Computational algorithm for inventory model with a service level constraint, lead time demand with the mixture of distributions and controllable negative exponential backorder rate

In this paper, let the backorder rate be a control variable to widen applications of Wu and Tsai's model [J.W. Wu, H.Y. Tsai, Mixture inventory model with back orders and lost sales for variable lead time demand with the mixtures of normal distribution, International Journal of Systems Science 32 (2001) 259-268.]. And, we also assume that the backorder rate is dependent on the length of lead time through the amount of shortages. In addition, we assume that the lead time demand follows a mixture of normal distributions, and then we relax the assumption about the form of the mixture of distributions for the lead time demand and apply the minimax distribution free procedure to solve the problem. Further, instead of having a stock-out term in the objective function, a service level constraint is added to the models. Finally, we develop two computational algorithms to find the optimal order quantity and the optimal lead time. Furthermore, two numerical examples are also given to illustrate the results.