Continuity of discounted values and the structure of optimal policies for periodic‐review inventory systems with setup costs

This paper proves continuity of value functions in discounted periodic‐review single‐commodity total‐cost inventory control problems with continuous inventory levels, fixed ordering costs, possibly bounded inventory storage capacity, and possibly bounded order sizes for finite and infinite horizons. In each of these constrained models, the finite and infinite‐horizon value functions are continuous, there exist deterministic Markov optimal finite‐horizon policies, and there exist stationary deterministic Markov optimal infinite‐horizon policies. For models with bounded inventory storage and unbounded order sizes, this paper also characterizes the conditions under which st,St$$ \left({s}_t,{S}_t\right) $$ policies are optimal in the finite horizon and an (s,S)$$ \left(s,S\right) $$ policy is optimal in the infinite horizon.