Stochastic Setup-Cost Inventory Model with Backorders and Quasiconvex Cost Functions

TL;DR

This paper analyzes a stochastic inventory model with setup costs, backorders, and quasiconvex costs, proving the existence and convergence of optimal policies and value functions under certain conditions.

Contribution

It establishes the validity of optimality equations and convergence of discounted thresholds to average-cost thresholds in a quasiconvex cost setting, including cases with positive lead times.

Findings

Optimality equations hold for discounted and average-cost problems.

Discounted lower thresholds converge to average-cost thresholds.

Convergence results extend previous knowledge to quasiconvex costs.

Abstract

In this paper we study a periodic-review single-commodity setup-cost inventory model with backorders and holding/backlog costs satisfying quasiconvexity assumptions. We show that the Markov decision process for this inventory model satisfies the assumptions that lead to the validity of optimality equations for discounted and average-cost problems and to the existence of optimal $(s,S)$ policies. In particular, we prove the equicontinuity of the family of discounted value functions and the convergence of optimal discounted lower thresholds to the optimal average-cost one for some sequences of discount factors converging to $1.$ If an arbitrary nonnegative amount of inventory can be ordered, we establish stronger convergence properties: (i) the optimal discounted lower thresholds $s_\alpha$ converge to optimal average-cost lower threshold $s;$ and (ii) the discounted relative value functions converge to average-cost relative value function. These convergence results previously were known only for subsequences of discount factors even for problems with convex holding/backlog costs. The results of this paper hold for problems with deterministic positive lead times.