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Single-Item Continuous-Review Inventory Models with Random Supplies

K. L. Helmes, R. H. Stockbridge, C. Zhu

Abstract

This paper analyzes single-item continuous-review inventory models with random supplies in which the inventory dynamic between orders is described by a diffusion process, and a long-term average cost criterion is used to evaluate decisions. The class of models have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. Random yield is described by a (probability) transition function which depends on the inventory-on-hand and the {\em nominal} amount ordered; it is assumed to be a distribution with support in the interval determined by the order-from and the nominal order-to locations of the stock level. Using weak convergence arguments involving average expected occupation and ordering measures, conditions are given for the optimality of an $(s,S)$ ordering policy in the general class of policies with finite expected cost. The characterization of the cost of an $(s,S)$-policy as a function of two variables naturally leads to a nonlinear optimization problem over the stock levels $s$ and $S$ and existence of an optimizing pair $(s^*,S^*)$ is established under weak conditions. Thus, optimal policies of inventory models with random supplies can be (easily) numerically computed. The range of applicability of the optimality result is illustrated on several inventory models with random yields.

Single-Item Continuous-Review Inventory Models with Random Supplies

Abstract

This paper analyzes single-item continuous-review inventory models with random supplies in which the inventory dynamic between orders is described by a diffusion process, and a long-term average cost criterion is used to evaluate decisions. The class of models have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. Random yield is described by a (probability) transition function which depends on the inventory-on-hand and the {\em nominal} amount ordered; it is assumed to be a distribution with support in the interval determined by the order-from and the nominal order-to locations of the stock level. Using weak convergence arguments involving average expected occupation and ordering measures, conditions are given for the optimality of an ordering policy in the general class of policies with finite expected cost. The characterization of the cost of an -policy as a function of two variables naturally leads to a nonlinear optimization problem over the stock levels and and existence of an optimizing pair is established under weak conditions. Thus, optimal policies of inventory models with random supplies can be (easily) numerically computed. The range of applicability of the optimality result is illustrated on several inventory models with random yields.
Paper Structure (12 sections, 19 theorems, 108 equations, 3 figures, 1 table)

This paper contains 12 sections, 19 theorems, 108 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formulation and Existence Result
  3. Formulation of the model
  4. Important functions
  5. Analysis of nominal $(s,S)$ Ordering Policies
  6. Expected Occupation and Ordering Measures
  7. The Auxiliary Function ${U}_0$
  8. Policy Class ${\cal A}_0$ and Optimality
  9. Examples
  10. Drifted Brownian motion inventory models
  11. Geometric Brownian motion storage models
  12. Logistic storage model

Key Result

Lemma 2.4

Assume Condition diff-cnd. Suppose $a$ and $b$ are natural boundaries and let $c_0(a)$ and $c_0(b)$ be as in Condition cost-cnds(a). Then the following asymptotic behaviours hold: implying $\lim_{y\rightarrow a} g_0(y) = -\infty$ when $c_0(a) > 0$ and $\lim_{v\rightarrow b} g_0(v) = \infty$ when $c_0(b) > 0$.

Figures (3)

  • Figure 1: Neighborhoods of the Boundary
  • Figure 2: The regions $F_1$, $F_2$ and $F_3$
  • Figure 3: Partition of $\overline{\cal R}$

Theorems & Definitions (38)

  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Remark 2.10
  • Theorem 2.11
  • ...and 28 more