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REVIEW 2 major objections 5 minor 72 references

Branching out: Prognostics-Based Replacement Policies for Series Systems

T0 review · 2 major / 5 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read A decision-tree replacement policy with no policy tuning beats optimized maintenance benchmarks by up to 35%.

desk verdict A solid, transparent extension of the doa framework to series systems, with impressive simulated gains that should be read as conditional on the benchmark optimization and simulated setting. read the letter →

arxiv 2607.27899 v1 pith:DIKI7AGI submitted 2026-07-30 math.OC

Daniel Koutas , Daniel Straub This is my paper
classification math.OC MSC 90B2590C40
keywords predictivemaintenancemulti-componentseriessystemsrenewaltheorydecisiontreesremainingusefullifeeconomicdependencereplacementheuristicslow-datarobustness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that predictive replacement decisions in a series system of components can be made with a transparent, low-data heuristic that outperforms tuned benchmark policies. It builds a one-step decision tree that scores every possible 'replace now / do nothing' action vector by expected cost, where future costs are priced through renewal theory as a per-component long-run cost rate. The only parameters of these discrete-option-assessment (doa) policies are those cost rates, and the paper initializes them from classic age-based replacement rather than by optimizing policy performance. On a simulated two-component system, the best doa variant cuts the long-run cost rate by up to 35% relative to optimized component-threshold and system-reliability policies (up to 65% for five components), and it remains stable when only a handful of run-to-failure samples are available. A sympathetic reader would care because interpretable maintenance rules that work with scarce data are precisely what many engineering settings need.

What carries the argument

The central object is the one-step doa decision tree: with M components, it enumerates 2^M action vectors (each component either preventively replaced or left alone) and, for each, 2^M possible failure/survival outcomes of the next interval Δt. Branch probabilities use each component's predicted RUL distribution, and branch costs are the sum of direct replacement/failure costs plus a renewal-reward term; the key identity is cF,i = cc − c∞,i·E[RUL_i | RUL_i ≤ Δt], which prices the value of keeping a component alive. The policy parameters are the component cost rates c∞,i, initialized by minimizing the renewal-theory age-based replacement cost function—not by tuning to policy performance. This

What would settle it

Run the same cost comparison with the unrejected, full normal TTF distribution so that fresh components can fail within Δt; if doa1's cost-rate advantage over the optimized benchmarks shrinks or reverses, the zero-failure-after-replacement assumption is what carries the result.

Watch

Extended reading notes

Core claim

The central claim, stated the way the authors would state it, is that a doa (discrete option assessment) policy—constructed from a one-step decision tree whose branch probabilities come from each component's predicted remaining-useful-life distribution and whose branch costs include a renewal-theory term valuing life extension—can be deployed with parameters initialized by age-based replacement and still achieve lower long-run maintenance cost per unit time than benchmark policies whose parameters are optimized with a genetic algorithm. The paper reports cost-rate reductions of up to 35% for a 2-component system and up to 65% for a 5-component system, with the largest gains at high fixed (se

Load-bearing premise

The load-bearing premise is that a newly replaced component cannot fail before the next decision point; the evaluation enforces this by discarding any simulated fresh component whose lifetime is shorter than the decision interval, so the cost comparisons hold only for systems whose infant-mortality risk within one interval is negligible.

Editorial extensions

If this is right

  • If the claim holds, maintenance planners can skip policy-performance optimization entirely: initializing component cost rates from age-based replacement is enough to beat tuned threshold policies, especially when fixed replacement costs are high.
  • The performance gap widens with system size: the reported reduction grows from 35% at two components to 65% at five components, so the approach scales in the direction where benchmarks struggle.
  • In low-data regimes (10–100 run-to-failure samples per component), doa policies show better median cost and lower variance than optimized benchmarks, because initialization is univariate and does not chase a noisy performance surface.
  • Operational changes—new replacement methods, added/removed components—only require recomputing the component cost rates and updating the decision-tree cost formulas, not re-optimizing a joint parameter vector.
  • Among the two variants, doa1 (replace-at-next-opportunity proxy) is recommended across the investigated range; doa2 becomes preferable only at extreme component-cost imbalances.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The reported cost reductions are computed under a truncated lifetime distribution: the simulations reject new components that fail within the first decision interval, so real systems with infant mortality would likely see a smaller advantage—the 35%/65% figures are an upper bound.
  • Because the doa policy's parameters depend only on time-to-failure samples, not on the prognostic model, its low-data robustness may persist even when RUL predictions are poorly calibrated—an interaction worth testing explicitly.
  • The 2^(2M)-branch decision tree limits the method to small numbers of components; the paper's suggested branch-pruning bounds could make the approach practical for larger M.
  • A hybrid variant that reintroduces economic dependence in doa2's continuous replacement times may close the remaining gap to doa1 and should be testable with the same simulator.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes 'discrete option assessment' (doa) heuristic policies for preventive replacement of M-component series systems with economic dependence. At each decision epoch, a one-step decision tree is constructed: action branches correspond to joint PR/DN choices, and consequence branches correspond to component survival/failure over the interval Δt. Branch probabilities are computed from predicted RUL distributions; branch costs include direct replacement/failure costs plus renewal-theory-based continuation costs via component cost rates c∞,i. Two variants are derived (doa1: survivors assumed replaced at the next decision opportunity; doa2: survivors assumed replaced at their conditional expected failure time). The parameters c∞,i are initialized from an age-based replacement objective rather than optimized against final policy performance. Numerical experiments on a virtual RUL simulator with TTF ~ N(225,40) report up to 35% lower long-run cost rates than GA-optimized threshold benchmarks for a 2-component system and up to 65% for a 5-component system, and better robustness in low-data regimes (10–100 samples).

