This article deals with an increasing total profit for inventory optimal ordered quantity and partial backlogging with the holding cost depending on the storage time period, and the rate of market demand is assumed to fluctuate as a function, based on level of stock and selling price. Thereafter, using the concept of a Hessian matrix, we have proved the concave nature of the profit function for the case where maximum cost is obtained. Finally, in order to validate the derived models, numerical examples and sensitivity analysis are explained. Through numerical test, we show that the proposed algorithms give quite satisfactory solutions. Hence, it can be concluded that the total profit can be increased by allowing shortage and partial backlogging.

This paper analyses an inventory model for non-instantaneous deteriorating items under a two-warehouse system with the effect of inflation and time value of money. We propose the model from the seller’s prospective by incorporating the fact that granting the trade credit from the seller to its buyer not only increases sales and revenue but also opportunity cost and default risk. Moreover, in this model, shortages are allowed and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. The purpose of this study is to determine the optimal credit period and the optimal order quantity such that the total profit of the seller is maximised. Some numerical examples are presented for illustrating the proposed inventory model. Furthermore, sensitivity analysis of the optimal solutions with respect to major parameters is carried out and some managerial inferences are obtained.