Multiple criteria optimization for stochastic systems with uncertain parameters

Abstract

The aim of this article is to attract researchers' attention to a “mixed” variant of Multiple Criteria Optimization (MCO) problem with incomplete information, namely to the case where stochastic distributions for some characteristics of an optimized system are known, but only up to the uncertain parameters. Applied examples are given to show how the combination of stochastic and maximin approaches makes it possible to reduce the problems of this type to MCO problems with uncertainty. Both problems are reduced to classical well studied Markowitz model (without uncertainty). Such elimination of uncertainty is possible due to only one criterion depends on uncertainty. Further general case where uncertainty affects the values of all the criteria in an arbitrary manner is considered. For one of possible reasonable concept of P-optimal solution sufficient conditions are presented. They reduce the optimal solution calculation to maximin problem. Constructiveness of the suggested approach is illustrated by an example of multiple criteria linear programming with uncertain coefficients of linear criteria. If these coefficients satisfy given set of linear constraints, the computation of the optimal solutions is reduced to linear programming problem. Finally, some possible directions for future research are discussed.