How to Determine the Order-Up-To Level when Demand is Gamma Distributed with Unknown Parameters

Abstract

Inventory models need information about the demand distribution. In practice, this information is not known with certainty and has to be estimated with often relatively few historical demand observations. Using these estimates leads to underperformance. This paper focuses on gamma distributed demand and a periodic review, order-up-to inventory control policy, where the order-up-to level satisfies a service equation. Under this policy the underperformance is quantified analytically under strong assumptions and with help of simulation if these assumptions are relaxed. The analytical results can be used to improve the attained service level, such that it approaches the desired service level more closely, even if the assumptions are not met. With help of simulation we show that in some cases this improvement results in reaching the desired service level. For the remaining cases, i.e., the cases in which the desired service level is not reached, the underperformance decreases; improvements range from almost 17% up to over 90%. Moreover, with help of simulation and linear regression further improvements can be obtained. The desired service level is reached in more cases and the underperformance in the other cases is decreased even more compared to using only the first improvement. These improvements range from 57% up to 99% compared to the base case (i.e., do not use analytical results) and from 35% up to over 90% compared to using the analytical results, except for a few cases in which the service hardly improved, but in those cases the attained service level was already very close to the desired one. Finally, the method developed in this paper is applied to real demand data using simulation. The total improvements in this case study range from 53% up to 96%.