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Sep 16, 2023Development of optimized ensemble machine learning-based prediction models for wire electrical discharge machining processes | Scientific Reports
Scientific Reports volume 14, Article number: 23299 (2024) Cite this article
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This paper proposes development of optimized heterogeneous ensemble models for prediction of responses based on given sets of input parameters for wire electrical discharge machining (WEDM) processes, which have found immense applications in many of the present-day manufacturing industries because of their ability to generate complicated 2D and 3D profiles on hard-to-machine engineering materials. These ensembles are developed combining predictions of the three base models, i.e. random forest, support vector machine and ridge regression. These three base models are first framed utilizing the training datasets, providing predictions for all the responses under consideration. Based on these predictions, two optimization problems are formulated for each of the responses, while minimizing root mean squared error and mean absolute error, for subsequent development of two optimized ensembles whose predictions are the weighted sum of the predictions of the base models. The prediction performance of all the five models is ascertained through nine statistical metrics, after which a cumulative quality loss-based multi-response signal-to-noise (MRSN) ratio for each model is computed, for each of the responses, where a higher MRSN ratio indicates greater accuracy in prediction. This study is conducted using two experimental datasets of WEDM process. Overall, the optimized ensemble models having higher MRSN ratios than the base models are indicated to deliver better prediction accuracy.
Wire electrical discharge machining (WEDM) is a very useful non-traditional machining process that removes material from different hard-to-cut work samples through an electro-thermal phenomenon of generation of sparks between the wire electrode (tool) and electrically conductive workpiece, on which conventional machining processes are difficult to apply1. They have found wide applications in mould, tool and die making, automotive, aerospace, medical, optical, dentistry and jewellery industries2. During WEDM process, the workpiece is immersed in a dielectric medium, like deionized or distilled water. The wire electrode, introduced into the machining zone in continuous motion, is separated from the workpiece by a small gap with the dielectric fluid in between3. The electrical energy generated in the form of continuous but distinct electric sparks is responsible for erosion of material through heating and subsequent melting of material from the workpiece4. In this process, minute amount of wire electrode is also removed, requiring introduction of new wire electrode in the machining zone, while the material removed is rinsed out by circulation of the dielectric fluid. Figure 1 demonstrates the working principle of a typical WEDM process.
Working principle of a WEDM process.
The process of material removal due to electro-thermal action in WEDM is complex, governed by the influence of multiple factors (process parameters). There exist complex interrelationships between those factors and outputs (responses), where the outputs have strong dependencies on the process parameters. Any slight change in the values of WEDM parameters, such as discharge current, voltage, pulse duration, wire electrode characteristics etc., can significantly influence the responses, like removal rate of workpiece material, surface conditions and other similar machining variables2. Application of WEDM processes thus enables the manufacturing industries to generate complex and intricate 2D and 3D shape profiles on many of the present-day engineering materials with high dimensional accuracy and close tolerance, thus requiring significant control over material removal and other machining indicators by adjusting the input parameters. As a result, prediction of the response values is ardently necessary to gauge the ideal or optimal setup of the process parameters, which would help the machine operators to explore the fullest machining potential of a WEDM process. It is of utmost importance to develop a model, which would include WEDM parameters and responses, and predict those responses, subsequently enabling to increase machining efficiency and process economy5. The modeling for prediction of responses can be achieved through application of suitable machine learning (ML) techniques.
Application of ML models/algorithms thereby form the basis for development of intelligent machining principles6,7, which can subsequently be helpful in prediction and optimization of machining processes, like WEDM6. In this process, responses, like tool condition, material removal rate (MRR), surface conditions, geometrical errors etc. can be efficiently predicted based on different input parameters with the help of suitable ML models. Aided by these models, response monitoring of WEDM processes has become a less difficult task, incorporating diversified demands of the end users and dealing with the external factors influencing the process7. These models can take the help of an existing machining dataset to learn the hidden patterns, which can then aid in prediction of the response values for newly introduced data. The ML algorithms also show great promise for constant improvement while automating the associated decision-making functions through their efficient operation over large dimensions of the available datasets8. Since the response values of most of the machining processes, like WEDM, are quantitative and continuous in nature, a supervised (since the data is labeled) regression technique must be implemented. Since their inception, numerous researchers have adopted ML techniques for accurate prediction of the responses for various machining processes. However, those prediction results are required to be as accurate as possible so that proper monitoring of the responses can be undertaken. Modification in the existing ML approaches can improve ability of those models to provide more accurate prediction results, one such modification being the concept of ensemble methods11,12,13.
Ensemble methods incorporate multiple ML algorithms to deliver enhanced prediction performance compared to that of any of the constituent models as a single entity9. Ensemble models can be broadly classified into two categories, e.g. homogeneous (aggregation of the same ML algorithm into a consolidated one) and heterogeneous (aggregation of different ML algorithms). Heterogeneous ensembles hold an advantage over homogeneous ensembles, as they can consider advantages of different ML algorithms in a consolidated manner, without multiplied effect of the drawbacks of the same ML algorithm as noticed in homogeneous ensembles. Since the responses of a machining process, like WEDM, are quantitative and continuous in nature, the applicable ensembles are regression models. The simplest regression heterogeneous ensemble model-based predictions are proposed while computing either the simple average or weighted average of the predictions of the base models.
