Performance Assessment of Static and Adaptive Maintenance Strategies Under Stochastic Conditions with Penalty-Augmented Objectives Using a Monte Carlo Simulation Approach to Reliability and Cost Optimization

In contemporary industrial systems, effective maintenance strategies are crucial for ensuring operational reliability, minimizing downtime, and optimizing cost-efficiency [1]. As industries become increasingly complex, with systems that demand high performance and safety, the ability to predict and prevent failures is paramount. Maintenance plays a pivotal role in safeguarding assets, prolonging operational lifespans, and maintaining system integrity [2]. In this context, condition-based maintenance (CBM) has emerged as a promising approach. CBM utilizes condition monitoring to assess the health of assets in real time, scheduling inspections and interventions based on the observed degradation state of the system. This methodology aims to replace traditional time-based maintenance with more efficient, data-driven strategies that respond to the actual condition of the equipment [3-6].

Despite the advantages offered by CBM, its implementation is fraught with challenges primarily arising from the stochastic nature of degradation processes and the inherent imperfections of condition monitoring [7-10]. Degradation is often driven by complex, random phenomena that cannot be captured by simple deterministic models. Additionally, condition monitoring provides imperfect, noisy data that may not fully reflect the true state of degradation. These uncertainties complicate the decision-making process, as maintenance actions must be based on incomplete and potentially inaccurate information [11]. Traditional CBM strategies, which rely on fixed inspection intervals and constant preventive maintenance thresholds, fail to account for these sources of variability, potentially leading to suboptimal performance. These static policies can either result in excessive maintenance activities or, conversely, delayed interventions that may allow failures to occur before detection [12-17].

To address these limitations, adaptive maintenance decision policies have been proposed. Adaptive strategies dynamically adjust inspection frequencies and preventive thresholds in response to observed degradation and updated system information [18-22]. By recalibrating maintenance decisions based on real-time condition data, adaptive policies aim to more precisely align maintenance activities with the evolving risk profile of the system. These adaptive policies offer the potential for more cost-effective and reliable maintenance, particularly in environments characterized by high uncertainty. However, the comparative performance of static versus adaptive strategies under realistic operational conditions remains underexplored, particularly in terms of long-term cost, reliability, and risk mitigation [23-25].

The objective of this study is to develop a robust framework for evaluating and comparing static and adaptive CBM strategies [26-30]. The framework incorporates a Wiener process degradation model (WP), representing the stochastic evolution of system degradation, alongside imperfect condition monitoring (MM) characterized by Gaussian noise. Inspection policies (IP) are formulated as either fixed intervals (ΔT_fix) or adaptive intervals (ΔT_k) based on residual useful life (RUL) estimates. Preventive maintenance thresholds (PM) are either static (L_m) or adaptive (L_m,k), with the latter being recalibrated based on the observed degradation state. The performance of these strategies is evaluated using a range of metrics, including the short-horizon failure probability (p_hit), which quantifies the likelihood of failure before the next inspection, and the Kaplan–Meier survival estimator (KM) for reliability estimation. Economic performance is assessed through the average cost rate (C_avg) and the PM/CM ratio, which compares preventive to corrective maintenance efforts. Additionally, a penalty-augmented objective function (J_λ) is introduced, incorporating both cost and reliability penalties to ensure a balanced trade-off between economic efficiency and operational safety [31].

This paper contributes to the field by presenting a comprehensive simulation framework that integrates these key components: degradation modeling, inspection scheduling, preventive maintenance threshold adjustment, risk quantification, and performance evaluation. The simulation framework is used to assess three maintenance policies: (i) the static policy with fixed inspections and fixed thresholds (ΔT, M), (ii) the semi-adaptive policy with fixed inspections and adaptive thresholds (ΔT, M_k), and (iii) the fully adaptive policy incorporating both dynamic inspections and adaptive thresholds (ΔT_k, M_k). The study provides a detailed comparative analysis of these strategies, highlighting their relative advantages and trade-offs in terms of cost, downtime, reliability, and risk [32].