Significance. If the quantitative claims hold, the paper offers a transparent, low-data predictive maintenance heuristic for series systems that explicitly incorporates economic dependence and renewal-reward costs, extending the single-component framework of [22]. The derivation is largely self-consistent, the simulator and optimization setup are described in unusual detail, and source code is provided. The main value is a theory-guided alternative to black-box or purely data-driven heuristics, with the potential for practical deployment where failure data are scarce. The two most consequential assertions—superiority over optimized benchmarks and robustness against overfitting—rest on the adequacy of the benchmark optimization and on the negligible-early-failure assumption, both of which need sharper support.

major comments (2)
  1. [§4.2, Figs. 9–10; Appendix A] The central quantitative claim (up to 35%/65% improvement over 'optimized' benchmarks) is only as strong as the benchmark optimization. The paper uses a single GA configuration (Table A.1) with no convergence diagnostics, multiple restarts, or comparison against alternative optimizers. For rh1* with M=5, the objective is 5-dimensional and noisy; 25 generations of 250 individuals may not yield the global optimum. If the benchmarks are under-optimized, the reported cost reductions are inflated. Please either (a) provide evidence of convergence (e.g., repeated GA runs, final diversity, grid/gradient comparisons for the abundant-data cases), or (b) qualify all claims as 'compared to benchmarks optimized with a fixed-budget GA' and adjust the abstract and concluding remarks accordingly.
  2. [§3.2.2, Eq. (16); §4.1, footnote 11] Both the derivation and the simulator enforce Pr(TF ≤ Δt) ≈ 0 for newly replaced components. For the chosen TTF distribution N(225,40) and Δt=10, the truncation probability is about 4×10⁻⁸, so the numerical results are internally consistent. However, the paper's broader conclusion that the doa policies 'outperform' benchmarks as general replacement heuristics is not tested for systems in which early failures are non-negligible (as the bathtub-curve discussion acknowledges). To support the general claim, add sensitivity experiments with, e.g., smaller mean TTF, larger Δt, or a Weibull TTF with non-negligible early-failure probability, and show how Eq. (16) would be amended. If the authors prefer to limit the scope, that limitation should be stated explicitly in the abstract and conclusion.
minor comments (5)
  1. [§3.2.5, after Eq. (22)] The sentence 'Again, the subtrahend in Equation (24) represents the value...' should reference Eq. (22), not Eq. (24).
  2. [§4.1, last sentence] The text says 'An example realization of RUL predictions obtained with this simulator is shown in Figure 1', but Figure 1 is the workflow diagram; either add the example-realization figure or correct the reference.
  3. [Table A.1] The table heading 'GA parameters for rh2' is misleading because the same GA is used for rh1*; rename it to something like 'GA parameters for benchmark parameter optimization'.
  4. [§3.2.1] The term 'hybrid policies' is introduced but not defined until later; consider giving an explicit definition (or at least a forward reference to the discussion) at first use.
  5. [§4.1, footnote 11] The phrase 'slightly altering' the TTF distribution could be quantified; stating the truncation probability (≈4×10⁻⁸) would reassure readers that the assumption is indeed negligible in this setting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: doa parameter initialization is independent of the policy evaluation metric, and the acknowledged TTF truncation is a validity limitation rather than a circular reduction.

