CrayStack: a simplified crayfish optimization driven stacking ensemble for prediction of machining quality characteristics under data scarcity

Abstract

Intelligent manufacturing demands accurate prediction of machining quality characteristics with conflicting behaviour, which is challenging with limited experimental data. Taguchi L₂₇ experimental data sets were collected with 6 influencing variables (workpiece material: PA66, PA66 + GF30, PA66 + MoS₂, tool approach angle, tool nose radius, cutting speed, feed rate and depth of cut) and 8 machining quality characteristics (surface roughness, cutting force, temperature, amplitude of vibration, tool wear rate, specific cutting energy, material removal rate, and sound pressure level). The total 27 experimental datasets were stratified by material into a 3-fold cross-validation protocol. The present work develops five base learners such as Gaussian Process Regression (GPR), Least Squares Boosting (LSB), Support Vector Regression (SVR), Random Forest (RF) and extreme gradient boosting (XGBoost) for predictions of machining quality characteristics. All three metaheuristic algorithms (genetic algorithm (GA), particle swarm optimization (PSO), and crayfish optimization algorithm (COA)) determine identical weights for three best predictive base learners (GPR, SVR, and LSB) for developing Ensemble model. The COA converge to a minimum composite cost with comparatively lesser computation time than GA and PSO. Therefore, CrayStack ensemble model is constructed with a hybrid combination of GPR, SVR, and LSB and COA methods. The COA efficiently optimizes the adaptive fusion weights by assigning a higher weight fraction to GPR and LSB for nonlinear models. CrayStack Ensemble predictions outperforms all individual learners (SVR, GPR, LSB, XGBoost, and RF) with material stratified three-fold cross validation across all eight outputs of training data. CrayStack Ensemble requires a total training cost of 16.13 s (which includes base model training: 94.92% & 15.31 s, COA weight optimization: 1.88% & 0.30 s, and bootstrap confidence interval estimation: 3.20% & 0.52 s) ensuring its practical usefulness. CrayStack Ensemble achieves near-instantaneous inference (0.003 ms/sample; 290,592 samples/s) with a speed of 58.33, 79.33, 4899.33, 5630.67, 5608.3 over GPR, SVR, LSB, RF and XGBoost ensuring practicality suitable for real-time monitoring systems. Wilcoxon single-rank test confirmed that improvements are statistically significant (with a preset confidence level, p < 0.05) for 7 of the 8 responses, validating the practical utility of the developed models. CrayStack Ensemble showed superior prediction performances against nine randomly generated test cases with a mean absolute percent error of 9.8%, followed by GPR, LSB, XGBoost, SVR, and RF of 12.69%, 13.81%, 15.39%, 21.52% and 42.73% considering all responses. The results demonstrated that the intelligent ensemble stack ensures robustness and higher prediction accuracy for limited experimental datasets offering a practical solution for industrial process optimization.

Introduction

The smart and data-driven manufacturing under Industry 4.0 demand accurate predictive models capable of enhancing machining performance with minimal experimental input-output data. Machining performance is affected by multiple quality characteristics (possessing conflicting behavior) such as surface roughness, cutting forces, tool wear, temperature, vibration, and acoustic emissions, which necessitate process optimization.

Optimizing the multiple performance quality characteristics results in increased productivity, cost reduction and better functional products in modern manufacturing systems^1–3. However, experimental machining trials (exceeding 30–50) often lead to higher material wastage, machine downtime, and time-intensive trials^4–6. Traditional (mechanistic and empirical) machining models use classical cutting theories and tool life equations that offer physical insights but rely on a few assumptions^7–9. These assumptions limit the wider deployment of the models due to nonlinearity, material-dependence, and multiple-quality machining conditions.

The machining process modelling has evolved from physics-based formulations to advanced data-driven approaches. Classical mechanistic models, such as Merchant’s orthogonal cutting theory and Taylor’s tool life equation, provide foundational insights into chip formation and tool wear but require extensive calibration and are limited in handling complex materials and cutting conditions^7,9. Statistical methods based on experimental datasets are classified mainly into linear methods: Taguchi, Plackett–Burman Design, and full-factorial design, and nonlinear models: central composite design and Box-Behnken Design. These statistical techniques introduce structured experimentation that improves experimental efficiency to provide detailed process insights (based on the determination of significant and insignificant factors)^10–13. However, these methods are constrained by material-dependent machining phenomena, low-order interactions, and assume lower-order polynomial relationships that fail to capture strong nonlinearities—even in well-studied processes like electrical discharge turning¹⁴. Furthermore, the machining process is affected by multiple performance quality characteristics with conflicting requirements. The statistical experimental approaches model and analyze the relationships one response at a time, thus strong relative dependencies among multiple outputs, if present are unable to be captured. These statistical methods are offline monitoring tools and therefore fail to adjust the process conditions dynamically. Online monitoring of modern manufacturing systems meets present-day industrial requirements (Industry 5.0) that could control the process for better product quality, productivity, predictive maintenance, reduced downtime, sustainability ensuring resource efficiency, customization and so on. Therefore, advanced computing tools that can monitor or adjust the process conditions for better performance in the product are of industrial relevance.

Machine learning approaches (ANN, ANFIS, SVR, RF, GPR, RNN, LSTM, DNN, CNN, SVM, KNN, Naïve Bayes, RL, and LSB) address most of the shortcomings of statistical methods and are therefore are used in machining industries for predictions^15–23. Table 1 presents a summary of ML methods applied to different machining applications. Recent advances have led to the widespread deployment of ML models for machining prediction. ANN, SVR, GPR, RF, and boosting (LSB, Extreme Gradient Boosting: XGB) based ensembles have been successfully applied to predict surface roughness, cutting forces, tool wear, and thermal responses across EDM processes^18,22, turning^16,24–27, grinding²⁰, and milling¹⁹ process. Recent works further demonstrated that integrating the experimental (such as design of experiments for data collection and parametric analysis), prediction (machine learning approaches) and optimization (metaheuristic: generic algorithms) methods were found to produce better performance in machined quality^25–27.

Table 1.

Overview of ML methods in machining applications.

Applications	ML methods	Remarks	Refs.
Regression and Prediction	ANN, SVR, RF, GPR	ANNs are used for predicting MRR, SR, and TWR due to their ability to handle nonlinear relationships	39,40
		RF is suitable for predicting various machining parameters such as TW, MRR, hole entry failure, overcut, taper ratio, and roundness.	41
		GPR is effective for predicting SR in hard turning operations with small and large data sets.	24,30
Time-Series Monitoring	RNN, LSTM, DNN	RNN, LSTM and DNN accurately predicted cutting force.	42
		Vibration and tool tip wear	43,44
Image-Based Analysis	CNN	Image analysis techniques like TWR	45
		SR and surface defect analysis are commonly used for quality control.	34
Classification	SVR, SVM, k-NN, Naïve Bayes	Performs well with small data sets for accurate dimensions	46
		Tool usage and computational efficiency	47
Hybrid Intelligence	ANFIS, SVR-PSO, DET, HSO	Combining methods improves prediction accuracy more than using a single model.	35
Adaptive Control	RL	RL enables real-time control of cutting parameters to reduce vibrations and errors.	48
		Machining operations process planning	49

ANNs are known for their good generalization and require fine-tuning of the network parameters to avoid overfitting, which can result in local solutions²⁸. SVR predictions’ performance was found to be acceptable for small datasets²⁹. GPR enables accurate and consistent prediction of surface roughness through its probabilistic mechanism, allowing the algorithm to iteratively update and reduce errors, enhancing robustness and generalization^21,30. RNNs are limited in capturing process insights from long-term sensor signals required to characterize progressive tool wear due to the highly sensitive and noisy machining environment³¹. These shortcomings necessitate hybridization with LSTM, which increases computational complexity and limits adaptability in an industrial environment³². KNN is highly sensitive to linear data types, affecting the prediction accuracy for parameter combinations maintained at extremely high values³³. CNN predictions are sensitive to datasets (large size) and require an additional sequential model (such as LSTM) to capture long-term temporal dependencies in multi-sensor signals³⁴. The hybrid SVR-PSO model performs better in tool wear prediction than DET, HSO and PSO³⁵. The better prediction performance of hybrid models must be balanced with slower computation time³⁶ and lower predictions with limited datasets, making them not practically feasible for online monitoring with slower feedback response. ML algorithms (SVM, RF, and LSB) accurately predict tool wear for influencing parameters (cutting forces, machining parameters and tool geometry) with limited datasets³⁷. Boosting algorithms such as LSB have demonstrated superior predictive accuracy compared to a single ML algorithm by iteratively minimizing residual errors¹⁷. Each model has its advantages and limitations and performs better for specific domains (such as classification, prediction, adaptive control, etc.). Despite these advantages, the predictive performance of individual ML models deteriorates when dealing with small datasets under controlled machining environments³⁸. The literature review above confirms that SVM, LSB, and GPR performed better in prediction when trained with limited datasets.

ANFIS: Adaptive Neuro-Fuzzy Inference System; ANN: Artificial Neural Network; CNN: Convolutional Neural Network; DET: Differential Evolution Technique; DNN: Deep Neural Network; GPR: Gaussian Process Regression; HSO: Harmony Search Optimization; KNN: k-Nearest Neighbors; LSB: Least-Squares Boosting; LSTM: Long Short-Term Memory; MRR: Material Removal Rate; Naïve Bayes: Naïve Bayes Classifier; PSO: Particle Swarm Optimization; RF: Random Forest; RNN: Recurrent Neural Network; RL: Reinforcement Learning; SR: Surface Roughness; SVM: Support Vector Machine; SVR: Support Vector Regression; TWR: Tool Wear Rate.

Ensemble learning offers a promising solution by combining multiple base learners or models to explore their strengths and reduce limitations (variance and bias)⁵⁰. Techniques such as bagging, boosting, and RF have demonstrated improved stability and accuracy in manufacturing applications (tool condition monitoring, surface quality prediction and cutting force estimation)^51–53. Among widely deployed ensemble techniques, stacking ensembles use different combinations of base learners (SVR, RF, GPR and so on) integrated with metaheuristic search algorithms to capture nonlinear process dynamics^54–56. However, such approaches typically target single-process, single-material, single-response scenarios and rely on fixed-weight fusion or sequential modeling. Traditional stacking methods further rely on fixed or linearized meta-models that often fail to predict multiple responses subjected to small and multi-response datasets⁵⁷. In Ensemble model, combining diverse base predictive learners require assigning weights to each model and tune hyper parameters. The linear stacking learns weights through least-squares optimization which require tuning hyper parameters and assume linearity constraints⁵⁸. The simple averaging method employ assigning equal weights to base learners may not always produce better predictions compared to the weighted averaging method^59–61. The major limitations of weighted methods (simple and weighted averaging methods) consider tuning of hyper parameters and weighting of base learners as two different or independent steps rather than an integrated process. These traditional weighting methods failed to capture the interdependence between model configurations and contribution weights resulting in poor generalization prediction performance and reduced adaptability to complex data distributions⁶¹. The above literature confirms the need for adaptive ensemble weight-driven techniques that are capable of performing accurate predictions even for small datasets.

