Data-Driven Maintenance Optimization: Aligning Prognostic Learning with Decision Objective

Zhuojun Xie; Adam Abdin; Yiping Fang

cs.AIJun 24, 2025

Data-Driven Maintenance Optimization: Aligning Prognostic Learning with Decision Objective

Authors: Zhuojun Xie, Adam Abdin, Yiping Fang

Organizations: Laboratoire Génie Industriel, CentraleSupélec, Université Paris-Saclay

Abstract

Estimation error in machine learning prognostic models can propagate into poor maintenance decisions. We study a data-driven predictive maintenance problem coupling health-condition estimation with a joint ordering-replacement model. We show that the conventional Estimate-Then-Optimize (ETO) approach yields inconsistent, suboptimal decisions for a broad class of such problems. To bridge the gap, we propose an Integrated Estimate-Optimize (IEO) framework that directly targets maintenance cost as the prognostic learning objective, bypassing surrogate statistical accuracy metrics such as Kullback--Leibler divergence or mean squared error. Theoretically, we prove out-of-sample performance guarantees of the IEO framework under certain conditions, including i.i.d. and sliding-window data-generating processes. Computationally, we develop a training procedure combining Monte Carlo dropout, central-difference gradient approximation, and an ETO warm-start to handle discrete maintenance decisions and nonlinear costs. Experiments on the four CMAPSS subsets demonstrate that, under the nominal setting, IEO reduces mean maintenance cost by 14.8--22.7% relative to ETO and by 7.2--11.8% relative to a decision-oriented baseline, while also reducing failure rates and cost variation. These improvements are obtained despite higher prediction errors. Increasing the ratio of unplanned-downtime costs to planned-downtime costs makes this tradeoff more pronounced: IEO shifts predictions further toward RUL underestimation, accompanied by greater cost savings and fewer failures. These findings support aligning prognostic learning with maintenance objectives when imperfect predictions must guide decisions under costly unplanned downtime.

Figures & tables

Figure 1: Target RLD $P_{0}$ , two estimates $P_{1}$ and $P_{2}$ with identical KL-div from $P_{0}$ , and their corresponding ordering, decommissioning, and replacement decisions in Example 1 . Although $P_{1}$ and $P_{2}$ have identical estimation accuracy, they induce different maintenance decisions.

Figure 2: Geometric illustration of the mismatch between RLD estimation accuracy and maintenance decisions. (a) Two close RLD estimates $P_{1}$ and $P_{2}$ can lie in adjacent decision regions and induce different optimal decisions, as characterized in Proposition 1 . (b) Two RLD estimates can have identical KL-div from the true RLD $P_{0}$ while lying in different decision regions and therefore inducing different maintenance decisions, as discussed in Remark 2 .

\hat{\xi}_{i,k}\leftarrow m(x_{i};\omega_{k},\mathbf{b}_{i,k}).

Algorithm 1 Central-Difference Surrogate Gradient Descent with Dropout

Figure 3: Architecture of the LSTM prognostic model used in the experiments. The final LSTM hidden state is passed through dropout and a fully connected prediction head containing a ReLU activation and a second dropout layer. $B$ , $W$ , and $n_{s}$ denote the batch size, sliding-window length, and number of retained sensors, respectively.

Approach	Training/Validation Criterion	Batch Size	Patience	Learning Rate
ETO	MSE/MSE	100	10 epochs	$\{1,2,5,10\}\times 10^{-4}$
DOT	MSE/MMC	100	10 epochs	$\{1,2,5,10\}\times 10^{-4}$
IEO	MMC/MMC	100	10 epochs	$\{1,2,5,10\}\times 10^{-4}$

Table 1: Training and model-selection configurations of ETO, DOT, and IEO.