full rationale

The derivation chain is not circular. The doa policy parameters c∞,i are initialized from an age-based replacement problem via Eq. (11)-(12), whose objective is the renewal-reward cost rate of a no-monitoring age-replacement policy, not the performance of the final doa policy. The paper explicitly distinguishes this from optimization: "the optimal values for the policy parameters are found by repeated application of the respective policy to a set of components, i.e., based on policy performance information. 'Parameter initialization', on the other hand, refers to some kind of selection rule of the parameters, which is not based on policy performance." The branch costs in Eqs. (17), (22), and (24) use these initialized cost rates, while the evaluation metric in Eqs. (3)-(4) is a Monte Carlo estimate of the policy's actual long-run cost rate. Thus the policy is not fitting its own evaluation metric. The self-citation [22] supplies a general one-step decision-tree methodology, but the multi-component extension is derived in this paper and the numerical comparisons are new; no load-bearing result is reduced to an unverified self-citation. The footnote 11 truncation of the TTF distribution (rejecting new components with failure time below Δt) is an acknowledged limitation that conditions the numerical results, but it is not circular: the benchmarks are evaluated under the same truncated distribution, and the paper explicitly notes that the decision tree can be amended to include the failure probability. The claimed cost reductions are therefore conditional on a stated assumption, but they do not reduce by construction to the inputs of the model.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The proposed doa policies introduce no new physical entities; their central parameters are the component cost rates c_infinity,i, initialized from age-based replacement, plus the fixed decision interval Delta t. The benchmark thresholds are free parameters optimized by a genetic algorithm and affect the strength of the comparison. The strongest auxiliary assumptions are the zero-failure probability of new components within Delta t and the 'replace later' proxies, both acknowledged in the text but load-bearing for the numerical claims.

free parameters (3)
  • c_infinity,i (component cost rates) = c0_infinity,i from Eq. (12); values depend on TTF training samples and costs
    Appear in cF,i (Eq. 17) and cI,i (Eqs. 22/24). Initialized by age-based replacement optimization (Eqs. 11-12), not by optimizing final policy performance; the paper labels this 'initialization' rather than 'optimization'.
  • Decision interval Delta t = 10 (fixed)
    Discretizes decision times and enters pF,i, E[RUL|...], and cI,i cost terms. Results are demonstrated only for Delta t = 10.
  • Benchmark thresholds p_thres,i, r_thres = values found by GA per setting
    Define the 'optimized benchmark' baselines; their quality directly affects the claimed outperformance. A single GA was used, with the authors acknowledging possible suboptimality.
assumptions (6)
  • domain assumption Failures are self-announcing; replacements are perfect, immediate and instantaneous; successive lifetimes are i.i.d. with finite mean.
    Sets up renewal-reward theory (Section 2.2) and Eq. (3).
  • ad hoc to paper Newly replaced components have zero probability of failure within the next decision interval Delta t.
    Used to zero out branches in Eq. (16); enforced in simulation by rejecting TF<Delta t samples (Section 4.1 fn. 11), altering the nominal TTF distribution.
  • domain assumption Predicted RUL-PDFs from the prognostic model are accurate enough for decision-making (only 'slightly biased').
    The policy input; if predictions are miscalibrated, branch probabilities in Eq. (15) are wrong.
  • ad hoc to paper For doa1, a surviving component is assumed preventively replaced at the next decision opportunity; for doa2, at its conditional expected failure time.
    These proxies define Eqs. (22)/(24) and are acknowledged as approximate/conservative/non-conservative.
  • domain assumption After replacement/failure, future maintenance costs can be represented by the component's long-run average cost rate c_infinity,i.
    Renewal-theory substitution in Eqs. (17) and (22)/(24); assumes population average applies to individual future.
  • domain assumption Component lifetimes/RULs are stochastically independent.
    Branch probabilities factor as products in Eq. (15); only economic dependence considered (Sections 2.1, 5.2.3).

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Cite this review

Pith. "Pith review of Branching out: Prognostics-Based Replacement Policies for Series Systems." pith.science (2026). https://pith.science/paper/DIKI7AGI

@misc{pith2026260727899,
  author       = {Pith},
  title        = {Pith review of: Branching out: Prognostics-Based Replacement Policies for Series Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DIKI7AGI}},
  note         = {Machine review of arXiv:2607.27899}
}
read the original abstract

We propose a hybrid planning method for deriving prognostics-based predictive maintenance policies. The method accounts for the available decision options, the information on the future state of the system provided by a prognostic model, and the costs of the underlying renewal-reward process. It results in policies defined by only a few parameters, which can be determined based on theoretical considerations or by optimization from run-to-failure data. We demonstrate the potential of the method in two separate predictive maintenance decision settings: preventive replacement and preventive ordering. Numerical investigations show that the derived policies rival the performance of optimized benchmark policies, while being significantly more efficient and robust against overfitting.

Figures

Figures reproduced from arXiv: 2607.27899 by the authors.

