Continuity of Discounted Values and the Structure of Optimal Policies for Periodic-Review Inventory Control with Setup Costs

TL;DR

This paper establishes the continuity of value functions and characterizes optimal policies in discounted periodic-review inventory control problems with setup costs, covering various constraints and horizons.

Contribution

It proves continuity of value functions and provides conditions for optimal $(s,S)$ policies in constrained inventory models with setup costs.

Findings

Value functions are continuous in finite and infinite horizons.

Existence of deterministic Markov optimal policies.

Conditions for optimal $(s,S)$ policies under constraints.

Abstract

This paper proves continuity of value functions in discounted periodic-review single-commodity total-cost inventory control problems with \revision{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 $(s_t, S_t)$ policies are optimal in the finite horizon and an $(s,S)$ policy is optimal in the infinite horizon.