While classical P–F interval and predictive maintenance (PdM) strategies have been widely studied, this work focuses specifically on the relative behavior of static versus adaptive CBM policies under identical stochastic conditions. The proposed Monte Carlo framework is intentionally generic and can later be extended to include P–F or predictive triggers, enabling systematic benchmarking in future research.

The remainder of this paper is organized as follows. Section 2 outlines the methodological framework, including the mathematical formulations for degradation, inspections, and maintenance decision rules. Section 3 describes the three maintenance strategies under consideration. Section 4 details the Monte Carlo simulation architecture, while Section 5 explains the simulation configuration and input parameters. Section 6 presents the results and their analysis, followed by a discussion of the engineering implications in Section 7. Finally, Section 8 concludes the paper and outlines directions for future research.

Adaptive maintenance decision policies constitute the operational core of the CMB work. They specify how inspection schedules and preventive maintenance thresholds are defined, either statically or adaptively, in response to observed system degradation. The three strategies examined in this study represent progressively increasing levels of adaptivity, ranging from a purely static baseline to a fully adaptive policy [51, 52]. This section details the formulation of each strategy, its governing equations, and its conceptual implications.

3.1 Strategy I: Fixed inspections and fixed threshold (ΔT, M)

The first policy, denoted (ΔT, M), represents the static benchmark (see Figure 1). Under this regime, inspections are executed periodically at constant intervals using Eq. (3).

While preventive maintenance is initiated when the observed degradation trajectory crosses a fixed threshold as you see in Eq. (6).

Eqs. (3) and (6) define a fully deterministic policy structure, in which neither the inspection cadence nor the intervention threshold is modified in response to evolving degradation states. This approach reflects long-standing industrial practice, particularly in sectors where condition monitoring is rudimentary or where regulatory frameworks mandate fixed maintenance intervals. However, it suffers from inherent inefficiencies: inspections may be redundant during periods of low degradation intensity, while they may be too sparse during accelerated deterioration phases. Likewise, the fixed threshold L_m may trigger preventive interventions either prematurely (if degradation progresses slowly) or too late (if stochastic variability accelerates failure).

Figure 1. Principe of maintenance strategy (∆T, M)

As such, the (ΔT, M) policy is primarily retained as a baseline reference for quantifying the incremental benefits of adaptivity.

3.2 Strategy II: Fixed inspections and adaptive threshold (ΔT, M_k)

The second policy, denoted (ΔT, M_k), preserves periodic inspections but introduces adaptivity in the preventive threshold (see Figure 2). Inspections continue to be scheduled at fixed intervals ΔT_fix, as in the baseline policy, but the preventive threshold is dynamically recalibrated at each inspection according to Eq. (7). Subject to the operational constraint of Eq. (8).

Figure 2. Principe of maintenance strategy (∆T, M_k)

Eq. (7) introduces a linear adaptation mechanism: the preventive threshold is progressively reduced as the system approaches the failure limit L_f. The adaptation coefficient α_L regulates the aggressiveness of this recalibration, while the lower bound L_min prevents excessively conservative interventions. The additional constraint Eq. (8) guarantees that preventive actions always maintain a non-negligible safety buffer from failure.

Conceptually, this strategy embodies a semi-adaptive philosophy. It retains the simplicity of periodic inspections, thereby minimizing the complexity of planning, but incorporates risk sensitivity at the intervention stage. Preventive actions become more likely as degradation intensifies, which reduces the probability of unplanned corrective maintenance. However, this responsiveness may increase the frequency of preventive interventions relative to the static baseline, potentially raising cumulative costs. The (ΔT, M_k) strategy thus represents a compromise: It augments reliability by modulating preventive aggressiveness, but does not alter the inspection burden.

