Issue: 2019/Vol.29/No.2, Pages 77-88

GENERATING A SET OF COMPROMISE SOLUTIONS OF A MULTI OBJECTIVE LINEAR PROGRAMMING PROBLEM THROUGH GAME THEORY

Mustafa Sivri^{}, Hale Gonce Kocken^{}, Inci Albayrak^{}, Sema Akin^{}

**Cite as: **M. Sivri, H. G. Kocken, I. Albayrak, S. Akin. Generating a set of compromise solutions of a multi objective linear programming problem through game theory. Operations Research and Decisions 2019: 29(2), 77-88. DOI 10.5277/ord190205

** Abstract**

Most of real-life problems, including design, optimization, scheduling and control, etc., are inherently characterized by multiple conflicting objectives, and thus multi-objective linear programming (MOLP) problems are frequently encountered in the literature. One of the biggest difficulties in solving MOLP problems lies in the trade-off among objectives. Since the optimal solution of one objective may lead other objective(s) to bad results, all objectives must be optimized simultaneously. Additionally, the obtained solution will not satisfy all the objectives in the same satisfaction degree. Thus, it will be useful to generate a set of compromise solutions in order to present it to the decision maker (DM). With this motivation, after determining a modified payoff matrix for MOLP, all possible ratios are formed between all rows. These ratio matrices are considered a two person zero-sum game and solved by linear programming (LP) approach. Taking into consideration the results of the related game, the original MOLP problem is converted to a single objective LP problem. Since there exist numerous ratio matrices, a set of compromise solutions is obtained for MOLP problem. Numerical examples are used to demonstrate this approach.

** Keywords: **multi-objective programming problem, game theory, compromise solution

**Received: **20 September 2018 **Accepted: ** 17 June 2019