Mixture Distributions for Modeling Lead Time Demand in Coordinated Supply Chains

This paper introduces a mixture distribution approach to modeling the probability density function for lead time demand (LTD) in problems where a continuous review inventory system is implemented. The method differs from the typical “moment-matching” approach by focusing on building up an accurate, closed-form approximation to the LTD distribution from its components by using mixtures of polynomial functions. First, construction of the lead time distribution is illustrated and the approach is compared to two other possible lead time distributions. This distribution is then utilized to determine optimal order policies in a situation where members of a two-level supply chain coordinate their actions. Introduction This objective of this paper is to describe and implement a mixture distribution method for modeling lead time demand (LTD) in continuous-review inventory problems, then utilize this distribution to jointly determine an optimal order quantity and reorder point. A common approach to finding a probability density function (PDF) for LTD involves modeling lead time (LT) and demand per unit time (DPUT) with standard PDFs. Based on the distributions assigned, a compound probability distribution is determined for demand during LT, or LTD. In some cases, analytical formulas for optimal reorder point, safety stock, or stockout costs are available in terms of the compound distribution’s parameters, while in other situations the values associated with certain percentiles of the compound LTD distribution are estimated to provide these values. While the problem of finding an appropriate LTD distribution has been well studied, researchers in recent years have continued to pursue methods that overcome unrealistic distributional assumptions (RuizTorres & Mahmoodi, 2010; Vernimmen, Dullaert, Willemé, & Witlox, 2008). There are many situations in which assigning a single, standard PDF as the compound distribution for LTD leads to a poor approximation. Tyworth (1992) observed that this is particularly true when LT is random and follows a non-standard, empirical distribution. This paper illustrates an approach for constructing a mixture distribution for LTD that can incorporate any discrete or continuous LT distribution while approximating the DPUT distribution with a normal PDF. The mixture distribution method for modeling the LTD distribution differs from the typical “moment-matching” approach, as it focuses on constructing the distribution from its components. Use of the mixture distribution technique was demonstrated by Cobb (2013) in a single item continuous-review inventory model for one buyer, and was further described by Cobb and Johnson (2013). Both of these papers