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Abstract

This study presents a novel optimization framework for condition-based maintenance using a discrete-time absorbing Markov chain to balance maintenance costs and system availability. The framework employs a multi-threshold policy, in which degradation thresholds dynamically determine non-periodic inspection intervals and maintenance actions. The degradation process follows a continuous-time Gamma process. Preventive maintenance is triggered when degradation exceeds a preventive threshold but remains below the failure limit, whereas corrective maintenance occurs at the failure limit. The optimization ensures cost efficiency while maintaining availability above a predefined target. Key performance metrics, including degradation-dependent inspection costs, maintenance cycle length, and quality loss, are analytically derived and evaluated. Importantly, the total cost per unit time explicitly integrates a degradation-dependent quality-loss component alongside inspection, maintenance, and downtime costs, highlighting quality loss as a key driver of decision-making trade-offs. The framework jointly optimizes the number of degradation phases, inspection intervals, and degradation limits to minimize the total cost per unit time subject to an availability constraint. The mixed-integer nonlinear programming problem is solved using a two-stage approach: a greedy search to select the number of degradation phases, followed by a constrained nonlinear program to refine the inspection intervals and thresholds. Numerical results demonstrate cost-efficient policies with high availability, providing practical insights for decision-makers. The work presents a robust decision-support tool for managing complex degrading systems.

Keywords

condition-based maintenance inspection control limit policy Markov chains quality optimization

Introduction

Effective maintenance management is essential for achieving operational reliability and availability across various industrial sectors. The increasing focus on maintenance planning and control is fueled by the continuous automation of production processes and the pressures of a highly competitive market. Maintenance-related costs account for 15%–70% of total production expenses in manufacturing, depending on industry type. Traditional maintenance strategies, such as reactive or time-based maintenance (TBM), rely on predetermined schedules based on age or usage.¹ However, these approaches often result in unnecessary interventions, underutilization of the remaining useful life, or unexpected failures, as they fail to fully leverage information about the asset’s degradation state.² Recent advancements in real-time monitoring technologies have shifted toward condition-based maintenance (CBM), which dynamically tailors maintenance decisions based on real-time degradation signals of assets, for example, temperature, vibration, or pressure.^3,4 Unlike conventional maintenance strategies, CBM leverages operational data to optimize maintenance timing, balancing preventive measures with the risk of failure. This paradigm has significantly reduced maintenance costs and improved system availability and lifecycle efficiency, as evidenced by several studies.^5–7 CBM has found increasing application in systems operating under dynamic environments with stochastic degradation processes, as highlighted by Sun et al.⁸ and Shen et al.⁹

Modeling degradation processes is a pivotal aspect of CBM policy development. Continuous-time degradation models, such as the Gamma process, have been widely employed due to their flexibility in capturing monotonic wear and degradation.^10,11 However, integrating degradation modeling with optimization frameworks remains computationally intensive, particularly for systems that require multi-threshold maintenance strategies. Traditional approaches often fail to adequately account for the dynamic interplay among inspection intervals, maintenance thresholds, and system reliability.¹² The integration of periodic inspections into CBM policies plays a crucial role in ensuring effective maintenance decisions while balancing operational costs and system degradation.¹³

This study introduces a novel CBM optimization policy that leverages a discrete-time absorbing Markov chain model to efficiently manage trade-offs between maintenance costs and system availability. Our framework incorporates a multi-threshold policy that dynamically adjusts inspection intervals and maintenance decisions based on the system’s degradation state. Non-periodic inspections are dynamically scheduled based on the system’s degradation state, providing timely insights into its trajectory. This degradation-dependent scheduling ensures that preventive and corrective maintenance actions are optimally deployed.

The proposed framework is formulated as a mixed-integernonlinear programming (MINLP) model and solved using a two-stage optimization approach. First, a greedy search determines the optimal number of degradation phases, followed by a constrained nonlinear programming algorithm that refines the inspection intervals and maintenance thresholds. Performance metrics, including maintenance cycle length, inspection costs, and degradation-dependent quality loss, are derived analytically, ensuring a comprehensive evaluation of the proposed policy under varying operational conditions.

Numerical experiments demonstrate the efficacy and robustness of the proposed approach in minimizing total maintenance costs while maintaining high system availability. Our method provides a robust decision-support tool for managing complex degrading systems by optimizing the interaction between inspection frequency and degradation thresholds. This work extends the state-of-the-art CBM optimization by addressing limitations in current approaches and highlighting the benefits of integrating multi-threshold policies within Markov-based modeling frameworks.

The remainder of this paper is organized as follows: The “Literature review” section examines the state-of-the-art in CBM and related optimization techniques. Next, the “Contributions” section outlines the targeted contributions of this study. This is followed by the “Notations and assumptions” section, which defines the notation and assumptions, and then introduces the problem formulation in the “Model analysis” section. Subsequently, the “Problem description” section presents the mathematical model, analysis, and optimization methodology. Numerical experiments and results are provided in the “Numerical results” section, along with practical guidelines to support implementation of the proposed methodology. Finally, the Conclusion Section summarizes key findings and identifies directions for future research.

Literature review

Condition-based maintenance (CBM) has emerged as an essential strategy in industrial maintenance, leveraging real-time data to optimize maintenance schedules and reduce costs.¹⁴ Unlike traditional reactive or time-based maintenance strategies, CBM focuses on monitoring the actual condition of assets to predict failures and schedule maintenance activities at the most opportune moments, reducing downtime, maintenance costs, and operational disruptions. CBM involves complex processes, including data collection, analysis, and decision-making, that must be systematically addressed to ensure cost-effectiveness. Recent advances in Big Data analytics and IoT have enabled real-time decision-making based on abundant data from various sources.¹⁵ The field of CBM encompasses four main research areas: theoretical foundations, implementation strategies, operational aspects related to inspection and replacement, and prognosis.^16,17 CBM applications span various domains, incorporating data mining, artificial intelligence, and statistics to create maintenance-smart systems. While CBM offers significant advantages, its practical implementation requires careful consideration of technical and financial factors.^18,19

Degradation modeling is a cornerstone of CBM, providing the tools to assess and predict asset condition over time. Degradation modeling is the process of anticipating how a system or product will behave over time as it ages or degrades. This understanding is critical in various applications, including product design, production, and maintenance. Accurate degradation models enable the estimation of remaining useful life (RUL), a critical factor in making timely and effective maintenance decisions.²⁰ Among the stochastic processes used for modeling degradation, the Wiener process, gamma process, and random coefficient models stand out for their applicability to different types of degradation phenomena.²¹ The Wiener process is particularly effective for modeling fatigue crack growth, as it accounts for random effects and measurement errors.^22,23 Its utility in modeling mission-oriented systems with imperfect preventive maintenance and Bayesian parameter estimation for remaining useful life (RUL) prediction is well demonstrated.^24,25 Gamma processes are more suitable for monotone-increasing irreversible degradation patterns, such as material fatigue or corrosion, due to their ability to capture progressive, non-decreasing degradation, aligning with real-world maintenance scenarios.²⁶ They are commonly applied in modeling systems subject to periodic inspection policies²⁷ and recently extended to incorporate imperfect preventive actions and statistical inference.^28,29 Gamma processes with random effects are especially valuable in CBM, as they can account for heterogeneous initial degradation levels and correlations between initial degradation and degradation rates.²¹ Additionally, the optimal design of degradation tests using gamma processes ensures precise lifetime predictions by determining appropriate sample sizes, inspection frequencies, and measurement counts under budget constraints.³⁰ Random coefficient models further enhance CBM strategies by explicitly accounting for the inherent heterogeneity in many real-world systems. Unlike traditional models that often assume homogeneity in degradation behavior across components, random coefficient models capture variability in initial degradation levels, degradation rates, and responses to environmental factors. This adaptability is crucial for systems composed of multiple units operating under varying conditions, where a one-size-fits-all approach may lead to suboptimal maintenance decisions, as demonstrated in Zhou et al.³¹ and Tang et al.,³² where dynamic monitoring intervals and control limits are incorporated to reflect variations in component-specific degradation rates.