Figure 1
Figure 1. General workflow3of multi-component predictive maintenance policies. At every time step tk, new SHM data is gathered for each component, fed into the component-specific prognostic models, which output corresponding RUL distributions. These are then collectively fed into a policy (π), which outputs a joint action vector, assigning each component either a “do nothing” or “maintain” decision. 2.3 Policy evaluation In p… view at source ↗
Figure 2
Figure 2. Long-running maintenance costs per unit time plotted over the number of time steps of length [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. General method for deriving PdM policies i. Construct a simplified decision tree based on the avail￾able maintenance options and their potential conse￾quences. ii. Assign probabilities to each branch based on the RUL￾PDFs obtained by the prognostic models. iii. Assign to each branch the expected costs resulting from a) the monitored components, b) the underlying renewal-reward processes. iv. Simplify the cost expres… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Decision tree for a deteriorating component that can be preventively replaced. Component states are [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Decision tree for a series system of two deteriorating components that can be preventively replaced. System [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Comparison of incurred costs preventive vs. corrective replacement of a component; taken from [22]. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Depiction of doa’s initialization target function for a component with cf = 10, cv = 10, cc = 100 computed with 10 (left) and 1, 000 (right) failure samples. 3.4.2 Parameter optimization with few training data Finding a heuristic’s optimal parameter values with respect…
Figure 8
Figure 8. Figure 8: rh2 performance (Cˆ∞) over the domain rthres ∈ [0, 1] shown for a set of 10 (left) and 1, 000 (right) training data samples per component. The specific setup is the same as in Section 4.3. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 8
Figure 8. Figure 8: Optimization of this noisy target function (especially in higher dimensions) is challenging. Due to the ubiquity of these problems in many scientific areas, various approaches for handling this noise have been developed over the past decades, such as Bayesian optimizat…
Figure 9
Figure 9. Figure 9: Comparison of the proposed doa policies with the benchmarks rh1 ∗ and rh2 ∗ for a 2-component system over a range of fixed costs cf . The following settings were used for training the optimized policies as well as for computing the system’s c∞ via Equation (4) with Mon…
Figure 10
Figure 10. Figure 10: Comparison of the proposed doa policies with the benchmarks rh1 ∗ and rh2 ∗ for a 5-component system over a range of fixed costs cf . The following settings were used for training the optimized policies as well as for computing the system’s c∞ via Equation (4) with Mo…
Figure 11
Figure 11. Figure 11: Comparison of the proposed doa policies with the benchmarks rh1 ∗ and rh2 ∗ for a 2-component system over a range of variable costs cv,2. The following settings were used for training the optimized policies as well as for computing the system’s c∞ via Equation (4) wit…
Figure 12
Figure 12. Figure 12: Boxplot comparison of the proposed doa policies with initialized parameters against the benchmarks rhi∗ with optimized parameters. A 5-component system is used with cc = 100, cf = 6, cv = [2, 4, 6, 8, 10]. The initialization/optimization was carried out for 10 and 100…

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Works this paper leans on

72 extracted references · 2 linked inside Pith

  1. [22]

    Leaf it to renewal: Improved predictive maintenance policies via renewal theory and decision trees.arXiv preprint arXiv:2509.20145, 2025

    Daniel Koutas and Daniel Straub. Leaf it to renewal: Improved predictive maintenance policies via renewal theory and decision trees.arXiv preprint arXiv:2509.20145, 2025

    arXiv 2025

  2. [1]

    A condition-based maintenance policy with non- periodic inspections for a two-unit series system.Reliability Engineering & System Safety, 87(1):109–120, 2005

    Bruno Castanier, Antoine Grall, and Christophe Bérenguer. A condition-based maintenance policy with non- periodic inspections for a two-unit series system.Reliability Engineering & System Safety, 87(1):109–120, 2005

    2005

  3. [2]

    Multi-level predictive maintenance for multi-component systems

    Kim-Anh Nguyen, Phuc Do, and Antoine Grall. Multi-level predictive maintenance for multi-component systems. Reliability Engineering & System Safety, 144:83–94, 2015

    2015

  4. [3]

    A Review of Multi-Component Maintenance Models with Economic Dependence.Mathematical Methods of Operations Research, 45(3):411–435, 1997

    Rommert Dekker, Ralph E Wildeman, and Frank A Van der Duyn Schouten. A Review of Multi-Component Maintenance Models with Economic Dependence.Mathematical Methods of Operations Research, 45(3):411–435, 1997

    1997

  5. [4]

    Optimal Maintenance of Multi-Component Systems: a Review.Complex System Maintenance Handbook, pages 263–286, 2008

    Robin P Nicolai and Rommert Dekker. Optimal Maintenance of Multi-Component Systems: a Review.Complex System Maintenance Handbook, pages 263–286, 2008

    2008

  6. [5]

    ISO 13306:2017 — Maintenance terminology

    International Organization for Standardization. ISO 13306:2017 — Maintenance terminology. Standard, 2017. ISO 13306:2017

    2017

  7. [6]

    Elsevier, 2002

    R Keith Mobley.An Introduction to Predictive Maintenance. Elsevier, 2002

    2002

  8. [7]

    Prognostics and Health Management of Electronics.Encyclopedia of Structural Health Monitoring, 2009

    Michael Pecht. Prognostics and Health Management of Electronics.Encyclopedia of Structural Health Monitoring, 2009

