A note on Tempelmeier's {\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. 
Roberto Rossi, S. Armagan Tarim, Onur A. Kilic
math.OC cs.SY