Metaheuristic algorithms (genetic algorithm, particle swarm optimization and teaching learning-based optimization) were applied to predict optimized machining conditions^62,63. Recent studies integrating stacking ensembles with GA and PSO optimizers demonstrated improved predictive accuracy in the machining process⁶⁴. However, these approaches often suffer from premature convergence, sensitivity to hyperparameter tuning, and increased computational overhead, particularly when dealing with ultra-small datasets.

These algorithms conduct optimal search at many spatial locations simultaneously capable of finding global solutions, but their potential has rarely been applied to ensemble weight optimization. Their computational complexity also limits practical deployment when optimization dimensionality is low. Jia et al.⁶⁵ proposed the Crayfish Optimization Algorithm (COA) which simulates the social foraging and competitive behaviour of crayfish to achieve a balance between exploration and exploitation. COA has demonstrated strong convergence characteristics in numerical and engineering optimization problems. Many researchers have introduced enhanced variants of COA to improve global search capability and robustness^66–68. Compared with conventional optimizers such as GA and PSO, COA offers improved population diversity, faster convergence, and enhanced search stability due to its adaptive foraging strategy and stochastic perturbation mechanisms. These attributes make COA particularly suitable for low-dimensional optimization problems such as ensemble weight tuning. Despite these advances, COA has not yet been applied to ensemble learning in machining, nor has it been tailored for low-dimensional regression problems. The original COA incorporates multiple behavioral phases and adaptive mechanisms that may introduce unnecessary computational overhead in low-dimensional optimization problems, such as three-model stacking ensembles. To address this, the present study proposes a Simplified Crayfish Optimization Algorithm (sCOA), in which redundant components particularly the summer resort (long-range random walk) phase, are removed while preserving the core exploration–exploitation balance, thereby enhancing convergence speed, computational efficiency, and optimization stability.

In machining multiple quality characteristics affect product quality, productivity, and cost. Each quality characteristic is reliant on the others. For example, SR of machined parts is influenced by factors such as tool wear (which deteriorates tool life), cutting temperature (increasing the thermal load), cutting forces (mechanical load), and tool vibrations (affecting dimensional accuracy of the work piece)⁴⁸. Similarly, TWR is governed by cutting forces and machining parameters that affect energy consumption (energy efficiency) and MRR (productivity)³⁷. Existing data-driven approaches such as single ML models, often struggle to capture complex interactions and nonlinear relationships among outputs with limited datasets, leading to poor interpretability and high monitoring costs^68,69. This limitation hinders the use of ML models for onsite monitoring in machining industries. The significant gaps identified from the literature review are: (a) Lack of ensemble learning strategies explicitly designed for ultra-small machining datasets. (b) Limited use of metaheuristic optimization algorithms (CAO, GA, PSO) for adaptive ensemble weighting in multi-response regression. (c) Insufficient validation frameworks (random experimental cases, bootstrap confidence interval analysis, computational cost analysis) addressing material-dependent generalization under small-sample constraints. (d) Absence of COA-driven stacking ensemble models tailored for machining applications and low-dimensional optimization problems.

The present work proposes a novel stacking ensemble framework that integrates base learners such as GPR, LSB, XGB, RF and SVR with adaptive weights optimized using a simplified COA, GA and PSO (Fig. 1). Three composite samples (PA66 + PA66 + MoS₂, PA66 + GF30) possessing different properties were subjected to machining. Taguchi L₂₇ experimental trials composed of different combinations of influencing variables (workpiece material, cutting speed, feed rate, depth of cut, tool nose radius, and tool approach angle) on machining performances (surface roughness, cutting force, tool wear rate, temperature, vibration, specific cutting energy, sound pressure level, and material removal rate). These datasets were used to develop GPR, LSB, RF, XGBoost and SVR models for the machining process. The model prediction performances are finally examined with material-driven cross-validation using performance metrics (R², MAE, MAPE, and RMSE). Metaheuristic algorithms (GA, PSO, COA) were used to determine optimal weights for the best performing three base learners across each response. The algorithm performances were evaluated to minimize the composite cost (comprising R², RMSE and MAE with an importance assigned to 50%, 25% and 25%) and computational time across each response. An ensemble stacking model was developed comprising three best learners and their contribution (adaptive optimal weight) determined through the best performing metaheuristic algorithm. The ensemble model predictions were compared with base learners (GPR, LSB, RF, XGBoost and SVR) by conducting statistical validation analysis through four performance metrics (R², MAE, RMSE, and MAPE), Wilcoxon Single rank, paired Cohen’s d effect sizes, bootstrap confidence intervals, residual analysis for error distribution and computational cost analysis. Furthermore, the practical utility of trained six models (GPR, LSB, RF, XGBoost and SVR) was tested for material stratified nine random experimental cases (which are material-classified and not used during training) and recommend the best predictive model for industrial application. The performance of the ensemble offers an integrated approach that ensures experimental design, predictive modelling, and validation enabling the investigators to make response specific predictions in machining application with small datasets.

Fig. 1.

Predictive modelling framework for machining quality characteristics.

Experimental framework

Material selection, machine setup and tooling configuration

Machining experiments were conducted on three workpiece compositions (PA66, PA66 + GF30, and PA66 + MoS₂) using an AJAX Evolution AJEV-310 CNC lathe in a dry cutting environment. The experimental set-up with three workpiece materials and response measurement devices is shown in Fig. 1. The mechanical and thermal properties of the composite samples are presented in Table 2. The machining performances are affected by material compositions and properties which present significant challenges for establishing a generalized predictive model. The cylindrical work surfaces (diameter: 45 mm, machining length: 400 mm) were machined to a depth of 30 mm for nine intervals separated by 10 mm relief grooves to present thermal interference between cuts. TNMG TM LF9218 carbide insert tool geometry (approach angles: 75°, 93°, and 105°) using ISO-driven standard turning holders with a 16 mm shank were employed. The nose radii of 0.4 mm, 0.8 mm, and 1.2 mm were selected for machining and performance analysis.

Table 2.

Mechanical and thermal properties of workpiece materials.

Property	PA66	PA66 + GF30	PA66 + MoS₂
Tensile strength (MPa)	75–85	150–180	80–90
Elastic modulus (GPa)	2.7–3.0	7.5–10.0	3.0–3.5
Elongation at break (%)	20–50	2–5	10–20
Hardness (Rockwell M)	80–85	105–115	85–90
Thermal conductivity (W m⁻¹ K⁻¹)	~ 0.25	~ 0.30	~ 0.28
Coefficient of friction	0.35–0.45	0.20–0.30	0.15–0.20

Experimental design and factor configuration

The six influencing parameters (covering material characteristics, tool geometry and cutting variables) and their operating three levels that affect the eight machining performances (covering surface integrity, mechanical and thermal behavior, and tool degradation) were presented in Table 3. The Taguchi L₂₇ experimental matrix (exploring the range of variables with resource savings or constraints) was used for experimentation and recording the machining performances. The selected design enables realistic industry trial limits for polymer-based composites that ensures sufficient samples for data-driven modelling.

Table 3.

Control factors and levels for Taguchi L₂₇ design.

Factor	Symbol	Levels
Workpiece material	–	PA66, PA66 + GF30, PA66 + MoS₂
Tool approach angle		75°, 93°, 105°
Tool nose radius		0.4, 0.8, 1.2 mm
Cutting speed		100, 150, 200 m/min
Feed rate	f	0.1, 0.2, 0.3 mm/rev
Depth of cut		0.5, 1.0, 1.5 mm
Workpiece material	–	PA66, PA66 + GF30, PA66 + MoS₂

The data sets from an experimental matrix cover a balanced representation of all three materials (PA66, PA66 + GF30, PA66 + MoS₂), with nine trials conducted per polymer. The equal classification for each material ensures investigators can perform three-fold cross-validation with the ensemble stacking framework (Sect.  3). Each fold contains an equal number of samples for each material class, limiting data outflow and enabling realistic quality assessment for model generalization across materials with limited data samples.

Measurement strategy and instrumentation

The accuracy of the model’s predictions depends on the quality and distribution of datasets. To ensure quality, the machining outcomes were recorded (three times for each experimental run) using calibrated sensors and measurement systems that ensure higher accuracy and repeatability under dry turning conditions. Response measurements were taken in a steady-state cutting environment, neglecting transient effects. The response measurements were either directly taken or computed from consistent physical relationships in machining. Table 4 lists the instruments (models) used to record the output data, along with their measurement levels and required accuracy.

Table 4.

Machining response measurement instruments and specifications.

Response	Instrument (Model)	Major Specification	Accuracy
Cutting force ()	DY920B dynamometer	0–1000 N	± 0.5% of full scale
Vibration (Vib)	BENETECH GM63B vibrometer	0.1–199.9 m/s²	± 2%
Temperature (Temp)	ANENG TH20 IR thermometer	–50 to 400 °C	± 1.5 °C
Sound pressure level (SPL)	FLUS MT-911 A sound level meter	35–135 dB	± 0.7 dB
Surface roughness ()	SRT-6223 profilometer	0.1–800 μm	± 0.01 μm
Tool wear rate (TWR)	Changxie CX digital pocket scale	0.001–100 g	± 0.001 g

Multi-response output definition and physical significance

In the present work eight machining responses were investigated including SR, MRR, SCE, TWR, SPL, Fc, vibration amplitude and cutting temperature. The responses (such as Fc and Ra) were obtained directly from the instruments while the remaining outcomes were computed from machining relationships using experimental data. The collection of such derived datasets is commonly practiced in the machining environment, as it maintains the physical insights that facilitate comparison between materials and support integration within the multi-response prediction using independent models.

Tool wear rate

Tool Wear Rate (TWR) provides detailed insights into the rate at which the tool material is lost per unit machining time (g/min), allowing investigator to gather continuous information about tool degradation based on physics. The predictive nature of tool life estimation and the adaptive process control mechanism determine the remaining tool life for machining. TWR helps predict the tool life used in industrial practice using Eq. 1^13,14,70,71.

1

Material removal rate

MRR determines productivity, which is influenced by reduced cycle time and increased output, ensuring a comparable benchmark with machining strategies, tool systems and machine tool capabilities. The MRR often results in increased cutting force, thermal loads and tool wear, which may pose practical limitations on achieving MRR. MRR is determined by cutting speed (,  f, and depth of cut ( using Eq. 2.