This paper adopts a novel approach for developing optimized weighted average ensembles for prediction of WEDM responses based on two past experimental datasets. At first, three ML algorithms, i.e. random forest (RF), support vector machine (SVM) and ridge regression are implemented on the considered WEDM datasets for predicting the corresponding response values. The weight (importance) to be assigned to each of the base models is identified through minimization of two of the mostly reported performance metrics of regression models, i.e. root mean square error (RMSE) and mean absolute error (MAE) after solving the two developed optimization models. Hence, two ensemble models are developed, one with the optimized RMSE and other with the optimized MAE, by effectively integrating the three base models. The performance of the optimized ensemble models is subsequently compared against the base models using some well-reported statistical metrics, like mean squared error (MSE), correlation coefficient (R), mean absolute percentage error (MAPE), root mean square percentage error (RMSPE), root mean square logarithmic error (RMSLE), relative absolute error (RAE) and root relative squared error (RRSE). As mentioned earlier, these ensemble models are developed based on two WEDM experimental datasets to demonstrate validity and applicability of the results in all such cases.
Through the proposed methodology for predicting WEDM process responses, this paper contributes the followings:
To the best of the authors’ knowledge, it is the first attempt in implementing a parallel heterogeneous ensemble learning model for prediction of WEDM responses. This is further highlighted through a review of past research works of application of various prediction models for WEDM processes. Furthermore, such ensembles are also rare in all types of machining processes.
Use of a mathematical programming-based optimization model for prediction error measurement while deriving weights of the base ML models is also a novel approach for the most accurate prediction of the WEDM responses. This is probably the first attempt in developing a mathematical programming-based optimization model for all types of the machining processes.
Few studies have applied metaheuristic-based optimization of a single error measure while proposing an optimized heterogeneous ensemble learning model to predict the response values. This paper improves upon those studies by developing prediction models while minimizing two of the most commonly reported error measures, i.e. RMSE and MAE, followed by a comparison of those models against the base models.
There is a lack of application of parallel heterogeneous ensemble learners for multiple machining responses. This is the first such endeavour in predicting multiple responses for any of the machining processes.
Implementation of the proposed approach to multiple experimental WEDM datasets exhibits its diverse range of potential applications in experiments in the domain of both traditional and non-traditional machining processes.
This paper is structured as follows: “Literature review” provides a concise review on the applications of different ML algorithms as prediction models mainly for WEDM processes. Details of the considered ML algorithms, developed ensemble models and statistical metrics are presented in “Proposed ensemble model”. The corresponding ensemble models are framed in “Development of ensemble models for WEDM processes” and “Conclusions” are drawn in Sect. 5 along with future research directions.
In recent years, the experimental data-based prediction modeling of WEDM processes has been established as a focal point of research. Numerous works regarding prediction modeling of WEDM processes are available in the literature considering diverse responses and input parameters. Nain et al.10 employed RF and M5P tree algorithms for development of prediction models for a WEDM process while accurately guessing values of MRR, surface roughness (SR) and dimensional deviation (DD), followed by sensitivity analysis and application of particle swarm optimization (PSO) technique for process optimization. Ulas et al.11 applied four ML algorithms, i.e. extreme learning machine (ELM), weighted extreme learning machine (W-ELM), support vector regression (SVR) and quadratic support vector regression (Q-SVR) for prediction of SR values during WEDM of Al7075 alloy, and observed W-ELM as the best-performing model with the highest R2, and lowest MAE and RMSE values. Lalwani et al.12 employed a back-propagation artificial neural network (ANN) to develop prediction models for MRR, SR and kerf width (KW), and compared the model performance against the results derived using response surface methodology (RSM). Finally, the WEDM process was optimized using non-dominated sorting genetic algorithm (NSGA-II).
Naresh et al.13 developed an adaptive neuro-fuzzy inference system (ANFIS), along with ANN models through back-propagation with three different algorithms, i.e. Elman regression, generalized regression and Levenberg-Marquardt, for prediction of MRR and SR values, and compared their prediction performance. Based on the considered statistical metrics, ANFIS had appeared as the best model having the highest accuracy for anticipation of WEDM responses. Dandge and Chakraborty14implemented two decision tree-based algorithms, i.e. classification and regression trees (CART) and chi-squared automatic interaction detection (CHAID) for prediction of four responses, i.e. machining rate, SR, wire wear ratio and DD, based on six WEDM parameters from an experimental dataset, followed by identification of the most important parameters influencing the responses. Shanmugasundar et al.15 applied three ML algorithms, i.e. linear regression (LR), RF and AdaBoost to the datasets of two non-traditional machining processes, including WEDM. AdaBoost had emerged out as the most favourable and deployable model to predict MRR and KW, being insensitive to the number of regressors in the model.
