Serial Inventory Systems with Markov-Modulated Demand: Derivative Bounds, Asymptotic Analysis, and Insights

In this paper we consider the inventory control problem for serial supply chains with continuous, Markovmodulated demand (MMD). Our goal is to simplify the computational complexity by resorting to certain approximation techniques, and, in doing so, to gain a deeper understanding of the problem. To this end, we analyze the problem in several new ways. We first perform a derivative analysis of the problem’s optimality equations, and develop general, analytical solution bounds for the optimal policy. Based on the bound results, we derive a simple procedure for computing near-optimal heuristic solutions for the problem. These simple solutions reveal a closer relationship with the primitive model parameters. Second, we perform asymptotic analysis with long replenishment lead time and establish an MMD central limit theorem. We further show that the relative errors between our heuristics and the optimal solutions converge to zero as the lead time becomes sufficiently long, with the rate of convergence being the square root of the lead time. Our numerical results reveal that our heuristic solutions can achieve near-optimal performance even under relatively short lead times. Third, we show that, by leveraging the Laplace transformation, the optimal policy becomes computationally tractable under the gamma distribution family. This enables us to numerically compare various heuristic solutions with the optimal solution, and to demonstrate that our heuristic outperforms existing heuristics in most cases. Finally, we observe that the internal fill rate and demand variability propagation in an optimally controlled supply chain under MMD exhibit behaviors different from those under stationary demand.