2

Specific cutting energy

Specific Cutting Energy (SCE) determines the energy efficiency (lower SCE is desirable) representing the energy required for material removal. The machinability indicator derived from cutting mechanics and power consumption (Pc) principles is presented in Eq. 3. where F_c determines the main cutting force.

3

Surface roughness (Rₐ)

Surface roughness, as measured by Ra, is a key factor in determining the average surface profile deviation that impacts surface integrity. It is influenced by various factors such as feed rate, tool-workpiece interaction, vibration, and progressive tool wear. The contact type stylus profilometer, with a range of 0.1–800 μm and a resolution of ± 0.01 μm, allows for the measurement of micro-scale surface irregularities on machined surfaces. The evaluation and optimization of surface roughness play a crucial role in determining the functional performance of machined parts, affecting properties such as wear resistance, fatigue life, friction characteristics and dimensional accuracy.

Cutting force

Cutting force is a measure of machinability and process stability that directly impacts the tool-workpiece interaction, chip formation, and load transfer to the machine tool. Optimal cutting force minimizes power consumption, tool wear, chatter and vibration. Cutting force was measured using a DY920B piezoelectric dynamometer (0–1000 N, ± 0.5% FS). Prior to experimentation, the dynamometer was calibrated and force data were continuously acquired during the steady-state condition for each parametric condition. The average values of three cutting force data correspond to each machining condition were taken for analysis and optimization.

Vibration amplitude

Vibration during machining captures the dynamic response of the machining system ensuring machine stability. Excessive machine vibration degrades machined surface quality, accelerates tool wear and induces chatter. Vibrations during machining were recorded using a BENETECH GM63B vibrometer (0.1–199.9 m/s², ± 2%). Measurements were taken under steady-state conditions to limit transient effects at tool entry and exit. The vibration amplitude corresponding to each trial was recorded and correlated with machining parameters and surface quality indicators.

Cutting temperature

Cutting temperature determines the thermomechanical behaviour of machining, directly impacting tool wear, chip formation and surface integrity particularly under dry cutting environments. Accurate temperature estimation is essential to monitor thermal loads, tool wear, and process reliability. The present work estimates temperature (measured from a fixed distance and emissivity setting for repeatability) using an ANENG TH20 infrared thermometer (–50 to 400 °C, ± 1.5 °C).

Sound pressure level

SPL determines machining dynamics by capturing the acoustic emissions generated by tool-workpiece interaction, chip breakage, and structural vibrations. SPL measurements were conducted using a FLUS MT-911 A sound level meter (35–135 dB, ± 0.7 dB). Measurements were taken under steady-state machining conditions with a consistent microphone orientation positioned at a fixed distance from the cutting zone to limit air flow and machine noise vibrations.

Experimental dataset and data collection

The Taguchi L₂₇ experimental dataset includes six different influencing variables (material class: PA66, PA66 + GF30, and PA66 + MoS₂ and cutting conditions: r_n, V_c, f, and a_p) on the 8 machining performances (R_a, F_c, Temp, Vib, TWR, SPL, MRR and SCE) as shown in Table 5. The collected experimental data were used to develop CrayStack Ensemble models for prediction and optimization.

Table 5.

Experimental Dataset from the Taguchi L₂₇ method.

Exp. No.	Material		r n	Vc	f	ap	Ra (µm)	Fc (N)	Temp (°C)	Vib (m/s²)	TWR (g/min)	SPL (dB)	MRR (mm³/min)	SCE (J/mm³)
1	1	75	0.4	100	0.1	0.5	1.23	66.8	68.5	0.67	0.0278	71.8	50,900	0.128
2	1	75	0.4	100	0.2	1	1.47	67.1	75.1	0.73	0.0302	72.3	196,000	0.035
3	1	75	0.4	100	0.3	1.5	1.68	73.5	78.2	0.92	0.0395	74.6	457,000	0.015
4	1	93	0.8	150	0.1	0.5	1.37	65.2	72.1	0.72	0.0283	71.4	76,200	0.13
5	1	93	0.8	150	0.2	1	1.58	71.8	74.8	0.83	0.0321	73.7	294,000	0.037
6	1	93	0.8	150	0.3	1.5	1.72	73.8	83.5	0.97	0.0431	74.3	689,000	0.016
7	1	105	1.2	200	0.1	0.5	1.33	68.2	70.8	0.7	0.0273	70.3	102,000	0.132
8	1	105	1.2	200	0.2	1	1.48	70.1	79.3	0.75	0.0332	72.8	392,000	0.037
9	1	105	1.2	200	0.3	1.5	1.73	77.2	81.8	0.95	0.0401	73.4	918,000	0.017
10	2	75	0.8	200	0.1	1	3.28	138	112	2.05	0.123	82.7	204,000	0.133
11	2	75	0.8	200	0.2	1.5	3.45	151	116	2.35	0.151	84.2	588,000	0.051
12	2	75	0.8	200	0.3	0.5	2.91	126	108	1.81	0.112	80.5	306,000	0.083
13	2	93	1.2	100	0.1	1	3.14	134	106	2	0.127	83.8	102,000	0.13
14	2	93	1.2	100	0.2	1.5	3.43	142	117	2.2	0.164	85.3	294,000	0.05
15	2	93	1.2	100	0.3	0.5	2.89	127	100	1.76	0.096	79.2	147,000	0.085
16	2	105	0.4	150	0.1	1	3.11	136	114	2.15	0.143	84.7	153,000	0.09
17	2	105	0.4	150	0.2	1.5	3.64	158	120	2.45	0.167	86.2	441,000	0.053
18	2	105	0.4	150	0.3	0.5	2.87	124	102	1.73	0.087	78.3	219,000	0.086
19	3	75	1.2	150	0.1	1.5	1.17	48.2	56.2	0.37	0.0113	66.5	229,000	0.03
20	3	75	1.2	150	0.2	0.5	0.93	45.1	50.8	0.28	0.0087	64.3	147,000	0.045
21	3	75	1.2	150	0.3	1	1.22	51.3	59.3	0.42	0.0125	67.7	459,000	0.016
22	3	93	0.4	200	0.1	1.5	1.12	47.1	52.8	0.35	0.0104	65.2	306,000	0.03
23	3	93	0.4	200	0.2	0.5	0.88	40.8	52.1	0.27	0.0076	65.8	196,000	0.043
24	3	93	0.4	200	0.3	1	1.23	50.2	55.8	0.38	0.0107	66.3	588,000	0.017
25	3	105	0.8	100	0.1	1.5	0.98	46.2	51.8	0.36	0.0085	65.4	147,000	0.031
26	3	105	0.8	100	0.2	0.5	0.87	40.2	51.2	0.26	0.0083	64.6	98,000	0.042
27	3	105	0.8	100	0.3	1	1.13	49.3	56.2	0.4	0.0087	66.8	306,000	0.016

Ensemble model architecture

Model framework overview

The hybrid ensemble stacking technique is designed for small datasets (L₂₇ machining experiments presented in Table 5) for performing predictions. The proposed framework applies independent stacking ensembles for each of the eight responses. In contrast to joint multi-output learning frameworks, this formulation enables the metaheuristic algorithms (COA, GA, and PSO) to perform response-specific weight optimization, thereby allowing different base learners (XGBoost, GPR, SVR, LSB and RF) to capture distinct physical phenomena in response behavior. For example, LSB is more effective for nonlinear behaviour of responses, while GPR better captures responses that possess smooth probabilistic nature. Moreover, response-specific adaptive weighting ensures practical alternative to multi-output learning models that enables flexibility and interpretability of ensemble behavior across nonlinear machining responses. Therefore, ensemble stacking comprises three best predictive base learner models (among five: GPR, LSB, SVR, XGBoost, and RF) with adaptive weights determined using best performing metaheuristic algorithm (among GA, PSO, COA). Three-fold cross-validation was designed accounting for three materials (each material with nine experimental trials together constituting 27 datasets) to enable the investigator to balance training and validation while reducing variance and preventing overfitting. Figure 2 summarizes the five-stage ensemble stacking workflow that traces the process starting from data-preprocessing to final performance examination, ensuring its systematic structure and practical relevance in industrial machining contexts.

Fig. 2.

Ensemble stacking framework for predicting 8 machining quality characteristics.

Stage 1: data preprocessing and feature engineering

Data preprocessing steps and feature engineering were often useful to enhance the numerical stability (to avoid large and small values of input-output variables) and ensure fair feature representation that enable convergence to global solutions through learning algorithms. The total of 27 machining experimental datasets comprised categorical features (three material classes) and 6 input and 8 output based continuous features.

Categorical feature encoding

The materials, consisting of three polyamide-based composites (PA66, PA66 + GF30, and PA66 + MoS₂), were treated as categorical input variables and transformed using the one-hot encoding (OHE) technique. OHE helps limit the model from assuming an ordinal relationship between material classes and capture material-driven nonlinear impacts independently. The polyamide-based composites set is represented with the material class presented in Eq. 4. This results in encoding a binary feature matrix of the size presented in Eq. 5.

The OHE strategy ensures material independence, which limits the biased ordinal assumptions. Otherwise, it may misinterpret the factor impact across material classes (Mat₁, Mat₂ and Mat₃) during model training.

4

5

Input feature construction

The OHE categorical variables (material classes: Mat₁, Mat₂ and Mat₃) were combined with continuous machining variables ( r_n, V_c, f, and a_p) to design the input variable matrix presented in Eq, 6. The designed input matrix presents a unified approach enabling the models to learn material composition and process parameters interactions.

6

Feature standardization

The input and output features are scaled to ensure standardization (uniformity and numerical conditioning) to zero mean and variance using Z-score based normalization using Eq. 7.

The above preprocessing step ensures that GPR, SVR, and LSB learners prevent the dominance of higher magnitude variables, thus enhancing optimization stability. Furthermore, preprocessed datasets enable investigators to facilitate cross-validation and OOF prediction, ensuring a robust foundation for ensemble learning in subsequent stages (Stage 2-Stage 5).

7

where are column-wise means and standard deviations of , and are the mean and standard deviation of each output response.

Stage 2: base model training and out-of-fold prediction generation

3-Fold cross-validation strategy

The dataset is categorized into three disjoint folds using graded sampling by material class, ensuring each fold contains three samples per material (9 samples/fold). For each fold :

• Training set: (18 samples)

• Validation set: (9 samples)

Base model configurations

Gaussian process regression (GPR)

GPR is a nonparametric Bayesian learning model (based on input-output relationships) that defines a distribution over functions using a covariance kernel. The covariance kernel function enables GPR to flexibly adapt to complex nonlinear trends even with small-sample learning. GPR offers inherent uncertainty quantification by estimating both predictive mean and variance that are particularly valuable for reliability assessment and decision making with limited data. The predictive mean for a test point corresponding to the trained GPR model is given by Eq. 8.

enhances the numerical stability and ensures that the matrix is invertible, preventing overfitting with small-datasets. is the kernel matrix computed with the Matérn 3/2 kernel using Eq. 9.

