From Trees to Closed Loops: Inventory Management in Treewidth-Bounded Supply Chain Networks

Abstract

We propose a new linear programming (LP) framework for the Guaranteed Service Model (GSM), one of the most widely applied approaches for optimizing safety stock placement in supply chain networks. Unlike existing approaches, our framework optimizes the GSM on any acyclic supply chain network of bounded treewidth — a graph-theoretic measure quantifying how "tree-like" a network is. The framework uses an exact LP reformulation of the GSM with size exponential in the network's treewidth but polynomial in other network parameters, such as the number of nodes. This makes it tractable for real-world supply chains, which typically exhibit low treewidth, and allows for the use of standard LP optimization software, unlike previous GSM optimization strategies, which rely on specialized algorithms or heuristics. It also enables new results and insights for the GSM, such as duality-based sensitivity analysis, principled upper and lower bounds, and the ability to incorporate operational constraints. We extend this framework to handle closed-loop supply chains with reverse flows, demonstrating that individual reverse flows only marginally increase computational complexity. This closed-loop GSM facilitates the analysis of how reverse flows impact safety stock placement in supply chains that involve recycling, remanufacturing, and product returns. Overall, our framework provides new theoretical foundations and practical tools for managing safety stock in complex modern supply chain networks. 

Keywords: supply chain management, inventory management, algorithms, computational complexity, circular economy