Turnpike Sets and Their Analysis in Stochastic Production Planning Problems

This paper considers optimal infinite horizon stochastic production planning problems with capacity and demand to be finite state Markov chains. The existence of the optimal feedback control is shown with the aid of viscosity solutions to the dynamic programming equations. Turnpike set concepts are introduced to characterize the optimal inventory levels. It is proved that the turnpike set is an attractor set for the optimal trajectories provided that the capacity is assumed to be fixed at a level exceeding the maximum possible demand. Conditions under which the optimal trajectories enter the convex closure of the set in finite time are given. The structure of turnpike sets is analyzed. Last but not least, it is shown that the turnpike sets exhibit a monotone property with respect to capacity and demand. It turns out that the monotonicity property helps in solving the optimal production problem numerically, and in some cases, analytically.