Evaluating re-order levels and probabilities of stockout during a lead time for some stock-control models

When several suppliers are concurrently used to replenish one stock item, it is often of interest to compute the mean and the variance of the effective lead time, i.e., the minimum of all lead times. However, in many cases, especially with non-identical lead time distributions, these quantities do not have closed form expressions and their calculation can be complicated. Difficulty in computation also arises in evaluating demand during lead time, even if there is only one supplier. This paper presents a numerical integration method using Gaussian quadrature to evaluate the effective lead time mean and variance, re-order level and the probability of stockout during the effective lead time. The Gaussian quadrature rules are easy to implement and several data sets are used in the paper to demonstrate the method.