A control limit policy is a key concept in condition-based maintenance (CBM), where maintenance decisions are based on the condition of the monitored equipment. This policy defines a threshold for replacement or maintenance actions to minimize the long-term average maintenance costs.³³ Dynamic control-limit policies, such as the inflow-dependent approach proposed by Qian and Wu³⁴ for hydroelectric generating units, adapt to time-varying conditions, underscoring the importance of accounting for external environmental factors. For multi-component systems, Nguyen et al.³⁵ proposed a predictive CBM approach that combines degradation information with importance measures to identify the most critical components. Their framework integrates predictive models of future degradation states with system-level importance indices, allowing maintenance actions to be prioritized based on both the likelihood of failure and the impact on overall system performance. Zhu et al.³⁶ introduced joint maintenance intervals to reduce high setup costs, demonstrating significant cost savings compared to failure- or age-based policies. Liu et al.³⁷ explored age- and state-dependent operating expenses, showing that optimal maintenance thresholds decrease with system age, thereby integrating economic and environmental considerations. Golmakani and Fattahipour³⁸ further emphasized the simultaneous optimization of control limits and inspection intervals to balance preventive and failure replacement costs effectively. Building on this, Golmakani and Pouresmaeeli³⁹ developed a framework to optimize replacement thresholds and inspection intervals for systems with non-decreasing failure-replacement costs and costly inspections. Their approach addresses practical scenarios where failure costs depend on the system’s degradation state or the time of failure, thereby demonstrating the value of detailed cost modeling in CBM. Advanced sensor and information technology, as stated by Zhu et al.,³⁶ facilitates the implementation of CBM by enabling remote data collection. Innovative mathematical frameworks, such as the Markov decision process-based approach by Havinga and de Jonge⁴⁰ and the Piecewise-Deterministic Markov Process (PDMP)-based model by Wang and Chen,⁴¹ highlight the potential of detailed condition monitoring in refining control-limit policies. However, the effectiveness of control-limit policies depends critically on the granularity of condition information, a point also highlighted in optimal replacement models for partially observable systems with dependent failure modes,⁴² as well as on the stability of degradation processes and the relative costs of corrective maintenance.⁴³

Building on single-control-limit policies, multi-threshold control policies extend the condition-based maintenance (CBM) concept by incorporating multiple degradation limits to refine maintenance decisions. These policies benefit systems in which degradation progresses through multiple, identifiable phases.⁴⁴ On the other hand, they developed a hybrid policy combining corrective, periodic, and condition-based maintenance. They implemented two thresholds: an opportunistic threshold to align maintenance with other planned interventions and an intervention threshold to prevent imminent failures. This policy minimized expected maintenance costs and downtime, showcasing the benefits of integrating multiple maintenance strategies. Zheng et al.⁴⁵ proposed a dynamic threshold policy using the proportional hazards model, incorporating both age and covariate information from condition monitoring. Their approach outperforms other CBM policies in minimizing long-run average costs. Peng et al.⁴⁶ examined CBM for systems with non-homogeneous degradation processes, considering both constant and non-periodic inspection intervals. They show that optimal preventive replacement thresholds decrease monotonically with system age. Al Hanbali et al.¹¹ proposed a two-threshold control limit policy for a corrosion-prone pressure vessel in the process industry. Their approach introduced preventive maintenance and switching limits, demonstrating annual cost savings of up to 12.3% compared to single-threshold policies. Recently, Wang et al.⁴⁷ introduced a prognosis-driven, multi-threshold inspection and replacement model for high-speed railway bearings, incorporating safety coefficients, and residual lifetime metrics to enhance the robustness of the decision-making process.

Integrated condition-based maintenance (CBM) modeling has significantly addressed system degradation, incorporated diverse optimization strategies, and refined maintenance policies to enhance cost efficiency. These advancements have led to the development of multi-faceted frameworks that integrate dynamic inspection intervals, state-dependent control limits, and non-homogeneous degradation models. Castanier et al.⁴⁸ investigated a maintenance policy for repairable systems, focusing on state-dependent costs and time under non-periodic inspections using semi-regenerative renewal theory. Their approach highlighted the potential for cost-effective CBM strategies through dynamic control limits but remained constrained to single-objective optimization. Xiang et al.⁴⁹ addressed the optimization of burn-in and CBM policies for heterogeneous populations, integrating imperfect maintenance and bi-objective frameworks to balance system reliability and cost. Poppe et al.⁴⁴ proposed a hybrid CBM policy integrating two thresholds: an opportunistic threshold, used to align maintenance with other planned interventions, and an “intervention” threshold, which prevents imminent failures. Their approach demonstrated the benefits of combining corrective, periodic, and condition-based maintenance strategies, achieving cost savings and reduced downtime through optimized intervention scheduling. Hao et al.⁵⁰ studied a CBM strategy under imperfect inspections for continuous degradation. They demonstrate that relying on imperfect readings can outperform perfect inspections in specific settings, and they propose a two-stage policy that combines low-cost imperfect checks with perfect checks near decision points to minimize the long-run cost rate. The study also explicitly highlights the roles of false positives and false negatives in the formulation of renewal costs. Zhang et al.⁵¹ presented a hierarchical maintenance framework that integrates partial and full inspections, leveraging degradation-state information to optimize maintenance costs and system reliability. Warranty-driven settings have further motivated joint optimization of inspection and condition-based maintenance for deteriorating products under extended warranty agreements. Zheng et al.⁵² developed a joint inspection–CBM policy in which deterioration follows a non-stationary gamma process and, at each inspection, the decision maker selects whether to repair and schedules the next inspection. Their objective is to minimize warranty servicing cost, solved via stochastic dynamic programming, with an optimal control-limit structure and an efficient backward-induction procedure. In a related warranty context, Darghouth⁵³ proposed an integrated two-dimensional warranty framework for second-hand equipment that explicitly accounts for past life through prior age and usage. The framework combines a reliability threshold-driven CBM policy with decisions on preventive maintenance efficiency and upgrade intensity to minimize the dealer’s expected warranty servicing cost. Wang et al.⁵⁴ proposed a condition-based framework that integrates inspection interval adjustments and component reallocation criteria, explicitly addressing the unbalanced degradation rates of two-component interchangeable series systems, such as railway rails on small radius curves.

Beyond single-process degradation, Cao et al.⁵⁵ modeled multi-state deterioration together with random shocks and the dependence between soft and hard failures. They coupled a semi-Markov framework for discrete deterioration with a compound Poisson shock process, designed a condition-based inspection policy that reduces inspection intervals in degraded states, and optimized a bi-objective problem balancing availability and cost using Monte Carlo search. Their results show that co-modeling deterioration and shocks avoids overestimating policy performance and yields more realistic availability–cost trade-offs than treating the processes separately. In the same spirit, Zheng et al.⁵⁶ proposed a multilevel preventive replacement policy for mission-oriented systems subject to internal deterioration and external shocks, where missions occur as a sequence of random types and the degradation state is observed at mission completion through condition monitoring. The policy relies on mission-type-dependent replacement thresholds, optimized to minimize the long-run expected average cost while accounting for corrective replacement costs and mission-failure penalties. Yao et al.⁵⁷ explored the maintenance modeling of systems governed by a non-homogeneous gamma degradation process. They considered non-periodic inspections and demonstrated the superiority of this approach in minimizing long-term costs compared to traditional periodic inspection models. Khatab et al.¹³ developed a condition-based selective maintenance model for multi-component systems operating over successive missions with scheduled breaks. Components degrade stochastically and may undergo imperfect maintenance or replacement during breaks, subject to time and reliability constraints. Their framework incorporates inspection-based failure detection, downtime penalties, and cost-effectiveness trade-offs, and determines the optimal subset of maintenance actions to minimize total expected mission cost. In a related work, Zheng⁵⁸ studied condition-based replacement for mission-oriented systems facing random mission types evolving as a Markov chain, where condition monitoring is performed at mission switches to support replacement decisions. They formulated the problem as a Markov decision process, considered cases in which the next mission type is known or unknown, and showed that the optimal decisions have a control-limit structure, leading to practical multilevel threshold policies. Recently, Hu and Sun⁵⁹ proposed a dynamic inspection and replacement policy for systems subject to continuous degradation and periodic shocks. Degradation is modeled as a drifted Wiener process, and shock increments are modeled as gamma-distributed. The joint inspection timing and replacement decisions are optimized via a Markov decision process without prespecifying the inspection structure. The framework is extended to multi-component systems with dependent degradation using multivariate Wiener and gamma shock models, and scalability is addressed through approximate dynamic programming supported by Monte Carlo simulation.

Despite these advances, the vast majority of CBM studies adopt single- or two-threshold policies and optimize either cost or reliability in isolation, with little attention to degradation-dependent inspection spacing or quality-loss effects.^48,60 Moreover, the few works that tackle non-periodic inspections typically rely on heavy numerical integration or value-iteration schemes,^61,62 which hinder analytical insight and rapid what-if analysis. These unresolved issues motivate the multi-threshold, degradation-aware framework developed in the remainder of this paper.

