Optimal inventory policy for a perishable item with demand function sensitive to price and time

We formulate a model for determining the optimal pricing, order quantity and replenishment period for perishable items with price-dependent and time-dependent demand. The items have a fixed shelf-life, and the demand rate decreases linearly in the selling price and polynomially over the time after replenishment, until it vanishes either at the reservation price or at expiration time. We prove that the three-variable profit maximization problem can be reduced into a single-variable problem, in which the variable is the duration of the replenishment period. We show that the profit function is strictly pseudo-concave and provide means of obtaining the optimal policy. Three numerical examples are presented to demonstrate the model accompanied by a sensitivity analysis.