Sharma et al.16 modeled WEDM of titanium alloy with the help of two ML algorithms, i.e. Gaussian process regression (GPR) and SVM to predict the corresponding SR values. Their efficacy as prediction models was also validated based on test datasets. Three prediction models, i.e. multilayer perceptron, ensemble neural network (ENN) and optimization-based evolving product-unit neural network (EPUNN) were developed by Gurgenc and Altay17 for SR prediction during WEDM of AZ91D magnesium alloy. The superiority of EPUNN model over the others was also established. Saha et al.18 first employed four different ML algorithms, i.e. LR, GPR, SVM and regression trees as effective prediction models for cutting rate and SR. The GPR model having the maximum computational efficiency among the tested models was later considered for data-driven uncertainty quantification and sensitivity analysis of the said WEDM process. Verma and Singh19implemented SVR with four different kernel functions, i.e. linear, polynomial, radial basis and sigmoid to predict SR and slicing speed during WEDM-based wafer slicing. Based on the computed statistical measures, SVR with radial basis function had appeared to provide the best results in comparison to other kernel functions. Ishfaq et al.20 attempted to predict cutting speed (CS) through application of three ML algorithms, like ANN, SVM and ELM. It was revealed that ANN model had performed comparatively better in predicting the considered response with respect to higher R2, and smaller MAE and RMSE values. Saha et al.21 conducted a study to investigate the influence of two clustering algorithms, i.e. grid partitioning (GP) and subtractive clustering (SC) on the performance of ANFIS-based prediction of MRR and SR values during WEDM of A286 superalloy. The predictive performance of ANFIS-GP model had appeared to be better than that of ANFIS-SC model.
In a recent study, Jithendra et al.22 augmented an ANFIS model by encrypting snake optimizer (SO) for predicting SR, MRR and residual stress of Monel 400 alloy. The predictive performance of ANFIS-SO model was later contrasted against other hybrid approaches, like ANFIS- beetle antennae search, ANFIS-reptile search algorithm and ANFIS-COOT algorithm, showcasing its excellent performance with minimum computational effort. Treating pulse duration and peak current as the input parameters, and MRR, SR, DD and tolerance errors as the responses, Natarajan et al.23 hybridized grey relational analysis (GRA) with ANFIS and ANN models. ANFIS-GRA had evolved out as an effective prediction model which could accurately evaluate the performance indicators of the WEDM process under consideration.
It can be unveiled from the above survey of the existing literature that most of the researchers have employed a single ML algorithm or some discretely chosen ML algorithms to develop the corresponding prediction models for WEDM processes. In multiple instances, ML algorithms are generally found to suffer from underfitting of data, i.e. they are not able to extract meaningful patterns from the training datasets, and thus, do not fit properly, making them unsuitable for prediction based on both the training and newer datasets. Conversely, ML algorithms may also find it difficult to generalize their predictive performance to newer datasets due to overfitting, i.e. fitting with the training dataset to such an extent that while prediction on the training data is highly accurate, the performance on newer datasets is dismal. Hence, the presence of bias and variance in the performance of those algorithms adversely affects their accuracy of prediction24. These issues can be resolved through modification of the existing ML algorithms, one such approach being the concept of ensemble methods. Ensemble methods minimize model variance, subsequently reducing overfitting, thus deriving more accurate prediction results24.
Some of the research works have included homogeneous ensemble models, like RF or AdaBoost as the prediction model. Homogeneous models, however, have a disadvantage as the inadequacies of a single algorithm may be compounded, making them unacceptable for accurate prediction of the machining responses25. Conversely, heterogeneous models can derive the advantages of different algorithms and counter the drawbacks of the individual algorithms, making them more robust and generalized. So, it can be concluded that heterogeneous ensemble models enjoy considerable advantages, and are better suited for modeling and prediction of WEDM processes. As mentioned earlier, the simplest heterogeneous ensembles are simple average and weighted average approaches. For both the methods, there are multiple base regression models, which are trained on the same dataset. The difference lies in the prediction on a new dataset, as in the simple average method, the final prediction is the average of the predictions of the base regression models, while in the weighted average method, more weightage is allocated to the model with a better prediction performance on the new dataset and a weighted average is computed to be the prediction of the ensemble. Usually, these weights are computed based on the comparative prediction performance of the base models. As a result, those models, whose predictions are usually closer to the actual values, are assigned higher weights, making the predictions more accurate. This paper computes the weights of the base models through optimization ensuring a higher level of accuracy. Therefore, to attain greater prediction accuracy and minimum error, this paper proposes development of a heterogeneous weighted average ensemble methodology, where weights are obtained through mathematical programming, with minimized RMSE and MAE values.
As mentioned earlier, this paper proposes a methodology to develop novel optimized weighted average ensemble models for prediction of responses for two WEDM processes based on past experimental datasets. At first, three ML algorithms, i.e. RF, SVM, and ridge regression are implemented on two separate WEDM datasets as the base models. RF is a decision tree-based homogeneous ensemble technique that has found applications in some research studies regarding modeling of WEDM processes. On the other hand, SVM, a kernel-based algorithm, is one of the most commonly implemented ML algorithms, employed for development of prediction models for WEDM processes, as established through the literature review. Ridge regression is an extension of the classic multiple regression modeling, involving regularization to reduce overfitting errors. Even though multiple regression analysis is the most commonly applied modeling technique for WEDM processes, there is no literature available that studies application of ridge regression for the said purpose. Hence, it is fairly apparent that these three models are distinctly different from each other, not only on the basis of their theoretical and statistical framework, but also on the frequency of their applications in research articles, and hence, they are brought together for the development of a comprehensive ensemble model.
