Optimization of Supply Chain Networks Using Integer and Convex Programming Techniques

Supply chain network design (SCND) decides where to open plants and distribution centers, which customers each facility serves, and how products flow between them, subject to cost, service, and risk criteria. Integer programming represents the discrete decisions (open or close, assign or not). Convex programming represents scale economies, risk pooling, risk measures, and robust constraints. This paper reviews how the two are combined. It covers the mixed-integer linear core of network design, decomposition methods that make large instances tractable, convex and conic extensions for inventory and uncertainty, and stochastic, robust, and reliability formulations. The evidence shows that integer programming has produced large documented savings in industrial settings, including a reported plant-count reduction of almost 20 percent at a major consumer-goods company. Benders and Lagrangian decomposition are the workhorses at scale. Convex structure matters most in three places: risk pooling in inventory, risk measures such as conditional value-at-risk, and budgeted robust constraints, which preserve linearity under the Bertsimas–Sim approach. The review also shows where convexity fails. Concave inventory and emissions terms produce non-convex problems that need piecewise-linear or conic reformulation. Two comparative tables map models to solution methods and reported evidence, and map limitations to evidence and mitigations. The paper then synthesizes a Benders-based framework in which a mixed-integer master selects the network and convex subproblems evaluate flows, risk, and robust capacity. This framework is a conceptual synthesis of the reviewed literature and has not been tested experimentally.