A Project on Improved Inventory Control for a Danish Company With Some Theoretical Spillovers

We report on a project on developing a base-stock policy for a Danish company. Based on historical demand records the time between order requests is fitted to an Erlang distribution, while the order sizes are fitted to a binomial, a Poisson or a negative binomial distribution depending on the interrelationship between the empirical mean and variance. The base stock level S is computed as the least value of S bringing the order fill rate (the fraction of orders where the whole order can be delivered instantaneously) at or above a pre-specified level. We then report on a theoretical study where we for a base stock policy compare the two service measures: order fill and volume fill rate (the fraction of the total volume that is delivered instantaneously also often just denoted fill rate). We study the case where order sizes follow a negative binomial distribution. Here we show that it is the shape parameter of the negative binomial distribution that solely determines the interrelationship between these two service measures. They are equal for the special case where the order sizes follow a geometric distribution, however this equality is only in mean values and not when considering sample paths as a simulation experiment reveals. Some of the demand patterns of the company revealed some degree of heterogeneity. This has sparked an investigation of a base stock policy with two customer groups, with each distinct demand patterns. For each customer group we develop expressions for the service measures: order fill and volume fill rate. Based on assumptions about first order stochastic dominance we prove when one customer class will get the best service. That theoretical result is validated through a series of numerical experiments which also reveal that it is quite robust.

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Robin Nicolai

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