Fill rate in a periodic review Order-Up-To policy under correlated normally distributed demand


We investigate the inventory customer service metric known as the fill rate. The fill rate is defined as the proportion of demand that is immediately fulfilled from inventory. However, the task of finding analytical solutions for general cases is difficult. In the literature two approximate approaches are proposed. The first of these approximations is the traditional fill rate (or p2 service measure) that is exact in the Order-Up-To replenishment policy with Minimum Mean Squared Error forecasting, zero lead-time and independent and identically distributed (i.i.d.) demand. However, when any of these assumptions is relaxed then the traditional fill rate measure is only a lower bound. A second approximation in the literature has been proposed by Sobel (2004) that is better able to cope with non-zero lead-times as it manages the double accounting of accumulated backlogs. Sobel’s approach still requires positive i.i.d. demands implying there is no correlation between demand and net stock. However the assumption of i.i.d. demand is unrealistic. Correlation may be introduced by auto-correlated demand or forecasting methods, amongst others. We propose a new fill rate measure that can handle correlated and possibly negative demand. We assume normally distributed demand, and treat negative demand as returns. The problems reduces to identifying the minimum of correlated bi-variate random variables. There is an exact solution, but it has no closed form. However, the solution is amenable to numerical techniques and we present a custom Microsoft Excel function for practical investigations.

Pre-prints of the 18th International Working Seminar of Production Economics, Innsbruck, Austria, 24th-28th Feburary, Vol. 1, 103–115