Optimizing price, order quantity, and backordering level using a nonlinear holding cost and a power demand pattern

•This paper builds an inventory model with shortages and these are completely backordered.•The demand for the product jointly combines the impact of the selling price and a time power function.•The holding cost is a power of the time that the product is held in storage.•An effective and efficient algorithm that obtains the optimal solution is provided.•The inventory model is a generalized model due to it contains some inventory models as special cases. It is well-known that the demand rate for some products depends on several factors, such as price, time, and stock, among others. Moreover, the holding cost can vary over time. More specifically, it increases with time since a long period of storage requires more expensive warehouse facilities. This research introduces an inventory model with shortages for a single product where the demand rate depends simultaneously on both the selling price and time according to a power pattern. Shortages are completely backordered. Demand for the product jointly combines the impact of the selling price and a time power function, which is performed as an addition. Furthermore, the holding cost is a power of the time that the product is held in storage. The main objective is to derive the optimal inventory policy such that the total profit per unit of time is maximized. For optimizing the inventory problem, some theoretical results are derived first to prove that the total profit function is strictly pseudo concave with respect to the decision variables. Next, an efficient algorithm that obtains the optimal solution is provided. The proposed inventory model is a general model because it contains several published inventory models as special cases. Some numerical examples are presented and solved to illustrate and validate the proposed inventory model. Also, a sensitivity analysis is conducted in order to highlight and generate significant insights.