A comparison of related works is provided in Table 2. This table highlights the methodologies, decision variables, and objectives addressed in previous studies and outlines the key differences and contributions of our framework.

Targeted contributions

This work advances CBM research on degrading systems in four complementary directions:

(i) Multi-threshold framework with degradation-dependent inspections: Unlike classical single- or two-limit CBM policies, the proposed framework allows both the inspection spacing and the intervention limits to evolve with the current degradation state, capturing the acceleration of failure risk near the end of life.

(ii) Recursive-form performance algebra via an absorbing Markov chain (AMC): By mapping the gamma-driven degradation path onto an upper-triangular block AMC, we obtain explicit formulas for expected cycle length, cost components, downtime, and steady-state availability, eliminating the nested integrals or value-iteration schemes still required by recent CBM models (e.g., Zhao et al.⁶⁰ and Yang et al.⁶¹).

(iii) Integrated cost-availability optimization posed as a tractable MINLP: The explicit derived formulas are embedded in a mixed-integer nonlinear program (MINLP) that minimizes unit-time cost while meeting a user-defined availability requirement. A two-stage greedy–COBYLA algorithm exploits the problem structure: the outer search fixes the number of phases, and the inner step jointly tunes inspection intervals and degradation limits to near-global optimality within industrial runtimes.

(iv) Incorporation of quality-loss economics and wear-dependent inspection effort: A Taguchi-style quality-loss curve and inspection times that grow with wear are integrated analytically. Such cost drivers are typically absent from conventional CBM studies (e.g., de Jonge and Scarf⁵ and Poppe et al.⁴⁴). Our case study demonstrates that they can shift the optimum and reduce total cost by up to 9%.

Assumptions and notations

Below are the main assumptions of the considered framework:

− The degradation level of the system over time is monitored non-periodically at discrete times $T_{k}$ .

− The inspections are perfect and reveal the current system’s degradation, for example, Castanier et al.⁴⁸ and Liao et al.⁶⁵

− The degradation process of the system follows a continuous-time Gamma process.

− Preventive and corrective maintenance actions return the system to an as-good-as-new state, for example, Zhou et al.⁶⁶ and Elwany et al.⁶⁷

− No degradation occurs for the system if it is under inspection, preventive, or corrective maintenance.

− When the system’s degradation exceeds a defined limit, the inspections are scheduled more frequently to reduce the chance of failure occurrence, see, for example, Dieulle et al.⁶⁸ and Barker and Newby.⁶⁹

− When the system’s degradation is found at an inspection to be above a preventive limit, a preventive maintenance action is undertaken.

− The inspection time is a function of the system’s degradation.

− The quality loss cost is a function of the system’s degradation. Typically, it increases as the system degrades.

− The probability distribution of degradation increments remains consistent across inspection intervals and is solely influenced by the phase-dependent inspection time and degradation limits.

− The degradation increments over successive intervals are independent, and the process has independent and stationary increments, consistent with the Gamma process properties.

− The durations of inspection, preventive maintenance, and corrective maintenance are deterministic and known.

The notations outlined below are utilized throughout this paper (Table 1):

Table 1.

Nomenclature.

AMC	Discrete-time finite state-space absorbing Markov chain modeling the degradation process.
$α$	Shape parameter of Gamma distribution.
$β$	Rate parameter of Gamma distribution.
$L$	System’s failure limit.
$y_{m}$	Degradation limit at which time between inspections is reduced, $m = 1, \dots, n$ .
$δ_{y}$	Minimum increment between successive degradation limits.
$τ_{m}$	Time between inspections in phase $m$ , i.e., degradation lies in $[y_{m - 1}, y_{m})$ .
$δ_{τ}$	Minimum decrement between successive inspection intervals.
$Δ X_{m}$	Degradation accumulated in an inspection interval in phase $m$ .
$c_{C}$	Corrective maintenance action cost.
$c_{I}$	Inspection cost.
$c_{P}$	Preventive maintenance action cost.
$c_{D}$	Downtime cost per unit of time.
$c_{Q}$	Quality loss cost per unit.
$t_{C}$	Corrective maintenance time.
$t_{I} (x)$	Inspection time as a function of degradation $x$ .
$t_{P}$	Preventive maintenance time.
$QCU (x)$	Quality loss per year as a function of degradation $x$ .
$ϵ$	Discretization constant of the degradation process.
$i_{m}$	Discrete degradation limit corresponding to $y_{m}$ .
$f$	Discrete value of failure limit.
$pr$	Preventive maintenance state.
$co$	Corrective maintenance state.
$p_{k} (m)$	Transition probability from state $l$ in phase $m$ to $l + k$ .
$p_{l}^{pr} (m)$	Transition probability to preventive maintenance absorbing state.
$p_{l}^{co} (m)$	Transition probability to corrective maintenance absorbing state.
$A_{mk}$	Transition probability matrix from phase $m$ to phase $k$ .
$v_{j}$	Expected number of visits to state $j$ before absorption.
$R_{m}$	$m$ th block of the AMC fundamental matrix.
$r^{m}$	First row of $R_{m}$ .
$e^{m}$	Ones-vector of size $i_{m} - i_{m - 1}$ .
$e_{l}^{m}$	Canonical vector of size $i_{m} - i_{m - 1}$ .
$CL$	Expected maintenance cycle length.
$IC$	Expected total inspection cost per cycle.
$IT$	Expected total inspection time per cycle.
$p_{C}$	Corrective maintenance probability in a cycle.
$MC$	Expected maintenance cost per cycle.
$MT$	Expected maintenance time per cycle.
$DC$	Expected downtime cost per cycle.
$DT$	Expected downtime duration per cycle.
$QC$	Expected quality loss cost per cycle.
$TCU$	Expected total cost per time unit.
$Av$	Limiting system’s availability.
$SCV$	Squared coefficient of variation.

Table 2.

Comparison of relevant CBM optimization approaches in the literature.

Reference	Limit	Inspection policy	Maintenance strategy	Component	Key feature	Methodology	Decision Variables/Action	Key performance indicators (KPIs)	Optimization objective	Solution approach
[48]	Multiple	Non-periodic	Imperfect, AGAN	Single	State-dependent cost and time	Semi-regenerative, RT	PM limits, ITs	CPT, Availability	Single	Volterra equations
[49]	Single	Periodic	Imperfect, AGAN	Multiple	Burn-in effect	RT	PM limit and IT	CPT, Availability	Bi-objective	Infinite sum of integrations
[44]	Two	Periodic	AGAN	Single	Hybride policy with PM, CM and CBM	RT	Opportunistic and intervention limits	CPT or MDT	Single	Closed-from
[61]	Single	Non-periodic	AGAN	Single	Soft and hard failures	RT	PM limit, ITs	CPT	Single	Bounded sum of double and triple integrations
[60]	Two	Non-periodic	AGAN	Single	Spare part ordering	RT	Ordering limit, PM limit, IT	CPT	Single	Infinite sum of integrations
[62]	Single	Periodic	AGAN	Single	Planning time	Absorbing MC, RT	PM limit	CPT	Single	Matrix inverse
[11]	Two	Non-periodic	AGAN	Single	Switching limit to frequent inspections	RT	switching limit, PM limit, and ITs	CPT, Availability	Single	Closed-form
[47]	Two	Periodic	AGAN	Single	Safety prognosis, two-phase process	RT	PM limits	CPT	Single	Infinite sum of integrations
[51]	Single	Periodic	AGAN, imperfect	Single	Partial $and$ full inspections	RT	PM limit, IT	CPT	Single	Sums and products of probabilities
[57]	—	Non-periodic	AGAN	Single	Non homogeneous process	Semi-MDP	IT, Replace	TC	Single	Value iteration algorithm
[63]	Single	Periodic	AGAN	Single	Inspection errors	RT	PM limit, IT	CPT	Single	Infinite sum of integrations
[64]	—	Periodic	AGAN	Multiple	Load reallocation	MDP	Replace, Reallocate	TC	Single	Value iteration algorithm
This study	Multiple	Non-periodic	AGAN	Single	Degradation-dependent inspections, quality loss	Absorbing MC, RT	PM limits, ITs	CPT, Availability	Single with constraint on Availablity	Explicit-form

AGAN: as-good-as-new; CPT: cost per unit of time; TC: total cost in planning horizon; MC: Markov chain; MDT: maintenance downtime time; RT: renewal theory; MDP: Markov decision process; IT: inspection interval.

