Turnpike sets in stochastic production planning problems

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Abstract
Optimal infinite-horizon stochastic production planning problems with capacity and demand are considered to be finite state Markov chains. Turnpike set concepts are introduced to characterize the optimal inventory levels. It is shown 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 described, and 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.