On Robust Solutions to Multi-objective Linear Programs

In multiple criteria linear programming (MOLP) any efficient solution can be found by the weighting approach with some positive weights allocated to several criteria. The weights settings represent preferences model thus involving impreciseness and uncertainties. The resulting weighted average performance may be lower than expected. Several approaches have been developed to deal with uncertain or imprecise data. In this paper we focus on robust approaches to the weighted averages of criteria where the weights are varying. Assume that the weights may be affected by perturbations varying within given intervals. Note that the weights are normalized and although varying independently they must total to 1. We are interested in the optimization of the worst case weighted average outcome with respect to the weights perturbation set. For the case of unlimited perturbations the worst case weighted average becomes the worst outcome (max-min solution). For the special case of proportional perturbation limits this becomes the conditional average. In general case, the worst case weighted average is a generalization of the conditional average. Nevertheless, it can be effectively reformulated as an LP expansion of the original problem.