Some New Intuitionistic Fuzzy Entropy Measures and Its Applications to Decision-Making Process

Abstract

Multi-attribute decision-making (MADM) problems are an important part of decision theory in which we choose the best one from the set of finite alternatives based on the collective information. Traditionally, it has been assumed that the information regarding accessing the alternatives is taken in the form of real numbers. However, uncertainty and fuzziness are big issues in real-world problems nowadays and can be found everywhere as in our discussion or the way we process information. To~deal with such a situation, the theory of fuzzy sets or extended fuzzy sets such as an~intuitionistic fuzzy set (IFS) or interval-valued IFS (IVIFS) are the most successful ones, which characterize the attribute values in terms of membership degrees. Among these methodologies, information measures play a significant role in processing the imperfect and uncertain information. From the existing literature, it can be worth noticed that similarity, distance, entropy and inclusion measures are important tools for measuring the uncertainty associated with FS and IFS. Out of these various measures, Entropy measure is basically known as the measure of information of a revolutionary discovery named as 'Information Theory' in communication system, originated from the fundamental paper ``\textit{The mathematical Theory of Communication}'' in $1948$ by Claude E. Shannon.

The objective of this work is to addresses some new entropy measures to quantify the degree of fuzziness of a set in the IFS environment. In the present thesis, different kinds of the entropy measures are addressed which makes the decision more flexible and reliable corresponding to different values of the parameters. Further, based on the proposed measures, different kinds of decision-making approaches are presented in details. In the presented approaches, the characteristics of the attribute weights are taken as either partially known or completely unknown.

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 MADM problems under IFS environment is presented in the first chapter. In \textbf{Chapter 2}, the basic and preliminaries related to the IFSs and the entropy measures are given.

\textbf{Chapter 3} presents a new generalized parametric entropy measures under the IFS environment. The present measures is based on the three parameters, $\alpha,\beta$ and $\gamma$ and it is shown that some of the existing measures are taken as a special case of it. Further, the behavior of these parameters are investigated in details. Further, we develop two approaches to deal with MADM problem with the illustration on a real-life decision making. A sensitivity analysis has also been done by taking different values of the parameters.

In \textbf{Chapter 4}, a new entropy measure based on R-norm under the intuitionistic fuzzy set environment. Further, based on the proposed measures, two MADM approaches have presented to solve the decision-making problems based on either the information of the attribute weights is completely known or partially known. Further, the applicability of the proposed measures are explained through an example.

In \textbf{Chapter 5}, we present a novel {$(R, S)$}-norm based information measure called the entropy to measure the degree of fuzziness of the IFSs. The validity of the proposed measure is tested on the linguistic variable to demonstrate it. Then, we utilized it to propose two decision-making approaches to solve the MADM problem by considering the attribute weights as either partially known or completely unknown. Finally, a~practical example is provided to illustrate the decision-making process.