Stochastic Integer Programming with Simple Recourse

We consider the expected value function of a stochastic integer linear programming problem with simple recourse. By making appropriate simplifications we reduce it to a separable function which allows us to study the one-dimensional functions g(x) = E ξ − x + and h(x) = E x− ξ + instead. We derive formulas for the functions g and h in terms of the cumulative distribution function F of the random variable ξ. We find conditions for finiteness, (Lipschitz-)continuity, differentiability and convexity of g and h. Since in general g and h are not convex functions, much attention is paid to finding their convex hulls, g∗∗ and h∗∗ respectively. We prove that g∗∗(x) = E(ζ − x) and h∗∗(x) = E(x− ψ), where the random variables ζ and ψ have some cumulative distribution function G and H . Moreover, if F belongs to a certain class, we find explicit formulas for g∗∗, G, h∗∗ and H . As examples we derive explicit formulas for the exponential and uniform distribution. ∗Department of Econometrics, University of Groningen, The Netherlands. †Institute for Actuarial Sciences and Econometrics, University of Amsterdam, The Netherlands. ‡Department of Econometrics, University of Groningen, The Netherlands. Supported by the Landelijk Netwerk Mathematische Besliskunde (LNMB).