    2009

Show all 72 references
  1. [8]

    A Survey of Predictive Maintenance: Systems, Purposes and Approaches.arXiv preprint arXiv:1912.07383, pages 1–36, 2019

    Yongyi Ran, Xin Zhou, Pengfeng Lin, Yonggang Wen, and Ruilong Deng. A Survey of Predictive Maintenance: Systems, Purposes and Approaches.arXiv preprint arXiv:1912.07383, pages 1–36, 2019

    1912 arXiv

  2. [9]

    Condition-based dynamic maintenance operations planning & grouping

    Keomany Bouvard, Samuel Artus, Christophe Bérenguer, and Vincent Cocquempot. Condition-based dynamic maintenance operations planning & grouping. Application to commercial heavy vehicles.Reliability Engineering & System Safety, 96(6):601–610, 2011

    2011

  3. [10]

    A dynamic predictive maintenance policy for complex multi- component systems.Reliability Engineering & System Safety, 120:39–50, 2013

    Adriaan Van Horenbeek and Liliane Pintelon. A dynamic predictive maintenance policy for complex multi- component systems.Reliability Engineering & System Safety, 120:39–50, 2013

    2013

  4. [11]

    A segmental hidden semi-Markov model (HSMM)-based diagnostics and prognostics framework and methodology.Mechanical systems and signal processing, 21(5):2248–2266, 2007

    Ming Dong and David He. A segmental hidden semi-Markov model (HSMM)-based diagnostics and prognostics framework and methodology.Mechanical systems and signal processing, 21(5):2248–2266, 2007

    2007

  5. [12]

    Remaining useful life estimation in prognostics using deep convolution neural networks.Reliability Engineering & System Safety, 172:1–11, 2018

    Xiang Li, Qian Ding, and Jian-Qiao Sun. Remaining useful life estimation in prognostics using deep convolution neural networks.Reliability Engineering & System Safety, 172:1–11, 2018

    2018

  6. [13]

    Prognostics and Health Management of Engineering Systems

    Nam-Ho Kim, Dawn An, and Joo-Ho Choi. Prognostics and Health Management of Engineering Systems. Switzerland: Springer International Publishing, 2017

    2017

  7. [14]

    Remaining useful life estimation–A review on the statistical data driven approaches.European Journal of Operational Research, 213(1):1–14, 2011

    Xiao-Sheng Si, Wenbin Wang, Chang-Hua Hu, and Dong-Hua Zhou. Remaining useful life estimation–A review on the statistical data driven approaches.European Journal of Operational Research, 213(1):1–14, 2011. 22 Branching OutA PREPRINT

    2011

  8. [15]

    Nailong Zhang and Wujun Si. Deep reinforcement learning for condition-based maintenance planning of multi- component systems under dependent competing risks.Reliability Engineering & System Safety, 203:107094, 2020

    2020

  9. [16]

    A deep reinforcement learning approach for maintenance planning of multi-component systems with complex structure.Neural Computing and Applications, 35(21):15549–15562, 2023

    Jiahao Chen and Yu Wang. A deep reinforcement learning approach for maintenance planning of multi-component systems with complex structure.Neural Computing and Applications, 35(21):15549–15562, 2023

    2023

  10. [17]

    A deep reinforcement learning approach for repair-based maintenance of multi-unit systems using proportional hazards model.Reliability Engineering & System Safety, 234:109179, 2023

    Seyedvahid Najafi and Chi-Guhn Lee. A deep reinforcement learning approach for repair-based maintenance of multi-unit systems using proportional hazards model.Reliability Engineering & System Safety, 234:109179, 2023

    2023

  11. [18]

    Guided probabilistic reinforcement learning for sampling-efficient maintenance scheduling of multi-component system.Applied Mathematical Modelling, 119:677–697, 2023

    Yiming Zhang, Dingyang Zhang, Xiaoge Zhang, Lemiao Qiu, Felix TS Chan, Zili Wang, and Shuyou Zhang. Guided probabilistic reinforcement learning for sampling-efficient maintenance scheduling of multi-component system.Applied Mathematical Modelling, 119:677–697, 2023

    2023

  12. [19]

    Optimal Adaptive Inspection and Maintenance Planning for Deteriorating Structural Systems.Reliability Engineering & System Safety, 215:107891, 2021

    Elizabeth Bismut and Daniel Straub. Optimal Adaptive Inspection and Maintenance Planning for Deteriorating Structural Systems.Reliability Engineering & System Safety, 215:107891, 2021

    2021

  13. [20]

    A metric for assessing and optimizing data-driven prognostic algorithms for predictive maintenance.Reliability Engineering & System Safety, 242:109723, 2024

