A note on defective units in an inventory model with sub-lot sampling inspection for variable lead-time demand with the mixture of free distributions

In a recent paper Wu and Ouyang assumed that an arriving order lot may contain some defective items and considered that the number of defective items in the sub-lot sampled to be a random variable. They derived a modified mixture inventory model with backorders and lost sales, in which the order quantity, re-order point, and the lead-time were decision variables. In their studies they assumed that the lead-time demand followed a normal distribution for the first model and relaxed the assumption about the form of the distribution function of the lead-time demand for the second model. When the demand of the different customers is not identical with regard to the lead-time, then one cannot use only a single distribution (such as Wu and Ouyang) to describe the demand of the lead-time. Hence, we extend and correct the model of Wu and Ouyang by considering the lead-time demand with the mixed normal distributions for the first model and the lead-time demand with the mixed distributions for the second model. And we also apply the minimax mixed distributions free approach to the second model. Moreover, we also develop an algorithm procedure to obtain the optimal ordering strategy for each case.