On the Optimality of State-Dependent Base-Stock Policies for an Inventory System with PH-Type Disruptions

Background: The management of inventory under realistic supply chain disruptions, which are often non-exponential, challenges classical control theory. This study addresses the critical question of whether the optimality of simple base-stock policies holds under the combined influence of non-exponential disruptions and random yield. Methods: We model the system as a Piecewise Deterministic Markov Process (PDMP) with impulse control, using Phase-Type (PH) distributions to capture non-memoryless event timings. The analysis involves proving the existence of a solution to the Average Cost Optimality Equation (ACOE) via a vanishing discount approach, and the framework is validated with a numerical experiment. Results: Our primary finding is a rigorous proof that a state-dependent base-stock policy is optimal, a significant generalisation of classical theory. We establish this by demonstrating the value function's convexity. The numerical experiment quantifies the significant cost penalties (over 12%) incurred by using simpler, memoryless models for supply disruptions. Conclusions: The study provides a crucial theoretical justification for the robustness of simple threshold-based control policies in complex, realistic settings. It highlights for managers the importance of modelling the variability of disruptions, not just their average duration, to avoid costly strategic errors.