A note on Tempelmeier's {\beta}-service measure under non-stationary stochastic demand

TL;DR

This paper critiques Tempelmeier's eta-service measure under non-stationary demand, showing that his formulation may produce sub-optimal policies due to a misinterpretation of the fill rate.

Contribution

It clarifies the discrepancy in Tempelmeier's eta-service measure and demonstrates potential sub-optimality through a numerical example.

Findings

Tempelmeier's eta-service measure does not align with the standard fill rate definition.

His MIP model may lead to sub-optimal replenishment policies.

The paper highlights the importance of correct service level measure interpretation.

Abstract

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.