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Piecewise deterministic Markov process for condition-based imperfect maintenance models

Weikai Wang, Xian Chen

TL;DR

This paper addresses maintenance planning for systems subject to both deterministic degradation and random shocks by formulating a condition-based imperfect maintenance problem as a piecewise deterministic Markov process (PDMP). It introduces an impulse-control framework where imperfect maintenance uses a Beta-distributed improvement factor and corrective maintenance resets the system, while inspections occur at fixed intervals with delayed execution. A coating-maintenance example demonstrates that the PDMDP-based policy can outperform threshold-based strategies and highlights how the optimal policy depends on the discount factor, inspection interval, and maintenance costs. The work extends the CBM literature by coupling PDMP with history-dependent imperfect maintenance and providing a practical computational approach for the value function via the Costa–Davis impulse-control theory, with implications for improved reliability and cost efficiency in industrial systems.

Abstract

In this paper, a condition-based imperfect maintenance model based on piecewise deterministic Markov process (PDMP) is constructed. The degradation of the system includes two types: natural degradation and random shocks. The natural degradation is deterministic and can be nonlinear. The damage increment caused by a random shock follows a certain distribution, and its parameters are related to the degradation state. Maintenance methods include corrective maintenance and imperfect maintenance. Imperfect maintenance reduces the degradation degree of the system according to a random proportion. The maintenance action is delayed, and the system will suffer natural degradations and random shocks while waiting for maintenance. At each inspection time, the decision-maker needs to make a choice among planning no maintenance, imperfect maintenance and perfect maintenance, so as to minimize the total discounted cost of the system. The impulse optimal control theory of PDMP is used to determine the optimal maintenance strategy. A numerical study dealing with component coating maintenance problem is presented. Relationship with optimal threshold strategy is discussed. Sensitivity analyses on the influences of discount factor, observation interval and maintenance cost to the discounted cost and optimal actions are presented.

Piecewise deterministic Markov process for condition-based imperfect maintenance models

TL;DR

This paper addresses maintenance planning for systems subject to both deterministic degradation and random shocks by formulating a condition-based imperfect maintenance problem as a piecewise deterministic Markov process (PDMP). It introduces an impulse-control framework where imperfect maintenance uses a Beta-distributed improvement factor and corrective maintenance resets the system, while inspections occur at fixed intervals with delayed execution. A coating-maintenance example demonstrates that the PDMDP-based policy can outperform threshold-based strategies and highlights how the optimal policy depends on the discount factor, inspection interval, and maintenance costs. The work extends the CBM literature by coupling PDMP with history-dependent imperfect maintenance and providing a practical computational approach for the value function via the Costa–Davis impulse-control theory, with implications for improved reliability and cost efficiency in industrial systems.

Abstract

In this paper, a condition-based imperfect maintenance model based on piecewise deterministic Markov process (PDMP) is constructed. The degradation of the system includes two types: natural degradation and random shocks. The natural degradation is deterministic and can be nonlinear. The damage increment caused by a random shock follows a certain distribution, and its parameters are related to the degradation state. Maintenance methods include corrective maintenance and imperfect maintenance. Imperfect maintenance reduces the degradation degree of the system according to a random proportion. The maintenance action is delayed, and the system will suffer natural degradations and random shocks while waiting for maintenance. At each inspection time, the decision-maker needs to make a choice among planning no maintenance, imperfect maintenance and perfect maintenance, so as to minimize the total discounted cost of the system. The impulse optimal control theory of PDMP is used to determine the optimal maintenance strategy. A numerical study dealing with component coating maintenance problem is presented. Relationship with optimal threshold strategy is discussed. Sensitivity analyses on the influences of discount factor, observation interval and maintenance cost to the discounted cost and optimal actions are presented.
Paper Structure (29 sections, 1 theorem, 31 equations, 18 figures, 2 tables)

This paper contains 29 sections, 1 theorem, 31 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Piecewise deterministic Markov processes
  3. Mathematical background
  4. Model assumptions
  5. Degradation-maintenance model
  6. Degradation process
  7. Maintenance process
  8. Inspection and waiting process
  9. Formulation of the piecewise deterministic Markov decision process model
  10. Actions
  11. Boundaries
  12. Deterministic process
  13. Random jumps
  14. Jumps at intervention times and maintenance costs
  15. Transition measure
  16. ...and 14 more sections

Key Result

Theorem 1

Suppose Assumptions A,B,C in Costa1 Section 3.2 are satisfied. For any $x\in\bar{X}$ we define the sequence of functions $\{ W_k \}_{k\in\mathbb{N}}$ as follows: for any $x\in\bar{X}$ where $K_A$ and $K_B$ are constants defined in Costa1 Lemma 5.3, $A_{\varepsilon_1} = \{ x\in X: t^*(x) > \varepsilon_1 \}$ and $K\ge \sup_{x\in X}\eta(x)$ is a constant. where $\rho>0$ is the discount factor. For a

Figures (18)

  • Figure 2.1: An illustration of the degradation-maintenance process.
  • Figure 3.1: The relationship between natural loss of coating thickness $w$ and days.
  • Figure 3.2: A simulated PDMP path of IMM. The above four figures show the degradation state $w$ of the system, the execution times of imperfect maintenance $n$, the change of observation time $\sigma$, and the planned optimal action $d$ at each time under this path respectively.
  • Figure 3.3: The left figure is the time of random shocks and damage increment $\varpi$ of a simulation path of PDMP model, the right figure is the cumulative discounted cost $V$ of that path.
  • Figure 3.4: A PDMP simulation path of CMM. The above four figures show the degradation state $w$, planned action $d$, the time of random shocks, damage increment $\varpi$, and cumulative discounted cost $v$ respectively.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Theorem 1