuzzy set theory has been used and then analyzed the system in the form of fuzzy membership functions by using the concept of distribution and complementary distribution functions. The advantage of the proposed generalized parabolic fuzzy number is that it gives compressed range of prediction. Another advantages of the proposed approach are that they do not need the computation of $\alpha-$ cut of the fuzzy number. Results obtained by using fuzzy numbers are practically much better than those obtained by classical methods.

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 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 trapezoidal fuzzy number is extended to parabolic fuzzy number. A generalized parabolic fuzzy number has been introduced here. Also, the definition of trapezoidal parabolic fuzzy number and arithmetic operations between two trapezoidal parabolic 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 major advantages of the technique are that they do not need the computation of $\alpha-$ cut of the fuzzy number and hence it becomes more powerful where the standard method fails. 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 expression of the defuzzification by the various methods such as center of area, the bisection of area, largest of maxima, smallest of maxima, regular weighted point etc., are computed by using trapezoidal fuzzy numbers and trapezoidal parabolic fuzzy numbers. Based on their results, it has been concluded that the defuzzified expression for triangular fuzzy numbers and triangular parabolic fuzzy numbers are special cases for our computed results.

Chapter 5 deals with the overall concluding observations of this study and a brief discussion on the scope for future work.