Inventory Placement in Acyclic Supply Chain Networks Thomas L. Magnanti, ∗ Zuo-Jun Max Shen, † Jia Shu, ‡ David Simchi-Levi § and Chung-Piaw Teo ¶ March 2004 Abstract The strategic safety stock placement problem is a constrained separable concave minimization problem and so is solvable, in principle, as a sequence of mixed-integer programming problems using a successive piecewise linear approximation approach. Unfortunately, direct implementation of this approach proves to be very time consuming, and so the research community has focused on other problem specific techniques, most notably a dynamic programming approach proposed by Graves and Willems (2000) for situations when the underlying network is a spanning tree. We examine a new successive piecewise linear approximation approach to this problem. By adding a set of redundant constraints to the formulation and by iteratively refining the piecewise linear approximations, we show that a commercial solver (CPLEX) is able to routinely solve moderate-size supply chain safety stock placement problems to optimality. For a random 100-stage sparse acyclic supply chain network, the algorithm typically finds a solution in under three minutes, achieving a speed-up of close to 90-100 over a mixed integer programming implementation using the traditional formulation. The speedup arises because the CPLEX solver is able to automatically generate stronger flow cover cuts using the added redundant constraints. 1 Introduction We consider a supply chain network with nodes representing stages of the production/assembly oper- ations of components. The arcs in the network specify the flow of components: an arc (i, j ) indicates that the production/assembly process of stage j requires a component produced at stage i. The source nodes in the network correspond to external supplies from exogenous suppliers, whereas the sink nodes correspond to finished goods. We assume further that the production/assembly operation is determin- istic, but finished good products (at the sink nodes) might experience (bounded) normally distributed ∗ School of Engineering and Sloan School of Management, Massachusetts Institute of Technology (supported in part by Singapore-MIT Alliance) † Dept. of Industrial and Systems Engineering, University of Florida ‡ High Performance Computation for Engineered Systems, Singapore-MIT Alliance § Dept. of Civil and Environmental Engineering and Division of Engineering Systems, Massachusetts Institute of Technology ¶ Dept. of Decision Sciences, National University of Singapore (supported in part by Singapore-MIT Alliance) 1