Problem description

The Gamma process is widely used in CBM, as it is suitable for modeling monotone degradation, such as crack growth, fatigue, and creep.⁷⁰ The system suffers from continuous degradation that accumulates over time. The accumulated degradation per unit of time follows the Gamma distribution with a probability density function expressed as follows:

f_{α, β} (x) = \frac{1}{Γ (α)} β^{α} x^{α - 1} e^{- β x} I_{x \geq 0},

(1)

where $α$ is the shape parameter, and $β$ is the rate parameter, and $I_{x \geq 0}$ is an indicator function $I_{x \geq 0}$ =1 if $x \geq 0$ and 0, otherwise. Furthermore, the amounts of degradation accumulated in two disjoint intervals are independent. The degradation process is modeled as a Gamma process. At timezero, the system’s degradation level is 0 (as-good-as-new), and it fails if its degradation level reaches the failure limit $L$ . In this case, a corrective maintenance action is performed.

The inspection reveals the system’s degradation state. Inspections are conducted at varying intervals depending on the current degradation level. We define $n$ degradation limits $0 = y_{0} \leq y_{1} \leq y_{2} \leq \dots \leq y_{n} < L$ , where $y_{n}$ denotes the preventive maintenance threshold. Inspections are initially scheduled every $τ_{1}$ units of time as long as the degradation remains in $[y_{0}, y_{1}]$ . Whenever the degradation enters $[y_{m - 1}, y_{m})$ , $m = 1, \dots, n$ , inspections are performed every $τ_{m}$ units of time with $τ_{m} \leq τ_{m - 1}$ . This reduction continues until the degradation reaches $y_{n}$ or $L$ , whichever occurs first. We refer to the period during which the degradation lies in $[y_{m - 1}, y_{m})$ as phase $m$ .

According to the problem at hand, the decision variables are the number of phases $n$ , the degradation limits ${y_{m}}_{m = 1}^{n}$ , and the inspection intervals ${τ_{m}}_{m = 1}^{n}$ . Our objective is to determine these variables to minimize the long-run expected total cost per unit time, denoted BY $TCU$ , subject to a minimum availability requirement. The formal optimization problem is stated in the “Performance optimization” section.

The random variable $Δ X_{m}$ represents the amount of degradation accumulated during an inspection interval in phase $m$ . Consequently, $Δ X_{m}$ follows a Gamma distribution with a shape parameter $α τ_{m}$ and a rate parameter $β$ .

The beginning of an inspection interval $k$ , where $k = 0, 1, \dots$ , is referred to as the $k$ th inspection epoch and denoted by $T_{k}$ . The degradation value, $X_{k + 1}$ , at the inspection epoch $T_{k + 1}$ , $k = 0, 1, \dots$ , is related to the degradation $X_{k}$ at the previous epoch $T_{k}$ as follows:

X_{k + 1} = X_{k} + \sum_{m = 1}^{n} Δ X_{m} I_{{y_{m - 1} \leq X_{k} < y_{m}}}

where $X_{0} = 0$ , and $I_{{y_{m - 1} \leq X_{k} < y_{m}}}$ is the indicator function that is equal to one if $y_{m - 1} \leq X_{k} < y_{m}$ . We note that the preventive limit $y_{n}$ is the degradation’s upper limit of the last phase $n$ and is a decision variable that does not vary with $k$ .

The system’s maintenance policy is as follows: if an inspection occurs at time $T_{k}$ and the system is operational, the inspection will be conducted, and a preventive maintenance decision will be made based on the degradation level at that time. The inspection provides an accurate assessment of the system’s degradation state. A preventive threshold limit, $y_{n}$ , is defined for preventive maintenance. At inspection epoch $T_{k}$ , if the degradation level is greater than or equal to $y_{n}$ but strictly less than $L$ , the system undergoes preventive maintenance. If the system fails by reaching the failure limit $L$ before the next scheduled inspection, that is, within the interval $[T_{k - 1}, T_{k})$ , it will be correctively replaced at the subsequent inspection, $T_{k}$ . In this scenario, when the degradation reaches the failure limit $L$ at some time $t \in [T_{k - 1}, T_{k})$ , the system is considered failed during the downtime interval $[t, T_{k})$ . Both preventive and corrective maintenance restore the system to an as-good-as-new state. The mean preventive maintenance time is denoted by $t_{P}$ , and the mean corrective maintenance time (following a failure) is represented as $t_{C}$ ( $t_{C} \geq t_{P}$ ). The mean time to complete an inspection at time $T_{k}$ depends on the system’s degradation state and is denoted $t_{I} (X_{k})$ .

The preventive maintenance cost is denoted by $c_{P}$ , while the corrective maintenance cost is $c_{C}$ ( $c_{C} > c_{P}$ ). The inspection cost remains fixed at $c_{I}$ .

The preventive maintenance requires an inspection, and its cost inherently includes the inspection cost, that is, $c_{P} > c_{I}$ . Similarly, the corrective maintenance cost includes the inspection cost needed to reveal the system’s failure. Additionally, during maintenance, a fraction of the last inspection interval contributes to system downtime. The downtime cost per unit time is denoted by $c_{D}$ . Finally, the system degradation level, $x$ , incurs an annual quality loss, calculated as follows^71,72:

QCU (x) = P_{R} c_{Q} (p_{0} + η (1 - \exp (- λ_{p} x^{γ_{q}}))),

(2)

where $P_{R}$ is the annual production rate, $c_{Q}$ is the loss cost per unit, $p_{0}$ is the defect rate for $x = 0$ , and $η$ , $λ_{p}$ , and $γ_{q}$ are parameters of the defect rate growth as a function of the degradation.

Model analysis

The continuous, non-decreasing degradation process is discretized by introducing $ϵ$ , a small degradation increment. This discretization transforms the process into a discrete-state, non-decreasing degradation process with state $I_{l} = [l ϵ, (l + 1) ϵ [$ , where $l$ is a nonnegative integer. For simplicity, when referring to the process being in state $I_{l}$ , we assume the degradation value is uniformly distributed within the interval, with a mean value of $(l + 1 / 2) ϵ$ .^62,73 Furthermore, the state $I_{l}$ will be referred to by its index $l$ .

When degradation is in phase $m$ and due to the properties of the gamma process, the probability of a one-step transition during the inspection interval $τ_{m}$ , from state $l$ to state $l + k$ is independent of $l$ . More precisely, the one-step transition probability is given $p_{k} (m) = P (k ϵ \leq Δ X_{m} < (k + 1) ϵ)$ , where $Δ X_{m}$ represents the amount of degradation accumulated during the inspection interval $τ_{m}$ in phase $m$ . Assuming a Gamma process, $Δ X_{m}$ follows a Gamma distribution with parameters $(α τ_{m}, β)$ .

In the subsequent analysis, the values of $y_{1}, \dots, y_{n}, L$ are assumed to be integer multiples of $ϵ$ , that is, there exist positive integers $i_{1}, \dots, i_{n}, f$ such that $y_{1} = i_{1} ϵ, \dots, y_{n} = i_{n} ϵ$ , and $L = f ϵ$ . Due to the following ordering of the limits $y_{1} \leq \dots \leq y_{n} < L$ , the corresponding discrete values are related as such $i_{1} \leq \dots \leq i_{n} < f$ . For the model implementation, the discrete value $i_{m}$ , $m = 1, \dots, n$ , can be selected as the smallest integer greater than or equal to $y_{m} / ϵ$ . The value of $ϵ$ can be arbitrarily small to minimize the error induced by this assumption.

Based on the system’s state $l$ , representing the degradation level at inspection epoch $T_{k}$ , one of the following decisions is made:

If $i_{n} \leq l \leq f - 1$ , the system undergoes preventive maintenance and is replaced preventively.

If $l \geq f$ , system failure has occurred, and the system is replaced correctively.

If $l \leq i_{n} - 1$ , no maintenance action is taken; however, the next inspection is scheduled after $τ_{m}$ units of time for states $i_{m - 1} \leq l \leq i_{m} - 1$ , for $m = 1, \dots, n$ .

To simplify the state space of the Markov chain, all preventive maintenance states, that is, states $l$ such that $l \in i_{n}, \dots, f - 1$ , are grouped into a single state denoted as $pr$ . Similarly, all corrective maintenance states with values $l \geq f$ are grouped into a single state denoted $co$ . Thus, the discrete-time Markov chain modeling the system’s degradation process has the following finite state-space $0, 1, \dots, i_{n} - 1, pr, co$ .

The focus is on the degradation process within a maintenance cycle that begins in an as-good-as-new state, that is, $X (0) = 0$ , and ends with a replacement due to either preventive or corrective maintenance, whichever occurs first. Accordingly, the states $pr$ and $co$ are modeled as absorbing states, while the remaining states $0, 1, \dots, i_{n} - 1$ are transient.