The weight to be assigned to each of the base models is estimated with the help of the corresponding optimization problem. To obtain the ensemble model with minimum RMSE or MAE value, an optimization problem is thus formulated for the proposed ensemble model with RMSE or MAE as the objective function, from which the predicted values of the ensemble model, and weights as well as RMSE and MAE values of the ensemble model are computed for each of the responses. Other statistical performance metrics are also computed from the predicted response values for each of the base models as well as RMSE-optimized and MAE-optimized ensemble models. Finally, among the considered prediction models, the best-performing model is identified based on the estimated multi-response signal-to-noise (MRSN) ratio. In case of an experiment involving multiple responses, as studied in this paper, these steps are reiterated for each of the responses. The proposed methodology is explained in Fig. 2 in the form of a flowchart. The details of the ML algorithms and optimization problem, along with the statistical metrics employed to compare prediction performance of the considered models and computation of MRSN ratios are presented here-in-under.
Flowchart of the proposed methodology.
As mentioned in the previous section, the ensemble approach in this paper includes RF, SVM and ridge regression as the base models. The theoretical fundamentals of these ML techniques are elucidated here through a brief description.
RF is a decision tree-based ensemble ML technique, which serves as an aggregator of the predictions made by several decision trees, where generation of the trees occurs on the basis of the values of an independent set of random vectors26. It is a special case of bagging ensemble learner, as it implements bootstrap samples to create datasets for training of the base learning decision trees27. Training datasets are generated randomly from the original dataset, with the same number of observations or cases as that of the original dataset. Training is performed through sampling with replacement, which implies that while some observations in the original dataset may occur more than once in any given generated dataset, some observations may not occur at all. Each of the constituent decision trees can then be trained with each of the generated datasets. However, RF differs from bagging with respect to randomized selection of variables or features. During development of each of the trees constituting the model, a subset of variables is chosen followed by the typical approach of split selection within the selected feature subset27. Selection of the subset of variables has an advantage as it decreases similarities among the constituent decision trees developed by training with different generated datasets, while a specific number of variables (usually considered one-third of the total number of variables in case of regression) would balance reduction in variance and computational cost28. Aggregation of the predictions follows after a sufficient number of constituent trees are trained. For continuous numeric responses of a WEDM process, like MRR or SR, the aggregation approach involves a simple average of the predictions of the constituent decision trees.
SVM is a kernel-based algorithm, generally used as a classification technique, and characterizes a decision boundary with the help of a subset of training data, which are called support vectors26. SVM functions through characterization of training data as spatial points by plotting the observations belonging to dissimilar categories or classes in order to maintain an established gap or decision boundary, dividing the points according to the categories. Thereafter, it makes predictions on a new dataset, by plotting the observations on the same space as above, where each observation is assigned a category based on the side of the boundary where that observation would be plotted29. The decision boundary is a hyperplane with its dimension equal to one less than the number of variables. In case of a dataset linearly separable by multiple hyperplanes, the one with the maximum possible gap on either side is selected as the decision boundary. This is obtained by moving a pair of hyperplanes away from the decision boundaries on either side, but parallel to each other until they coincide with a point on each side. The hyperplane with the maximum gap (called maximum margin) is finally selected.
However, in case of linearly inseparable data points, i.e. no linear decision boundary can completely separate the data points belonging to dissimilar categories, two approaches can be considered. One of them requires construction of soft margins, i.e. a hyperplane-based decision boundary with a considerable amount of margin which would not be able to classify a few points to either side, even if there exists a decision boundary with a lesser margin being able to completely separate the data points belonging to different categories. This would help in avoiding a case of overfitting which may arise in the latter case. For development of an SVM model with soft margins, there is a compromise between the margin width and classification error with the training dataset26. The other approach can be applicable for non-linearly separable data points and involves the kernel trick, i.e. conversion of the original data space to a high-dimensional space to enable a linear decision boundary to effectively separate the points29.
As mentioned above, the responses of a WEDM process are quantitative and continuous in nature, requiring regression models for prediction of their values. SVM can be extended to a regression problem for the same where the central concept, based around Vapnik’s ε-insensitivity approach, concerns utilization of data points with absolute residual values less than some constant value ε, which is the width of a region symmetrically surrounding the fitted function represented as a hyperplane30. The ε-insensitive region around the function is also known as the ε-tube, which is designed to have a minimal width while including the maximum possible number of data points inside. The data points inside the tube, on either side of the fitted hyperplane, are ignored and not penalized while the ones outside the tube receive penalizations based on their separation from the fitted hyperplane. For regression model, the tube region around the hyperplane function with continuous values is obtained through a compromise between complexity of the model and error in prediction. The model development includes minimization of the convex ε-insensitive loss function and obtaining the tube with minimal width containing maximum possible number of data points. This necessitates multi-objective optimization of the loss function and tube geometry30.
