The cost of lead-time variability: The case of the exponential distribution

We present closed-form solutions for the reorder point, z, the order quantity with non-zero lead time, Q, and the total relevant cost, TRC(z, Q, K′), where the lead time is doubly truncated (at r=z/D and r+q=z/D+Q/D, D being a constant demand rate), using the concept of pseudo-setup cost, K′. We aim to provide a paradigm for quantifying the effect lead-time variability on cost, as such paradigms are lacking in the inventory literature. From our analysis, we conclude that to achieve zero inventory (ZI) in a stochastic lead-time setting, both the setup cost and the lead-time variability would have to be eliminated. Furthermore, it is the variance and not the mean that affects the total relevant cost in a stochastic lead-time model. Besides the statistical analysis, we examine the economic impact of the truncation and determine model accuracy by means of a heuristic that is robust to parameter changes and that allows the convenience of substituting in the analysis the doubly truncated Exp(λ) by an untruncated Exp(λ+1), with a loss of accuracy of no more than 5 percent for all numerical examples tested.