Demand forecasting under fill rate constraints—The case of re-order points

Abstract

We consider computation of demand forecasts fulfilling an assumption about the level of the P2 service measure in continuous review inventory systems with fixed order quantities and re-order points as decision variables. First, we introduce a family of loss functions whose expectations are uniquely minimized at the functional of demand of interest here. Then, concentrating on the linear-quadratic loss and assuming some regularity conditions, we provide consistency and asymptotic normality conditions for the direct M-estimator of the appropriate decision levels in the case of iid data. The large sample properties of this direct estimator are examined, under different distributional assumptions, in comparison with the two-step estimator based on the maximum likelihood principle. Furthermore, we also study the small sample properties of these two estimators (and also another one utilizing a nonparametric kernel density estimation) and study the small sample coverage of confidence intervals for the decision levels constructed according to the derived asymptotic formulas. Finally, we also discuss possible extensions of this estimation framework to the case of dependent data and introduce certain classes of dynamic models for forecasting under the P2 constraint.