Inventory model involving controllable backorder rate and variable lead time demand with the mixtures of distribution

In the model of Ouyang and Chuang [Comput. Ind. Eng. 40 (2001) 339], they assume that the backorder rate is dependent on the length of lead time through the amount of shortages and let the backorder rate is a control variable. But, since they only assumed a single distribution for the lead time demand, when the demand of the different customers are not identical in the lead time, then we cannot use a single distribution (such as [Comput. Ind. Eng. 40 (2001) 339]) to describe the demand of the lead time. Hence, in our studies, we first assume that the lead time demand follows a mixtures of normal distribution, and then we relax the assumption about the form of the mixtures of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. We develop an algorithm procedure, respectively, to find the optimal order quantity and the optimal lead time. Furthermore, two numerical examples are also given to illustrate the results.