Figure 1 illustrates the transition diagram of the discrete-time finite state-space absorbing Markov chain, referred to as (AMC). The transition probabilities from state $l$ to state $l + n$ for $l \in i_{m - 1}, \dots, i_{m} - 1$ , representing the degradation process in phase $m$ , are defined as:

P_{l, l + k} = {\begin{matrix} p_{k} (m), & 0 \leq k \leq i_{n} - l, \\ p_{l}^{pr} (m), & i_{n} - l \leq k < f - l, \\ p_{l}^{co} (m), & k \geq f - l . \end{matrix}

where

\begin{matrix} \begin{matrix} p_{k} (m) = P (k ϵ \leq Δ X_{m} < (k + 1) ϵ), \\ p_{l}^{pr} (m) = P ((i_{n} - l) ϵ \leq Δ X_{m} < (f - l) ϵ), \\ p_{l}^{co} (m) = P (Δ X_{m} \geq (f - l) ϵ) . \end{matrix} \end{matrix}

(3)

and $l \in {i_{m - 1}, \dots, i_{m} - 1}$ , $Δ X_{m}$ follows a Gamma distribution with parameters $(α τ_{m}, β)$ , and $ϵ > 0$ . Based on $P_{l, l + k}$ , it is evident that the transition probabilities between transient states, as well as from transient states to the absorbing states, depend on the phase $m$ to which state $l$ belongs.

Figure 1.

Transition diagram of the discrete-time finite state-space Markov chain (AMC) modeling the system’s degradation process.

AMC transition and fundamental matrices

In this section, we analyze the transition probability matrix of the absorbing Markov chain modeling the system’s degradation over time. Let $P = [P_{ij}]$ denote the transition probability matrix of the AMC, with states ordered lexicographically as ${0, 1, \dots, i_{n} - 1, pr, co}$ . The matrix $P$ can be partitioned into the following canonical form:

P = (\begin{matrix} A & B \\ 0 & I_{2} \end{matrix})

where, $A$ is an upper-triangular $i_{n}$ -by- $i_{n}$ matrix representing the transition probabilities between the transient states, that is, ${0, 1, \dots, i_{n} - 1}$ , $B$ is a $i_{n}$ -by- $2$ matrix representing the transition probabilities from transient states to absorbing states, $0$ is a zero matrix of size $2$ -by- $i_{n}$ , and $I_{2}$ is theidentity matrix of size 2-by-2. Given the multi-phase structure of $AMC$ , the matrix $A$ , an upper-triangular block matrix, and the matrix $B$ have the following form:

A = (\begin{matrix} A_{11} & A_{12} & \dots & A_{1 n} \\ 0 & A_{22} & \dots & A_{2 n} \\ ⋮ & ⋱ & ⋱ & ⋮ \\ 0 & \dots & 0 & A_{nn} \end{matrix}), B = (\begin{matrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ ⋮ & ⋮ \\ b_{n 1} & b_{n 2} \end{matrix})

$A_{mm}$ is a square matrix of size $(i_{m} - i_{m - 1}, i_{m} - i_{m - 1})$ representing the transitions between the states in phase $m$ , an upper-triangular square Toeplitz matrix of entry equal to $p_{j - i} (m)$ for $i, j \in {i_{m - 1}, \dots, i_{m} - 1}$ and $j \geq i$ , $A_{mk}, k \geq m + 1$ , is a matrix of size $(i_{m} - i_{m - 1}, i_{k} - i_{k - 1})$ representing the transitions from phase $m$ to phase $k$ with $(i, j)$ -entry equal to $p_{j - i} (m)$ for $i \in {i_{m - 1}, \dots, i_{m} - 1}, j \in {i_{k - 1}, \dots, i_{k} - 1}$ , $b_{l 1} = p_{l}^{pr} (m)$ for $l \in {i_{m - 1}, \dots, i_{m} - 1}$ , and $b_{l 2} = p_{l}^{co} (m)$ for $l \in {i_{m - 1}, \dots, i_{m} - 1}$ .

The fundamental matrix of AMC is defined as $(I - A)^{- 1}$ , where $I$ is an identity matrix of the same size as $A$ . Given that an absorbing Markov chain starts initially in state $i$ , the $(i, j)$ -entry of the fundamental matrix represents the expected number of times state $j$ is visited before the Markov chain is absorbed,⁷⁴ Th. 3.2.4.

In our case, AMC initially starts in state $0$ , the as-good-as-new state. Therefore, the first row of the fundamental matrix is of primary interest. Let $v_{j}$ denote the $j$ th-entry of the first row of the fundamental matrix for $j = 0, \dots, i_{n} - 1$ .

In the following, we derive a recursive-form expression for the first row of the fundamental matrix, $(I - A)^{- 1}$ . Leveraging the upper-triangular block structure of $I - A$ , the first row of blocks of the fundamental matrix, denoted as $(R_{1}, \dots, R_{n})$ , can be shown to satisfy the following recursion:

R_{m} = (\sum_{l = 1}^{m - 1} R_{l} A_{lm}) (I - A_{mm})^{- 1}, m = 2, \dots, n

(4)

where, $R_{1} = (I - A_{11})^{- 1}$ . The matrix $(I - A_{mm})^{- 1}$ , $m = 1, \dots, n$ , can be interpreted as the fundamental matrix of an absorbing Markov chain consisting of phase $m$ states as transient states and the remaining states of AMC as absorbing. Note, $(I - A_{mm})$ , $m = 1, \dots, n$ , is an upper-triangular Toeplitz matrix defined by its first row that is equal to $(1 - p_{0} (m), - p_{1} (m), \dots, - p_{i_{m} - i_{m - 1} - 1} (m))$ , see equation (3). Therefore, $(I - A_{mm})^{- 1}$ is a Toeplitz matrix defined by its first-row of entries $(1 - p_{0} (m))^{- 1} (a_{0} (m), \dots, a_{i_{m} - i_{m - 1} - 1} (m))$ .⁷⁵ The entries $a_{k} (m)$ , for $m = 1, \dots, n$ , are computed recursively as follows:

\begin{matrix} \begin{matrix} a_{k} (m) = \frac{1}{1 - p_{0} (m)} \sum_{j = 1}^{k} p_{j} (m) a_{k - j} (m), \end{matrix} \\ k = 1, \dots, i_{m} - i_{m - 1} - 1 . \end{matrix}

(5)

where $a_{0} (m) = 1$ . Finally, the expected number of visits $v_{j}$ to state $j$ before absorption in the AMC, starting from state 0, is given by:

v_{j} = (e_{1}^{m})^{t} R_{m} e_{j}^{m} = r^{m} e_{j}^{m}, for j = i_{m - 1}, \dots, i_{m} - 1,

(6)

$e_{l}^{m}$ represents a column vector of size $i_{m} - i_{m - 1}$ , with the $l$ th entry equal to 1 and all remaining entries equal to 0, $(a)^{t}$ denotes the transpose of vector $a$ , $R_{m}$ is defined in equation (4), and $r_{m}$ is the first row of $R_{m}$ , given by:

r_{m} = (e_{1}^{m})^{t} R_{m}, for m = 1, \dots, n .

(7)

Performance metrics

In this section, we derive the various cost and time factors during a maintenance cycle, as well as the expected total cost per unit of time and the limiting system’s availability. A maintenance cycle is defined as the time period during which the AMC starts in state 0 (as-good-as-new) and ends with maintenance completion following its absorption into the $pr$ or $co$ state.

- Expected inspection cost and time per cycle: Let $c_{I}$ denote the cost per inspection. During a maintenance cycle, the expected total inspection cost is given by:

IC = c_{I} \sum_{j = 0}^{i_{n} - 1} v_{j} = c_{I} \sum_{m = 1}^{n} r^{m} e^{m}

(8)

where $e^{m}$ is a column vector of ones with size $i_{m} - i_{m - 1}$ . Given that the AMC is in state $j$ , the inspection time is represented by $t_{I} (j)$ . Consequently, the expected total inspection time over a maintenance cycle is given by the following expression.

IT = \sum_{j = 0}^{i_{n} - 1} t_{I} (j) v_{j} = \sum_{m = 1}^{n} r^{m} \sum_{j = i_{m - 1}}^{i_{m} - 1} t_{I} (j) e_{j}^{m} = \sum_{m = 1}^{n} r^{m} t_{I}^{m}

(9)

where, $t_{I}^{m}$ is a column vector of entries $(t_{I} (i_{m - 1}), \dots, t_{I} (i_{m} - 1))$ .

