On the average-cost optimality equations and convergence of discounted-cost relative value functions for inventory control problems with quasiconvex cost functions

Average-cost optimality inequalities imply the existence of stationary optimal policies for Markov Decision Processes with average costs per unit time, and these inequalities hold under broad natural conditions. Additional conditions are required for the validity of the average-cost optimality equations. Recently Feinberg and Liang [10, Theorem 3.2] showed that the equicontinuity of value functions for discounted costs is sufficient additional condition for the validity of average-cost optimality equations for problems with weakly continuous transition probabilities and with possibly unbounded one-step costs, and this condition holds for setup-cost inventory control problems with backorders and convex holding/backlog costs. This paper studies periodic-review setup-cost inventory control problem with backorders and with quasiconvex cost functions and general demands. It is shown that such problems satisfy the equicontinuity condition. Therefore, optimality inequalities hold in the form of equalities with a continuous average-cost relative value function for this problem. In addition, this implies that average-cost optimal (s, S) policies for the inventory control problem can be derived from the average-cost optimality equation. With the additional assumption on the monotonicity of the cost function, we establish the convergence of discounted-cost optimal ordering threshold s_a and convergence of discounted-cost relative value functions, when the discount factor converges to 1, to the corresponding optimal threshold and optimal relative value function for the average-cost problem.