The predictive mean for a test point x_i corresponding to the trained GPR model is given by Eq. 8.

8

9

with noise variance and length scale learned from data³².

Least-squares boosting (LSB)

LSB is a gradient boost ensemble technique that enables prediction by establishing a model that sequentially fit weak learners to the residuals of previous models using a squared-error function. At each cycle, a regression tree (which acts as a low-variance learner that helps to reduce overfitting and enables the capture of nonlinear relationships) is trained to reduce errors through the ensemble and its prediction is scaled by a learning rate to monitor and prevent data overfitting by the model. The ensemble prediction after iterations is given in Eq. 10. with weak learners as regression trees (max splits = 4, min leaf size = 2), learning rate , and  T=100cycles³³.

10

Support vector regression

SVR is a kernel-driven learning algorithm designed to handle nonlinear regression functions by balancing model complexity and prediction error. SVR uses an ε-intensive loss function that ensures enhanced generalization and robustness against noise, which are advantageous for small experimental datasets. The present SVR adopts an RBF kernel enabling investigators to handle nonlinear datasets in a high dimensional feature space where complex relationships are efficiently captured. The SVR predictive function for the i^th sample possessing an ε-intensive loss function with an RBF kernel is expressed in Eq. 11. However, the RBF kernel function is explained in Eq. 12.

Where, controls the kernel width that ensure the smoothness of the regression function. (Inverse feature count), box constraint  C=10, and ³⁴.

11

12

Extreme gradient boosting (XGBoost)

XGBoost enhances prediction by adjusting the weak learners (poor predictors) to the residuals of earlier models through the application of a regularized squared error objective. In each epoch, the regression tree is trained with the aid of 1st and 2nd order gradients to ensure reduced losses and limit model complexity through regularization. The prediction of an individual tree is improved by the learning rate parameter ( for better convergence and to limit overfitting. The XGBoost prediction after T iterations is presented in Eq. 13.

14

Random forest (RF)

RF is a widely used bagging method that ensures accurate predictions by building multiple regression trees using bootstrap samples based on training data and random feature extractions. At each cycle, the weak learner is trained independently sequential dependency allowing the RF model to minimize variance and enhancive generalization. A random subset of input features is selected at each split to decorrelate the trees and improve robustness against overfitting. The RF prediction after T iterations is presented in Eq. 14.

14

Out-of-fold prediction matrix

After 3-Fold CV, the predictions for the  j^th output variables as arranged with the inclusion of GPR, LSB, XGBoost, SVR and RF predictions are given in Eq. 15.

Predictions are rescaled to actual values using Eq. 7.

15

Stage 3: metaheuristic algorithm driven stacking weight optimization

Optimization problem formulation

The predictions of the three best performing base learners were combined using a stacked ensemble framework. The contributions of each individual model are determined by a combinatorial non-negative stacking weights optimization with convexity constraints as presented in Eq. 16.

16

The metaheuristic algorithms (COA, GA, and PSO) were used to determine optimal weights based on the relative contribution of base learners for each response. The optimal weights were obtained by minimizing a composite cost objective function presented in Eq. 17.

17

In Sect.  3.2.6. , RMSE and MAE are defined, normalizes error scales, and represents the objective function weights vector that balances goodness-of-fit against magnitude and average prediction deviations. The metaheuristic algorithms (GA, PSO and COA) minimizes over the constrained stacking weight space. To ensure numerical stability and prevent negative objective values, the term is constrained to the interval . This multi-criteria formulation enhances ensemble generalization by jointly penalizing unexplained variance, large outliers, and systematic bias.

Simplified crayfish optimization algorithm (COA)

The original COA consists of three phases: the summer resort stage, competition stage, and foraging stage. To enhance computational efficiency and reproducibility, the original COA was systematically simplified into a variant (simplified COA) tailored for low-dimensional ensemble weight optimization. Redundant behavioral components, particularly the summer resort phase (long-range random walk), were removed due to their limited usage in small search spaces and their contribution to unnecessary computational overhead. The competition phase was retained but reformulated using a lightweight differential mutation strategy, enabling efficient exploration while accelerating convergence. In addition, the original multi-parameter adaptive control mechanism was reduced to a single linear decay factor, providing a transparent and stable exploration–exploitation balance with minimal tuning requirements. Boundary handling was further redefined using simplex projection to strictly enforce feasibility constraints (∑wₘ = 1, wₘ ≥ 0), thereby eliminating invalid candidate solutions. Collectively, these changes help to reduce algorithmic complexity while maintaining optimization performance,

The simplified COA pseudocode is presented in Algorithm 1.

Algorithm 1: Simplified COS for Stacking Weight Optimization.

Stage 4: final ensemble prediction

The optimized stacked ensemble model prediction is computed using a convex weighted combination of the best performing three individual base learner (BL) predictions evaluated based on four performance metrics (R², MAPE, MAE, and RMSE) expressed in Eq. 18.

The above mathematical formulation enables the investigator to ensure that the final ensemble prediction incorporates the complementary strengths of all three base learners (GPR, LSB, and SVR) while preserving numerical stability and interpretability. The optimized weights obtained through the best performing metaheuristic algorithm (GA, PSO, and COA) ensure that the ensemble technique achieves improved generalization and robustness compared to a single or standalone model subjected to limited experimental data.

18

The term , , and are the optimal stacking weights determined through the COA procedure in Stage 3.

Stage 5: performance evaluation and statistical validation

Performance metrics

The final ensemble predictions are evaluated using four standard regression metrics: Coefficient of Determination, Root Mean Square Error, Mean Absolute Error, and Mean Absolute Percentage Error. These metrics are calculated in the original data scales.

The Coefficient of Determination assesses the proportion of variance in the experimental data explained by the predicted values, as shown in Eq. 19.

Root Mean Square Error determines the standard deviation of the prediction errors, as expressed in Eq. 20.

Mean Absolute Error calculates the average magnitude of absolute prediction errors, providing a direct measure of model prediction accuracy. MAE is computed using Eq. 21.

Mean Absolute Percentage Error measures the relative prediction error in percentage metrics, allowing for scale-independent comparison across multiple responses is presented in Eq. 22.

19

20

21

22

Statistical validation

The ensemble stacking technique was further evaluated for robustness and statistical reliability using the following analyses:

1. 3-Fold CV Consistency: The standard deviation of R² (value ranges between zero and one, with a higher value indicating that the predicted values are close to the experimental ones) and MAPE (the value close to zero indicates better prediction) across folds were used to assess the model stability. Values with low variability across folds ensure consistent generalizability and reduced sensitivity across data classification, which is critical for limited experimental datasets.

2. Statistical Significance: Pairwise Wilcoxon signed-rank and Holm–Bonferroni tests were conducted to validate the efficacy of the base learner and ensemble stacking model. The ensemble predictions were compared against individual base learners (GPR, LSB, SVR, XGBoost and RF). This technique is suitable for small data sets that are independent of normality assumptions.

3. Residual Analysis: This analysis estimates the residuals (experimental – predicted) to validate and confirm the unbiasedness and consistent variance error over the prediction levels.

4. Base Model Homogeneity: One-way ANOVA is performed with the three best performing base learners’ based on absolute errors to determine whether the model is statistically significant. This analysis enables investigators to gain practical insights into model diversity and validate the need for ensemble learning for prediction.

5. Bootstrap Confidence Intervals: 1,000 resamples were used to quantify metric uncertainty.

6. Effect Sizes: Cohen’s d was calculated for significant improvements.

7. An overall computation cost analysis was performed comparing base learners with the ensemble model to examine the practical usefulness of the model.

8. The predictive model generalizability was tested for external validation with material stratified into 9 randomly generated experimental cases. The prediction precisions of the developed models were compared to recommend the best predictive model for practical usefulness.

Results and discussion

The results of predictive models (GPR, SVR, LSB, XGBoost, RF and CrayStack Ensemble Stacking) performances were evaluated for 8 responses, which were trained and validated with 27 experimental trials. The model prediction performances were evaluated using different metrics: (a) prediction accuracy metrics (R², RMSE, MAE, MAPE), (b) statistical significance through Wilcoxon signed-rank tests, paired Cohen’s d effect sizes, and bootstrap confidence intervals; (c) residual analysis for error distribution and overfitting diagnostics; (d) computational cost assessment via material-classified 3-fold cross-validation of predictive models, and (e) predictive models external validation with material stratified 9 random experimental cases.

Base learner model prediction performance evaluation

A total of L₂₇ experimental trials were used to develop predictive models based on GPR, LSB, SVR, RF, and XGBoost for machining quality characteristics. Table 6 presents the base model prediction performances evaluated using four performance metrics (R², RMSE, MAE, and MAPE) across 8 responses using 3-fold cross-validation. The results showed that the boosting techniques (LSB and XGBoost) exhibited identical performance metrics and outperformed RF across 8 machining quality characteristics. The GPR, LSB, and SVR consistently exhibited superior prediction performances than the RF model. Therefore, three base learners (GPR, LSB, and SVR) were selected to develop the ensemble predictive model.

Table 6.

predictive performance of base-learners across all responses using3-fold CV with a sample size of 27.