- Expected maintenance cost and time per cycle: $c_{P}$ denotes the preventive maintenance cost and $c_{C}$ represents the corrective maintenance cost. A maintenancecycle ends with corrective maintenance with a probability $p_{C}$ , which corresponds to the probability that the AMC is absorbed into the state $co$ . Thus, the corrective maintenance probability for a maintenance cycle is given by Kemeny and Snell⁷⁴ Th. 3.3.7:

p_{C} = \sum_{j = 0}^{i_{n} - 1} v_{j} b_{j 2} = \sum_{m = 1}^{n} r^{m} \sum_{j = i_{m - 1}}^{i_{m} - 1} e_{j}^{m} p_{j}^{co} (m) = \sum_{m = 1}^{n} r^{m} p_{m}^{co}

(10)

Equations (3) and (4) are used to derive the first equality. Here, $p_{m}^{co}$ is a column vector with entries $(p_{i_{m - 1}}^{co} (m), \dots, p_{i_{m} - 1}^{co} (m))$ , representing the one-step transition probabilities from the states in phase $m$ to the absorbing state co. Let $t_{P}$ denote the preventive maintenance time and $t_{C}$ the corrective maintenance time. Consequently, the expected maintenance cost ( $MC$ ) and time ( $MT$ ) per cycle are expressed as follows:

\begin{matrix} MC = (1 - p_{C}) c_{P} + p_{C} c_{C} \\ = c_{P} + p_{C} (c_{C} - c_{P}) \end{matrix}

(11)

\begin{matrix} MT = t_{P} + p_{C} (t_{C} - t_{P}) \end{matrix}

(12)

- Expected downtime and cost per cycle: A cycle finishes with downtime if the AMC ends up in the absorbing state $co$ . The downtime is experienced for a fraction of the time between inspections $τ_{m}$ , when the AMC transitions to the absorbing state $co$ from a state in phase $m$ . We approximate this fraction as $1 / 2$ .¹¹ This can be derived by approximating the distribution of the degradation overshoot in the last inspection period that crosses the failure limit by the equilibrium excess distribution of the degradation in that period.⁷⁶ Consequently, the expected downtime length ( $DT$ ) and cost ( $DC$ ) per cycle are given, respectively, by:

\begin{matrix} DT = \frac{1}{2} \sum_{m = 1}^{n} τ_{m} r^{m} p_{m}^{co} \end{matrix}

(13)

\begin{matrix} DC = c_{D} DT \end{matrix}

(14)

Here, $p_{m}^{co}$ is the column vector defined in equation (10) previously and $c_{D}$ represents the downtime cost per unit of time.

- Expected quality loss per cycle: $QCU (x)$ represents the cost incurred for quality loss per time unit when the system degradation is equal to $x$ . Given that the AMC is in state $l$ at the inspection epoch $T_{k}$ , the system’s degradation has a mean value of $X_{k} = (l + 0.5) ϵ$ . Thus, the expected quality loss cost per cycle can be expressed as follows:

\begin{matrix} QC = \sum_{m = 1}^{n} τ_{m} r^{m} \sum_{j = i_{m - 1}}^{i_{m} - 1} e_{j}^{m} QCU ((j + 0.5) ϵ) \\ = \sum_{m = 1}^{n} τ_{m} r^{m} {qc}_{m} . \end{matrix}

(15)

where, ${qc}_{m}$ is a column vector of entries $(QCU ((i_{m - 1} + 0.5) ϵ), \dots, QCU ((i_{m} - 0.5) ϵ))$ and $QCU (\cdot)$ is defined in equation (2).

- Expected maintenance cycle length: The maintenance cycle is the sum of the AMC time to absorption, the inspection time, and the maintenance time. Therefore, the expected maintenance cycle length ( $CL$ ) is equal to:

CL = \sum_{m = 1}^{n} τ_{m} r^{m} e^{m} + MT + IT

(16)

where, $IT$ and $MT$ are given in equations (9) and (12), respectively.

- Expected total cost per unit of time: According to the Renewal Reward Theorem, the long-term expected maintenance cost per unit of time is the total expected cost per cycle divided by the expected cycle length, expressed as:

\begin{matrix} TCU = \frac{IC + MC + DC + QC}{CL} \\ = \frac{\sum_{m = 1}^{n} r^{m} [c_{I} e^{m} + (c_{C} - c_{P}) p_{m}^{co}]}{\sum_{m = 1}^{n} r^{m} [τ_{m} e^{m} + t_{I}^{m} + (t_{C} - t_{P}) p_{m}^{co}] + t_{P}} \\ + \frac{\sum_{m = 1}^{n} r^{m} [\frac{c_{D}}{2} τ_{m} p_{m}^{co} + τ_{m} {qc}_{m}] + c_{P}}{\sum_{m = 1}^{n} r^{m} [τ_{m} e^{m} + t_{I}^{m} + (t_{C} - t_{P}) p_{m}^{co}] + t_{P}} . \end{matrix}

(17)

where, the last equality follows from $IC$ in equation (8), $MC$ in equation (11), $DC$ in equation (14), $QC$ in equation (15), and $CL$ in equation (16).

- Limiting system’s availability: The system is unavailable if it is under inspection, maintenance, or down. Therefore, the limiting system’s availability ( $Av$ ) is expressed as:

\begin{matrix} Av = 1 - \frac{MT + IT + DT}{CL} \\ = \frac{\sum_{m = 1}^{n} τ_{m} r^{m} [e^{m} - \frac{1}{2} p_{m}^{co}]}{\sum_{m = 1}^{n} r^{m} [τ_{m} e^{m} + t_{I}^{m} + (t_{C} - t_{P}) p_{m}^{co}] + t_{P}} . \end{matrix}

(18)

The expected total cost per unit of time and the system’s availability are nonlinear functions of the inspection intervals and the degradation limits.

Performance optimization

Our objective is to minimize the expected cost per unit of time, $TCU$ , as defined in equation (17), by determining the number of phases $n$ , the inspection intervals for each phase $τ_{m}$ ( $m = 1, \dots, n$ ), and the degradation limits $y_{m}$ ( $m = 1, \dots, n$ ). The system’s availability, $Av$ , given in equation (18), must satisfy the constraint $Av \geq lb$ , where $lb$ represents a predefined lower bound. The inspection interval $τ_{m}$ in phase $m$ is non-negative, decreases with $m$ , and must satisfy $τ_{m - 1} - τ_{m} \geq δ_{τ}$ for $m = 2, \dots, n$ . Similarly, the degradation limit $y_{m}$ in phase $m$ is non-negative,increases with $m$ , and must satisfy $y_{m} - y_{m - 1} \geq δ_{y}$ for $m = 2, \dots, n$ . Furthermore, the preventive maintenance limit $y_{n}$ must be less than the failure limit $L$ . The number of phases $n$ is a positive integer that cannot exceed $n_{\max}$ .

The input parameters $δ_{τ}$ , $δ_{y}$ , and $n_{\max}$ are introduced to reflect practical limitations and ensure feasibility. The optimization problem is formulated as follows.

τ_{m} \geq 0, y_{m} \geq 0, m = 1, \dots, n .

(19)

The optimization problem under consideration is a mixed-integer nonlinear programming (MINLP) problem. The proposed solution methodology follows a two-stage approach. In the first stage, a greedy search is performed over the number of phases $n$ (integer decision variable), with the stopping criterion based on the increase in the total cost per time unit ( $TCU$ ), as detailed in Algorithm 2. In the second stage, the optimal values of $TCU$ and the decision variables $τ_{m}$ and $y_{m}$ for all phases, $m = 1, \dots, n$ , are determined based on the value of $n$ obtained from the first stage. The second-stage optimization problem is a constrained nonlinear programming (NLP) problem.

To solve the NLP problem, we tested several iterative algorithms, including constrained optimization by linear approximation (COBYLA), sequential least squares programming (SLSQP), and the trust region-constrained algorithm (trust-constr). Among these, the COBYLA algorithm demonstrated the best performance in terms of both objective function value and computational speed for a fixed maximum number of iterations. The COBYLA algorithm was implemented using a multi-start approach, as described in Algorithm 1.

Algorithm 1.

Exploration-exploitation algorithm for the second stage NLP problem with multi-start.