Multiple linear regression (MLR) is one of the most basic prediction models for continuous response variables, and is easy to understand and interpret. However, an MLR model, developed based on the principle of least squares, underperforms with poorly-conditioned data in regards to the size of the model and prediction accuracy31. Such a regression model, developed to fit a training dataset, fails to maintain good prediction accuracy for a new dataset. The least square regression model also cannot deal with high dimensional data, i.e. data where the number of variables is greater than the number of observations. These drawbacks can be overcome through regularized regression models, like ridge regression. It improves estimation of the parameters while increasing bias up to an admissible level. Ridge regression introduces a biased estimator \(\:{\beta\:}_{R}\), as shown in the following equation:
where \(\:X\) is the design matrix, \(\:Y\) is the regressor (dependent variable), \(\:I\) is the identify matrix and \(\:\lambda\:\) is the biasing parameter having value greater than or equal to zero. If \(\:\lambda\:\:=\:0\), the above equation would be reduced to a least square estimation. The biasing parameter is responsible for regularization through controlling the size of the coefficients. The coefficients, which when unconstrained, may result in large variances, and are shrunk through penalization bringing them close to a value of zero32. If the value of the biasing parameter is high, bias in the estimator would then increase, while the variance would decrease. In a ridge regression model, the value of the biasing parameter is assigned so that the decrease in variance is equal to the square of the bias33.
As aforementioned, the weight to be assigned to each of the constituent base models of the ensemble model is estimated after solving the corresponding optimization problems. Two ensemble models are obtained, one with minimum RMSE and another with minimum MAE values. They are developed with RMSE or MAE of the proposed ensemble as the objective function, which is required to be minimized. The objective function is subjected to two sets of constraint. One of them is the set of predicted response values based on the test data by the ensemble model, which is expressed as the weighted sum of the prediction of the constituent ML models. The other is the sum of the weights, which is always equal to one. The unknown values in the optimization problem are thus the response values predicted by the ensemble model, as well as weights of the constituent models, which are obtained after solving the said problem. The corresponding RMSE and MAE values are also obtained as solutions to the problems. As a result, predictions for the test data, and RMSE and MAE values based on the test data are computed for five different models, i.e. SVM, RF, ridge regression, RMSE-optimized ensemble model and MAE-optimized ensemble model. Equations (2)–(4) describe the optimization problem for RMSE or MAE-optimized ensemble model.
Or
Subject to
whereis the ith observed value for the particular response, is the th predicted value for that response using the ensemble model, which is unknown and can be determined solving the optimization problem. Two sets of values are obtained, one by minimising the ensemble model RMSE and the other by the minimization of the ensemble model MAE. In each of the cases, is calculated as the weighted sum of the predictions of the base models, where wj is the weight assigned to jth base model andis the predicted value for ith observation by jth base model. These weights are also unknown and can be computed through the developed optimization problems. Like the predicted response values of the ensemble models, two sets of weights are thus derived from this computation. The number of observations for which these ensemble model-based predictions can be obtained is n, which is equal to the number of observations in the test dataset. The two sets of values, computed for the RMSE-optimized and MAE-optimized ensembles, are thereby considered for comparison based on some statistical performance metrics. For further explanation, the codes written in LINDO for optimization of RMSE and MAE of the ensemble models for a particular WEDM response are provided in Appendix A.
As evident from above, two ensemble models are developed in this paper through minimization of RMSE and MAE values. The mathematical expressions for RMSE and MAE are provided as below:
Apart from the fact that these are the two most commonly reported metrics in similar studies, there is a fundamental difference in their concept. While \(\:RMSE\) deals with squared error, MAE considers absolute error. Hence, the comparative results obtained through optimization of these metrics are likely to be different. While comparison can be made between the prediction models through the values of RMSE and MAE only, but, for a comprehensive study, it is also necessary to include other metrics to examine performance of the ensemble models and base models. For this purpose, values of \(\:MSE,\:R,\:MAPE,\:RMSPE,\:RMSLE,\:RAE\) and \(\:RRSE\) are calculated here. Besides \(\:R\), which is a larger-the-better metric, all the other metrics are of smaller-the-better type. Along with \(\:RMSE\) and \(\:MAE\), these seven metrics are adopted in this paper to compare performance of the considered models through calculation of MRSN ratio values, whose preliminaries are presented in the next sub-section. The expressions of the seven performance metrics are shown through Eqs. (7)–(13)34:
The S/N ratio is a measure of quality characteristics for any set of observations obtained using Taguchi’s orthogonal array-based experiments and the data derived from them. It was proposed as an acceptable indicator of performance, considering output characteristic of a process, and combining both the desirable and undesirable components. The desirable component is called ‘signal’ indicating the mean characteristic value, while the undesirable component, termed ‘noise’, indicates variability of the characteristic. The mathematical computation of S/N ratio is carried out on the basis of three situations, i.e. nominal-the-best, smaller-the-better and larger-the-better. In this paper, the metrics, based on which S/N ratios are calculated, are predominantly smaller-the-better, with one metric being larger-the-better type. In case of a multi-objective problem, where there are multiple contrasting parameters or objectives, a single overall S/N ratio is computed. This paper includes a mixture of smaller-the-better and larger-the-better metrics as output characteristics and hence, an overall S/N ratio, called MRSN ratio, is computed in the following steps35.
