Robust solutions to multi-objective linear programs with uncertain data

• We consider multi-objective linear programming problems in the face of data uncertainty.
• The uncertainty affects both the objective function and the constraints.
• We give a formula for radius of robust feasibility guaranteeing constraint feasibility of the robust counterpart under affine data parametrization.
• We characterize robust weakly efficient solutions that are immunized against objective matrix rank-one uncertainty.
• We examine some classes of commonly used constraint data uncertainty sets under which the robust weakly efficiency of robust feasible solutions can be numerically checked.

In this paper we examine multi-objective linear programming problems in the face of data uncertainty both in the objective function and the constraints. First, we derive a formula for the radius of robust feasibility guaranteeing constraint feasibility for all possible scenarios within a specified uncertainty set under affine data parametrization. We then present numerically tractable optimality conditions for minmax robust weakly efficient solutions, i.e., the weakly efficient solutions of the robust counterpart. We also consider highly robust weakly efficient solutions, i.e., robust feasible solutions which are weakly efficient for any possible instance of the objective matrix within a specified uncertainty set, providing lower bounds for the radius of highly robust efficiency guaranteeing the existence of this type of solutions under affine and rank-1 objective data uncertainty. Finally, we provide numerically tractable optimality conditions for highly robust weakly efficient solutions.