Set input variables

Numberiterations

ξ

, and

n

Randomstarts \leftarrow

maximum of 1 and the integer value of

Numberiterations \times ξ

l \leftarrow 1

repeat ▹ Exploration step
Set a randomly initial feasible solution

x_{0}

TCU (l) \leftarrow

objective function value by solving OP in (19) with

x_{0}

as initial solution and for

n

x (l) \leftarrow

optimal solution found

l \leftarrow l + 1

until

l \geq Randomstarts

x^{*} \leftarrow

solution of the smallest

TCU (l), l = 1, \dots, Randomstarts

repeat▹ Exploitation step
Set

x_{0}

x^{*}

plus a random error

TCU (l) \leftarrow

objective function value by solving OP in (19) with

x_{0}

as initial solution and for

n

x (l) \leftarrow

optimal solution found
if

x (l) < x^{*}

then

x^{*} \leftarrow x (l)

end if

l \leftarrow l + 1

until

l \geq Numberiterations

Return the smallest

TCU (l)

for

l = 1, \dots, Numberiterations

and its solution

Algorithm 1 employs a two-phase strategy to enhance solution quality. In the exploration phase, the solution space is sampled by running the COBYLA algorithm with multiple randomly generated initial solutions. In the exploitation phase, the best solution from the exploration phase is perturbed randomly to generate new initial points, and the COBYLA algorithm is executed multiple times with these perturbed solutions. The total number of initial solutions is partitioned between the exploration and exploitation phases according to a predefined proportion $ξ$ . Finally, the best solution obtained across both phases is returned to the first stage for further refinement.

Algorithm 2.

Two-stage solution approach.

n \leftarrow 1

TCU (1) \leftarrow

Return the best objective function value found using Algorithm 1 for

n = 1

repeat

n \leftarrow n + 1

TCU (n) \leftarrow

Return the best objective function value found using Algorithm 1 for

n

until

TCU (n) \geq TCU (n - 1)

n = n_{\max}

Numerical results

In this section, we demonstrate the applicability and effectiveness of the proposed multi-threshold CBM policy by applying it to the numerical case introduced by Al Hanbali et al.¹¹ (see Table 3). The Gamma process parameters can be estimated using the method of moments and the maximum likelihood method; see, for details, van Noortwijk.⁷⁰ As a key difference from the original setup, we consider a multi-threshold policy in which the inspection time depends on the current degradation level, and the quality-loss cost is non-negligible. Specifically, in the base case, we model the inspection time as a linear function of degradation, given by $t_{I} (x) = 0.05 + 0.05 \frac{x}{L}$ , where $x$ denotes the degradation at the time of inspection, and $L$ is the failure limit. The quality loss parameters in equation (2) are adopted from the numerical case study in Cheng and Li,⁷² in which the parameters of the quality loss function are estimated through the maximum likelihood method based on historical data (Table 4). Regarding the selection of the discretization constant $ϵ$ , the smaller the value of $ϵ$ , the larger the state space of the Markov chain, increasing the computational complexity. For a small numerical setting, we varied the value of $ϵ$ between the expected degradation per unit of time, that is, $α / β$ , and its tenth value. We found that the limiting system’s availability and the expected cost per unit of time quickly converge. However, computational time increases exponentially, especially for very small values of $ϵ$ .

Table 3.

Model parameters.¹¹

$α$	1.1 per year
$β$	9.091 per mm
$L$	11.33 mm
$ϵ$	0.001
$t_{P}$	1 week
$t_{C}$	4 weeks
$t_{I} (x)$	$0.05 + 0.05 \frac{x}{L}$ weeks
$c_{C}$	$500,000
$c_{P}$	$50,000
$c_{I}$	$10,000
$c_{D}$ (per year)	$5,000,000

Table 4.

Quality loss parameters.⁷²

$c_{Q}$ (per unit)	$43
$p_{0}$	0.004
$η$	0.08
$λ_{p}$	0.005
$γ_{q}$	1.16
$P_{R}$ (units per day)	320

The downtime cost per year, $c_{D}$ , is estimated as the full loss of production per year that is approximately equal to the product of the number of operational days per year (assumed equal to 365 days), production rate per day, $P_{R}$ , and quality loss cost per unit, $c_{Q}$ . In Table 5, we list the optimization parameters found as results of trying several values for the number of random starts, the maximum number of objective function evaluations, and the split factor between exploration and exploitation, $ξ$ . The bounding factors $δ_{y}$ , $δ_{τ}$ , $lb$ , and $n_{\max}$ were estimated by considering the practical limitations of the multi-threshold CBM policy implementation. For example, significant differences between the threshold value and the inspection duration are desirable for a slowly degrading system, such as the system considered in the paper, with a mean degradation of $α / β = 0.121$ mm/year. Based on the obtained results, practical implications are thus derived.

Table 5.

Optimization algorithm parameters.

Number of random starts	20
Number of objective function evaluations	500
$ξ$	0.5
$δ_{y}$	0.1 mm
$δ_{τ}$	1 month
$lb$	90%
$n_{\max}$	5

Benchmarking of the multi-threshold CBM policy

In this section, we benchmark the optimal multi-threshold CBM policy, where degradation levels and the time between inspections are decision variables, against three baseline scenarios. The first scenario is the two-threshold maintenance policy. The second (resp. the third) scenario is the optimal multi-threshold policy, in which the degradation levels (resp. the time between inspections) are the only decision variables in the two-phase solution approach.

In addition to the total cost per unit of time ( $TCU$ ), various performance indicators are included. Namely, we consider the limiting availability ( $Av$ ), the expected yearly maintenance cost ( $MCU : = MC / CL$ ), the expected yearly inspection cost ( $ICU : = IC / CL$ ), the expected yearly quality loss cost ( $QCU : = QC / CL$ ), and finally the expected yearly downtime cost ( $DCU : = DC / CL$ ). Table 6 shows that the optimal multi-threshold policy, with degradation levels and the time between inspections as decision variables, outperforms the other three scenarios by at least 20% and up to 60%. This is achieved by reducing the annual quality-loss cost, the largest cost component in this case.

Table 6.

Comparison of the optimal multi-threshold versus three baseline scenarios.

	Two-threshold	Multi-threshold (fixed $τ_{m}$ )	Multi-threshold (fixed $y_{m}$ )	Optimized multi-threshold
$TCU$	411.00	209.63	202.60	161.31
$Av$	99.95%	99.95%	99.83%	99.86%
$MCU$	0.60	1.17	0.63	0.60
$ICU$	1.58	0.71	9.21	6.67
$QCU$	408.58	207.75	192.76	153.82
$DCU$	0.25	0.00	0.00	0.21

The cost factors are expressed in $1,000/year.

Impact of quality loss parameters on system performance

Figure 2 illustrates the impact of varying the parameters of the quality loss model ( $p_{0}$ , $η$ , $λ_{p}$ , and $γ_{q}$ ) as defined in equation (2). The parameter values are adjusted within the sets ${0.004, 0.008}$ , ${0.08, 0.16}$ , ${0.005, 0.01}$ , and ${1.16, 2.32}$ , respectively.

Figure 2.

Optimal cost per unit of time $TCU$ for different values of $p_{0}$ , $η$ , $λ_{p}$ , and $γ_{q}$ quality loss parameters.

The findings show that the parameters $p_{0}$ , $λ_{p}$ , and $γ_{q}$ have a relatively minor influence on the optimal $TCU$ . For instance, when $η$ is fixed at $0.08$ and the other parameters are varied within their respective sets, the optimal $TCU$ fluctuates within the range $[157.29, 168.94]$ , corresponding to a relative change of 6.89%. In contrast, the parameter $η$ exerts a significant influence on the optimal $TCU$ . For example, when $(p_{0}, η, λ_{p}, γ_{q})$ equals $(0.008, 0.16, 0.01, 2.32)$ and $(0.008, 0.08, 0.01, 2.32)$ , the optimal $TCU$ decreases from $301.91$ to $159.97$ , representing a reduction of 47.01%. These findings underscore the critical importance of accurately estimating $η$ , as its value substantially impacts the optimal cost and overall system performance.

Sensitivity analysis of Gamma process parameters and their impact on maintenance costs

Table 7 presents the sensitivity of the optimal total cost per year ( $TCU$ ) for various values of the degradation process shape parameter $α$ and rate parameter $β$ . The expected degradation per year is given by $α / β$ mm/year, with a squared coefficient of variation (SCV) equal to $1 / α$ .

Table 7.

Optimal $TCU$ ( $1000 $ / year$ ) for different values of the Gamma process parameters $α$ and $β$ .

