Computational algorithmic procedure of optimal inventory policy involving a negative exponential crashing cost and variable lead time demand

When the demand of the different customers are not identical in the lead time, we cannot use only a single distribution (such as Ouyang et al. (1996) [L.Y. Ouyang, N.C. Yeh, K.S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of the Operational Research Society 47 (1996) 829–832] using normal distribution) to describe the demand of the lead time. Hence, in this paper, we extend the models of Ouyang et al. (1996) and Ouyang and Wu (1998) [L.Y. Ouyang, K.S. Wu, A minimax distribution free procedure for mixed inventory model with variable lead time, International Journal of Production Economics 56–57 (1998) 511–516] by considering the mixture of normal distributions and the mixture of free distributions (see Everitt and Hand (1981) [B.S. Everitt, D.J. Hand, Finite Mixture Distribution, Chapman and Hall, London, NY, 1981]), respectively. Moreover, we quote the continuous model which the total crashing cost is related to the lead time by a negative exponential function (such as Ben-Daya and Raouf (1994) [M. Ben-Daya, A. Raouf, Inventory models involving lead time as decision variable, Journal of the Operational Research Society 45 (1994) 579–582]). Finally, we give two algorithmic procedures to find the optimal inventory policy and two numerical examples to illustrate the results.