Step 1: At first, quality losses are computed for both the larger-the-better and smaller-the-better quality characteristics, described for kth trial in \(\:it\)h experiment and jth response, as shown below36:
where and \(\:{L}_{ij-l{arg}er}\) are the corresponding quality losses for smaller-the-better and larger-the-better quality characteristics, respectively for n experimental trials, whose response values are denoted as .
Step 2: This step involves scaling of the loss function by min-max normalization procedure, which is expressed using Eq. (16).
where is the scaled loss function, and \(\:{L}_{j}^{min}\) and are the minimum and maximum loss function values for \(\:j\)th response, respectively.
Step 3: The total quality loss for th experiment is calculated as a weighted sum of all the scaled loss function values, as given in Eq. (17):
where is the number of responses and \(\:{\omega\:}_{j}\) is the weight assigned to jth response.
Step 4: In the final step, MRSN ratio for \(\:i\)th experiment is computed using the following equation:
In this paper, \(\:i\)th experiment corresponds to \(\:i\)th model being studied, while\(\:\:j\)th response denotes \(\:j\)th model performance metric (out of \(\:p\) number of metrics). Here, the number of trials per experiment or model and per response or performance metric is equal to 1.
Based on Box-Behnken design plan, Kumar et al.37 randomly conducted 54 experiments (to avoid chance of any systematic error) on stir cast Al/SiCp metal matrix composites (MMCs) using a sprintcut CNC-WEDM setup (Electronica make), considering pulse-on time (\(\:{T}_{on}\)), pulse-off time (\(\:{T}_{off}\)), spark gap voltage (\(\:SV\)), peak current (\(\:{I}_{p}\)), wire tension (\(\:WT\)) and wire feed (\(\:WF\)) as the independent (process) variables, each set at three different levels (shown in Table 1), and CS, SR and spark gap (SG) as the dependent variables (responses). The dimensions of the MMC specimens were 5 × 5 × 15 mm. Brass wire having diameter 250 μm was used as the tool electrode and deionized water was utilized as the dielectric fluid. After each WEDM experiment, the average value of CS was estimated by measuring the total area machined and the corresponding machining time. On the other hand, a surface roughness measuring instrument (Mitutoyo make, Model S J 301) and a spark gap measuring device (TESAMASTER micrometer with least count = 0.001 mm) were employed to record SR and SG values, respectively. Furthermore, Table 2summarily presents the average, standard deviation, maximum and minimum values of each of the responses for the said WEDM process, based on the observations of 54 experiments, conducted by Kumar et al.37.
From the experimental results, as shown in Table 2, the corresponding training and testing datasets are randomly considered based on an 80:20 split ratio, leaving 43 observations in training dataset and 11 observations in testing dataset. The split is randomly created using a function in R software v4.4.1, called ‘createDataPartition’, which maintains an approximate proportion between the training and testing datasets. All the three base models, i.e. RF, SVM and ridge regression, are developed on the training data, and later validated on the testing data. Those base models are framed using R software v4.4.1, applying the function train() (belongs to ‘caret’ package). The train() function assesses the influence of the model tuning parameters on the prediction performance, followed by selection of the optimal model based on those parameters and gauging its performance employing the training dataset. It also checks performance of the model at different values of the tuning parameters and provides results for those tuning parameters at which the model delivers the lowest error metric (usually RMSE) or highest accuracy. At first, the base models are developed treating CS as the dependent variable. Development of the RF model involves consideration of the number of randomly chosen predictors or input variables at each split as the tuning parameter. The program calculates its value as 4, having the minimum RMSE. It is followed by the development of an SVM model with the linear kernel on the same training dataset, with the cost parameter (tuning parameter for SVM model) equal to (1) Development of the ridge regression model involves selection of λ value between 0.0001 and (2) Twenty different values of λ are tested (with a gap of 0.105258 in between each value of λ), and the one with the minimum RMSE is considered for the final model. In this case, the value of λ = 1.052679 is chosen and the corrseponding ridge regression model is developed accordingly. For the other responses, i.e. SR and SG, the RF model tuning parameter, i.e. (the number of input variables randomly chosen at each split) is set to 4, the cost parameter of the SVM model is equal to 1, while λ = 0.0001 is observed to provide the lowest RMSE for the ridge regression model.
All the three models developed on the training dataset are now implemented on the test dataset, where they provide predictions for each of the responses. These predictions as well as actual experimental observations are subsequently utilized for formulation of the corresponding optimization problem in the framework, as shown in Eqs. (2)–(4). As mentioned earlier, two optimization problems are developed for each of the responses, one minimizing RMSE and the other minimizing MAE. Therefore, a total of six optimization problems are framed in this example (two for each of the three responses) and their solutions are derived using LINDO software (v15). The outputs would provide the respective objective function values (RMSE and MAE) along with the predicted responses for RMSE-optimized and MAE-optimized ensemble models, and weight of each of the base models. The weights of the corresponding base models and predicted values for each of the responses (based on the test dataset) are exhibited in Tables 3 and 4, respectively. It can be observed that for MAE-optimized models, ridge regression, SVM and RF have the highest weightage for the responses CS, SR and SG, respectively. However, in case of RMSE-optimized models, SVM has the highest weightage for responses CS and SR, while RF has the maximum weight for SG response.
