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Multi-machine preventative maintenance scheduling with imperfect interventions: a restless bandit approach

Diego Ruiz-Hernandez, Jesús María Pinar-Pérez, David Delgado-Gómez

TL;DR

This work formulates multi-machine preventive maintenance with imperfect interventions as a restless-bandit problem and derives a Whittle-index-like policy. By establishing indexability and closed-form $W$-indices, the authors enable scalable online allocation of limited repair crews across deteriorating machines, accounting for stochastic intervention outcomes. Numerical experiments show the index policy achieves near-optimal performance in small instances (often optimal) and robust superiority over myopic or threshold schemes in larger systems. The approach provides a practical, theoretically grounded tool for dynamic resource allocation in maintenance contexts where interventions are uncertain, with potential extensions to broader independent-element systems.

Abstract

In this paper we address the problem of allocating the efforts of a collection of repairmen to a number of deteriorating machines in order to reduce operation costs and to mitigate the cost (and likelihood) of unexpected failures. Notwithstanding these preventive maintenance interventions are aimed at returning the machine to a so-called as-good-as-new state, unforeseeable factors may imply that maintenance interventions are not perfect and the machine is only returned to an earlier (uncertain) state of wear. The problem is modelled as a restless bandit problem and an index policy for the sequential allocation of maintenance tasks is proposed. A series of numerical experiments shows the strong performance of the proposed policy. Moreover, the methodology is of interest in the general context of dynamic resource allocation and restless bandit problems, as well as being useful in the particular imperfect maintenance model described.

Multi-machine preventative maintenance scheduling with imperfect interventions: a restless bandit approach

TL;DR

This work formulates multi-machine preventive maintenance with imperfect interventions as a restless-bandit problem and derives a Whittle-index-like policy. By establishing indexability and closed-form -indices, the authors enable scalable online allocation of limited repair crews across deteriorating machines, accounting for stochastic intervention outcomes. Numerical experiments show the index policy achieves near-optimal performance in small instances (often optimal) and robust superiority over myopic or threshold schemes in larger systems. The approach provides a practical, theoretically grounded tool for dynamic resource allocation in maintenance contexts where interventions are uncertain, with potential extensions to broader independent-element systems.

Abstract

In this paper we address the problem of allocating the efforts of a collection of repairmen to a number of deteriorating machines in order to reduce operation costs and to mitigate the cost (and likelihood) of unexpected failures. Notwithstanding these preventive maintenance interventions are aimed at returning the machine to a so-called as-good-as-new state, unforeseeable factors may imply that maintenance interventions are not perfect and the machine is only returned to an earlier (uncertain) state of wear. The problem is modelled as a restless bandit problem and an index policy for the sequential allocation of maintenance tasks is proposed. A series of numerical experiments shows the strong performance of the proposed policy. Moreover, the methodology is of interest in the general context of dynamic resource allocation and restless bandit problems, as well as being useful in the particular imperfect maintenance model described.
Paper Structure (15 sections, 6 theorems, 53 equations, 6 figures, 5 tables)

This paper contains 15 sections, 6 theorems, 53 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. The Multi-Machine Maintenance Problem with Imperfect Interventions
  4. Mathematical Formulation
  5. Indices and Indexability Analysis
  6. Cost of immediate intervention: the $\mathfrak{B}\left(x,W\right)$-policy
  7. Cost of postponing intervention: the $G\left(x\right)$-policy
  8. W-Indices
  9. Closed-form expressions for the $W$-indices
  10. Numerical Assessment
  11. Computation of the W-Indices
  12. Suboptimality Assessment of the Index Policy
  13. Index Policy Performance in Large Systems
  14. An alternative index policy
  15. Conclusions

Key Result

Proposition 1

Quantity $\mathfrak{B}\left(x,W\right)$ is continuous and strictly increasing in $W$.

Figures (6)

  • Figure 1: Illustrating example: five machines and two technicians.
  • Figure 2: Machine's evolution under $\mathfrak{B}$-policy
  • Figure 3: Machine's evolution under G-policy
  • Figure 4: Determination of $W$-index from policies $\mathfrak{B}\left(x,W\right)$ (dashed) and $G\left(x\right)$ (solid).
  • Figure 5: Illustrating example: index policy.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 1: Indexability
  • Definition 2: Whittle Indices
  • Definition 3: Index Policy
  • Proposition 1
  • Definition 4
  • Definition 5
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Corollary 1
  • ...and 1 more