tailieunhanh - Characterization of substantially and quasi-substantially efficient solutions in multiobjective optimization problems

In this paper, we study the notion of substantial efficiency for a given multiobjective optimization problem. We provide two characterizations for substantially efficient solutions: The first one is based on a scalar problem and the second one is in terms of a stability concept. | Turk J Math (2017) 41: 293 – 304 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Characterization of substantially and quasi-substantially efficient solutions in multiobjective optimization problems Latif POURKARIMI∗, Masoud KARIMI Department of Mathematics, Razi University, Kermanshah, Iran Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we study the notion of substantial efficiency for a given multiobjective optimization problem. We provide two characterizations for substantially efficient solutions: the first one is based on a scalar problem and the second one is in terms of a stability concept. Moreover, this paper introduces the notion of quasi-substantial efficiency. Similar to those of substantial efficiency, two characterizations for quasi-substantially efficient solutions are obtained. Key words: Substantial efficiency, scalarization function, stable problem,quasi-substantial efficiency 1. Introduction and preliminaries In many real problems of economics, management science, engineering, and industry, decisions are characterized by many criteria and usually these criteria cannot be brought to a common scale by some utility functions. These problems are referred to as multiobjective optimization problems. Multiobjective optimization is one of the most important areas in optimization, which is of great interest because of the large variety of applications. From the large amount of relevant publications about multiobjective optimization, we mention three books [3, 4, 15]. Because of the conflict between objective functions, often there is no solution that optimizes all objective functions simultaneously. Hence, efficient solutions are considered as primary solutions of multiobjective optimization problems. An efficient problem is a feasible solution in which improvement of no objective function is possible without impairing at