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Optimization of inventory and capacity in large-scale assembly systems using extreme-value theory

Mirjam S. Meijer, Dennis Schol, Willem van Jaarsveld, Maria Vlasiou, Bert Zwart

Abstract

High-tech systems are typically produced in two stages: 1) Production of components using specialized equipment and staff; 2) System assembly/integration. Component production capacity is subject to fluctuations, causing a high risk of shortages of at least one component, which results in costly delays. Companies hedge this risk by strategic investments in excess production capacity and in buffer inventories of components. To optimize these, it is crucial to characterize the relation between component shortage risk and capacity and inventory investments. We suppose that component production capacity and produce demand are normally distributed over finite time intervals, and we accordingly model the production system as a symmetric fork-join queueing network with $N$ statistically identical queues with a common arrival process and independent service processes. Assuming a symmetric cost structure, we subsequently apply extreme value theory to gain analytic insights into this optimization problem. We derive several new results for this queueing network, notably that the scaled maximum of $N$ steady-state queue lengths converges in distribution to a Gaussian random variable. These results translate into asymptotically optimal methods to dimension the system. Tests on a range of problems reveal that these methods typically work well for systems of moderate size.

Optimization of inventory and capacity in large-scale assembly systems using extreme-value theory

Abstract

High-tech systems are typically produced in two stages: 1) Production of components using specialized equipment and staff; 2) System assembly/integration. Component production capacity is subject to fluctuations, causing a high risk of shortages of at least one component, which results in costly delays. Companies hedge this risk by strategic investments in excess production capacity and in buffer inventories of components. To optimize these, it is crucial to characterize the relation between component shortage risk and capacity and inventory investments. We suppose that component production capacity and produce demand are normally distributed over finite time intervals, and we accordingly model the production system as a symmetric fork-join queueing network with statistically identical queues with a common arrival process and independent service processes. Assuming a symmetric cost structure, we subsequently apply extreme value theory to gain analytic insights into this optimization problem. We derive several new results for this queueing network, notably that the scaled maximum of steady-state queue lengths converges in distribution to a Gaussian random variable. These results translate into asymptotically optimal methods to dimension the system. Tests on a range of problems reveal that these methods typically work well for systems of moderate size.

Paper Structure

This paper contains 35 sections, 23 theorems, 159 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Extreme value analysis.
  3. Demand fluctuations.
  4. New extreme-value limit.
  5. Numerical experiments.
  6. Limitations of simulation.
  7. Summary of results.
  8. New insights.
  9. Overview.
  10. Literature Review
  11. Model and preliminaries
  12. Model.
  13. Brownian fork-join queue.
  14. Base-stock level and capacity.
  15. Transient inventory levels and backorders.
  16. ...and 20 more sections

Key Result

Lemma 3.2

Given $(Q_{i}(t,\beta),i\leq N)$ with $Q_{i}(t,\beta)$ defined in eq: transient queue length, we have that $(Q_{i}(t,\beta),i\leq N) \stackrel{d}{\rightarrow} (Q_{i}(\infty,\beta),i\leq N)$ with In particular,

Figures (1)

  • Figure 1: Fork-join queue

Theorems & Definitions (29)

  • Definition 3.1
  • Lemma 3.2: Steady-state of backlogs.
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Lemma 3.9
  • Corollary 3.10
  • ...and 19 more