Supply chain optimization : formulations and algorithms

Abstract

In this thesis, we develop practical solution methods for a supply chain optimization problem: a multi-echelon, un capacitated, time-expanded network of distribution cen­ters and stores, for which we seek the shipping schedule that minimizes total inventory, backlogging, and shipping costs, assuming deterministic, time-varying demand over a fixed time horizon for a single product. Because of fixed ordering and shipping costs, this concave cost network flow problem is in a class of NP-hard network design problems. We develop mathematical programming formulations, heuristic algorithms, and enhanced algorithms using approximate dynamic programming (ADP). We achieve a strong mixed integer programming (MIP) formulation, and fast, reliable algorithms, which can be extended to problems with multiple products. Beginning with a lot-size based formulation, we strengthen the formulation in steps to develop one which is a variation of a node-arc formulation for the network design problem. In addition, we present a path-flow formulation for the single product case and an enhanced network design formulation for the multiple product case. The basic algorithm we develop uses a dynamic lot-size model with backlogging together with a greedy procedure that emulates inventory pull systems. Four re­lated algorithms perform local searches of the basic algorithm's solution or explore alternative solutions using pricing schemes, including a Lagrangian-based heuristic. We show how approximate dynamic programming can be used to solve this sup­ply chain optimization problem as a dynamic control problem using any of the five algorithms. In addition to improving all the algorithms, the ADP enhancement turns the simplest algorithm into one comparable to the more complex ones. Our computational results illustrate that our enhanced network design formula­tion almost always produces integral solutions and can be used to solve problems of moderate size (3 distribution centers, 30 stores, 30 periods). Our heuristic methods, particularly those enhanced by ADP methods, produce near optimal solutions for truly large scale problems.