    Antonios Kamariotis, Konstantinos Tatsis, Eleni Chatzi, Kai Goebel, and Daniel Straub. A metric for assessing and optimizing data-driven prognostic algorithms for predictive maintenance.Reliability Engineering & System Safety, 242:109723, 2024

    2024

  14. [21]

    Rui He, Zhigang Tian, Yifei Wang, Mingjian Zuo, and Ziwei Guo. Condition-based maintenance optimization for multi-component systems considering prognostic information and degraded working efficiency.Reliability Engineering & System Safety, 234:109167, 2023

    2023

  15. [23]

    Renewal Theory From the Point of View of the Theory of Probability.Transactions of the American Mathematical Society, 63(3):422–438, 1948

    Joseph L Doob. Renewal Theory From the Point of View of the Theory of Probability.Transactions of the American Mathematical Society, 63(3):422–438, 1948

    1948

  16. [24]

    Renewal Theory and Its Ramifications.Journal of the Royal Statistical Society Series B: Statistical Methodology, 20(2):243–284, 1958

    Walter L Smith. Renewal Theory and Its Ramifications.Journal of the Royal Statistical Society Series B: Statistical Methodology, 20(2):243–284, 1958

    1958

  17. [25]

    Opportunistic replacement of a single part in the presence of several monitored parts.Management Science, 10(1):70–84, 1963

    Roy Radner and Dale W Jorgenson. Opportunistic replacement of a single part in the presence of several monitored parts.Management Science, 10(1):70–84, 1963

    1963

  18. [26]

    Maintenance Scheduling for Multicomponent Equipment.AIIE Transac- tions, 9(3):297–305, 1977

    Roger C Vergin and Michael Scriabin. Maintenance Scheduling for Multicomponent Equipment.AIIE Transac- tions, 9(3):297–305, 1977

    1977

  19. [27]

    An Optimum Group Maintenance Policy.Naval Research Logistics Quarterly, 30 (4):667–674, 1983

    K Okumoto and EA Elsayed. An Optimum Group Maintenance Policy.Naval Research Logistics Quarterly, 30 (4):667–674, 1983

    1983

  20. [28]

    A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System.Journal of the Operational Research Society, 42(4):331–341, 1991

    Shey-Huei Sheu. A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System.Journal of the Operational Research Society, 42(4):331–341, 1991

    1991

  21. [29]

    A survey of maintenance models for multi-unit systems.European Journal of Operational Research, 51(1):1–23, 1991

    Danny I Cho and Mahmut Parlar. A survey of maintenance models for multi-unit systems.European Journal of Operational Research, 51(1):1–23, 1991

    1991

  22. [30]

    Availability optimization of systems subject to competing risk.European Journal of Operational Research, 202(3):781–788, 2010

    Yada Zhu, Elsayed A Elsayed, Haitao Liao, and Ling-Yau Chan. Availability optimization of systems subject to competing risk.European Journal of Operational Research, 202(3):781–788, 2010

    2010

  23. [31]

    A review on condition-based maintenance optimization models for stochastically deteriorating system.Reliability Engineering & System Safety, 157:54–63, 2017

    Suzan Alaswad and Yisha Xiang. A review on condition-based maintenance optimization models for stochastically deteriorating system.Reliability Engineering & System Safety, 157:54–63, 2017

    2017

  24. [32]

    A review on maintenance optimization.European Journal of Operational Research, 285(3):805–824, 2020

    Bram De Jonge and Philip A Scarf. A review on maintenance optimization.European Journal of Operational Research, 285(3):805–824, 2020

    2020

  25. [33]

    System Maintenance Scheduling With Prognostics Information Using Genetic Algorithm.IEEE Transactions on Reliability, 58(3):539–552, 2009

    Fatih Camci. System Maintenance Scheduling With Prognostics Information Using Genetic Algorithm.IEEE Transactions on Reliability, 58(3):539–552, 2009

    2009

  26. [34]

    Group machinery intelligent maintenance: Adaptive health prediction and global dynamic maintenance decision-making.Reliability Engineering & System Safety, 252:110426, 2024

    Li Yang, Shihan Zhou, Xiaobing Ma, Yi Chen, Heping Jia, and Wei Dai. Group machinery intelligent maintenance: Adaptive health prediction and global dynamic maintenance decision-making.Reliability Engineering & System Safety, 252:110426, 2024

    2024

  27. [35]

    Optimisation of preventive main- tenance grouping strategy for multi-component series systems: Particle swarm based approach.Computers & Industrial Engineering, 102:440–451, 2016

    N Chalabi, Mohammed Dahane, Bouziane Beldjilali, and Abdelkader Neki. Optimisation of preventive main- tenance grouping strategy for multi-component series systems: Particle swarm based approach.Computers & Industrial Engineering, 102:440–451, 2016

    2016

  28. [36]

    A Survey of Maintenance and Replacement Models for Maintainability and Reliability of Multi-Item Systems.Reliability Engineering, 16(4):297–309, 1986