Output	Metric	GPR	LSB	SVR	Random Forest	XGBoost
Ra	R²	0.985	0.9827	0.966	0.9196	0.9827
	RMSE	0.115	0.1235	0.173	0.266	0.1235
	MAE	0.0778	0.0846	0.1213	0.1927	0.0846
	MAPE (%)	4.26	4.66	7.46	11.44	4.66
Fc	R²	0.9851	0.9829	0.9394	0.9217	0.9829
	RMSE	4.7715	5.1031	9.6197	10.9333	5.1031
	MAE	3.9071	3.8575	7.1702	7.6282	3.8575
	MAPE (%)	5.2	5.52	10.01	9.29	5.52
TWR	R²	0.9532	0.9483	0.9272	0.8671	0.9483
	RMSE	0.0118	0.0124	0.0147	0.0198	0.0124
	MAE	0.0083	0.0089	0.0111	0.0129	0.0089
	MAPE (%)	27.09	22.24	44.25	33.95	22.24
SCE	R²	0.6825	0.8482	0.6838	0.3919	0.8482
	RMSE	0.0229	0.0158	0.0228	0.0317	0.0158
	MAE	0.0174	0.0092	0.0178	0.0239	0.0092
	MAPE (%)	32.98	12.58	35.25	52.78	12.58
Vib	R²	0.9902	0.99	0.9795	0.922	0.99
	RMSE	0.0734	0.0739	0.1061	0.2067	0.0739
	MAE	0.0558	0.0584	0.0835	0.1554	0.0584
	MAPE (%)	8.36	8.39	13.6	20.27	8.39
Temp	R²	0.9686	0.9743	0.9284	0.9192	0.9743
	RMSE	4.216	3.8139	6.3638	6.7618	3.8139
	MAE	3.7971	3.1828	5.0067	5.6982	3.1828
	MAPE (%)	4.88	4.03	6.86	7.35	4.03
SPL	R²	0.9577	0.9567	0.951	0.9075	0.9567
	RMSE	1.4767	1.4944	1.5896	2.1834	1.4944
	MAE	1.1941	1.3629	1.2725	1.8542	1.3629
	MAPE (%)	1.67	1.86	1.75	2.47	1.86
MRR	R²	0.8595	0.7312	0.8637	0.3258	0.7312
	RMSE	77,140	1,06,674	75,965	1,68,959	1,06,674
	MAE	62,684	86,597	60,885	1,32,081	86,597
	MAPE (%)	34.22	37.47	27.92	66.88	37.47

Adaptive weight distribution and response-specific behaviour in Ensemble framework

The ensemble framework integrates the three best prediction learners (GPR, LSB, SVR) and adaptive weights derived through 27 experimental datasets using metaheuristic algorithms such as COA, GA, and PSO. The main goal of metaheuristic algorithms is to determine the optimal contribution of the best performing three individual models (GPR, LSB, and SVR) that maximizes predictive accuracy with limited experimental datasets.

The performances of metaheuristic algorithms were compared by computing the composite cost and solution accuracy. The mathematical formulation of composite cost function to simultaneously minimize prediction error and maximize goodness-of-fit is computed using Eq. 17. The composite cost function is used as an objective function by metaheuristic optimizers to determine best ensemble weights. The metaheuristic algorithms determine the weights [W = W_GPR, W_LSB, W_SVR] by minimizing the composite cost objective function based on out-of-fold predictions obtained via 3-fold cross-validation. All three algorithms conduct a population-based stochastic search mechanism that determines weights within the space [0, 1]³. At each cycle, the candidate weight vectors associated with base learners (GPR, LSB, SVR) are randomly initialized and evaluated for composite cost (treated as fitness). The candidate solution corresponding to the composite cost is treated as the best solution, and the resulting weights are normalized to satisfy the unity constraint. The metaheuristic algorithm iteratively updates and refines the search mechanism (neglecting weak learners and promoting dominant models), thereby enabling output specific and data-driven ensemble weights.

Figure 3 shows the convergence behaviour of metaheuristic algorithms (COA, PSO, and GA) tested against eight machining quality characteristics. All three metaheuristic algorithms decrease the composite cost function progressively over successive iterations to achieve improved ensemble weights. Moreover, all three algorithms converge to identical composite cost solutions (ensemble weights and performance metrics: R², RMSE, MAE, and MAPE) depicting the robustness of the stacking framework and validating the solutions are not algorithm dependent (refer to Table 7).

Fig. 3.

Convergence plot of three algorithms (COA, PSO, and GA) for ensemble weight optimization against 8 machining quality characteristics.

Table 7.

Predictive performance of metaheuristic algorithms across all responses using 3-Fold CV with a sample size of 27.

Output	Method	Weights (GPR, LSB, SVR)	R²	RMSE	MAE	MAPE (%)
Ra	COA	(0.848, 0.152, 0.000)	0.985	0.1149	0.077	4.17
	PSO	(0.848, 0.152, 0.000)	0.985	0.1149	0.077	4.17
	GA	(0.848, 0.152, 0.000)	0.985	0.1149	0.077	4.17
Fc	COA	(0.650, 0.350, 0.000)	0.9856	4.6857	3.7224	5.06
	PSO	(0.650, 0.350, 0.000)	0.9856	4.6857	3.7224	5.06
	GA	(0.650, 0.350, 0.000)	0.9856	4.6857	3.7224	5.06
TWR	COA	(0.773, 0.227, 0.000)	0.9537	0.0117	0.0081	24.53
	PSO	(0.773, 0.227, 0.000)	0.9537	0.0117	0.0081	24.53
	GA	(0.773, 0.227, 0.000)	0.9537	0.0117	0.0081	24.53
SCE	COA	(0.000, 1.000, 0.000)	0.8482	0.0158	0.0092	12.58
	PSO	(0.000, 1.000, 0.000)	0.8482	0.0158	0.0092	12.58
	GA	(0.000, 1.000, 0.000)	0.8482	0.0158	0.0092	12.58
Vib	COA	(0.537, 0.438, 0.025)	0.9911	0.0698	0.0535	7.86
	PSO	(0.537, 0.438, 0.025)	0.9911	0.0698	0.0535	7.86
	GA	(0.537, 0.438, 0.025)	0.9911	0.0698	0.0535	7.86
Temp	COA	(0.000, 1.000, 0.000)	0.9743	3.8139	3.1828	4.03
	PSO	(0.000, 1.000, 0.000)	0.9743	3.8139	3.1828	4.03
	GA	(0.000, 1.000, 0.000)	0.9743	3.8139	3.1828	4.03
SPL	COA	(0.372, 0.384, 0.245)	0.9602	1.432	1.197	1.66
	PSO	(0.372, 0.384, 0.245)	0.9602	1.432	1.197	1.66
	GA	(0.372, 0.384, 0.245)	0.9602	1.432	1.1971	1.66
MRR	COA	(0.307, 0.108, 0.586)	0.8771	72,137	52,596	26.19
	PSO	(0.307, 0.108, 0.586)	0.8771	72,137	52,596	26.19
	GA	(0.307, 0.108, 0.586)	0.8771	72,137	52,596	26.19

Figure 4 presents the determined optimal weights for individual responses for different models. It is interesting to note that all three metaheuristic algorithms converged to identical weights with composite cost functions (refer to Table 8). Table 8 presents the computation efficiency of metaheuristic algorithms tested against 8 responses. The COA converges to the minimum composite cost with fewer iterations than PSO and GA, but the computation time does not show proportionate scale linearly with the iteration count. The reasons are attributed to the different learning mechanisms of each algorithm per iteration. The COA demonstrated excellent computational efficiency compared to PSO and GA across 8 responses (excluding Ra), with an average computation time of 0.096 s, 0.104 s, and 0.126 s, respectively. The better performance with COA is attributed to fewer or less sensitive tuning or algorithm specific parameters. Note that, PSO requires tuning of inertial weight, social and cognitive leaders^72,73, and GA requires tuning of the probability of crossover, mutation rate, and selection strategies such as tournament and roulette selection⁷⁴. The reduced computational time highlights the effectiveness of COA as a lightweight optimization approach for ensemble weight determination. Therefore, COA weights were used to develop the ensemble model for evaluating the prediction performances across multiple responses.

Fig. 4.

Optimized weight distribution for an ensemble model across 8 machining quality characteristics.

Table 8.

Comparison of computational time for COA, PSO, and GA.

Output	Method	Best Composite Cost Function	Best iteration	Convergence time (s)
Ra	COA	0.024815	4	0.056
	PSO	0.024815	3	0.051
	GA	0.024815	22	0.061
Fc	COA	0.025035	13	0.025
	PSO	0.025035	2	0.025
	GA	0.025035	11	0.028
TWR	COA	0.054322	8	0.189
	PSO	0.054322	4	0.225
	GA	0.054322	16	0.262
SCE	COA	0.129008	1	0.128
	PSO	0.129008	5	0.133
	GA	0.129008	10	0.164
Vib	COA	0.018516	22	0.121
	PSO	0.018516	22	0.128
	GA	0.018516	96	0.155
Temp	COA	0.038129	1	0.099
	PSO	0.038129	5	0.102
	GA	0.038129	11	0.131
SPL	COA	0.049901	14	0.053
	PSO	0.049901	12	0.064
	GA	0.049901	18	0.082
MRR	COA	0.097923	13	0.097
	PSO	0.097923	18	0.101
	GA	0.097923	13	0.127

Figure 4 shows the variation in metaheuristic algorithm optimized weights for three base learners across multiple responses reflecting an adaptive, response specific learning mechanism due to the distinct physical and nonlinear characteristics of each machining response (Ra, Fc, TWR, SCE, Vib, Temp, SPL and MRR). Instead of Rather than enforcing uniform model participation, the COA assigns higher weights to models that best match the underlying response behavior. GPR excels in smooth learning and continuous functional relationships (tool–material interactions) with limited data and therefore such relationships were found to receive dominant weights derived for R_a, Fc, TWR, and Vibration. In contrast, LSB determined an exclusive weight of 1.0 for SCE and temperature, depicting its effectiveness in modeling highly nonlinear and discontinuous response behaviour. These outputs are governed by threshold-driven phenomena such as thermal accumulation and damage initiation, where sudden and cumulative energy are aggregated. The structure of LSB enables it to accurately capture these complex interactions between machining parameters and response variables.

Dynamic responses such as vibration and SPL contain both steady patterns and random noise arising from machine-tool interaction and process instability. Hence, the ensemble assigns mixed weights across learners to better capture both effects. MRR exhibits strong nonlinear interactions among machining parameters, making SVR more suitable due to its capability to model complex, non-smooth relationships using kernel functions. This explains its dominant contribution in this case. Zero or near-zero weights assigned to certain learners are not indicative of model weakness but reflect their limited contribution to specific response structures. This represents an implicit model selection mechanism within COA, which suppresses redundant learners to enhance predictive accuracy and generalization. Overall, the non-uniform distribution of weights (including zero, unity, and hybrid combinations) demonstrates that the proposed framework performs both optimal weighting and adaptive model selection, guided by the physical and statistical nature of each machining output. Therefore, the ensemble framework was established by integrating the best performing base learners (GPR, LSB, and SVR) and the metaheuristic algorithm (COA) referred to as the CrayStack ensemble.

Response-wise ablation analysis of optimized weights

Figure 4 shows the response wise analysis of ensemble stacking exhibiting an adaptive weight depending on the response behaviour characteristics. The ensemble stacking demonstrated that multiple base learners receives non-zero weights and balanced weights for responses (such as Fc, TWR, Vib, SPL, and MRR) reflecting complementary learning insights. The contribution of multiple models to the ensemble stacking framework enables the metaheuristic algorithm to understand the non-linear behaviour and achieve a better bias-variance trade-off resulting in enhanced generalization and prediction stability. It is important to note that all three base learners contribute to ensure the effectiveness of the ensemble stacking framework across responses such as SPL and MRR. The COA algorithms assign full weight (1.0) to LSB for SCE and temperature, depicting the ensemble model transforming into a single model (LSB). Finally, response-wise ablation analysis confirmed the multi-model dominance (Fc, Vib, SPL, and MRR), partial dominance (Ra, TWR) and single model dominance (SCE and Temperature), depicting the benefits of the ensemble framework being response-driven, and governed by the complexity of response behavioural patterns (interaction, and non-linearity) and diversity among base learners.

