Chance-Constrained Equivalents of Some Stochastic Programming Problems

This paper shows that chance-constrained equivalents exist for several stochastic programming problems that are concerned with selecting a decision vector which will optimize the expectation of a random loss function, sometimes subject to deterministic constraints. The equivalent chance-constrained problems are concerned with selecting a decision vector that will optimize a deterministic function subject to one or more probability constraints of the form pr[ y i ≧ ξ i ] ≧ α i , i ∈ I , and sometimes also subject to deterministic constraints. Equivalence is shown for stochastic programming problems with either linear or nonlinear random loss functions, including many stochastic scheduling and inventory problems. Such stochastic programming problems may be converted to equivalent chance-constrained problems that can be solved in their certainty equivalent form, for example, by the Horizon Method of Stochastic Scheduling.