3.3 Strategy III: Adaptive inspections and adaptive threshold (ΔT_k, M_k)

The third strategy, denoted (ΔT_k, M_k), is the most comprehensively adaptive among those analyzed (see Figure 3). It introduces responsiveness into both inspection scheduling and preventive threshold setting. At each inspection epoch, the RUL is estimated in Eq. (4).

And the subsequent inspection interval is recalibrated dynamically using Eq. (5).

Eqs. (4)-(5) formalize a fully adaptive inspection policy: inspections accelerate as the estimated RUL shortens, thereby intensifying monitoring during high-risk phases, and relax when the system is deemed healthy. The preventive threshold is simultaneously governed by Eqs. (7)-(8), ensuring that interventions are also more conservative when failure proximity increases.

This policy represents a risk-driven paradigm in which both monitoring intensity and intervention aggressiveness evolve dynamically. By design, it minimizes exposure to undetected failures, since inspection and intervention cadence tighten as the system approaches criticality. However, the dual adaptivity increases operational variability: Inspection workloads fluctuate, preventive interventions occur with higher frequency under rapid degradation, and planning complexity is elevated. While these characteristics may increase cost volatility, they also deliver superior reliability assurance.

Figure 3. Principe of maintenance strategy (∆T_k, M_k)

The performance of the three integrated maintenance strategies ((ΔT, M), (ΔT, M_k), and (ΔT_k, M_k)) was evaluated using a Monte Carlo simulation framework specifically designed to capture the stochastic variability of degradation processes, the imperfect nature of inspection data, and the asymmetric costs and downtimes associated with preventive and corrective interventions. The simulation was implemented with careful consideration of both statistical robustness and engineering realism, ensuring that the comparative results reflect practical operating conditions rather than theoretical simplifications.

The operational horizon was set to T_op = 2 × 10⁴ time units, which corresponds to a sufficiently long service life to observe multiple cycles of degradation and maintenance within each simulation run. This value was chosen to ensure that both preventive and corrective actions had the opportunity to occur repeatedly, thereby providing statistically meaningful estimates of costs, downtime, and reliability. To reduce stochastic noise in the results, the analysis was based on N_sim = 1000 independent replications. This sample size is large enough to stabilize Monte Carlo estimates while remaining computationally feasible. Each degradation trajectory was simulated by discretizing the governing stochastic differential equation using the Euler–Maruyama scheme with a time increment of Δt = 1. The choice of this step size reflects a compromise: it is small enough to accurately capture stochastic dynamics yet sufficiently large to avoid unnecessary computational expense. For interpretability, a subset of fifteen trajectories was retained for graphical representation, providing insight into the variability of degradation paths and maintenance outcomes across different strategies.

The underlying degradation process was modeled as a Wiener process with drift, which is widely recognized in the literature as a parsimonious yet effective representation of cumulative damage subject to random fluctuations. The governing dynamics were expressed in Eq. (1).

The drift coefficient was fixed at μ = 0.01, representing a relatively slow but steady accumulation of wear or damage over time, while the diffusion parameter was set to σ = 0.12, reflecting significant environmental and operational variability that causes degradation to deviate unpredictably from its mean trajectory. The system was assumed to start from an as-good-as-new condition, X₀ = 0, after installation or major overhaul. Failure was defined as the first hitting time of the degradation trajectory with the critical threshold L_f = 5. This failure threshold can be interpreted as a physical or functional limit, such as a maximum allowable crack length, wear depth, or loss of thickness in a structural element. The chosen values generate degradation trajectories that remain within a realistic industrial range, balancing the competing influences of predictable wear and stochastic shocks. These parameter values were intentionally selected to generate a high-variance degradation environment, serving as a stress-test for the maintenance policies. The resulting low reliability levels are therefore not indicative of model failure, but rather reflect the intended difficulty of the simulation scenario, designed to expose the sensitivity and robustness of static and adaptive CBM strategies under severe stochastic variability.

