ON SOME MULTIOBJECTIVE MATHEMATICAL PROGRAMMING PROBLEMS

Abstract

The work being presented in the thesis is devoted to the study of optimality and duality for some multiobjective mathematical programming problems. a brief account of the work chapter-wise is given below. Chapter 1 consists of introduction to multiobjective programming, definitions notations and pre-requisites for the present work. The relevant literature has been reviewed and topics such as efficiency, weak efficiency and proper efficiency have been revisited. In Chapter 2 we consider Wolfe and Mond-Weir type multiobjective symmetric dual problems of Weir and Mond [56] without nonnegativity constraints. Duality relations are proved assuming invexity/generalized invexity. Moreover, if the kernel function is skew symmetric, the multiobjective problems are shown to be self dualb. In Chapter 3, weak and strong duality results are established for the symmetric dual multiobjective fractional programming problems of Jeyakumar and Mond [32] without nonnegativity constraints under pseudo-invexity. Self duality is also discussed. In Chapter 4, we formulate Wolfe type symmetric dual multiobjective variational problems without nonnegativity constraints. Duality relations are proved using invexity. A self duality theorem is also validated. It is then indicated that our results present a multiobjective version of the problems of Smart and Mond [49] and are dynamic generalizations of the corresponding (static) Wolfe type symmetric and self duality theorems of chapter 2. Chapter 5 deals with Mond-Weir type symmetric dual for multiobjective variations problems. Weak, strong and self duality theorems are established assuming generalized convexity. A relationship between these variational problems and the symmetric dual multiobjective problems of Weir and Mond [56] is incorporated. In Chapter 6, we present a pair of Mond-Weir type symmetric dual fractional variational problems. For this pair, duality relations are proved and a self duality theorem is validated under generalized convexity assumptions. These problems can be considered as dynamic version of the symmetric dual fractional problems of Weir [55). In Chapter 7, we study a multiobjective control problem. Fritz John and Kuhn-Tucker type necessary optimality conditions for an efficient solution of the problem are obtained by first reducing the multiobjective control problem to a system of single objective control problems, and then using the necessary optimality conditions of Chandra, Craven and Husain [9]. As an application of Kuhn-Tucker type optimality conditions, Wolfe and Mond-Weir type dual multiobjective control programming problems are formulated and duality relations are established under invexity/generalized invexity. In the appendix a pair of scalar symmetric dual fractional problems is considered. These problems are obtained by ignoring nonnegativity constraints in the problems studied by Chandra, Craven and Mond [10].This enables us to replace convexity hypotheses of [10] by the corresponding invexity assumptions. A self duality theorem is also obtained. We then state the corresponding results for symmetric dual continuous fractional programming without nonnegativity constraints of Chandra and Husain [12).