Following prediction of the responses using all the five models, performance of each of them is now ascertained using the corresponding statistical metrics. Thus, all the nine metrics, i.e. MAE, RMSE, MSE, R, MAPE, RMSPE, RMSLE, RAE and RRSE are computed for each model, for the three responses. These metrics, except R, are computed in R software v4.4.1, using the package ‘MLmetrics’. Consequently, MRSN ratio is also calculated for each model, based on the computed values of all the metrics. These metrics are treated as multiple responses and each model is considered equivalent to an experimental setting. Hence, the number of trials per experiment in this paper would be equal to one, since there is only one value for a model-metric combination. The quality loss for each value of the metrics is calculated, followed by scaling of the quality loss, and computation of the total quality loss and MRSN ratio of each model for each response. Here, the weights assigned to the performance metrics are considered equal to each other, and hence, the total quality loss is computed as the average of all the scaled losses for a particular model. Tables 5, 6 and 7 show the performance metric values, total quality loss and MRSN ratio of all the five models, for CS, SR and SG respectively. Similarly, Figs. 3, 4 and 5 exhibit the scaled quality losses for CS, SR and SG respectively.
From Tables 5, 6 and 7, it is evident based on the cumulative MRSN ratio values that the accuracies of RMSE-optimized ensemble models are greater than the other models for all the responses. It is followed by MAE-optimized ensemble models, which have the second-largest MRSN ratio values. Figures 3, 4 and 5 reiterate these observations, as the two ensemble models have the lowest scaled quality losses. For responses CS and SR, RF models have the lowest MRSN ratio value, followed by ridge regression and SVM models. However, for response SG, ridge regression has the lowest MRSN value, followed by SVM and RF models. It is also apparent from the tables and illustrated figures that the two developed ensemble models distinctly outperform the individual base models. The response SR has the closest gap between the inferior ensemble model and best-performing individual base model, where MRSN ratio of MAE-optimized ensemble model is distinctly greater than that of SVM model by a margin of 2.998.
It can be noted that RMSE-optimized ensemble models perform best with respect to those metrics involving computation of the squared error. In basic regression, the aim is to minimize the loss function, i.e. sum of squared errors, through adjustment of the model parameters. Furthermore, while absolute errors provide equal weightage to all the errors, squaring an error magnifies the larger error more than the smaller error. Penalizing large errors by squaring them is necessary for improvement of performance of a model38. Keeping it in mind, multiple squared error performance metrics are thereby included in this paper.
Scaled quality losses for response CS.
Scaled quality losses for response SR.
Scaled quality losses for the response SG.
Comparing values of the non-error-based metric included in the paper, i.e. R, which is the correlation between the actual and predicted values of a response, it is observed that RMSE-optimized ensemble models are consistently better than the individual base models. They are also observed to be better than MAE-optimized ensemble models in two of the three cases (for SR and SG), while for response CS, both ensemble models are almost equal in performance. So, RMSE-optimized ensemble models are proven to be the most versatile and accurate among all the considered prediction models.
Kumar et al.39 conducted 54 WEDM experiments on CP-Ti-G2 alloy (a bio-compatible material), with \(\:{T}_{on}\), \(\:{T}_{off}\), \(\:SV\), \(\:{I}_{p}\), \(\:WT\) and wire speed (\(\:WS\)) as the input variables, and MRR and SR as the responses. Each of the WEDM parameters was varied at three different levels, as shown in Table 8. While performing WEDM experiments, CuZn37 HH brass wire with diameter 250 μm was utilized as the tool electrode, while the dielectric pressure was maintained at 7 kg/cm2. The workpiece dimension was considered as 10 × 10 × 26 mm. Furthermore, Table 9provides a summary of each of the responses for the said WEDM process based on the research work of Kumar et al.39. From these experimental results, 45 observations are considered as training data, while 9 observations as testing data. As implemented in the previous example, the split is randomly created by ‘createDataPartition’ function, maintaining an approximate proportion between the training and testing datasets. The change in the number of observations in training and testing data, although derived randomly, also helps to demonstrate that the developed ensemble models are applicable to varying proportions of training and testing datasets.
Likewise, the previous example, three base models, i.e. RF, SVM and ridge regression, are developed on the training data, and subsequently validated on the testing data. It is followed by development of two weighted average ensemble models, obtained through minimization of RMSE and MAE values. The base models are first developed with SR as the dependent variable. The number of randomly selected predictors (input variables) at each split, at which the corresponding RMSE value is observed to be minimum, is estimated as 4. The cost parameter of the linear kernel SVM model is set as 1. Among the 20 equidistant λ values from 0.0001 to 2, the value of λ = 0.0001, corresponding to minimum error, is chosen during development of the ridge regression model. On the other hand, during development of the base models for MRR, the number of input variables selected randomly at each split for the RF model is determined as 6. The cost parameter of the SVM model is observed to be equal to 1, while λ value considered for ridge regression is 0.0001.