Inspection processes were modeled as noisy observations of the true degradation state, consistent with practical monitoring systems where sensors and non-destructive tests introduce measurement errors. Observations at inspection times were expressed in Eq. (2), where the measurement error ε followed a Gaussian distribution with standard deviation σ_meas = 0.05. This value reflects the relatively high accuracy of modern condition monitoring systems, while acknowledging that no measurement system is free from noise. In the fixed inspection policy, inspections occurred at regular intervals of ΔT_fix = 120. This interval represents a compromise between frequent monitoring, which increases costs and downtime, and sparse monitoring, which increases the risk of undetected failures. In adaptive inspection strategies, the interval between inspections was recalibrated dynamically according to estimates of RUL, such that ΔT_k is expressed using Eq. (5). The adaptation coefficient was set to α_T = 0.4, which provides a moderate sensitivity to the estimated remaining life. To prevent unrealistic extremes, inspection intervals were bounded between ΔT_max = 360. The minimum interval ensures that inspections cannot occur more frequently than is practically possible (e.g., due to manpower or logistical constraints), while the maximum interval guarantees that inspections are not postponed excessively, which would expose the system to unacceptable risks.

Preventive maintenance was governed by threshold-based rules. In the baseline policy, the preventive threshold was fixed at L_m = 3.5, a value chosen to strike a balance between premature preventive interventions, which inflate costs, and delayed interventions, which increase the risk of corrective maintenance. A lower admissible bound of L_min = 2.5 was imposed to ensure that adaptive strategies would not recalibrate thresholds to excessively conservative values, which could result in unsustainable preventive workloads. In adaptive-threshold strategies, the trigger was updated according to Eq. (7), with α_L = 0.3. This value was selected to introduce moderate responsiveness to the proximity of failure while avoiding destabilizing oscillations in preventive policy. In addition, a mandatory safety margin was imposed such that $\mathrm{L}_{\mathrm{m}, \mathrm{k}} \leq \mathrm{L}_{\mathrm{f}}-\delta \mathrm{L}_{\text {safety}}$, with $\delta \mathrm{L}_{\text {safety}}=0.4$. This safety margin guarantees that preventive maintenance is always initiated at least 0.4 units below the failure threshold, thereby ensuring that no adaptation places the system at excessive risk of immediate failure.

The economic and operational impacts of interventions were explicitly quantified. Preventive maintenance was assigned a unit cost of C_PM = 100 and a downtime of t_PM = 30. These values capture the relatively modest economic and operational impact of preventive actions, which typically involve planned component replacements or adjustments. Corrective maintenance was modeled as substantially more expensive, with a unit cost of C_CM = 5000 and a downtime of t_CM = 800. This strong asymmetry reflects the realities of industrial practice, where corrective actions often require unplanned shutdowns, significant resource mobilization, and potential collateral damage. Inspections were comparatively inexpensive, with a cost of C_insp = 5 and a downtime of t_insp = 5, but their cumulative effect becomes non-negligible when inspection frequency is high. These parameters provide a realistic economic framework in which strategies must carefully balance inspection, preventive, and corrective actions.

To ensure compliance with engineering requirements, reliability and risk constraints were embedded into the evaluation. A minimum survival probability of R_min = 0.95 was imposed, with reliability estimated at a reporting horizon of R_time = 2000 using Kaplan–Meier survival analysis. This requirement reflects the expectation that in industrial settings, systems should maintain high levels of reliability over typical production cycles. In addition, a short-horizon constraint was specified, requiring that the probability of failure between consecutive inspections not exceed β = 0.05. This constraint acknowledges that frequent risks of interim failure are unacceptable, even if long-term survival targets are nominally achieved.

