The stochastic location model with risk pooling

In this paper, we present a stochastic version of the location model with risk pooling (LMRP) that optimizes location, inventory, and allocation decisions under random parameters described by discrete scenarios. The goal of our model (called the stochastic LMRP, or SLMRP) is to find solutions that minimize the expected total cost (including location, transportation, and inventory costs) of the system across all scenarios. The location model explicitly handles the economies of scale and risk-pooling effects that result from consolidating inventory sites. The SLMRP framework can also be used to solve multi-commodity and multi-period problems.We present a Lagrangian-relaxation–based exact algorithm for the SLMRP. The Lagrangian subproblem is a non-linear integer program, but it can be solved by a low-order polynomial algorithm. We discuss simple variable-fixing routines that can drastically reduce the size of the problem. We present quantitative and qualitative computational results on problems with up to 150 nodes and 9 scenarios, describing both algorithm performance and solution behavior as key parameters change.