OPTIMIZATION AND UNCERTAINTY IN FUZZY DECISION MAKING

Abstract

Decision-making unfolds as a non-deterministic process, generating conflicting situations in various facets of real-life scenarios, thereby giving rise to uncertainty and imprecision. It’s commonplace to grapple with vaguely defined data in real-world contexts, necessitat ing the utilization of mathematical frameworks such as fuzzy sets and fuzzy matrices to navigate through these conflicting events. The imprecision inherent in decision-making problems can manifest in multiple forms, notably fuzziness, which is effectively addressed through the application of fuzzy set theory and fuzzy matrix theory within decision the ory. Fuzzy matrices are particularly adept at scrutinizing and resolving real-world issues represented by matrices with varying degrees of vagueness. Conventional or fuzzy quan titative methods often fall short of representing the inherently ambiguous nature of human activities and decisions. Employing a fuzzy approach becomes imperative in such sce narios, engendering the need for computational methods involving uncertainty. Various computation models, stemming from the optimization and uncertainty model, have been introduced in the literature to contend with this approach. Decision-making is intricately linked with strategic information exchange through language, albeit in a manner character ized by rigor and stylization. Despite the efforts towards formalization, uncertainty remains entrenched in the conceptual tools employed in optimization and uncertainty theories, pos ing a fundamental challenge. Our study delves into addressing linear programming prob lems and matrices imbued with imprecise information, striving to augment their practical utility. This endeavor contributes to enhancing the process of decision analysis within an uncertain qualitative milieu. The thesis, titled "Optimization and Uncertainty in Fuzzy Decision-Making Problems", comprises six chapters, followed by a summary and delin eation of future research directions. Motivated by existing computational models designed to mitigate uncertainty in decision-making, our work extensively scrutinizes the literature. The primary objective is to tackle the prevailing vagueness and imprecision inherent in decision-making problems and matrix quandaries, particularly in the context of fuzzy vari ables. The thesis concludes with a comprehensive bibliography and a list of publications, v underscoring the breadth and depth of exploration in the field. The introductory Chapter 1 presents an overview of the fuzzy sets, fuzzy matrix, im precise models, and their elementary applications anticipated in the transportation prob lem (TP) and assignment problem (AP), followed by their implementation in the distinct decision-making problems. Further, this chapter discusses the notion of fuzziness involved in qualitative concept of matrix. Some basic concepts used throughout the thesis have been defined along with the motivation of the research work. Thus, the current chapter creates a background for this thesis’s work and motivates the work carried out in this the sis. The Chapter 2 entitled, “Transportation problem under interval-valued Pythagorean fuzzy and spherical fuzzy environment” establishes the basis for a theory of TPs. In literature, the TPs with Pythagorean fuzzy and picture fuzzy models are considered and solved. However, the theory of TPs having interval-valued Pythagorean fuzzy sets (IVPyFS) and spherical fuzzy set (SFS) is pristine and yet to be explored. The use of IVPyFS and SFS to represent practical transportation situations has shown to be a powerful approach. The chapter is based on two research papers entitled, “Solution of transportation problem using interval-valued Pythagorean fuzzy approach”, published in Advanced Engineering Opti mization Through Intelligent Techniques: Select Proceedings of AEOTIT 2022 (pp. 359-368), Springer, 10 (1), 2199368 (2023) and “Solution of transportation problem under spherical fuzzy set”, published in 2021 IEEE 6th International Conference on Comput ing, Communication and Automation (ICCCA) (pp. 444-448). , IEEE. The chapter mentioned above constitutes a transportation problem with IVPyFS and SFS belonging to an uncertain parameter set where all plausible imprecise descriptors provided by experts have a symmetric and uniform distribution. In practical life decision problems, the experts may prefer another special type of TP model called the “Assign ment Problem” model. Several computational models are established in literature to deal with assignment problems with imprecise parameters like cost, condition, road condition, etc. Therefore, in Chapter 3 entitled, “A novel similarity measure and score function of Pythagorean fuzzy sets and their application in assignment problem,” we propose a newly constructed methodology to handle assignment problems with uncertain parameters. To handle the uncertainty in practical applications of assignment problems (AP), a method for solving the Pythagorean fuzzy assignment problem (PyFAP) has been proposed us ing a similarity measure and a proposed score function. Numerical examples are given to explain the methodology. Hence, this chapter also discusses and solves the decision vi matrix of AP with uncertain parameters. The chapter is based on the research paper ti tled, “A novel similarity measure and score function of Pythagorean fuzzy sets and their application in assignment problem”, published in Economic Computation and Economic Cybernetics Studies and Research, (SCIE, Impact Factor: 0.9). Chapter 4 entitled, “Interval-valued picture fuzzy matrix: basic properties and applica tion” proposes a novel concept of the matrix based on interval-valued picture fuzzy sets. Further, based on the defined concept, the several key definitions and theorems for the interval-valued picture fuzzy matrix (IVPFM) and present a procedure for calculating its determinant and adjoint. Using composition functions, a new algorithms to identify the greatest and least eigenvalue for the defined problem is developed. The proposed ap proach can be perceived as a convenient technique for multiple criteria decision-making (MCDM) problems by the proposed distance measure. The chapter is based on a re search paper titled, “Interval-valued picture fuzzy matrix: basic properties and application,” published in Soft Computing, Springer (SCIE, Impact Factor: 3.1). In Chapter 5 entitled, “Interval-valued spherical fuzzy matrix and its applications in multi attribute decision-making process”, the concept of matrix with interval-valued spherical fuzzy concept is proposed in which each row of the matrix may correspond to an element, while each column represents a different dimension or attribute of membership, neutrality and non-membership degree in interval number instead to a single point of a real number. The theory of the interval-valued spherical fuzzy matrix (IVSFM) represents more flexibly uncertain and vague information. In this context, we establish significant definitions and theorems about the given matrices. Further, introduces the methodology for determin ing the determinant and adjoint of IVSFM. Finally, proposes a new score function for the interval-valued spherical fuzzy sets and prove its validity with the help of basic properties and the application of the decision-making problems for a career placement assessment. The chapter is based on a research paper titled, “Interval-valued spherical fuzzy matrix and its applications in the multi-attribute decision-making process,” published in Maejo International Journal of Science and Technology.,(SCIE, Impact Factor: 0.8). Chapter 6 entitled, “Interval-valued fermatean fuzzy matrix and its application” presents an Interval-valued fermatean fuzzy matrix in which the membership and non-membership degrees of the fermatean fuzzy matrix in continuous form (interval number). The methodol ogy presented in this chapter manipulates imprecise and uncertain information in decision making in interval-valued fermatean fuzzy set theory. The chapter is based on the research paper titled, “Interval-valued fermatean fuzzy matrix and its application” (communicated in vii “Cybernetics and Systems"). Chapter 6 is followed by the summary of the research work carried out in this thesis. In addition, the future scope of the thesis has been discussed briefly. Finally, the thesis ends with the bibliography and list of publications.