A 1.79-approximation Algorithm for a Continuous Review Lost-Sales Inventory Model

Abstract

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing inventory systems. Recent numerical results have suggested that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation, and why such policies generally perform so well remains a major open question. In this paper, we provide a theoretical foundation for this phenomenon in two classical inventory models with lead times. First, in a continuous review lost-sales inventory model with lead times and Poisson demand, we prove that a so-called capped base-stock policy has a worst-case performance guarantee of 1.79, by conducting an asymptotic analysis under a large penalty cost and lead time. Second, we extend the analysis to a more complex continuous review dual-sourcing inventory model with general lead times and Poisson demand, and also prove that a so-called capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. Our results provide a deeper understanding of the superior numerical performance of capped policies, and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.