Lead Time Demand Modeling in Continuous Review Supply Chain Models

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 truncated exponential (MTE) functions. First, construction of the LTD is illustrated and the approach is compared to two other possible LTDs. This distribution is then utilized to determine optimal order policies in cases where a buyer makes its decisions alone, and later in a situation where members of a two-level supply chain coordinate their actions. Introduction Numerous probability models have been suggested for representing uncertain demand during lead time (LT) in continuous-review inventory management systems when both LT and demand per unit time (DPUT) are variable. A common approach to finding a distribution for lead time demand (LTD) involves modeling LT and DPUT with standard probability density functions (PDFs). Based on the distributions assigned, a compound probability distribution is determined for demand during lead time, or LTD. The latter distribution is used to determine reorder point and safety stock policies, and may be used to estimate inventory costs. 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. Although the problem of finding an appropriate LTD distribution has been well studied, papers written in recent years have continued to pursue methods that overcome unrealistic distributional assumptions (RuizTorres & Mahmoodi, 2010; Vernimmen, Dullaert, Willimé, & Witlox, 2008). This paper illustrates an approach for constructing a mixture distribution for LTD that allows the LT and DPUT distributions to be state-dependent. This method also allows input distributions that take any standard or empirical form. Use of the mixture distribution technique is first demonstrated in the context described by Cobb (2013), which is a singleitem continuous-review inventory model for one buyer. For single-firm operating in a continuous-review inventory system, the mixture distribution method for modeling the LTD distribution differs from the typical “moment-matching” approach. The method focuses on