Continuous inventory models of diffusion type: Long-term average cost criterion

Abstract

This paper establishes conditions for optimality of an (s, S) ordering policy for the minimization of the long-term average cost of one-dimensional diffusion inventory models. The class of such models under consideration have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to decrease the inventory level. Characterization of the cost of a general (s, S) policy as a function F of two variables naturally leads to a nonlinear optimization problem over the ordering levels s and S. Existence of an optimizing pair (s * , S * ) is established for these models. Using the minimal value F * of F , along with (s * , S * ), a function G is identified which is proven to be a solution of a quasi-variational inequality provided a simple condition holds. At this level of generality, optimality of the (s * , S * ) ordering policy is established within a large class of ordering policies such that local martingale and transversality conditions involving G hold. For specific models, optimality of an (s, S) policy in the general class of admissible policies can be established using comparison results. This most general optimality result is shown for the classical drifted Brownian motion inventory model with holding and fixed plus proportional ordering costs and for a geometric Brownian motion inventory model with fixed plus level-dependent ordering costs. However, for a drifted Brownian motion process with reflection at {0}, a new class of non-Markovian policies is introduced which have lower costs than the (s, S) policies. In addition, interpreting reflection at {0} as "justin-time" ordering, a necessary and sufficient condition is given that determines when just-in-time ordering is better than traditional (s, S) policies.MSC 2010 Classifications: 93E20, 90B05, 60H30

A weak convergence approach to inventory control using a long-term average criterion

“…When b is a natural boundary, the requirement that M[y, b) < ∞ for each y ∈ I is imposed in this paper as this is required for our general approach and does not hold in general. Lemma 2.4 of Helmes et al (2017) shows that this condition follows from the more restrictive assumptions in that paper.…”

“…In this section we briefly establish the models under consideration, which are more general than those in Helmes et al (2017). The approach is very similar to that taken in Sections 2 and 3 of that paper, so we refer the reader to those sections for more details.…”

“…The boundary point b is either natural or entrance and is unattainable from the interior in both cases. Following the approach in Helmes et al (2017), we define the state space of possible inventory levels to be the interval E which excludes any natural boundary point, and includes a when it is attainable and b when it is entrance. Since orders typically increase the inventory level, define R = {(y, z) ∈ E 2 : y < z} in which y denotes the pre-order and z the post-order inventory levels, respectively.…”

“…In this paper we further develop our examination of the long-term average cost criterion for inventory models of diffusion type in which the only control over the inventory levels is through the action of ordering additional stock; see Helmes et al (2017) for our previous investigation. It identifies weak sufficient conditions for optimality of an (s, S) ordering policy in the general Long-term average inventory control 1033 class of admissible policies as well as providing some sufficient conditions for nonexistence of an optimal (s, S) policy.…”

“…In particular, we concentrate on the more relaxed conditions in the current paper and we provide some new results that are required for the weak convergence analysis. Some results are common to both papers; we refer to Helmes et al (2017) for those results whose proofs remain valid, but we provide a complete proof in Appendix A for Theorem 2.1 since the previous proof is no longer valid for the general models of this paper. Section 2 is designed to briefly set some of the common foundations in the two papers so as to allow the reader to quickly access the new ideas and approaches in succeeding sections of this paper.…”