Performance was therefore assessed using a penalty-augmented objective function that integrates direct costs with penalties for violations of reliability and risk requirements. The augmented cost was defined in Eq. (13), where C_avg represents the long-run average cost rate, R_Rtime is the reliability at the reporting horizon, and 1_{phit>β} is an indicator capturing whether the between-inspection risk exceeded the tolerance level. Penalty weights were fixed at λ_R = 104 and λ_hit = 103, reflecting the engineering principle that reliability and risk compliance are non-negotiable, and that any policy violating these requirements should be deemed unacceptable regardless of its cost efficiency.

Finally, random number control was implemented to ensure reproducibility and comparability of results. A master seed of s_master = 42 was employed for the pseudo-random number generator. When common random numbers were disabled, independent random streams were generated for each strategy by applying offsets to this seed. This approach allowed fair comparisons under synchronized stochastic conditions, while also providing realistic variability in independent replications.

We evaluated convergence using 10 replication batches of 100 runs. Batch-mean trajectories for $\mathrm{C}_{\text {avg }}, \mathrm{R}(\mathrm{t}=2000)$, and $\operatorname{Pr}\left(\mathrm{p}_{\text {hit }}>\beta\right)$ showed no drift and tight overlapping CIs. Relative standard error of $\bar{C}$ was $<0.7 \%$; absolute half-widths for R and $\operatorname{Pr}\left(\mathrm{p}_{\text {hit }}>\beta\right)$ were $\leq 0.03$. The qualitative conclusions (constraint violations for all policies; negligible differences in means; minor tail-risk mitigation) were unchanged when reestimating with the first 600,800 , and full 1000 replications.

These convergence diagnostics confirm that our findings are not sensitive to the number of replications and that $\mathrm{N}_{\text {sim }}=$ 1000 ensures decision-grade precision.

With observed dispersion of $\hat{\sigma}\left(C_{\text {avg}}\right) \approx 0.84-0.89$, the standard error at $\mathrm{N}=1000$ is $\leq 0.028$, giving a 95% CI halfwidth $\leq 0.055$ ($<1.3 \%$ of $\bar{C} \approx 4.5$). For reliability $\mathrm{R} \approx 0.16$, a binomial proxy yields a half-width $\approx 0.023$. For $\operatorname{Pr}\left(\mathrm{p}_{\text {hit}}>\beta\right) \approx$ 0.64 , the half-width is $\approx 0.03$. Thus $\mathrm{N}=1000$ satisfies our precision targets and provides conservative margin; sequential checks indicated adequacy by $\mathrm{N} \approx 600-800$. Common random numbers were applied across strategies to improve comparative efficiency.

In summary, the simulation configuration combined a Wiener-based degradation model, noisy inspection measurements, adaptive or fixed preventive thresholds, explicit cost–downtime accounting, and penalty-augmented performance evaluation within a Monte Carlo environment of 1 replication. The chosen parameter values were calibrated to reflect realistic industrial conditions, ensuring that strategies are evaluated not only on cost performance but also on their ability to satisfy stringent reliability and safety requirements. This framework thus provides a rigorous and credible experimental basis for assessing the comparative merits of integrated CBM policies.

The collective findings from the global evaluation (Section 6.1), the short-horizon risk analysis (Section 6.2), and the penalty-augmented assessment (Section 6.3) converge toward a clear and compelling conclusion: although the three maintenance strategies (ΔT, M), (ΔT, M_k), and (ΔT_k, M_k) exhibit nearly indistinguishable cost performance, they are uniformly inadequate from a reliability standpoint. What initially appears as a stable and economically efficient set of policies is revealed, upon deeper scrutiny, to be structurally flawed, since reliability deficits and short-horizon risks overwhelm any apparent economic advantage.

