An inventory model with price- and stock-dependent demand and time- and stock quantity-dependent holding cost under profitability maximization

This paper focuses on inventory models with a broad framework for the storage cost and the demand rate. The cumulative storage cost is modelled with a power function, depending on both time and stock quantity, by using two elasticity coefficients. Similarly, the demand rate has an isoelastic dependence on sale price and stock quantity, modelled with another two elasticity coefficients. These four elasticity coefficients allow many real practical situations to be modelled. A reference price is used to measure the effect of the sale price on the demand rate. The goal is to maximize the income expense ratio (IER), and the sale price, the order level and the reorder point are the decision variables. The operating expense ratio (OER) of the system, defined as the quotient cost/income, is used to solve the problem. The optimum values are obtained with explicit expressions, which is an interesting result for inventory managers. Under the optimum policy, the reorder point is always equal to zero and the order quantity depends on the replenishing cost, the purchase price and the four elasticity coefficients. However, the optimum ordering policy does not depend on the scale parameters of the storage cost and the demand rate. A complete sensitivity analysis for most of the model parameters is performed. A numerical example is used to compare the optimum policies for the maximum income expense ratio and the maximum profit per unit time. Finally, some managerial insights derived from the results are given.