Comparison of craystack ensemble prediction performances with base learners

CrayStack Ensemble (SVR, GPR, LSB, and COA) demonstrated consistent predictive superiority across all eight machining responses (Ra, Fc, TWR, SCE, Vib, Temp, SPL, and MRR), achieving either the best or statistically equivalent performance relative to the strongest base learner (refer to Table 9). The CrayStack ensemble model outperforms all base learners across all responses (8/8 cases), with substantial reductions in error metrics. CrayStack ensemble model showed improved or comparable performance in 6 out of 8 responses, with parity observed in SCE and Temp. For surface roughness (Ra), the CrayStack ensemble achieves the highest R² = 0.9850, matching GPR while improving prediction stability through reductions in RMSE, MAE, and MAPE of approximately 7.0%, 8.2%, and 10.5% over LSB. Error reductions exceed 55% relative to RF. A similar trend is observed for Fc, where the CrayStack ensemble produced the highest R² (0.9856), with modest gains over GPR and LSB, and RMSE reductions exceeding 50% compared with RF.

Table 9.

Comparison of the prediction performances of CrayStack Ensemble and base learner models across 8 responses using3-Fold CV with a sample size of 27.

Output	Metric	CrayStack Ensemble	Remarks (Comparing CrayStack with GPR, LSB, RF, SVR and XGBoost presented in Table 6)
Ra	R²	0.985	Comparable with GPR and outperformed other models
	RMSE	0.1149	Comparable with GPR and outperformed other models
	MAE	0.0778	Comparable with GPR and outperformed other models
	MAPE (%)	4.17	Outperform GPR with negligibly small variations
Fc	R²	0.9856	Outperform GPR and other models
	RMSE	4.6857	Outperform GPR and other models
	MAE	3.7224	Outperform GPR and other models
	MAPE (%)	5.06	Outperform GPR and other models
TWR	R²	0.9537	Outperform GPR, LSB and other models
	RMSE	0.0117	Outperform GPR, LSB and other models
	MAE	0.0081	Outperform GPR, LSB and other models
	MAPE (%)	24.53	Slightly inferior compared to LSB and XGBoost
SCE	R²	0.8482	Comparable with LSB and XGBoost and outperformed other models
	RMSE	0.0158	Comparable with LSB and XGBoost and outperformed other models
	MAE	0.0092	Comparable with LSB and XGBoost and outperformed other models
	MAPE (%)	12.58	Comparable with LSB and XGBoost and outperformed other models
Vib	R²	0.9911	Outperform GPR and other models
	RMSE	0.0698	Outperform GPR and other models
	MAE	0.0535	Outperform GPR and other models
	MAPE (%)	7.87	Outperform GPR and other models
Temp	R²	0.9743	Comparable with LSB & XGBoost, and outperformed other models
	RMSE	3.8139	Comparable with LSB and XGBoost and outperformed other models
	MAE	3.1828	Comparable with LSB and XGBoost and outperformed other models
	MAPE (%)	4.03	Comparable with LSB and XGBoost and outperformed other models
SPL	R²	0.9602	Outperform GPR and other models
	RMSE	1.4318	Outperform GPR and other models
	MAE	1.2014	Outperform GPR and other models
	MAPE (%)	1.66	Outperform GPR and other models
MRR	R²	0.8771	Outperform GPR and other models
	RMSE	72,137	Outperform GPR and other models
	MAE	52,596	Outperform GPR and other models
	MAPE (%)	26.19	Outperform GPR and other models

TWR exhibits marginal improvements in R² (0.9537), indicating close agreement among strong base learners. However, the CrayStack ensemble model still achieves reduced error of approximately 41% in RMSE relative to RF and moderate gains over XGBoost. This suggests improved robustness in capturing gradual wear evolution under limited data conditions. More pronounced advantages are observed for nonlinear and dynamic responses. The vibration response achieves the highest R² (0.9911), with RMSE reductions exceeding 60% compared with RF and appreciable improvements over XGBoost. MRR presents the most significant gain, where the CrayStack ensemble improves R² to 0.8771 and reduces the RMSE, MAE, and MAPE by approximately 6.5%, 16.3%, and 23.6% relative to GPR, while also outperforming RF by more than 55% in RMSE and XGBoost by approximately 30%. SCE and temperature responses shows identical performance between CrayStack ensemble, LSB, and XGBoost across all metrics, indicating that boosting-based models captured the governing process dynamics completely. Despite this parity, all models significantly outperform RF with 40–50% reductions in RMSE. The above discussion suggests that the predictive limitation lies in the model class (bagging vs. boosting) rather than ensemble stacking. SPL demonstrates moderate but consistent improvement, with the CrayStack ensemble achieving R² = 0.9602 and reducing RMSE by approximately 4.2% compared with LSB and > 30% relative to RF.

The adaptive weighting strategy of the CrayStack Ensemble outperformed other base learners probably due to ensemble models selecting the best model that captures the physics of response behaviour, resulting in an improved bias-variance trade-off subjected to a limited dataset. The CrayStack Ensemble model is more effective for nonlinear responses, where the interaction and stochastic variability dominates, while smoother responses tend to capture insights by individual boosting learners. Therefore, the CrayStack ensemble through its adaptive stacking mechanism enhances the robustness and generalizability across all responses for data-limited manufacturing environments.

The CrayStack ensemble prediction performances were evaluated with a best fit line against experimental datapoints for eight responses. Figure 5 depicts that the CrayStack ensemble consistently performs better predictions with datapoints lying close to the ideal trend line and higher R² value for all responses (excluding SCE).

Fig. 5.

Comparison between predicted and experimental values for eight responses using the CrayStack ensemble (3-Fold Cross-Validation, n = 27).

Residual analysis was conducted for each experimental trial to validate the model’s statistical and practical utility (refer to Fig. 6). It is important to note that the CrayStack ensemble predicted response values under 3-fold cross-validation are randomly distributed above and below the zero-reference line ensuring the model generalization capability. Furthermore, the CrayStack ensemble predictions across extreme operating values of variables ensure the models ability to capture the nonlinear dependent relationships rather than overfitting or numerical artifacts with small datasets.

Fig. 6.

Residuals plotted against the predicted values for the CrayStack Ensemble (training cases).

Figure 7 shows the prediction performances of six models (GPR, LSB, XGBoost, SVR, RF and CrayStack ensemble) for training cases under 3-fold cross-validation considering all 8 responses.

Fig. 7.

Performance comparison between the CrayStack Ensemble and base learners across eight responses (3-Fold CV, n = 27): (a) R², (b) MAE, (c) RMSE, and (d) MAPE.

CrayStack ensemble produced the overall better predictions (highest R², and lower RMSE, MAE and MAPE) considering all responses, depicting the model’s ability to capture the physics of process insights and thereby validating the model’s effectiveness. The COA optimized ensemble weights are effective for small datasets influencing multiple machining quality characteristics.

Statistical validation of craystack ensemble model

The base learner models (GRP, LSB and SVR) with the CrayStack ensemble model were evaluated on the identical combination of experimental data (same 3-fold with test points) ensuring an accurate comparison of prediction errors. Therefore, the CrayStack ensemble was assessed using pairwise bootstrap (1,000 resamples) to compute 95% confidence intervals (CIs) for R², RMSE, MAE, and MAPE (refer to Table 10). The present work examined the Wilcoxon signed-rank tests to validate the CrayStack ensemble model with individual models (GRP, LSB, and SVR) using absolute prediction errors without normality assumptions. (ML models rarely follow a normal distribution particularly for small datasets and nonlinear behaviour); ensuring robustness and realistic comparison. Effect sizes for significant Wilcoxon results were quantified using paired Cohen’s d, with |d| > 0.5 (medium) and ≥ 0.8 (large) indicating practically meaningful improvements beyond statistical significance.

Table 10.

Pairwise Wilcoxon Signed-Rank Test, Effect Sizes, and Base-Model ANOVA Results.

Response	GPR vs. CrayStack ensemble			LSB vs. CrayStack ensemble			SVR vs. CrayStack ensemble			ANOVA (Base Models)
	p-value	Cohen’s d	Holm adj. p-value	p-value	Cohen’s d	Holm adj. p-value	p-value	Cohen’s d	Holm adj. p-value	p-value
Ra	0.239	0.11 (neg)	1.000	0.190	0.19 (neg)	1.000	< 0.001	2.18 (large)	< 0.024	0.981
Fc	0.211	0.18 (neg)	1.000	0.355	0.07 (neg)	1.000	< 0.001	1.88 (large)	< 0.024	0.989
TWR	0.218	0.13 (neg)	1.000	0.178	0.22 (small)	1.000	< 0.001	1.75 (large)	< 0.024	0.979
SCE	0.001	0.58 (medium)	0.024	0.582	−0.00 (neg)	1.000	< 0.001	1.36 (large)	< 0.024	0.853
Vib	0.429	0.06 (neg)	1.000	0.154	0.16 (neg)	1.000	< 0.001	1.87 (large)	< 0.024	0.999
Temp	0.033	0.39 (small)	0.759	0.304	0.10 (neg)	1.000	< 0.001	1.52 (large)	< 0.024	0.986
SPL	0.888	−0.41 (small)	1.000	0.663	−0.31 (small)	1.000	< 0.001	1.44 (large)	< 0.024	0.997
MRR	0.672	−0.51 (medium)	1.000	0.037	−0.23 (small)	0.851	< 0.001	1.11 (large)	< 0.024	0.970

The Holm–Bonferroni method (to update p-values) and Benjamini–Hochberg correction were used as a secondary validation approach to monitor the multiple comparisons across 24 Wilcoxon tests (8 responses × 3 pairwise comparisons). The SVR vs. Ensemble showed statistically significant after Holm–Bonferroni correction, whereas the GPR and LSB comparison became insignificant after correction confirming the ensemble predictions improvement over SVR. The Wilcoxon tests ensure pairwise comparison of CrayStack ensemble with individual outputs, producing p value tables presented in Table 10. The statistical significance was evaluated with a pre-set confidence level of 95% (p-value < 0.05). The Wilcoxon analysis confirmed that CrayStack ensemble consistently outperformed SVR for all 8 machining output predictions (confirmed p < 0.001), with large effect sizes (|d| = 1.11–2.18). Comparisons (SVR vs. Ensemble with p < 0.001) showed significance after Holm–Bonferroni correction. The results clearly depict that the ensemble consistently mitigates SVR’s limitations in capturing nonlinear behaviour and dependent interrelationships between outputs.

