Optimal ordering and pricing policy for demand functions that are separable into price and inventory age

We formulate and analyze two models for determining the optimal pricing, order quantity and replenishment period for items whose demand function is separable into components of price and inventory age. The first model assumes a multiplicative demand function. We provide conditions, which are satisfied by most common price-dependent demand functions, to reduce the three-variable profit maximization problem into a single-variable problem, which can be solved using an efficient line-search method. Next, we show that a genuine additive model cannot exist, and instead suggest and analyze a pseudo-additive model. However, this model is more limited than the multiplicative model in its ability to incorporate various combinations of price and inventory age effects, and reduction of the maximization problem into a single-variable problem is more complicated, except in the case of a linear price effect, which is further analyzed. For both models, we show that the optimal solution satisfies the first-order condition for equilibrium under a monopoly, with a modification that includes inventory holding costs. We solve numerical examples to illustrate the solution procedures.