At the global scale, the strategies achieve a deceptive stability. Average cost rates remain tightly clustered around 4.46–4.49, inspection frequencies converge to approximately 154 per horizon, and the preventive-to-corrective ratio stabilizes at about 1.6. On the surface, such convergence could be interpreted as robustness: the strategies appear to yield consistent economic outcomes across stochastic realizations of degradation. Yet this robustness is illusory. Kaplan–Meier survival probabilities collapse to approximately 0.16 across all policies, far below the prescribed requirement of R_min = 0.95. In other words, cost stability is purchased at the expense of catastrophic reliability degradation. The low survival probabilities (R ≈ 0.16) observed across all strategies do not signify deficiencies in model calibration, but rather confirm that the simulated system operates under deliberately challenging stochastic conditions. This setting enables the framework to diagnose the breakdown of conventional CBM decision rules when degradation volatility and monitoring uncertainty are high. Consequently, the observed reliability shortfalls represent a meaningful test outcome that validates the stress-testing capability of the proposed methodology.

The short-horizon analysis clarifies the mechanisms behind this failure. Between-inspection failure probabilities remain unacceptably high, with means near 0.25 and upper quantiles approaching unity. More than 64% of inspection intervals exceed the tolerance level β = 0.05, regardless of strategy. Preventive thresholds, even when adaptively adjusted, remain confined to a narrow operational band (≈ 2.5–3.2), providing insufficient conservatism to intercept degradation paths before failure. The adaptive interval strategy (ΔT_k, M_k) yields marginally fewer corrective events, but the improvement is trivial and inconsistent. Threshold adaptation (ΔT, M_k) performs even less effectively, generating outcomes that are statistically indistinguishable from the baseline fixed-interval strategy. Thus, the adaptive mechanisms, as parametrized, are too conservative to exert meaningful influence over short-term risk dynamics.

The penalty-augmented evaluation crystallizes these insights into a single metric of strategic adequacy. Once penalties for reliability shortfalls and excessive short-horizon risks are incorporated, the apparent cost efficiency of all policies is eclipsed by orders of magnitude. Augmented objectives reach values near 8500, driven overwhelmingly by reliability penalties of ≈ 7900 units and short-horizon risk penalties of ≈ 640 units. Subtle ranking differences emerge ((ΔT, M) performs marginally better than (ΔT_k, M_k), which in turn outperforms (ΔT, M_k)) yet these distinctions are numerically insignificant compared with the sheer dominance of the penalty terms. The message is unambiguous: none of the strategies, in their current formulation, can be considered viable in reliability-sensitive contexts.

From an engineering perspective, these findings underscore several key implications. First, the adaptation coefficients employed in this study (α_T = 0.40, α_L = 0.30) are too conservative to produce material shifts in system behavior. They adjust inspection intervals and preventive thresholds incrementally, but such adjustments are drowned out by the stochastic volatility of degradation trajectories. Stronger, more aggressive adaptive rules (or even hybrid control strategies with discontinuous or state-contingent responses) will be necessary to produce substantive improvements. Second, the reliance on frequent inspections (≈ 154 per horizon) highlights a structural inefficiency: inspection activities consume considerable downtime yet fail to preempt failures effectively. Unless inspection information is leveraged more decisively to trigger preventive interventions, increasing inspection cadence merely inflates downtime without improving reliability. Third, the dominance of corrective maintenance, which continues to account for roughly 40% of all interventions, demonstrates that any strategy that tolerates frequent failures is inherently unsustainable, regardless of its direct cost performance.

At the managerial level, the results emphasize the inadequacy of cost-centered decision-making. Superficially low and stable cost rates conceal the true operational risks borne by the system. When penalties for unreliability are considered, it becomes clear that the “hidden costs” of downtime, production loss, and safety hazards vastly outweigh any savings achieved by conservative preventive scheduling. For decision-makers, the implication is unequivocal: maintenance strategies must be judged not only on economic grounds but also on their capacity to satisfy reliability and risk constraints. Failure to adopt such a perspective risks endorsing policies that are economically efficient yet operationally catastrophic.

