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( $\lambda$ ) by an untruncated Exp( $\lambda +1$ ), with a loss of accuracy of no more than 5 percent for all numerical examples tested.