Figure 4: Distributions of test-set metrics over 50 random data splits under the nominal setting $c_{f}/c_{p}=5$ . Panels report (a) MAE, (b) $\Delta$ CRPS, (c) MMC, and (d) MFR for ETO, DOT, and IEO across the four CMAPSS subsets. Each box spans the inter-quartile range, the horizontal line denotes the median, and the whiskers extend to the most extreme observations within $1.5$ inter-quartile ranges of the lower and upper quartiles; observations beyond the whiskers are shown individually as outliers.

Subset	Approach	MAE	CRPS	$\Delta$ CRPS	MMC	MFR
FD001	ETO	$\mathbf{7.86\pm 0.50}$	$\mathbf{5.96\pm 0.47}$	$-0.01\pm 1.30$	$86.12\pm 11.17$	$0.28\pm 0.05$
	DOT	$8.62\pm 0.92$	$6.25\pm 0.59$	$1.91\pm 1.39$	$77.90\pm 9.85$	$0.20\pm 0.05$
	IEO	$9.36\pm 0.90$	$7.69\pm 0.92$	$5.70\pm 1.43$	$\mathbf{72.29\pm 5.01}$	$\mathbf{0.11\pm 0.03}$
FD002	ETO	$\mathbf{11.31\pm 0.85}$	$\mathbf{9.08\pm 0.85}$	$0.48\pm 2.06$	$125.49\pm 17.67$	$0.34\pm 0.07$
	DOT	$12.46\pm 1.79$	$10.10\pm 1.65$	$4.26\pm 2.34$	$113.01\pm 16.57$	$0.24\pm 0.06$
	IEO	$13.48\pm 1.43$	$11.51\pm 1.48$	$8.75\pm 2.16$	$\mathbf{99.70\pm 7.85}$	$\mathbf{0.13\pm 0.04}$

Table 2: Test-set performance under the nominal setting

c_{f}/c_{p}=5

. Entries report mean

\pm

standard deviation across 50 random data splits.

Figure 5: Effects of the cost ratio $c_{f}/c_{p}$ on prognostic accuracy and maintenance performance across the four CMAPSS subsets. Relative to ETO, the panels report MAE increase, MMC reduction, and $\Delta$ CRPS shift for DOT and IEO, together with the MFR of all three approaches. Detailed summary statistics are reported in Table 3 .

		MAE Increase (%)		MMC Reduction (%)		$\Delta$ CRPS Shift		MFR (%)
Subset	$\boldsymbol{c_{f}/c_{p}}$	DOT	IEO	DOT	IEO	DOT	IEO	ETO	DOT	IEO
FD001	2	$2.6\pm 6.0$	$3.1\pm 5.4$	$0.7\pm 5.7$	$2.1\pm 5.6$	$0.9\pm 1.0$	$2.4\pm 0.8$	$35.7\pm 6.6$	$32.3\pm 6.0$	$25.8\pm 5.7$
	5	$9.9\pm 12.2$	$19.4\pm 11.6$	$8.8\pm 12.2$	$14.8\pm 11.6$	$1.9\pm 1.1$	$5.7\pm 1.0$	$27.6\pm 5.4$	$20.1\pm 5.3$	$10.6\pm 3.4$
	10	$15.9\pm 13.2$	$37.4\pm 15.1$	$19.3\pm 10.8$	$29.3\pm 13.7$	$2.1\pm 1.2$	$7.9\pm 1.2$	$23.2\pm 5.9$	$14.7\pm 5.0$	$5.0\pm 2.5$
	20	$18.1\pm 12.5$	$61.0\pm 16.6$	$31.4\pm 18.1$	$47.0\pm 12.1$	$2.1\pm 1.4$	$10.3\pm 1.2$	$20.3\pm 4.7$	$11.5\pm 4.2$	$2.4\pm 1.3$
FD002	2	$-0.2\pm 4.7$	$1.7\pm 5.6$	$3.7\pm 5.1$	$4.8\pm 6.0$	$1.5\pm 1.5$	$3.5\pm 1.5$	$40.6\pm 6.7$	$35.3\pm 6.3$	$29.1\pm 6.3$

Table 3: Effects of the cost ratio

c_{f}/c_{p}

on prognostic and maintenance performance across the four CMAPSS subsets. Entries report mean

\pm

standard deviation across 50 random data splits. MAE Increase and MMC Reduction are relative percentage changes with respect to ETO,

\Delta

CRPS Shift is the difference relative to ETO, and MFR is reported as a percentage.

