MILP reformulation of the multi-echelon stochastic inventory system with uncertain demands

In this article we present an effective mixed-integer linear programming (MILP) formulation for design of a multi-echelon stochastic inventory system with uncertain customer demands. In You and Grossmann [2010] a three-echelon supply chain with inventories under uncertainty is presented. In that supply chain, the location of the plants and the customer demand zones (CDZ) are known. Potential distribution centers (DC) are given and the objective is to decide which DCs to install in order to minimize total costs, which include transportation cost, installation cost as well as inventory holding costs. The formulation also determines the service times for each DC, and what the size of the safety stock should be at all DCs and CDZs. This model uses single sourcing, which is often the case for supply chains in industrial gases or specialty chemicals. That means that all DCs are served by only one plant and each CDZ is served by only one DC. The detailed model can be found in You and Grossmann [2010]. In this short note we first reformulate the mixed-integer nonlinear programming (MINLP) model presented in You and Grossmann [2010] and You and Grossmann [2011], using an alternative linearization scheme leading to a model that is significantly smaller in size, and tighter. We also present a successive piecewise linear approximation with which we can solve the model with a sufficiently small optimality gap without the need of a global optimization solver.