Some Strategical Methods for Solving Decision-Making Problems under Fuzzy/Intuitionistic Fuzzy Set Environment

Abstract

Multiattribute group decision-making (MAGDM) problems are the imperative part of modern decision theory where a set of alternatives has to be assessed against the multiple influential attributes before the best alternative is selected. Indeed in ordinary life, ``to do or not to do'' is one of the foremost riddles that a person faces before jumping to action. The whole decision making (DM) process is subordinate upon the proper data being accessible to the proper individuals at the correct times. In general, in order to evaluate the given objects, a decision-maker may set some characteristic or criteria which need to be fulfilled/satisfied to select the best one(s) during solving the problems. Based on the criteria, DM problems are classified into two types, $(1)$ decision based on the single criteria; $(2)$ decision based two or more attributes known as multi-attribute decision-making (MADM). Due to the increasing complexity of the socioeconomic environment and the lack of knowledge, it is difficult for the decision maker to give the exact decision as there is always an imprecise, vague or uncertain information. To deal with this, the theory of the fuzzy sets (FSs) or its extensions such as intuitionistic FSs (IFSs), interval-valued IFSs (IVIFSs), Soft sets, etc., are widely used by the researchers so as to minimize the uncertainty level. In the last few decades, several types of research paid more attention to MADM or MAGDM problems in various fields. However, one of the most important factors to access the best one(s) is the considered environment, under which the decision-maker(s) have to evaluate the given alternatives. The environment considered during the DM problems may be quantitative and qualitative according to the situation of real-life problems. To address it, a concept of a linguistic variable (LV) and hence their corresponding approaches are developed by the researchers to analyze the information by using various information measures and the aggregation operators (AOs).

After this pioneering work, researchers have been engaged in extensions and applications to different disciplines. However, the most important task for the decision-maker is to rank the objects so as to obtain the desired object(s). An altar to these theories, in 1989, an uncertainty analysis theory, by combining dialectical thinking and mathematical tools, was developed by Zhao known as ``set pair analysis'' (SPA) and is differ from the traditional probability and fuzzy set theory in terms of considering both certainty and uncertainty as one consolidate certain-uncertain system. SPA theory studies the internal relationship between the parts of a system. The core idea of SPA is to define a set pair for two related sets and characterized them in terms of constructing the connection number (CN), which consists of ``identity'', ``discrepancy'', and``contrary'' degrees. For any two dependent sets $A$ and $B$ of a given problem $W$, a set pair between them is denoted by $H(A, B)$ and having $N$ characteristics. The CN corresponding to set pair $H$ denoted by $\mu(H,W)$ is defined asAfter this pioneering work, researchers have been engaged in extensions and applications to different disciplines. However, the most important task for the decision-maker is to rank the objects so as to obtain the desired object(s). An altar to these theories, in 1989, an uncertainty analysis theory, by combining dialectical thinking and mathematical tools, was developed by Zhao known as ``set pair analysis'' (SPA) and is differ from the traditional probability and fuzzy set theory in terms of considering both certainty and uncertainty as one consolidate certain-uncertain system. SPA theory studies the internal relationship between the parts of a system. The core idea of SPA is to define a set pair for two related sets and characterized them in terms of constructing the connection number (CN), which consists of ``identity'', ``discrepancy'', and``contrary'' degrees. For any two dependent sets $A$ and $B$ of a given problem $W$, a set pair between them is denoted by $H(A, B)$ and having $N$ characteristics. The CN corresponding to set pair $H$ denoted by $\mu(H,W)$ is defined as

\mu = a+bi + cj

where, $a(=S/N), b(=F/N),$ and $c(=P/N)$ represent the degrees of ``identity'', ``discrepancy'', and``contrary'' respectively, $i\in[-1,1]$ is the coefficient of ``discrepancy degree'', and $j=-1$ is the coefficient of ``contrary degree'' and $j=-1$. Here $S,F,P$ denotes the identity, discrepancy, contrary characteristics respectively. It is clearly seen that $0 \leq a,b,c \leq 1$ and $a+b+c=1$.

