This article deals with an inventory problem for non-instantaneous deteriorating items with stock-dependent demand when supplier offers an all-unit quantity discount. In the model, shortages are allowed and the backorder rate is a random variable. The purpose of this study is to determine an optimal ordering policy for minimizing the expected total relevant inventory cost for the retailer. We have developed some useful theorems to characterize the optimal solution and provide an easy and useful algorithm to find the optimal order quantity and replenishment time. Furthermore, several numerical examples are given to illustrate the solution procedure for the mathematical model developed. Finally, we also implement sensitivity analysis of the optimal solution with respect to major parameters.