On Complex Extensions and Information Measures of Neutrosophic Hesitant and Picture Fuzzy Sets in Decision Making

Abstract

The objective of this thesis entitled, \On Complex Extensions and Information Measures of Neutrosophic, Hesitant & Picture Fuzzy Sets in Decision Making" is tostudy the concept of fuzzy extensions on real and complex planes along with variousapplications in detail. The work presented in this thesis has been carried out inorder to fulfill the objective to propose the notion of cohesive fuzzy set, complexneutrosophic matrix, energy of picture fuzzy graphs with their various importantoperations & applications in the field of decision-making.

In literature, the notion of fuzzy sets and its generalized extensions have madea large amount of contribution in the progress of scientific and engineering researcharea. It has large number of applications in the areas (theoretical as well as practical) related to engineering, arts,humanities, computer science, health sciences,life sciences, physical sciences etc due to its ability of dealing with the uncertaintyfactor. In the current work, these concepts have been explained in detail and briefstructure of the format of the presented work is presented below:

We have presented the fundamental background of hesitant, neutrosophic, pictureand complex fuzzy sets with their mathematical form, definitions, operations andliterature survey in Chapter 1.

In Chapter 2, a novel concept of Cohesive fuzzy set (CHFS) has been proposed as asynchronized generalization from innovative notions of complex fuzzy set and hesitant fuzzy set. We have also studied the relationship and connections between theCohesive Fuzzy Set and Complex Intuitionistic Fuzzy Set along with the validationof the obtained results. Based on the proposed notion, various properties, operationsand identities have been established with their necessary proof. The applications ofCHFS in the process of filtering the signals for getting the reference signal using thenecessary Fourier cosine transform (FCT)/inverse FCT and identifying maximum number of sunspots in a particular interval under a solar activity have been suitablydiscussed with illustrative numerical examples. Some advantages of incorporatingthe proposed notion have also been tabulated for the sake of better understanding.

In Chapter 3, a new concept of the complex neutrosophic matrix has been introduced to solve different problems related to uncertainties. Based on the proposedmatrix, we have provided various algebraic operations like addition, subtraction,union many others which will be of great help in establishing the fundamental concepts. The matrix norm convergence of the proposed matrix has also been studiedfor the necessary foundation of the complex neutrosophic matrix. The two differenttypes of new similarity measure matrices for complex neutrosophic matrices havebeen proposed and validated the axiomatic definition of the similarity measure. In addition to this, a new similarity measure has also been proposed for complex fuzzymatrices along with detailed explanatory numerical example. The application in thearea of identification of reference signal has also been described.

In Chapter 4, four new similarity measures in their exponential form have beenproposed for the case of single valued neutrosophic set. Numerical examples for theclassification problem and the decision-making problem have also been presented andcompared the obtained results with the well established existing approaches. Later,a novel concept of single valuedneutrosophic information measure based on utilitydistribution and probabilistic randomness has also been proposed. The proposedconcept has been obtained by integrating the uncertainties caused by neutrosophicinformation, useful information (utility based) and probabilistic information. Further, in a similar integrating way, the divergence measure of the `useful' informationhas also been proposed for the study of applicable mutual information. Consequently, the hybrid ambiguity and neutrosophic information improvement measures have been studied with the help of the proposed `useful' information measures.

In Chapter 5, the notion of energy and Laplacian energy of Picture fuzzy graph anddirected Picture fuzzy graph have been proposed with the help of adjacency matrix.and the results on lower and upper bounds. On the basis of the proposed energy ofpicture fuzzy graph, a methodology for the ranking in a decision-making problemof site selection has been proposed. In order to illustrate the implementation ofthe proposed methodology, a hydro-power plant site selection problem has beenconsidered. The novelty of the proposed approach, comparative analysis, advantageshave also been studied.

Finally, the proposed work has been concluded in Chapter 6 of the current thesis.