MODELLING AND OPTIMIZATION OF VARIOUS PROBLEMS IN UNCERTAIN ENVIRONMENTS WITH APPLICATIONS

Abstract

The thesis investigates the modelling and development of solution methodologies with practical applications of some optimization problems in uncertain environments. It includes multiobjective transportation, convex/ concave inexact quadratic and fractional quadratic programming problems. Chapter 1 is introductory in nature. The brief chapter wise description is as follows: Chapter 2 deals with a balanced multiobjective transportation problem having all the parameters and variables as intuitionistic fuzzy numbers. An algorithm is then proposed to solve the problem in which the intuitionistic fuzzy constraints related with each objective are handled using linear, exponential and hyperbolic membership functions. Further, various theorems are proved to show the equivalence between the original problem and its various crisp counterparts. Numerical examples are also presented to clarify the steps involved in the solution procedure. Finally, based on the results obtained, a comparison among di↵erent membership functions is drawn. In Chapter 3, a convex quadratic programming problem, including mixed terms of the type xixj, i 6= j, under uncertainty is discussed. After splitting the problem into two subproblems, a duality based methodology is described to evaluate lower i ii and upper bounds of the problem. Further, a modified method, in lieu of duality based algorithm, is also developed. The modified approach significantly reduces the number of variables as well as constraints and hence, computationally efficient than duality approach. Numerical illustrations are also exhibited using the proposed modified method. In addition, an application of inexact quadratic programming in tea industry is formulated and then solved using the modified method. Finally, this approach is also applied to tackle a fuzzy quadratic programming problem using (↵, r)-cut to yield a numerical solution. Chapter 4 is devoted to the study of a new class of generalized concave interval quadratic programming problems over linear constraints. In the first part of this chapter, a method is developed to solve the problem and an example is demonstrated to give a clear understanding of the steps involved in the proposed approach. Further, as a practical application of the problem, a garbage disposal model is developed and analyzed for economic and ecological sustainability through prospective expansion plans. Moreover, it has also been shown that the proposed model removes the limitation of garbage segregation at source level. The second part of the chapter deals with a deficiency detected in the execution part on waste management application in Kong et al. [91]. The methodology discussed in [91] to solve convex quadratic programming problems, however, was applied to a concave optimization problem. Finally, the inconsistency is resolved using a special case of the algorithm discussed in the first part of the chapter. In the first part of the Chapter 5, an analytic approach is proposed to solve fully iii fuzzy quadratic programming problems (FFQPP) with all fuzzy equality constraints using two methods. After the defuzzification process, one method directly gives solution using a quadratic solver, however Karush-Kuhn-Tucker conditions are used in the second method. In this approach, the problem is reduced to a crisp quadratic programming problem and solution is obtained in the form of fuzzy variables. The second part of the chapter provides a numerical solution to solve general FFQPP. A weighted sum algorithm is developed after applying ↵-cut on the objective function as well as constraints. These ↵-cuts and weights help in exploring the solution of the problem as a mesh instead of mere lower and upper bounds of the objective function. The theorems have also been proved to establish the results of the algorithm. Further, numerical illustrations are exemplified and a real life application in dairy farming sector is formulated and then solved. The objective of Chapter 6 is to study a class of fractional fuzzy quadratic programming problems. A hybrid method is developed to provide a solution without compromising with the fuzzy characteristics. The original problem is converted into a crisp multiobjective problem in which constraints are tackled using variation of parameter technique. Further, goal programming is applied to reduce it to a single objective optimization problem. The solution methodology is also elaborated with the help of an example. Additionally, it has been shown by citing an example, that the same algorithm can be used to handle both the balanced and unbalanced transportation problems.