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Projected Inventory Level Policies for Lost Sales Inventory Systems: Asymptotic Optimality in Two Regimes

Willem van Jaarsveld, Joachim Arts

Abstract

We consider the canonical periodic review lost sales inventory system with positive lead-times and stochastic i.i.d. demand under the average cost criterion. We introduce a new policy that places orders such that the expected inventory level at the time of arrival of an order is at a fixed level and call it the Projected Inventory Level (PIL) policy. We prove that this policy has a cost-rate superior to the equivalent system where excess demand is back-ordered instead of lost and is therefore asymptotically optimal as the cost of losing a sale approaches infinity under mild distributional assumptions. We further show that this policy dominates the constant order policy for any finite lead-time and is therefore asymptotically optimal as the lead-time approaches infinity for the case of exponentially distributed demand per period. Numerical results show this policy also performs superior relative to other policies.

Projected Inventory Level Policies for Lost Sales Inventory Systems: Asymptotic Optimality in Two Regimes

Abstract

We consider the canonical periodic review lost sales inventory system with positive lead-times and stochastic i.i.d. demand under the average cost criterion. We introduce a new policy that places orders such that the expected inventory level at the time of arrival of an order is at a fixed level and call it the Projected Inventory Level (PIL) policy. We prove that this policy has a cost-rate superior to the equivalent system where excess demand is back-ordered instead of lost and is therefore asymptotically optimal as the cost of losing a sale approaches infinity under mild distributional assumptions. We further show that this policy dominates the constant order policy for any finite lead-time and is therefore asymptotically optimal as the lead-time approaches infinity for the case of exponentially distributed demand per period. Numerical results show this policy also performs superior relative to other policies.

Paper Structure

This paper contains 26 sections, 14 theorems, 60 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Brief Literature Review
  3. Model
  4. Long lead time asymptotics
  5. Dominance of PIL policies over COP policies
  6. Asymptotic optimality as $\tau\to\infty$
  7. Penalty cost asymptotics
  8. Comparison to back-order system
  9. Asymptotic optimality as $p\to\infty$
  10. Computational aspects
  11. Inventory Projection
  12. Optimization of the projected inventory level policy
  13. Low variance simulation estimator
  14. Numerical Results
  15. Long lead times
  16. ...and 11 more sections

Key Result

Lemma 1

For any given PIL policy, it is possible to place a non-negative order in each period $t\geq0$ to attain the projected inventory level $U\geq0$ provided that it is possible to do so in period 0 (i.e. provided $\mathbf{x}_0$ satisfies $\mathbb{E}[J_{\tau-1}|\mathbf{x}_0]\leq U$).

Figures (1)

  • Figure 1: Optimized Cost rate as a function of lead time for different policies for Mixed Erlang demand with mean 100 and coefficient of variation ($cv$) of 0.5 (sub-figures a, c, and e) and 1.5 (sub-figures b, d, and f) for penalty costs of 4, 9 and 19 respectively and holding cost rate 1.

Theorems & Definitions (29)

  • Lemma 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 5
  • ...and 19 more