As the complexities of the system increase day-by-day, so there is a need to plan/adopt suitable methodologies to solve the DM problems which can handle the information in a more accurate and certain manner.

The objective of this research work is to develop some new methodologies under the IFS or IVIFSs by utilizing the feature of the CNs of the SPA theory. To do it, we define the various measures and the AOs for solving the MADM and MAGDM problem where the information related to each alternative is expressed in terms of CNs from IFSs or IVIFSs. The various form of the connection number sets is defined under the different to handle the uncertain and imprecise information. The various desirable relations between the proposed measures and operators are studied. Later, based on the proposed measures, an efficient method is developed to solve the DM problems in which information related to each alternative is assessed under the consideration of the group of experts. Several real-life practical examples are taken to demonstrate the approach and compared their performance with some of the existing studies.

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

A brief account of the related work of various authors in the evaluation of DM approaches by using several approaches is presented in the first chapter.

In Chapter 2, the basics and preliminaries related to the intuitionistic fuzzy sets, information measures, AOs, etc., are given.

Chapter 3 presents the novel MADM method under IVIFSs environment by integrating a TOPSIS method. For it, firstly we construct the CN of the SPA theory for each IVIFS and then based on its, we define some exponential distance measures to compute the degree of discrimination between the two IVIFSs. The supremacy of the proposed measure is also discussed. Afterward, to solve the DM problems, a novel TOPSIS method based on the proposed distance measures is developed, illustrated with a numerical example and compared their results with the several existing studies.

In Chapter 4, we present some axioms of the distance measures based on Hamming, Euclidean, and Hausdorff metrics whose preferences related to the attributes are made in the form of CN. Several desirable relations between the proposed measures are investigated. Later, we develop a MADM approach based on the proposed distance measures to investigate the DM problem. The effectiveness of the approach has been demonstrated through a case study and compared their studies with several existing measures.

Chapter 5 presents some novel similarity measures to measure the relative strength of the different IFSs after pointing out the weakness of the existing measures. For it, a CN set (CNS) is formulated and hence based on it, some new similarity measures between them are defined. A comparative analysis of the proposed and existing measures are formulated in terms of the counter-intuitive cases for showing the validity of it. Finally, an illustrative example is provided to demonstrate it.

Chapter 6 presents a novel correlation coefficient and weighted correlation coefficients formulation to measure the relative strength of the different IFSs in the form of the CNs. The limitations of certain existing measures are highlighted and overcome by the proposed measure. Afterward, a DM approach is presented based on the developed measures. Two illustrative examples related to pattern recognition and medical diagnosis are taken to validate the effectiveness and applicability of the proposed method.

In Chapter 7, we constructed the CNs for the IFSs as well as IVIFSs and hence based on it, we constructed the TOPSIS method for solving the DM problems. The basic feature of the TOPSIS method is chosen the alternative which has the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. Afterward, based on the proposed CNs, we define two algorithms to solve the DM problems where the features are extracted either in IFNs or in IVIFNs. The validity of these proposed algorithms is tested with an example and compared it with several existing results.

Chapter 8 enhanced the LIFS with the SPA theory and hence defined the linguistic connection number (LCN) and studied their various operational laws. Based on it, we have developed various AOs namely, LCN weighted geometric (LCNWG), LCN ordered weighted geometric (LCNOWG), and LCN hybrid geometric (LCNHG) operators with LIFS environment. Also, the shortcoming of the existing operators under LIFS environment have been highlighted and overcomes by the proposed operators. Few properties of these operators have been also investigated. Further, a group DM approach has been presented, based on these operators, which has been illustrated by a numerical example to show the effectiveness and validity of the proposed approach.

Chapter 9 presents the group DM approach under the linguistic intuitionistic fuzzy (LIF) set environment. For it, we first propose a new ranking method named as possibility degree measures to compare the different LIF numbers. Further, in order to aggregate the different LIF numbers, some weighted and ordered weighted averaging AOs are proposed by using Einstein t-norm operations. The prominent characteristics of these operators are also investigated. Afterward, a MAGDM approach, based on proposed operators and the possibility degree measure, is developed under the LIFSs environment. A numerical case is taken to manifest the practicability and feasibility of the proposed group DM method.