    LC Thomas. A Survey of Maintenance and Replacement Models for Maintainability and Reliability of Multi-Item Systems.Reliability Engineering, 16(4):297–309, 1986

    1986

  29. [37]

    Study of a multi-component system with failure interaction.European Journal of Operational Research, 21(3):330–338, 1985

    DNP Murthy and DG Nguyen. Study of a multi-component system with failure interaction.European Journal of Operational Research, 21(3):330–338, 1985. 23 Branching OutA PREPRINT

    1985

  30. [38]

    Oxford University Press, 2020

    Geoffrey Grimmett and David Stirzaker.Probability and Random Processes. Oxford University Press, 2020

    2020

  31. [39]

    Mean and Variance of Ratio Estimators Used in Fluorescence Ratio Imaging

    GMP Van Kempen and LJ Van Vliet. Mean and Variance of Ratio Estimators Used in Fluorescence Ratio Imaging. Cytometry: The Journal of the International Society for Analytical Cytology, 39(4):300–305, 2000

    2000

  32. [40]

    John Wiley & Sons, 2000

    Howard Raiffa and Robert Schlaifer.Applied Statistical Decision Theory, volume 78. John Wiley & Sons, 2000

    2000

  33. [41]

    MIT press, 2015

    Mykel J Kochenderfer.Decision Making Under Uncertainty: Theory and Application. MIT press, 2015

    2015

  34. [42]

    Dynamic Programming.Science, 153(3731):34–37, 1966

    Richard Bellman. Dynamic Programming.Science, 153(3731):34–37, 1966

    1966

  35. [43]

    Anytime Point-Based Approximations for Large POMDPs

    Joelle Pineau, Geoffrey Gordon, and Sebastian Thrun. Anytime Point-Based Approximations for Large POMDPs. Journal of Artificial Intelligence Research, 27:335–380, 2006

    2006

  36. [44]

    Planning and acting in partially observable stochastic domains.Artificial Intelligence, 101(1-2):99–134, 1998

    Leslie Pack Kaelbling, Michael L Littman, and Anthony R Cassandra. Planning and acting in partially observable stochastic domains.Artificial Intelligence, 101(1-2):99–134, 1998

    1998

  37. [45]

    MIT press Cambridge, 1998

    Richard S Sutton, Andrew G Barto, et al.Reinforcement Learning: An Introduction, volume 1(1). MIT press Cambridge, 1998

    1998

  38. [46]

    The Complexity of Decentralized Control of Markov Decision Processes.Mathematics of Operations Research, 27(4):819–840, 2002

    Daniel S Bernstein, Robert Givan, Neil Immerman, and Shlomo Zilberstein. The Complexity of Decentralized Control of Markov Decision Processes.Mathematics of Operations Research, 27(4):819–840, 2002

    2002

  39. [47]

    Reinforcement Learning: A Survey.Journal of Artificial Intelligence Research, 4:237–285, 1996

    Leslie Pack Kaelbling, Michael L Littman, and Andrew W Moore. Reinforcement Learning: A Survey.Journal of Artificial Intelligence Research, 4:237–285, 1996

    1996

  40. [48]

    John Wiley & Sons, 2009

    Stewart N Ethier and Thomas G Kurtz.Markov Processes: Characterization and Convergence. John Wiley & Sons, 2009

    2009

  41. [49]

    Approximate solution of Markov renewal programs with finite time horizon.SIAM Journal on Control and Optimization, 37(2):502–520, 1999

    Karl Hinderer and Karl-Heinz Waldmann. Approximate solution of Markov renewal programs with finite time horizon.SIAM Journal on Control and Optimization, 37(2):502–520, 1999

    1999

  42. [50]

    MIT press, 2022

    Mykel J Kochenderfer, Tim A Wheeler, and Kyle H Wray.Algorithms for Decision Making. MIT press, 2022

    2022

  43. [51]

    Risk-based optimal inspection strategies for structural systems using dynamic Bayesian networks.Structural Safety, 76:68–80, 2019

    Jesus Luque and Daniel Straub. Risk-based optimal inspection strategies for structural systems using dynamic Bayesian networks.Structural Safety, 76:68–80, 2019

    2019

  44. [52]

    Jannie Sønderkær Nielsen and John Dalsgaard Sørensen. Computational framework for risk-based planning of inspections, maintenance and condition monitoring using discrete Bayesian networks.Structure and Infrastructure Engineering, 14(8):1082–1094, 2018

    2018

  45. [53]

    A new dynamic predictive maintenance framework using deep learning for failure prognostics.Reliability Engineering & System Safety, 188:251–262, 2019

    Khanh TP Nguyen and Kamal Medjaher. A new dynamic predictive maintenance framework using deep learning for failure prognostics.Reliability Engineering & System Safety, 188:251–262, 2019

