Technical Note—Production Management with General Demands and Lost Sales

We consider continuous-review, single-product inventory systems with a constant replenishment rate, Lévy demand, general inventory holding cost, and general lost-sales penalty. The Lévy demand encompasses various demand dynamics used in the inventory literature. We obtain optimal replenishment rates that minimize the time-average cost and expected discounted costs. We can solve this problem explicitly for the optimal replenishment rate by utilizing the renewal theorem for the time-average cost objective. For a more complex expected discounted cost minimization problem, we first obtain the Laplace transform of the cost objective in terms of the unique positive root of the corresponding Lundberg equation. Then, we devise a Fourier-cosine scheme to numerically compute the original cost objective together with a detailed error analysis to determine the optimal production rate. In particular cases, we obtain closed-form expressions of the optimal replenishment rates. The numerical examples further illustrate our numerical method’s accuracy, stability, and robustness. Finally, our Fourier-cosine method can be applied to compute risk analytics, including but not limited to the stockout probability and expected shortfall of the production-inventory system. Funding: S. P. Sethi acknowledges financial support from the Eugene McDermott Chair Professorship. C. C. Siu acknowledges financial support from the Research Grants Council of Hong Kong [Grant “Generalized Sethi Advertising Model and Extensions” with Project UGC/FDS14/P02/20]. S. C. P. Yam acknowledges financial support from [Grant HKGRF-14301321 with Project “General Theory for Infinite Dimensional Stochastic Control: Mean Field and Some Classical Problems” and Grant HKGRF-14300123 with Project “Well-Posedness of Some Poisson-Driven Mean Field Learning Models and Their Applications”]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.0191 .