Subset	ETO	DOT	IEO
FD001	69	26	159
FD002	81	31	173
FD003	73	26	162
FD004	63	31	154

Table 4: Average total training time per random data split over all four candidate learning rates and all cost ratios. The reported time for IEO includes its ETO warm-start.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Dataset	Method	MAE			CRPS			MMC
Dataset	Method	$H=100$	$H=125$	$H=150$	$H=100$	$H=125$	$H=150$	$H=100$	$H=125$	$H=150$
FD001	ETO	$7.86\pm 0.53$	$\mathbf{7.83\pm 0.55}$	$7.90\pm 0.49$	$5.99\pm 0.49$	$\mathbf{5.95\pm 0.55}$	$6.00\pm 0.47$	$86.20\pm 11.94$	$86.35\pm 12.78$	$\mathbf{85.56\pm 10.99}$
	DOT	$8.63\pm 0.85$	$8.62\pm 0.74$	$\mathbf{8.48\pm 0.81}$	$6.33\pm 0.65$	$6.34\pm 0.50$	$\mathbf{6.19\pm 0.52}$	$77.80\pm 9.75$	$79.17\pm 8.37$	$\mathbf{76.97\pm 9.44}$
	IEO	$9.38\pm 0.89$	$9.39\pm 0.83$	$\mathbf{9.30\pm 0.63}$	$7.76\pm 0.89$	$7.73\pm 0.84$	$\mathbf{7.62\pm 0.61}$	$72.14\pm 5.27$	$\mathbf{72.07\pm 5.10}$	$72.16\pm 5.39$
FD002	ETO	$11.36\pm 0.76$	$11.44\pm 0.94$	$\mathbf{11.28\pm 0.84}$	$9.11\pm 0.78$	$9.21\pm 0.93$	$\mathbf{9.03\pm 0.85}$	$129.21\pm 19.64$	$127.48\pm 17.88$	$\mathbf{124.64\pm 13.89}$
	DOT	$12.45\pm 1.39$	$\mathbf{12.29\pm 1.22}$	$12.40\pm 1.41$	$10.07\pm 1.37$	$\mathbf{9.96\pm 1.18}$	$10.04\pm 1.39$	$113.05\pm 13.70$	$113.40\pm 12.92$	$\mathbf{111.30\pm 13.21}$

Appendix

Table 5: Sensitivity analysis of

H

. Results are reported as mean

\pm

standard deviation over 50 runs. Bold values indicate the best performance.

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Predictive maintenance requires reliable remaining useful life (RUL) estimation. Existing methods mainly follow two paradigms: wear-based aging models that capture cumulative degradation and sensor-driven data models that reflect instantaneous health conditions, each providing only partial information. In this work, we propose a probabilistic fusion framework that integrates wear-based and sensor-based prognostic components through failure probability distributions. Based on explicit structural assumptions linking wear, latent health, sensor observations, and failure, we derive a principled combination rule that enables uncertainty-aware integration with adaptive weighting of the components. Experimentally, we assess this combination rule by learning the wear-based component using a parametric survival model and the sensor-based component using a 1D convolutional neural network (1D-CNN) with a post-hoc uncertainty model. Evaluation on multiple N-CMAPSS datasets demonstrates that the fused model improves point accuracy, preserves the C-index, and produces narrower yet well-calibrated prediction intervals compared to either component alone. The results highlight the complementary roles of wear-based survival model and sensor-based deep learning model, and show that their probabilistic integration provides a structured pathway toward more robust and consistent prognostics over the life-time.

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