Finally, the methodological implications are equally important. The convergence of outcomes across all three strategies demonstrates the limitations of incremental adaptation. Adjusting intervals or thresholds by modest factors cannot overcome the inherent variability of stochastic degradation processes. Future research must therefore explore hybrid maintenance policies that embed risk constraints directly into the decision logic—for example, enforcing upper bounds on p_hit, dynamically tightening preventive thresholds when survival probabilities fall below a critical margin, or integrating predictive models that anticipate degradation volatility. Moreover, multi-objective optimization frameworks that balance cost, availability, and risk in a unified manner are necessary to design strategies that are not only cost-efficient but also reliability-compliant.

In summary, the synthesis of Sections 6.1–6.3 demonstrates that while the three CBM strategies converge toward stable economic outcomes, they all fail when evaluated against reliability requirements and short-horizon risk constraints. Adaptation, as presently parameterized, is too weak to alter this trajectory. For practical deployment in reliability-critical systems, maintenance strategies must either adopt much stronger adaptive mechanisms or undergo a fundamental redesign to integrate risk and reliability as primary objectives, rather than treating them as secondary considerations. Only through such innovations can CBM policies achieve a balanced and sustainable compromise between economic efficiency, operational continuity, and system safety.

Our penalty-dominance bounds (λ_R > 4.5/0.79, λ_hit > 4.5/0.64) and the multi-order sensitivity sweep demonstrate that conclusions are robust to wide variations in penalty weights. Equivalently, a strictly constrained formulation would flag all three strategies as infeasible, leading to the same engineering judgment independent of penalty magnitudes.

While the numerical differences among the three strategies may appear modest, this outcome is itself significant. It indicates that mild adaptive adjustments of inspection intervals and preventive thresholds (α_T = 0.4, α_L = 0.3) are insufficient to overcome the intrinsic stochastic variability of degradation. This finding delineates the practical limits of conventional adaptive CBM frameworks and underscores the need for more aggressive or risk-informed adaptation mechanisms. Thus, the contribution of this study resides not only in demonstrating superior performance, but in revealing the structural conditions under which adaptive maintenance ceases to offer tangible gains.

The parameter-sensitivity extension confirms that the findings are not contingent on a particular calibration of α_T, α_L, or σ_meas. Instead, the convergence of performance across strategies reflects a structural limitation of current CBM adaptation mechanisms rather than parameter tuning.

The current study’s scope was limited to static and adaptive CBM policies to isolate the intrinsic effect of adaptivity under stochastic uncertainty. Although no direct comparison was made with P–F or predictive maintenance strategies, the proposed simulation architecture is inherently compatible with such approaches. For instance, the adaptive inspection rule (Eq. (5)) can be reformulated as a P–F-based or prognostic trigger, and the preventive threshold (Eq. (7)) can be linked to probability-of-failure predictions from data-driven models. Future work will integrate these strategies to establish a unified performance comparison between classical, adaptive, and predictive paradigms.

To provide practical guidance for engineers, the results of the sensitivity study have been translated into concrete recommendations for parameter tuning and strategy enhancement. Moderate adaptive coefficients (α_T ≈ 0.6–0.8,α_L ≈ 0.4–0.6) offer an effective compromise between reliability and cost, while excessive adaptation (α_T>0.9) yields diminishing returns due to inspection overhead. When sensor noise is high (σ_meas > 0.07), filtering techniques and more conservative preventive thresholds are advised. Furthermore, implementing a hybrid policy that explicitly constrains the between-inspection failure probability (p_hit ≤ β) ensures risk compliance and enhances reliability robustness. These prescriptions provide actionable guidelines for deploying adaptive CBM strategies in practice.

We gratefully acknowledge the invaluable support and guidance provided by the Department of Mechanical Engineering, Energetic team, Mechanical and Industrial Systems (EMISys), Mohammadia School of Engineers, Mohammed V University, Rabat, Morocco. We also extend our appreciation to the anonymous reviewers for their insightful feedback.