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Stable Spare Parts Pooling for Military Weapon Systems

Loe Schlicher, Marco Slikker, Willem van Jaarsveld

TL;DR

The paper addresses when Departments of Defence should form a joint spare parts pool and how pooling type affects stability. It jointly models stocking decisions with discrete-time Markov decision processes and analyzes cost allocations with cooperative game theory, proving that threshold pooling yields a non-empty core while full pooling can fail under heterogeneous demands and costs. The core results rely on reducing a multi-dimensional DP to a one-dimensional space and applying balancedness arguments (Bondareva–Shapley), providing a rigorous link between dynamic inventory control and cooperative stability. Practically, the findings inform defense logistics and coalition design by showing that threshold-based pooling strategies promote durable, incentive-compatible shared inventories, whereas unrestricted full pooling may lead to shirking or dissolution of the grand coalition under certain configurations.

Abstract

We study under which circumstances Departments of Defenses should be willing to deploy a joint parts part pooling program for their major weapon systems. Using cooperative game theory and Markov Decision Processes, we demonstrate that the type of pooling strategy plays a crucial role in the success of such a joint spare parts pool. More precisely, we show that a joint spare parts pool may not last long -- or even not arise -- if full pooling is applied, while it is stable under threshold pooling.

Stable Spare Parts Pooling for Military Weapon Systems

TL;DR

The paper addresses when Departments of Defence should form a joint spare parts pool and how pooling type affects stability. It jointly models stocking decisions with discrete-time Markov decision processes and analyzes cost allocations with cooperative game theory, proving that threshold pooling yields a non-empty core while full pooling can fail under heterogeneous demands and costs. The core results rely on reducing a multi-dimensional DP to a one-dimensional space and applying balancedness arguments (Bondareva–Shapley), providing a rigorous link between dynamic inventory control and cooperative stability. Practically, the findings inform defense logistics and coalition design by showing that threshold-based pooling strategies promote durable, incentive-compatible shared inventories, whereas unrestricted full pooling may lead to shirking or dissolution of the grand coalition under certain configurations.

Abstract

We study under which circumstances Departments of Defenses should be willing to deploy a joint parts part pooling program for their major weapon systems. Using cooperative game theory and Markov Decision Processes, we demonstrate that the type of pooling strategy plays a crucial role in the success of such a joint spare parts pool. More precisely, we show that a joint spare parts pool may not last long -- or even not arise -- if full pooling is applied, while it is stable under threshold pooling.

Paper Structure

This paper contains 18 sections, 17 theorems, 87 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Time Markov Decision Processes
  4. Cooperative Game Theory
  5. Balancedness of games
  6. Joint spare parts pooling situation and two associated games
  7. Joint spare parts pooling situation
  8. Full Pooling and Threshold Pooling Games
  9. MDP Formulation
  10. State and action space of threshold pooling game
  11. Uniformization and state transitions of threshold pooling game
  12. MDP for full-pooling
  13. Two examples: the full-pooling game and threshold pooling game
  14. Core non-emptiness of threshold pooling games
  15. Step 1: Reduction to one-dimensional space
  16. ...and 3 more sections

Key Result

Theorem 1

Let $(T,\mathscr{Y},\mathscr{A},\mathscr{C},p)$ be an MDP. If $(i)$ there exists a stationary policy $f$ inducing an irreducible and positive recurrent Markov chain on $\mathscr{Y}$, and satisfying $J_{f}(y) < \infty$ for all $y \in \mathscr{Y}$, and $(ii)$ there exists an $\epsilon >0$ such that $\ and moreover, there exists an optimal stationary policy.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • example 1
  • example 2
  • Lemma 3
  • Corollary 1
  • Definition 1
  • Definition 2
  • ...and 19 more