Extended Multi-Objective Optimization Problems

One another important feature of mathematical programming is the existence of uncertainties. These uncertainties may be due to a lack of information or to the internal error of measures, and evaluations in a fuzzy environment. All these features will condition the formulation for a MOO problem. A MOO problem may be a standard continuous or combinatorial programming problem, a robust optimization problem or a fuzzy optimization problem. Real-life optimization problems involve more complex situations including two types of difficulties, namely the existence of nonlinearities and uncertainties. Bimatrix games with associated quadratic programming problems and geometric programming illustrate the first type of real-world problems. The second type of programming models under uncertainties includes robust programming and fuzzy programming. Robust programming has been developed to increase the quality and reliability of engineering processes. Fuzzy programming problems refer to situations where decision-makers face with incorrect or uncertain data. The fuzzy environment includes uncertain preferences, fuzzy objectives, fuzzy constraints, fuzzy data. In such problems, decision-makers maximize their degree of satisfaction in a specified decision set. The extension to multiple objectives uses numerical examples.

Keywords: Bilinear programming, Bimatrix game, Equivalence Theorem, Fuzzy decision set, Fuzzy goal programming, Fuzzy programming, Geometric programming, KKT necessary optimality conditions, Matrix game, Maximin operator, Membership function, Mixed strategies, Multi-objective bimatrix game, Nash equilibrium, Payoff matrix, Pivot algorithm, Robustness, Robust optimization, Strategy space, Symmetric method.