Methods for Solving Decision-Making Problems Under Neutrosophic Environment

Abstract

Multiple-criteria decision-making (MCDM) 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. In a decision-making (DM) process, an important problem is how to express the preference value. Due to the increasing complexity of the socioeconomic environment and the lack of knowledge or the data about the DM problems, 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 fuzzy sets (IFSs), interval-valued intuitionistic fuzzy sets (IVIFSs), type-2 fuzzy sets (T2FSs), etc., are widely used by the researchers so as to minimize the uncertainty level. During the last decades, the researchers are paying more attention to these theories and have successfully applied it to the various situations in the DM process. Nevertheless, neither the FS nor IFS theory is able to deal with indeterminate and inconsistent information. For instance, we take a person giving their opinion about an object with 0.5 being the possibility that the statement is true, 0.7 is the possibility that the statement is false and 0.2 being the possibility that he or she is not sure. To resolve this, Smarandache in 1998, introduced a new component called the ``indeterminacy-membership function'' and added the ``truth membership function'' and ``falsity membership function'', all which are independent components lying in ] 0^-, 1^+ [, and hence the corresponding set is known as a neutrosophic set (NS), which is the generalization of the IFS and FS. However, without specification, NSs are difficult to apply to real-life problems. Thus, a particular case of the NS called a single-valued NS (SVNS) and the interval neutrosophic set (INS) has been proposed by the researchers.

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. For this, researchers have made efforts to enrich the concept of information measures as well as aggregation operators in neutrosophic environments. Among these, an aggregation operator is an important part of the DM which usually takes the form of mathematical function to aggregate all the input individual data into a single one. However, an information measure such as the distance and similarity measures, complementary to each other, are defined to differentiate between the two or more objects. Thus in order to handle the information in a more accurate and certain manner, there is a need to plan/adopt suitable methodologies to solve the DM problems.

The objective of this research work is to develop some new methodologies under the environment of the NS by utilizing available information and uncertain data. For it, we define several information measures such as distance measures, similarity measures, divergence measures as well as aggregation operators for solving the DM problems. The various desirable relations between the proposed measures and operators are studied. Later, we developed some new concept of new theory named as Possibility linguistic NSs by embedding the features of the possibility degrees to the linguistic features of the NSs. Afterward, based on all the developed theories, we present an efficient method to solve the DM problems in which information related to each alternative is assessed under the consideration of the experts' features. 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 eleven 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 NSs are given.

Chapter 3 presents some new aggregation operators by hybridizing the averaging and geometric aggregation operators with an attitude character parameter. The influence of this parameter will also be analyzed and shows its property. The advantages of adding the parameter are to control the effects of the decision parameters towards the biased ones. Further, some properties of proposed operators are investigated. To address more features, an algorithm is presented to addressed the problems using SVNS and INS features into the DM problem. Last, a numerical example is presented to show its superiority with respect to the several existing approaches.

In Chapter 4, we developed some new aggregation operators based on the Frank t-norm operational law under the SVNS environment. The Frank t-norm operation has an additional parameter which can give a flexible environment to the decision makers to choose their decisions, according to their desired goals. Based on the proposed aggregation operators, an algorithm for solving the MCDM approach is presented and illustrated with a numerical example to show its applicability.

In Chapter 5, by taking the features of the Muirhead mean (MM) to consider interrelationships among any number of arguments assigned by a variable vector, we introduce two new prioritized MM aggregation operators, such as the single-valued neutrosophic (SVN) prioritized MM (SVNPMM) and SVN prioritized dual MM (SVNPDMM) under SVN set environment. In addition, some properties of these new aggregation operators are investigated and some special cases are discussed. Furthermore, we propose a new method based on these operators for solving the MCDM problems and illustrate with a numerical example.

In Chapter 6, we developed a non-linear programming model based TOPSIS approach for solving MCDM problems with incomplete weight information. The relative closeness coefficient (RCC) degree of the TOPSIS method is formulated based on the distance measures. Further, the importance of the attribute weights is taken in the form of interval numbers rather than a real single number. Several special cases of the proposed approach are discussed in detail. In the developed non-linear programming model of RCC, a Charles-Chooper approach is utilized to solve the problem. The applicability of this presented approach is demonstrated with a numerical example of a power generation project and computed their results with several existing studying results.

Chapter 7 presents an axiomatic definition of divergence measure for SVNSs. The properties of the proposed divergence measure have been studied. Further, we develop a novel technique for order preference by similarity to ideal solution (TOPSIS) method for solving single-valued neutrosophic multi-criteria DM with incomplete weight information. Finally, a numerical example is presented to verify the proposed approach and to present its effectiveness and practicality.

In Chapter 8, we developed some bi-parametric generalized distance measure between the pairs of SVNSs. The presented measures utilized the concept of $L_{p}$ norm and the degree of uncertainties. The various properties and relationships between them are investigated. Later on, the applicability of these measures is explained with the help of pattern recognition and medical diagnosis problems.

Chapter 9 discusses the new concept of the logarithm operational laws for the pairs of SVNNs. Further, based on these proposed laws, some new aggregation operators named as logarithmic weighted averaging and geometric operators for SVNSs are defined and investigated their properties. To further strengthen their applicability, we developed an algorithm based on these operators for solving MCDM problems and demonstrate it with a numerical example. Finally, the influence of the logarithm is examined to show the behavior of the decision makers towards their optimal decision.

Chapter 10 deal with DM problems to address qualitative information rather than quantitative information. For it, a concept of linguistic SVNS is utilized with linguistic variables to represent the data. Further, to add prioritized factor during analyzing the data, we developed some new laws and hence based on it, some new prioritized weighted averaging and geometric aggregation operators are developing to address the problems with linguistic SVN (LSVN) information. The technique corresponding to them is demonstrated with a numerical example and compared with the existing studies.

In Chapter 11, we present a new theory named as Possibility LSVNS (PLSVNS) for evaluation of DM problems with the qualitative preference values. The presented idea considered the degree of the possibility value towards each linguistic feature of SVNS. Based on the features of PLSVNS, theories of COPRAS method and weighted averaging and geometric aggregation operators have been defined where the weight vector of the attributes is computed with some information measures. The applicability as well feasibility of the developed methods are explained with a numerical example. Finally, the advantages of the presented concepts are explained.