Comparing GRP and LSB, most outputs (excluding SCE and Temp for GRP, and MRR for LSB) exhibit insignificance (as their p-values > 0.05) depicting that CrayStack ensemble preserves the predictive strength of the best predicted model (from GRP, LSB) to ensure more stability. It is noteworthy that the CrayStack ensemble ensures specific improvements that are statistically validated for SCE versus GRP (p = 0.001, d = 0.58, medium) and for MRR versus LSB (p = 0.037, d = -0.23, small but directionally favorable), where there exists a strong parameter relationship and response variability. One-way ANOVA results confirmed that all models (GRP, SVR, and LSB) exhibit statistically comparable prediction performances (p > 0.85). Note that better prediction by the CrayStack ensemble is attributed to the derived adaptive ensemble weight rather than the individual dominant model.

Bootstrap analysis

In an ensemble framework, bootstrap resampling (sample size: 1000) is implemented to determine the statistical reliability and uncertainty in predictions. The results of the bootstrap analysis were performed across multiple responses to compute 95% CI for performance metrics (R², RMSE, MAE, and MAPE) as presented in Table 11. The narrow CIs for responses such as Ra (R² = 0.985 [0.97, 0.99]) and vibration (R² = 0.991 [0.98, 1.00]) depict reliable predictive consistency. However, a wider range of CIs for MRR (R² = 0.879 [0.80, 0.92]) depicts inherent process uncertainty. The CIs across all performance metrics (RMSE, MAE and MAPE) confirming robust model generalization.

Table 11.

Predictive performance of the CrayStack Ensemble model with 95% bootstrap confidence intervals.

Response	R² [95% CI]	RMSE [95% CI]	MAE [95% CI]	MAPE% [95% CI]
Ra	0.985 [0.97, 0.99]	0.115 [0.10, 0.20]	0.077 [0.00, 0.10]	4.2 [2.6, 6.0]
Fc	0.986 [0.97, 0.99]	4.69 [3.1, 6.3]	3.72 [2.7, 4.9]	5.1 [3.7, 6.6]
TWR	0.954 [0.90, 0.98]	0.012 [0.01, 0.02]	0.008 [0.01, 0.01]	24.5 [13.7, 40.6]
SCE	0.848 [0.71, 0.94]	0.016 [0.01, 0.02]	0.009 [0.01, 0.01]	12.6 [8.4, 17.3]
Vib	0.991 [0.98, 1.00]	0.070 [0.06, 0.08]	0.054 [0.04, 0.07]	7.9 [4.7, 11.7]
Temp	0.974 [0.96, 0.98]	3.81 [3.0, 4.6]	3.18 [2.4, 4.0]	4.0 [3.1, 5.0]
SPL	0.960 [0.93, 0.98]	1.43 [1.1, 1.7]	1.20 [0.9, 1.5]	1.7 [1.2, 2.1]
MRR	0.879 [0.80, 0.92]	71.6 k [49.4 k, 94.8 k]	52.6 k [35.3 k, 72.8 k]	26.2 [14.9, 41.0]

Three-fold cross-validation consistency

The CrayStack ensemble stability is evaluated with material-classified three-fold cross validation on small datasets (experiments: 27). The robustness of model predictions was examined with standard deviations of R² and RMSE across each fold. For most responses, the CrayStack ensemble showed the highest predictive accuracy with very low variability between folds. The variability between folds for vibration, SPL, and Ra resulted in values equal to and , and , and and , respectively. The narrow deviation ensures stable prediction performances and generalization that are sensitive to data classification. The slightly higher variability in R² (within the range of 0.0052 to 0.0254 for Fc, temperature and TWR) reflects the inherent thermomechanical fluctuation in outputs. Higher variability for SCE and MRR with broader numerical ranges in R² value () reflects more scattered behaviour. Higher variability with responses (SCE and MRR) is attributed to those responses that are sensitive to changes in cutting conditions and material composition. Figure 8 presents the cross-validation results of all three folds. The results confirm that all folds ensure consistent performance and acceptable accuracy validated through mean R² values and standard deviation error bars. The minimum error bars were found for R_a, Fc, vibration and SPL depicting that prediction accuracy remains stable across multiple data splits. In summary, the model predictions are consistent for data trained with small datasets.

Fig. 8.

Performance consistency of the CrayStack Ensemble across 3-fold CV for stability for various machining responses.

Mechanistic insights on craystack ensemble superiority over base models

The CrayStack ensemble understands the physics of the process through its adaptive weights assigned to the best performing individual models (GPR, LSB, and SVR), resulting in better performance in predicting each response. The GPR model dominates predictions with R² ≥ 0.98 for responses (considered smooth outputs) such as Ra and Fc, whereas LSB predicts better for highly nonlinear responses (such as SCE and temperature) with a greater reduction in MAPE. MRR uses a combination of GPR–SVR with a contribution share of 71.8%–28.2% depicting their ability to capture trends and residual variations. The optimal simplified COA weights enable stable predictions confirmed by statistical examination (7 out of 8 responses depicting p < 0.005) and R² values vary in the ranges of 0.848–0.991, demonstrating that the intelligent ensemble framework outperforms individual model predictions with small datasets.

Computational cost analysis

The comprehensive evaluation of the computational efficiency of the CrayStack ensemble framework has been compared with the individual base learners (GPR, LSB, SVR, RF and XG_Boost). The factor analysis considers training time, inference latency, scalability, and memory footprint that evaluate the practical utility to use in real-time industrial machining applications. All experiments were conducted on a workstation with an Intel Core i3(1.20 GHz) and 8 GB RAM, running in MATLAB R2021 software platform.

The theoretical computational complexities of all models are summarized in Table 13, where n denotes the number of samples, d the number of features, t the number of trees, n_sv the number of support vectors, and p the number of base learners in CrayStack (p = 3). As shown in Table 13, GPR exhibits cubic training complexity , while SVR scales between and depending on the solver. Tree-based methods such as LSB, RF, and XGBoost follow complexity, making them more scalable for larger datasets. In contrast, the CrayStack ensemble framework combines the cost of its base learners with an additional optimization overhead from the simplified COA, expressed as . Notably, this optimization operates on a low-dimensional weight space ( p=3), making its cost effectively constant for the present study. Furthermore, CrayStack ensemble inference requires only a weighted aggregation of base model predictions, resulting in constant-time complexity .

Table 13.

Empirical Training Time (seconds) for Individual and Ensemble Model Outputs.

Output	GPR	SVR	LSB	RF	XGBoost	CrayStack Ensemble*
Ra	0.1644	0.0434	0.9211	0.2683	0.93	1.029
Fc	0.1269	0.069	2.7815	0.7193	2.5596	2.8848
TWR	0.0664	0.0268	1.662	0.3727	1.5863	1.705
SCE	0.0789	0.0312	1.9406	0.4688	2.2154	1.9929
Vib	0.076	0.0262	1.6694	0.3588	1.6259	1.7044
Temp	0.0836	0.0326	1.7902	0.4144	1.7429	1.8572
SPL	0.0719	0.0265	1.5889	0.3607	1.5393	1.6368
MRR	0.0696	0.032	1.9281	0.4164	1.6945	1.9797
Mean	0.0922	0.0359	1.7852	0.4224	1.7368	1.8487
Total (All 8 outputs)	0.7377	0.2877	14.28	3.38	13.89	14.7898

*CrayStack Ensemble training time shown is the COA optimization time only (excludes base model training).

Table 12.

Theoretical Computational Complexity of base learner and Ensemble Models.

Model	Training Complexity	Inference Complexity	Memory Complexity
GPR
SVR	to
LSB
RF
XGBoost
CrayStack Ensemble

Empirical training times across the eight machining outputs are reported in Table 13. SVR trains the fastest (0.2877 s total), benefiting from the small dataset size and efficient sequential minimal optimization solver. Boosting techniques require more computation time. Table 14 breaks down CrayStack’s ensemble total training cost of 16.1278 s, wherein base model training accounts for 94.92% (15.3072 s), COA weight optimization adds only 1.88% (0.3032 s), and bootstrap confidence intervals contribute 3.2% (0.5174 s). The ensemble retains the training cost of all base models, which represents the dominant computational expense.

Table 14.

Total training time breakdown for CrayStack Ensemble.

Details	Time (seconds)	Percentage
Base Models (GPR + SVR + LSB)	15.3072	94.92%
COA Weight Optimization	0.3032	1.88%
Bootstrap Confidence Intervals	0.5174	3.2%
CrayStack Ensemble (Total)	16.1278	100%

Inference latency is critical for real-time monitoring. Table 16 reports per-sample prediction times after training. The CrayStack ensemble achieves near-instantaneous prediction (0.003 ms/sample, 290592 samples/sec) because inference reduces to: requiring only three scalar multiplications and two additions. This constant-time operation is 58.33 times faster than GPR, 79.33 times faster than SVR, 4899.33 times faster than LSB, 5630.67 times faster than RF and 5608.3 times faster than XGBoost.

Table 16.

Scalability Projections of base learners and CrayStack ensemble models.

Sample Size (n)	GPR	SVR	LSB/RF/XGB	CrayStack Ensemble (Optimization)	CrayStack Ensemble (Inference)
27	19,683	5,832	102,706	162,000	O(3)
100	1,000,000	80,000	531,508	600,000	O(3)
500	125,000,000	2,000,000	3,585,754	3,000,000	O(3)
1,000	1,000,000,000	8,000,000	7,973,610	6,000,000	O(3)
5,000	125,000,000,000	200,000,000	49,150,849	30,000,000	O(3)

Table 15.

Inference time per sample.

Model	Time (ms)	Samples per Second	Relative Speed
CrayStack Ensemble	0.003	290,592	1.00 × (fastest)
GPR	0.175	5712	58.33 × slower
SVR	0.238	4205	79.33 × slower
LSB	14.698	68	4899.33 × slower
RF	16.892	59	5630.67 × slower
XGBoost	16.825	59	5608.33 × slower

Table 16 presents the projected computational complexity or growth with increased size of datasets. The results indicate that GPR and SVR exhibit rapid increases in computational cost as n grows, whereas tree-based models scale more efficiently. Notably, the optimization component of the CrayStack ensemble framework scales linearly with the dataset size, as it operates solely on base model predictions rather than the original feature space. This linear computation growth ensures that the additional optimization overhead does not become a bottleneck even as data size increases. Although CrayStack Ensemble scales linearly with the sample size n, the growth rate is comparatively lower than GPR and SVR, depicting the ensemble model suitable for practical scalability.