    2019

  46. [54]

    Probabilistic forecasts, calibration and sharpness

    Tilmann Gneiting, Fadoua Balabdaoui, and Adrian E Raftery. Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society Series B: Statistical Methodology, 69(2):243–268, 2007

    2007

  47. [55]

    Can models fitted to small data samples lead to maintenance policies with near-optimum cost?IMA Journal of Management Mathematics, 6(1):3–12, 1995

    RD Baker and PA Scarf. Can models fitted to small data samples lead to maintenance policies with near-optimum cost?IMA Journal of Management Mathematics, 6(1):3–12, 1995

    1995

  48. [56]

    A Tutorial on Bayesian Optimization.arXiv preprint arXiv:1807.02811, 2018

    Peter I Frazier. A Tutorial on Bayesian Optimization.arXiv preprint arXiv:1807.02811, 2018

    2018 arXiv

  49. [57]

    Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning.Discrete Optimization, 19:79–102, 2016

    David R Morrison, Sheldon H Jacobson, Jason J Sauppe, and Edward C Sewell. Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning.Discrete Optimization, 19:79–102, 2016

    2016

  50. [58]

    A Genetic Algorithm Tutorial.Statistics and Computing, 4(2):65–85, 1994

    Darrell Whitley. A Genetic Algorithm Tutorial.Statistics and Computing, 4(2):65–85, 1994

    1994

  51. [59]

    Adaptive Population Sizing Schemes in Genetic Algorithms

    Fernando G Lobo and Claudio F Lima. Adaptive Population Sizing Schemes in Genetic Algorithms. InParameter Setting in Evolutionary Algorithms, pages 185–204. Springer, 2007

    2007

  52. [60]

    IOP Publishing, 2000

    Thomas Bäck, David B Fogel, and Zbigniew Michalewicz.Evolutionary Computation 1: Basic Algorithms and Operators. IOP Publishing, 2000

    2000

  53. [61]

    A Comparative Analysis of Selection Schemes Used in Genetic Algorithms

    David E Goldberg and Kalyanmoy Deb. A Comparative Analysis of Selection Schemes Used in Genetic Algorithms. InFoundations of Genetic Algorithms, volume 1, pages 69–93. Elsevier, 1991

    1991

  54. [62]

    Parameter Control in Evolutionary Algorithms: Trends and Challenges.IEEE Transactions on Evolutionary Computation, 19(2):167–187, 2014

    Giorgos Karafotias, Mark Hoogendoorn, and Ágoston E Eiben. Parameter Control in Evolutionary Algorithms: Trends and Challenges.IEEE Transactions on Evolutionary Computation, 19(2):167–187, 2014

    2014

  55. [63]

    MIT press, 1998

    Melanie Mitchell.An Introduction to Genetic Algorithms. MIT press, 1998

    1998

  56. [64]

    Genetic Algorithms in Search, Optimization & Machine Learning.Addison-Wesley, 1989(102): 36, 1989

    David E Golberg. Genetic Algorithms in Search, Optimization & Machine Learning.Addison-Wesley, 1989(102): 36, 1989

    1989

  57. [65]

    Simulated Binary Crossover for Continuous Search Space

    Kalyanmoy Deb, Ram Bhushan Agrawal, et al. Simulated Binary Crossover for Continuous Search Space. Complex Systems, 9(2):115–148, 1995. 24 Branching OutA PREPRINT

    1995

  58. [66]

    A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II.IEEE Transactions on Evolutionary Computation, 6(2):182–197, 2002

    Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and TAMT Meyarivan. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II.IEEE Transactions on Evolutionary Computation, 6(2):182–197, 2002

    2002

  59. [67]

    parent” candidates from which “offspring

    Melanie Mitchell, James P Crutchfield, and Peter T Hraber. Evolving Cellular Automata to Perform Computations: Mechanisms and Impediments.Physica D: Nonlinear Phenomena, 75(1-3):361–391, 1994. 25 Branching OutA PREPRINT A Parameter optimization via genetic algorithms The basic...

    1994

  60. [68]

    Draw a failure sampleT (i) F from the underlying time-to-failure distribution

  61. [69]

    Compute the true RUL values at each time step viaRU L(i) k =T (i) F −t k

  62. [70]

    ,ln(ϵ(i) n )]from the MVN distribution in Equation (33)

    Sample an error vector[ln(ϵ (i) 1 ), . . . ,ln(ϵ(i) n )]from the MVN distribution in Equation (33)

  63. [71]

    Compute the mean values of the prediction via Equation (32):µ (i) ln(RU Lk) = ln(RU L(i) true,k) + ln(ϵ(i) k )

  64. [72]

    The predicted RUL-PDF is then a lognormal distribution, i.e.,ln RU L(i) k ∼ N µ(i) ln(RU Lk), σln(ϵk) . 28

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.