Properties of two-stage stochastic multi-objective linear programs

Abstract We consider a two-stage stochastic multi-objective linear program (TSSMOLP) that is a natural generalization of the well-studied two-stage stochastic linear program (TSSLP) allowing modelers to specify multiple objectives in each stage. The second-stage recourse decision is governed by an uncertain multi-objective linear program (MOLP) whose solution maps to an uncertain second-stage nondominated set. The TSSMOLP then comprises the objective, which is the Minkowski sum of a linear term plus the expected value of the second-stage nondominated set, and the constraints, which are linear. Since the second-stage nondominated set is a random closed set, its expected value is defined through the selection expectation. We prove properties of TSSMOLPs, which are stochastic set optimization problems, and the multifunctions that arise therein. Specifically, we use a nondominance-equivalent reformulation to show that (i) the global Pareto set of a TSSMOLP is cone-convex, (ii) solving a parameterized TSSLP produces a weakly efficient point for the TSSMOLP, and (iii) a TSSMOLP in which the uncertainty is governed by an atomic probability measure with finite support can be reformulated as an MOLP. Thus, we unify existing work on TSSMOLPs under a single formulation allowing both atomic and nonatomic probability measures. Finally, we demonstrate our results through a numerical example.