Planning-Informed Metric (PI-Metric)

• The PI-Metric is a unified scalar measure that evaluates predictive maintenance by quantifying cost-rate degradation relative to an ideal policy baseline.
• It integrates renewal-reward theory and decision-theoretic foundations to assess maintenance policies in replacement and ordering scenarios.
• The metric supports end-to-end learning by optimizing model hyperparameters and threshold policies, thereby reducing average maintenance costs.

The Planning-Informed Metric (PI-Metric) is a quantitative performance measure for predictive algorithms and maintenance policies, designed to directly assess the downstream cost impact of algorithmic decisions under uncertainty. Its formalization provides a unified framework for evaluating and tuning Prognostic Health Management (PHM) and Predictive Maintenance (PdM) pipelines, with a focus on realistic decision settings involving replacement and inventory management. The PI-Metric also enables optimization of model hyper-parameters and maintenance policy thresholds directly against the planning objective, connecting statistical prediction and operational utility in a single, decision-oriented scalar quantity (Kamariotis et al., 2023).

1. Decision-Theoretic Foundations and Policy Contexts

The PI-Metric is defined within the context of operational maintenance planning, predicated on discrete-epoch decisions under monitored component degradation. Two key decision settings are addressed:

• Replacement Planning: At each discrete time $t_k = k\Delta T$, the decision maker chooses between preventive replacement (PR) and doing nothing (DN), absent inventory or ordering constraints.
• Ordering plus Replacement Planning: Decisions include advance ordering of spares with a lead time $L$ and subsequent replacement action; lead times and inventory introduce additional cost components such as downtime (unavailability) and holding costs.

PdM policies in these settings include both heuristics (e.g., probability thresholds on predicted remaining useful life) and full-distribution cost-minimizing strategies, as well as corrected formulations that account for the renewal nature of maintenance cycles (Kamariotis et al., 2023).

2. Formal Definition of the PI-Metric

The PI-Metric is built atop the renewal-reward theorem, which relates long-run average costs to expected costs and expected life-cycle durations. Its core quantities are:

• Empirical cost rate:

$R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$

where $E[C_m]$ is the expected total maintenance cost per renewal and $E[T_{\mathrm{lc}}]$ is the expected interval between renewals.

• Perfect policy baseline: When true failure times $T_F$ are known, the best action sequence eliminates corrective replacements entirely, yielding the lowest achievable cost rate $R_{\rm perfect}$.
• Relative cost-rate degradation (PI-Metric):

$$M = \frac{R - R_{\rm perfect}}{R_{\rm perfect}} \geq 0$$

Here, $M=0$ corresponds to perfect downstream planning; increasing $M$ quantifies cost inflation due to prognostic or policy-induced suboptimality (Kamariotis et al., 2023).

3. Estimation Procedure from Run-to-Failure Data

Calculation of the PI-Metric proceeds by simulating or observing complete run-to-failure cycles under the proposed PdM policy, with or without prediction noise:

• For each component $L$0, record total incurred costs $L$1 and life-cycle durations $L$2.
• Estimate policy cost rate and perfect baseline:

$L$3

• Compute the estimated PI-Metric:

$L$4

• Statistical uncertainty in $L$5 is dominated by the sample size $L$6 of run-to-failure experiments; first-order variance approximations can be employed (Kamariotis et al., 2023).

4. Policy and Model Optimization Using the PI-Metric

The PI-Metric serves as both an evaluative and optimization criterion for maintenance policies and prognostic algorithms. Specific applications include:

• Threshold Policy Optimization: Thresholds $L$7 in heuristic policies can be directly tuned via:

$L$8

• End-to-End Learning: Model hyper-parameters $L$9 in RUL predictors can be optimized by minimizing $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$0 as a loss:

$R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$1

• Tuning against $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$2 explicitly drives the learning and policy process to minimize downstream operational costs rather than intermediate prediction errors (Kamariotis et al., 2023).

5. Empirical Evaluation: Numerical and Real-World Scenarios

Empirical analyses demonstrate the unique properties and utility of the PI-Metric:

• Numerical Experiments: Virtual RUL simulations with controlled noise parameters ($R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$3) and cost ratios highlight that full-distribution policies outperform naive thresholding at high uncertainty, while threshold optimization achieves lowest $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$4 at the cost of potential overfitting when $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$5 is small.
• Turbofan Engine Case Study: On the CMAPSS FD001 run-to-failure set, various prognostic algorithms (LSTM classifier, Gaussian naive Bayes, decision tree, Bayesian exponential degradation regressor) are compared using both policy types. The LSTM classifier achieves consistently lowest $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$6, with policy and threshold optimization yielding up to 30% reduction in $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$7 values for cost-effective inventory settings (Kamariotis et al., 2023).

Policy Type	Data Requirement	Complexity
Threshold-based	Sufficient $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$8; higher uncertainty	Low
Full-distribution	Large $R = \frac{E[C_m]}{E[T_{\mathrm{lc}}]}$9 beneficial	High

6. Guidelines, Interpretation, and Limitations

The PI-Metric is decision-oriented, focusing uniquely on integration of prognostic uncertainty, model class (classifier or regressor), and operational costs. Key guidelines include:

• Engineering cost parameters must be supplied, as $E[C_m]$0 is sensitive to $E[C_m]$1, $E[C_m]$2, $E[C_m]$3, $E[C_m]$4, $E[C_m]$5, and $E[C_m]$6.
• Comparison across model classes is facilitated provided either Pr(RUL $E[C_m]$7) or full RUL PDFs are available.
• Model-free and data-driven: robust to modeling errors, provided sufficient empirical run-to-failure data are available. Small sample sizes ($E[C_m]$8) exacerbate uncertainty in $E[C_m]$9 and destabilize threshold optimization.
• Simple heuristic PdM policies are easily optimized via $E[T_{\mathrm{lc}}]$0, while full-distribution approaches are more robust at high uncertainty or when renewal theory corrections (policy 3) are warranted (Kamariotis et al., 2023).

7. Significance and Scope of the PI-Metric

The Planning-Informed Metric provides a unifying scalar performance measure that bridges predictions, actionable maintenance planning, and long-term cost rates. It enables direct, data-driven comparison and optimization of PdM policies and prognostics in a manner aligned with the operational objective of minimizing average maintenance costs per unit time. The metric has been demonstrated to be practically useful for empirical evaluation, policy selection, algorithm benchmarking, and “end-to-end” learning for PHM system design (Kamariotis et al., 2023).