This study considers a mixture inventory model with backorders and lost sales, and the backorder rate is a random variable. It also investigates the effect of quantity discounts on the inventory model, in which order quantity, reorder point and lead time are viewed as decision variables. In this paper, we first assume that the demand of lead time follows a normal distribution, then relax the assumption and only assume that the mean and variance ofthe lead timedemand areknown. For each case, wedevelop an algorithm to find the optimal ordering policy. Furthermore, the sensitivity analysis is performed and two numerical examples are given.