Dynamic Pricing and Periodic Ordering for a Stochastic Inventory System with Deteriorating Items

      We consider a problem of the dynamic pricing and the periodic ordering for deteriorating items with a stochastic inventory level depending on the stock-dependent demand and the selling price. In the model to be established, both shortages and remains are allowed, and the backlogging rate is variable and dependent on the waiting time for the next replenishment. Combined with the dynamic pricing, a stochastic dynamic optimization model, which maximizes the total profit, is developed. Based on the dynamic programming principle, the stochastic control model is transformed into a Hamilton–Jacobi–Bellman (HJB) equation. We show that there exists an optimal replenishment cycle length with an optimal inventory level at start of cycle. Moreover, the optimal pricing strategies are given. In addition, the finite difference scheme with the semi-smooth Newton method is proposed to solve the HJB equation numerically, and some numerical simulations are presented to illustrate the theoretical results. The sensitivity analysis of the optimal solution with respect to the major parameters is carried out and some managerial insights are proposed.