	$β = 4.545$	$β = 9.091$	$β = 18.182$
$α = 0.55$	173.00	156.58	143.77
$α = 1.1$	180.20	161.31	146.79
$α = 2.2$	187.17	159.46	149.98

The results indicate that the optimal $TCU$ generally increases with the expected yearly degradation rate, though the rate of increase is relatively modest. For instance, when $(α, β) = (2.2, 18.182)$ and $(α, β) = (2.2, 4.545)$ , the expected yearly degradation is $0.121$ mm and $0.484$ mm (four times higher), respectively, yet the optimal $TCU$ increases by only 24.8%. Furthermore, for a constant expected yearly degradation (diagonal values in Table 7), the optimal $TCU$ decreases as $α$ increases, albeit at a low rate. This reduction can be attributed to the lower SCV of degradation, which is inversely proportional to $α$ . For example, with $(α, β) = (0.55, 4.545)$ and $(α, β) = (2.2, 18.182)$ , the SCV of yearly degradation decreases from $1.818$ to $0.454$ (a fourfold reduction), yet the optimal $TCU$ decreases by only 13.9%. These findings demonstrate that the multi-threshold CBM policy effectively accommodates changes in the degradation process parameters by dynamically adjusting the thresholds and inspection intervals.

Effect of cost factors on maintenance strategy optimization

Figure 3 illustrates the sensitivity of the optimal $TCU$ to variations in corrective maintenance cost ( $c_{C}$ ), preventive maintenance cost ( $c_{P}$ ), inspection cost ( $c_{I}$ ), downtime cost per year ( $c_{D}$ ), and quality loss per unit ( $c_{Q}$ ). We observe that $c_{C}$ , $c_{P}$ , $c_{I}$ , and $c_{D}$ cost factors have a minor impact on the optimal $TCU$ . Among these four factors, $c_{I}$ exhibits the most impact on the optimal $TCU$ . For example, for ( $c_{C}$ , $c_{P}$ , $c_{I}$ , $c_{D}$ , $c_{Q}$ )= $(500, 50, 10, 5000, 0.043)$ and (500, 50, 20, 5000, 0.043), the cost is 161.31 and 166.67, respectively. Therefore, we infer that the multi-threshold CBM policy can mitigate $c_{C}$ , $c_{P}$ , $c_{I}$ , and $c_{D}$ cost factors increase, by appropriately changing the degradation limits and the time between inspections.

Figure 3.

Optimal cost per unit of time $TCU$ for different values of $c_{C}$ , $c_{P}$ , $c_{I}$ , and $c_{D}$ cost factors expressed in ($ 1000/year).

By contrast, the optimal $TCU$ is sensitive to the quality loss cost per unit $c_{Q}$ . For example, for ( $c_{C}$ , $c_{P}$ , $c_{I}$ , $c_{D}$ , $c_{Q}$ )=(500, 50, 10, 5000, 0.043) and (500, 50, 10, 5000, 0.086), the cost is 161.31 and 312.35, respectively, which means the optimal $TCU$ increases at a similar rate to the increase in $c_{Q}$ . This indicates the critical role of remanufacturing in reducing quality loss and overall costs.

Additionally, we examined the effects of varying time factors $t_{C}$ , $t_{P}$ , and $t_{I} (0)$ . These parameters were varied in the sets ${4, 8}$ , ${1, 2}$ , and ${0.025, 0.05, 0.1}$ weeks, respectively. The results indicate a minor impact on both $TCU$ and system availability ( $Av$ ), with the former ranging in [161.27, 161.65] (1000 $\ year) and the latter in [99.70, 99.92] (%).

Practical implications

The numerical results yield the following practical implications for the implementation and adoption of the proposed multi-threshold CBM policy in industrial settings:

Switching from the standard two-threshold policy to the optimized multi-threshold policy reduces the total annualized cost from $411,000 to $161,310 (a 60.8% decrease), while maintaining virtually unchanged availability (99.95% → 99.86%). This highlights that the optimization effort is financially worthwhile even under strict uptime requirements.

In the benchmark case, quality loss represents approximately 95% of the total cost ($153,820 out of $161,310). The policy, therefore, trades a modest increase in inspection expenditure (+$5090/year) for a 62% reduction in quality loss. Sensitivity analysis confirms that doubling the quality-loss penalty almost doubles the total cost, whereas doubling corrective, preventive, or downtime costs increases the total cost by less than 4%. This suggests that investing in high-resolution inspection and monitoring technologies is particularly advantageous when poor-quality outputs are costly, such as in aerospace, pharmaceutical, or precision manufacturing sectors.

Quadrupling the mean degradation rate leads to only a 25% increase in total cost, while halving the degradation variance reduces the total cost by about 14%. This indicates that the optimized policy remains robust and effective even when the degradation model is not perfectly calibrated, providing a practical advantage in real-world environments where degradation behavior is uncertain.

The model schedules shorter inspection intervals (e.g., 0.6 months) only during the final degradation phase, while maintaining longer intervals during the early life stages. This dynamic adjustment reduces latent downtime: the expected downtime cost per year is approximately $210 under the optimized policy compared to $250 under the two-threshold policy. In practice, this means that rapid inspection should be prioritized only when the system approaches the end-of-life degradation window, and unnecessary inspections should be avoided during early operational phases.

Nevertheless, under the standard assumptions of perfect inspections and as-good-as-new maintenance, the results suggest that, from a practical standpoint, concentrating inspection effort as the asset approaches critical degradation and explicitly accounting for quality-loss penalties can provide substantial economic benefits. In real deployments, inspection outcomes may be noisy, and maintenance actions may lead to partial restoration, which can affect the optimal inspection sequence and the effective threshold values.

Conclusion

In this paper, we propose a multi-threshold condition-based maintenance (CBM) policy for continuously degrading systems modeled as the Gamma process. Our model is characterized by key features that determine the inspection time and the quality-loss cost functions associated with system degradation. The expected maintenance cost per unit of time and the system’s availability are derived explicitly by leveraging a matrix-analytic approach to the absorbing Markov chain and modeling the system as a multi-state system. We propose a two-stage approach to solve the MINLP problem. The first stage involves a greedy search over the number of degradation limits, and the second stage reduces the problem to a constrained nonlinear programming (NLP) problem, which is solved using the COBYLA algorithm with multi-start. A case study retrieved from the literature was analyzed to demonstrate the applicability and effectiveness of the proposed approach, showcasing the superiority of the multi-threshold CBM policy in a comparative analysis. A comprehensive cost, time, and variability sensitivity analysis underscores the importance of accurate estimation of the quality loss cost per unit and of key variables, such as the mean and variance of degradation growth per time unit, in determining the optimal maintenance policy. This analysis serves as a crucial tool for decision-makers and maintenance engineers to better understand the impact of these factors on the overall maintenance strategy.

For future extensions, the model could be extended to consider other degradation processes beyond the Gamma process used in this work. For example, the Wiener process is another common degradation model in which degradation can increase or decrease over time. This would result in a Toeplitz block structure for the transient probability matrix, rather than the upper-triangular form derived here. This type of matrix was encountered and analyzed in the context of generalized queueing systems.

Additionally, the random-coefficient model is another interesting alternative, in which the degradation growth across consecutive inspection periods is not independent. In this case, the system’s state would need to be augmented to include the random coefficient values, thereby introducing additional complexity to the analysis. Exploring these alternative degradation processes could provide insights into how the maintenance policy, optimal limits, and time between inspections would need to be adapted for different types of degrading systems.

Finally, the proposed analysis relies on the standard assumptions of perfect inspections and as-good-as-new maintenance actions and focuses on a single-component system. In practical settings, inspection outcomes may be subject to misclassification errors, maintenance actions may result in partial restoration rather than full renewal, and several interacting components may need to be coordinated, all of which can affect the optimal inspection sequence and degradation thresholds. Extending the framework to account for imperfect inspections, imperfect maintenance, and multi-component systems with economic, structural, or stochastic dependence is therefore a natural and promising direction for future research. Another interesting research direction is when knowledge of the degradation probability distribution is limited, as it is typically uncertain. Techniques from robust optimization under uncertainty would be best applied in this situation; see, for example Sun et al.⁸

Footnotes

ORCID iDs

Ahmad Al Hanbali

Mohamed Noomane Darghouth

Awsan Mohammed

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing does not apply to this article, as no new data were created or analyzed in this study.

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Optimizing inspection and maintenance decisions in degrading systems with quality-loss

Abstract

Keywords

Introduction

Literature review

Targeted contributions

Assumptions and notations

Problem description

Model analysis

AMC transition and fundamental matrices

Performance metrics

Performance optimization

Numerical results

Benchmarking of the multi-threshold CBM policy

Impact of quality loss parameters on system performance

Sensitivity analysis of Gamma process parameters and their impact on maintenance costs

Effect of cost factors on maintenance strategy optimization

Practical implications

Conclusion

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

Data availability statement

References