Some Investigations on Fuzzy Relational Optimization Models and Their Possible Solutions

Abstract

This thesis explores different types of fuzzy relation equations (FRE) and optimization models associated to FRE. Various real world applications can be formulated in terms of FRE or optimization problems subject to FRE. In this work, investigation is done on solution structure of different types of FRE and methods are proposed for resolution of FRE. Methods based on coded and noncoded techniques are designed to deal with optimization problems subject to FRE as constraints. Chapter 1 presents the importance of fuzzy relations as an effective tool of modelling various real world problems. The research work done so far in this area has been reviewed in this chapter. Chapter 2 gives the relevant preliminaries to this work. General FRE, its solution structure and general model of fuzzy relational optimization problems is defined in this chapter. In Chapter 3, an algorithm is proposed for solving max-Archimedean interval-valued FRE, using the concept of covering. The proposed algorithm finds the set of all tolerable solutions efficiently, which are useful in fuzzy control problems. The algorithm proposed in Chapter 3 is extended to solve the system of max-Archimedean bipolar FRE in Chapter 4. It is proved that the solutions of the system are equivalent to the solutions of the covering problem. In Chapter 5, a linear optimization problem subject to max-and#321;ukasiewicz bipolar FRE as constraints is considered. This type of optimization problem is NP-hard, but it can be converted to 0-1 integer programming problem in polynomial time. A simple and efficient binary coded genetic algorithm is proposed for solving this optimization problem. In Chapter 6, a real coded genetic algorithm is proposed for solving a nonlinear optimization problem subject to max-Archimedean bipolar FRE as constraints. Using the structure of the solution set of max-Archimedean bipolar FRE, the initial population is generated. A feasibility algorithm is designed for making infeasible solution feasible. The algorithm is applied on some test problems for analysis. In Chapter 7, a real coded genetic algorithm is proposed for solving multi-objective optimization model constrained with max-Archimedean bipolar FRE. The algorithm finds the set of Pareto optimal solutions which are efficient than compromised solutions. newline