Strong Inequalities for Chance–Constrained Programming

As an essential substructure underlying a large class of chance-constrained programming problems with finite discrete distributions, the mixing set with 0 − 1 knapsack has received considerable attentions in recent literature. In this study, we present a family of strong inequalities that subsume known inequalities for this set. We also find many other inequalities that can be explained by lifting on 0 − 1 knapsack with continuous variable restrained in piecewise intervals. To develop stronger inequalities for the original chance-constrained model, earlier studies consider a relaxation of the intersection of multiple mixing sets by aggregating individual mixing sets into a single mixing set. Then, the inequalities can be generated from this resulting single mixing set. Nevertheless, it is unknown that if this procedure will derive facet-defining inequalities for the intersection of multiple mixing sets. Different from such traditional formulation aggregation strategy, we propose a new blending procedure that can produce strong inequalities by combining valid inequalities of all individual mixing sets. We further show that the inequalities are facet–defining under certain conditions. In the computational experiments, we illustrate the efficacy of the proposed inequalities with static probabilistic lot-sizing problems.