A note on Tempelmeiers {beta}-service measure under non-stationary stochastic demand

Tempelmeier (2007) considers the problem of computing replenishment cycle policy parameters under non-stationary stochastic demand and service level constraints. He analyses two possible service level measures: the minimum no stock-out probability per period ({\alpha}-service level) and the so called “fill rate”, that is the fraction of demand satisfied immediately from stock on hand ({\beta}-service level). For each of these possible measures, he presents a mixed integer programming (MIP) model to determine the optimal replenishment cycles and corresponding order-up-to levels minimizing the expected total setup and holding costs. His approach is essentially based on imposing service level dependent lower bounds on cycle order-up-to levels. In this note, we argue that Tempelmeier’s strategy, in the {\beta}-service level case, while being an interesting option for practitioners, does not comply with the standard definition of “fill rate”. By means of a simple numerical example we demonstrate that, as a consequence, his formulation might yield sub-optimal policies.

💡 Research Summary

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Tempelmeier (2007) addressed the problem of determining optimal replenishment cycles and order‑up‑to levels for inventory systems facing non‑stationary stochastic demand. He considered two service‑level measures: the α‑service level (the probability of no stock‑out in a period) and the β‑service level, commonly called “fill rate”, i.e., the fraction of demand that can be satisfied immediately from on‑hand inventory. For each measure he proposed a mixed‑integer programming (MIP) model that minimizes expected total setup and holding costs while imposing service‑level‑dependent lower bounds on the order‑up‑to levels.

The present note focuses on the β‑service level formulation. In the standard definition, the fill rate is the ratio of total demand satisfied immediately to total demand, which can be expressed as

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