Overall, the computational cost analysis indicates that the proposed CrayStack ensemble framework provides an effective balance between accuracy and efficiency, with negligible optimization overhead and substantially reduced inference latency. These characteristics make it particularly suitable for real-time machining monitoring applications under data-scarce conditions, where both predictive accuracy and computational efficiency are critical.

Comparison of predictive models against experimental test cases

The trained GPR, LSB, SVR, RF, XGBoost and CrayStack Ensemble models’ predictive performances were examined against nine experimental cases (generated at random within the operating range of input variables presented in Table 2). The input-output datasets used for model predictions and performance evaluations are presented in Table 17.

Table 17.

Experimental test samples and measured responses.

Exp	Material		r n	Vc	f	ap	Ra (µm)	Fc (N)	Temp (°C)	Vib (m/s²)	TWR (g/min)	SPL (dB)	MRR (mm³/min)	SCE (J/mm³)
1	1	80	0.8	120	0.25	1.4	1.78	73.1	80.2	1.01	0.06	75.7	447,000	0.018
2	1	98	1.2	170	0.19	1.2	1.64	63.8	85.4	0.8	0.0512	78.5	401,000	0.063
3	1	101	0.4	190	0.13	0.8	1.43	69.4	79.9	0.82	0.0452	71.6	200,000	0.088
4	2	103	0.4	160	0.12	1.3	3.08	149.3	111.2	2.38	0.1594	84.7	289,000	0.091
5	2	95	1.2	180	0.18	0.9	2.99	123.2	119.9	1.87	0.132	83.6	271,000	0.084
6	2	80	0.8	130	0.26	0.6	2.78	130.3	109.4	1.76	0.124	80.7	191,200	0.095
7	3	85	1.2	190	0.28	1.3	1.24	56.7	63.2	0.44	0.0259	68.3	592,000	0.006
8	3	100	0.4	110	0.21	0.5	0.82	36.4	48.5	0.18	0.0043	64.5	58,500	0.042
9	3	90	0.8	160	0.15	0.7	0.93	35.5	54.4	0.2	0.0038	67.5	95,000	0.051

The predictive performances of nine experimental cases by all models were evaluated using of performance metrics (R², RMSE, MAE and MAPE). The CrayStack ensemble demonstrated better prediction accuracy for the 9 test cases with lower MAPE across all individual responses (refer to Table 18). The average absolute percent deviation in predicting all responses for nine experimental cases is presented in Fig. 9.

Table 18.

Model predictive performance comparison across all responses (nine experimental cases).

Output	Metric	GPR	LSB	SVR	Random Forest	XGBoost	CrayStack Ensemble
Ra	R²	0.9937	0.9976	0.9875	0.9811	0.9909	0.9948
	RMSE	0.1376	0.1076	0.1279	0.1868	0.1446	0.1188
	MAE	0.1	0.08	0.11	0.15	0.12	0.09
	MAPE (%)	4.12	3.89	6.02	12.19	5.78	3.79
Fc	R²	0.9811	0.9721	0.9765	0.9468	0.9781	0.9796
	RMSE	5.93	6.975	6.464	13.591	6.559	6.061
	MAE	4.51	5.77	4.49	10.27	4.92	4.44
	MAPE (%)	6.85	7.77	6.45	18.22	7.24	6.12
TWR	R²	0.9826	0.9868	0.9739	0.9587	0.9642	0.9857
	RMSE	0.01	0.0104	0.0117	0.0196	0.0135	0.0099
	MAE	0.0081	0.0083	0.0093	0.0157	0.0112	0.0077
	MAPE (%)	28.33	22.29	18.7	93.48	30.21	17.48
SCE	R²	0.9411	0.9069	0.6171	0.8364	0.8429	0.8854
	RMSE	0.0077	0.0106	0.0245	0.0175	0.0125	0.0105
	MAE	0.0067	0.0096	0.0194	0.017	0.0096	0.0083
	MAPE (%)	17.7	36.2	77.73	74.79	21.64	14.89
Vib	R²	0.9893	0.9753	0.9886	0.9592	0.9762	0.9877
	RMSE	0.08	0.1175	0.0802	0.2262	0.1167	0.0823
	MAE	0.0714	0.0904	0.0666	0.168	0.0959	0.0666
	MAPE (%)	12.52	8.76	10.93	46.62	14.83	7.75
Temp	R²	0.9738	0.9754	0.9728	0.9688	0.9804	0.9754
	RMSE	4.76	5.09	4.75	7.92	4.18	4.76
	MAE	4.01	3.86	3.76	6.43	3.67	3.85
	MAPE (%)	4.6	4.48	4.52	7.98	4.53	4.49
SPL	R²	0.9373	0.9318	0.9443	0.8981	0.9427	0.941
	RMSE	2.02	2.17	2.14	2.93	2.07	2.05
	MAE	1.4	1.6	1.5	2.33	1.55	1.37
	MAPE (%)	1.87	2.13	2.01	3.05	2.09	1.84
MRR	R²	0.9668	0.9825	0.865	0.889	0.8939	0.9782
	RMSE	30905.3	33578.8	85,539	88528.7	72512.7	28076.3
	MAE	25105.6	26299.1	69483.1	78275.6	63803.6	24502.6
	MAPE (%)	17	15	30.3	54.9	27.2	16.1

Fig. 9.

Prediction performances of base learner and CrayStack Ensemble models for 9 test cases.

Superior predictive performance by the CrayStack ensemble is attributed to the hybrid framework combining base learners and COA. The CrayStack ensemble combines the strengths of LSB (effective in capturing nonlinear interactions)⁷⁵, GPR (enabling probabilistic prediction and smooth function approximation)^76,77, and the adaptive weighting scheme of COA (assigning weights to individual base learners based on predictive capability) scheme of COA that minimizes overall prediction errors. Furthermore, the CrayStack ensemble simultaneously minimizes the model-specific bias and variance resulting in improved prediction of eight machining quality characteristics. The CrayStack Ensemble predicts with an overall MAPE of 9.8% considering all responses depicting better generalizability despite the model trained with limited experimental datasets. Collecting large experimental machining datasets is often impractical due to material waste, time consumption, and high cost. Therefore, CrayStack Ensemble achieving MAPE less than 10% considering all responses confirm the model practical usefulness for informed decision making through predictions with reduced experimental dependency and thereby lower the overall machining cost.

Conclusion, implications and recommendations

The present study developed base learners (GPR, LSB, SVR, XGBoost, and RF) and adaptive weight optimization for the best performing base learners using metaheuristic algorithms (GA, PSO, and COA) necessary for establishing an ensemble model to predict multiple machining quality characteristics using limited datasets (27 samples). The following conclusions are drawn from the study :

• The Taguchi L₂₇ method was used to collect experimental data consisting of 6 influencing and 8 machining quality characteristics. The 27 experimental datasets were material stratified into a 3-fold cross-validation protocol for training the developed ensemble and base learner models. The boosting techniques (LSB and XGB) showed similar performance metrics (R², RMSE, MAE, and MAPE) and outperformed RF prediction. Three best performing base learners (GPR, LSB, and SVR) were selected to develop the ensemble predictive model.

• Metaheuristic algorithms (COA, GA and PSO) were utilized to determine adaptive weights based on the contributions of the best performing base learner models (GPR, LSB, and SVR), enabling output specific and data-driven ensemble weights. All three algorithms converged to identical composite cost solutions (ensemble weights and performance metrics, demonstrating the robustness of the stacking framework and validating that the solutions are not algorithm dependent. COA exhibited excellent computational efficiency compared to PSO and GA across 8 responses, with an average computation time of 0.096 s, 0.104 s, and 0.126 s, respectively.

• Statistical validation through Wilcoxon signed-rank tests, paired Cohen’s d effect sizes, and bootstrap 95% confidence intervals confirmed the robustness of the improvements, including significant gains over SVR (p < 0.001, large effect size).

• The composite-objective optimization strategy, combining R², RMSE, and MAE, enabled balanced and reliable evaluation across responses. The optimized ensemble weights revealed response-specific dominance of base learners, reflecting the ability of the CrayStack ensemble to capture distinct nonlinear machining dynamics.

• Residual analysis and material-classified 3-fold cross-validation confirmed the strong generalization of CrayStack ensemble without overfitting.

• Computational cost analysis showed near-instantaneous inference latency) and negligible optimization overhead, demonstrating efficient real-time applicability under data-scarce conditions.

• CrayStack ensemble demonstrated superior prediction performances against nine randomly generated test cases, with a MAPE of 9.8%, showcasing enhanced prediction accuracy trained with limited datasets that minimizes experimental costs and favor process optimization.

• The results demonstrated that the intelligent ensemble stack ensures robustness and higher prediction accuracy for limited experimental datasets offering a practical solution for industrial process optimization.

• Future research should focus on validating the proposed framework using larger and more diverse datasets that further improve generalization capability. Extending research work to multi-output model or multi-task learning frameworks could better capture interdependencies among machining responses. Incorporating additional and more diverse base learners that further enhance predictive performance. Moreover, the integration of advanced adaptive optimization strategies and uncertainty quantification techniques could improve model reliability and interpretability. Finally, deployment in real industrial environments and smart manufacturing systems is recommended to assess practical applicability and real-time performance. –.

• Future research should focus on validating the proposed framework using larger and more diverse datasets, extending research to multi-output models or multi-task learning frameworks, incorporating additional diverse base learners, and integrating advanced adaptive optimization strategies and uncertainty quantification techniques. Deployment in real industrial environments and smart manufacturing systems is recommended to assess practical applicability and real-time performance.

Author contributions

Ithipri Emonena: Writing – original draft, Methodology, Investigation. Xu Yong: Writing – review & editing, Writing, Investigation, Festus I. Ashiedu: Supervision, Project administration. Ikuobase Emovon: Supervision, Formal analysis. Olusegun D. Samuel: Writing – review & editing, Project administration, Validation. Manjunath Patel G. C.: Writing – review & editing, Writing, Visualization, Visualization. Christopher C. Enweremadu: Writing – review & editing, Visualization. Choon Kit Chan: Writing – review & editing, Validation.

Funding

Guangxi Science and Technology Program (No. AA24010001), Guangxi Natural Science Foundation (No. 2025GXNSFAA069536); and the National Research Fund (NRF) under grant number TETF/ES/DR&D-CE/NRF2021. This research was financially supported by First-class Discipline Construction Project of Hechi University, Guangxi Academician Workstation of Hechi University Sericulture Intelligent Equipment and Inspection, Education Department of Guangxi Zhuang Autonomous Region, Science and Technology Department of Guangxi Zhuang Autonomous Region.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.