This paper considers that the number of defective units in an arrival order to be a binominal random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity and lead time are decision variables. In our studies, we also 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. In addition, we assume that the lead time demand follows a mixtures of normal distribution, and then relax the assumption about the form of the mixtures of distribution function of the lead time demand and apply the minimax distribution free procedure to solve the problem. Furthermore, we develop an algorithm procedure to obtain the optimal ordering strategy for each case. Finally, two numerical examples are also given to illustrate the results.