SOLUTION OF OPTIMIZATION PROBLEMS IN CRISP, FUZZY AND STOCHASTIC ENVIRONMENTS

Abstract

Optimization problems arise in several fields such as System Engineering, Telecommunication, Agriculture, Biochemical Engineering, Business Management, Engineering Design and Manufacturing Systems etc. In fact, the newly developed optimization techniques are now being extensively used in various spheres of human activity where decisions have to be taken in some complex situations that can be represented by mathe►natical models. Optimization can thus be viewed as an art of decision making, or more specifically, as one of the major quantitative tools in the network of decision making in which decisions have to be taken. Mathematical models of real life optimization problems can be linear as well as non-linear. These may be single objective •or multiobjective. These models may have been developed in crisp, fuzzy or stochastic environments. Even though a number of methods have been proposed in literature for solving various types of optimization problems, there is still need for efficient and reliable computational algorithms for obtaining realistic solutions of different types of optimization problems. Whereas simplex based methods and Karmarkar's type algorithms can efficiently solve linear programming problems, none of the available algorithms can guarantee efficient, solution of every non-linear programming problem. Moreover most of these algorithms guarantee at the most a local optimal solution. However in many real life situations a global rather than a local optimal solution is desired. Even though some techniques are currently available in literature which claim to search for a global rather than a local optimal solution still there is a scope for developing more efficient and reliable algorithm of this category. Therefore the first objective of the present work has been to develop efficient and reliable techniques for solving global optimization problems. Certain computational algorithms based on random search, controlled random search and annealing concept have been developed for solving global optimization problems and their performance tested on test problems taken from literature. Classical mathematical programming model is insufficient in many real world situations. The nature of many real life problems requires taking into account multiobjectives on the one hand and various kinds of uncertainties on the other hand. These uncertainties are being incorporated these days in mathematical models of real life problems using stochastic and fuzzy set concepts. Certain interactive methods have been recently proposed in literature for solving multiobjective problems in fuzzy. and stochastic environments. However, most of these generally solve multiobjective linear programming problems in fuzzy or stochastic environment. The second objective of the present work has been to propose interactive methods for solving multiobjective linear as well as certain types of non-linear optimization problems in fuzzy/stochastic or mixed fuzzy-stochastic environments wherein decision variables can have real, integer or mixed integer values. The chapter wise summary of the work presented in the thesis is. given below. Chapter 1 is introductory in nature. Besides brief introduction to linear as well as non-linear models of optimization problems in crisp, fuzzy, stochastic and mixed fuzzy stochastic environments, some definitions and prerequisites for the present work are also presented in this chapter. Available literature on the subject has been briefly reviewed. Chapter closes with a summary of the work presented in the thesis. Mohan and Shanker [1994] developed a computational algorithm (named RST2 algorithm) based on controlled random search for solving global optimization problems. This algorithm uses quadratic approximation in the local phase of each iteration to search for a possible candidate for global minima. We propose in chapter 2 a controlled random search computational algorithm which uses in the local phase cubic approximation in place of quadratic approximation to yield a possible candidate for global minima. The algorithm has been named CRSTC algorithm, The reliability and efficiency of this algorithm has been tested on over twenty five test problems taken from literature. A comparative study has been also carried out to compare the relative performance of the proposed algorithm vis-a-vis the original controlled random search RST2 algorithm of Mohan and Shanker [1994]. Random search techniques are considered to be a powerful tool for solving global. optimization problems. - However, they generally take unusually long time to converge. Recently simulated annealing methods have also proven to be valuable tool for global optimization problems particularly for integer and combinatorial optimization problems. In chapter 3, we consider the effectiveness of incorporating simulated annealing concept in a random search method for solving global optimization problems. With this objective in view two computational -algorithms have been developed in this chapter. These are a random search algorithm (RS) and a random search algorithm incorporating simulated annealing concept (RSAN). The performance of both these algorithms has been tested on thirty four test problems taken from literature. In chapter 4, a controlled random search algorithm incorporating annealing concept has been developed for solving global optimization problems. This algorithm incorporates annealing concept of Bohachevsky et al. [1986] in a controlled random search method which relies on steepest descent approach to search for the possible candidates of global (iv) minima in the local phase. The algorithm has been named RSANSDC_ algorithm and its reliability and effectiveness tested on thirty four test problems taken from literature. The performance of the algorithm has been also compared with the performance of RS algorithm and RSAN algorithms developed in chapter 3 as well as RST2 and RST2AN algorithms earlier available in literature. Encouraged by the performance of RSANSDC algorithm developed in chapter 4 in solving global optimization problems of general nature in which the decision variables are permitted to have real values, a computational algorithm named RSANSDCU has been developed in chapter 5 for solving integer and mixed integer global optimization problems. This algorithm is essentially a suitably modified version of RSANSDC algorithm of chapter 4 specifically designed for efficiently . solving integer and mixed integer programming problems. The reliability and efficiency of RSANSDCU algorithm have been demonstrated on thirty three integer and mixed integer optimization problems taken from literature. The performance of this algorithm has been also compared with the performance of controlled random search, algorithm RST2ANU of Thanh [1996] and the standard simulated annealing algorithm (SSA)...