Joint production and pricing inventory system under stochastic reference price effect

Abstract

We investigate a manufacturer’s joint dynamic production and pricing problem under stochastic reference price effect over an infinite horizon in a continuous-time framework. The demand for products, which depends on their price and a memory-based reference price, is uncertain and characterized by a stochastic process. The reference price effect varies across consumers and subjects to some randomness characterized by a stochastic process. A joint dynamic production and pricing problem to maximize the total expected profit is modeled as a stochastic optimal control problem. We give the sufficient conditions of the existence of the optimal solutions and characterize the solutions for the general convex cost structure. In addition, we show that the strategies in a linear feedback form of the state variables (i.e., inventory and reference price) are optimal for a strictly convex cost structure, and provide sufficient conditions of stability and monotone convergence properties for the expected steady-state inventory and reference price. Numerical examples and sensitivity analysis are performed to provide several important managerial insights. We find that the optimal production rate and sales price are each negatively related to the inventory stock while positively to the reference price level. In expected steady-state, demand uncertainty has no influence on optimal strategies while a higher reference price uncertainty would result in a lower production rate as well as shortage level but a higher (reference) price. Moreover, the manufacturer is worse off when demand uncertainty is increased, whereas he can generate more profit by taking advantage of the randomness of the reference price. Finally, we show that the expected steady-state production rate decreases, but price increases as the reference effect factor or customers’ memory parameter increases.