An optimal control of inventory under probablistic replenishment intervals and known price increase

The unit purchase price of products may increase or decrease for various reasons which influences the seller's and buyer's inventory control policies. This paper develops an inventory control model when replenishment intervals are probabilistic and partial backordering in which an announced price increase happens and the buyer can make a special order before the price increase initiates. Inventory at the beginning of the period is not equal to zero and can be positive or negative amounts. The main aim of this paper is to determine the optimum amount of replenish-up-to level in special sale offer. This model is analyzed in two cases according to the inventory level in which the price increase happens and both cases are formulated to maximize the total saving function for placing a special order or not. A closed-form solution is obtained for the optimum amount of a special order. Two special cases with the replenishment intervals following uniform or exponential distributions are studied with sensitivity analysis. The results indicate that a buyer can determine the optimal values of the maximum level of inventory and order quantity when the time between two consecutive replenishments is a probabilistic variable and the buyer experiences known price increase for which the total profit is maximized.