An inventory control model for obsolete items with consideration of all unit quantity discount and price-dependent on order quantity

In this study, a single-item obsolescence inventory model with one retailer and with all-unit quantity discounts has been investigated. The retailer purchases items with a possibility of future obsolescence based on the purchase price which is dependent into the order quantity. It is assumed in this paper that there are dependencies between different decisions on the amount of order quantity, considering the probability of obsolescence at any given moment and the purchase price of items by taking into account the use of discount. The aim of the model is to minimize the retailer’s inventory cost through the finding optimal purchase price and order replenishment. Although discount models have been developed for deteriorating items with a constant or varying rate of deterioration, but the use of these models, for obsolete items, in which the demand of item falls suddenly in the future based on a probability distribution, has not been so far investigated. While, these are the real conditions that many retailers are facing and in such situations the retailers need to encourage the customers to buy more. In this model, the length of obsolescence time is considered as exponential function and the obsolescence is occurred suddenly. The model has been developed mathematically and then the optimal solution and convexity are resulted from the first and second derivatives of objective function respectively. Finally, for the sake of clarity of the proposed model, a numerical example is presented and sensitivity analyses are performed on critical parameters of the model.