These three base models, developed on the training dataset, are now validated using their predictions on the test dataset containing 9 observations for both the SR and MRR responses. Like the previous example, predictions of the base models and actual experimental observations form the basis for formulation of two optimization problems, one minimizing RMSE and the other minimizing RAE, for SR and MRR, resulting in a total of four such problems. For both the responses, solutions to the problems include the respective objective function values (RMSE and MAE), predicted response values for both the ensemble models and weights for the base models. The weights of the base models and predicted response values based on the base models as well as ensemble models are depicted in Tables 10 and 11, respectively.
It can be noticed that for response SR, SVM has the maximum weight for both the MAE-optimized and RMSE-optimized ensemble models. On the other hand, in case of MRR, RF has the maximum weight for both the ensemble models. Values of the nine metrics, i.e. MAE, RMSE, MSE, R, MAPE, RMSPE, RMSLE, RAE and RRSE, describing performance of the corresponding models based on their predictions, are computed for each of the five models. Finally, a cumulative univariate metric, in the form of MRSN ratio, is derived for each of the models for both the responses through estimation of quality loss, scaled quality loss and total quality loss values. Tables 12 and 13 depict those performance metrics, total quality loss and MRSN ratio of all the models, for both SR and MRR. The scaled quality losses for SR and MRR, are shown in Figs. 6 and 7, respectively.
From Tables 12 and 13, it can be revealed that the cumulative measures of errors for RMSE-optimized ensemble models are the maximum among the five models, for both SR and MRR. While for MRR, MAE-optimized weighted average ensemble model performs almost as accurately as RMSE-optimized model, and it is outperformed by SVM base model in case of SR. The MAE-optimized ensemble model is slightly better than SVM model in case of those measures not involving squared errors, while the opposite is observed for squared error measures. Simultaneously, for response SR, RF has the lowest MRSN ratio, followed by ridge regression, while SVM shows the highest cumulative level of inaccuracy in prediction of MRR values, followed by ridge regression. The only metric that does not consider the prediction errors, R, shows that the predictions of response values by the two ensemble models are closely correlated to the actual values than the base models, for both the responses.
Scaled quality losses for response SR for example 2.
Scaled quality losses for the response MRR for example 2.
Ensemble models prove to be an effective means of predicting response values of WEDM processes, based on input parameters. A single ML-based prediction model may exhibit high bias or variability in its prediction. The combination of the predicted values by multiple ML models in ensemble models provides aggregated results, compensating for both bias and variance in prediction. In a heterogeneous ensemble, the base ML models employed are different from each other, integrating advantages of those models, and making the prediction framework more generalized and robust. In this paper, two heterogeneous ensemble models are developed for prediction of responses of WEDM processes based on two experimental datasets. It is followed by comparison of prediction performance of the base models and ensemble models for both the cases.
After splitting the experimental data into corresponding training and testing sets, three base models, i.e. RF, SVM and ridge regression, are developed on the training datasets, providing predictions based on the input parameters in the testing datasets. From the predictions on the testing datasets, two optimization problems are developed where the objective functions involve minimization of RMSE and MAE values. Based on the derived solutions, two optimized weighted average ensemble models with minimum MAE and RMSE values are formulated, where the predictions of the two ensemble models are the weighted sum of predictions of the base models. Besides RMSE and MAE, other performance metrics, e.g. MSE, R, MAPE, RMSPE, RMSLE, RAE and RRSE are also calculated for all the models. A comprehensive MRSN ratio is finally computed based on quality losses for all the models, aiding in comparison between the models. Based on MRSN ratio values, it becomes evident that RMSE-optimized ensemble model has the best prediction performance for all the responses under consideration, while MAE-optimized ensemble, barring some exceptions, generally outperforms the base models.
The relationships between the responses and input parameters of a WEDM process are complex, where the interdependence can be linear or non-linear. Ensemble models, especially heterogeneous ensembles, are able to deal with such relationships, due to deployment of a combination of different models. As previously mentioned, heterogeneous ensembles assimilate benefits of different unique models, while negating their drawbacks, making the prediction more robust and generalized. These models usually have higher prediction accuracy, especially when the weights, allocated to the predictions of base models, are optimized. This combination of base model predictions balances the amount of bias and variance, thus making those models less susceptible to underfitting and/or overfitting. Consequently, the ensemble model-based predictions are stable, containing minimum noise. Based on these considerations, such ensemble models can be developed as an acceptable tool for modeling of WEDM processes and prediction of the responses of such machining processes. Future studies may include integration of more diverse ML algorithms into ensemble modeling, like ANN, ANFIS, K-nearest neighbours, AdaBoost, etc., as well as extension of such studies to other traditional and non-traditional machining processes.
The data presented in this study are available through email upon request to the corresponding author.
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This article was supported by the project SP2024/087 from Ministry of Education Youth and Sport of Czech Republic and VSB-TUO.
Department of Industrial and Systems Engineering, Indian Institute of Technology, Kharagpur, India
Baneswar Sarker
Department of Production Engineering, Jadavpur University, Kolkata, India
Shankar Chakraborty
Department of Machining, Assembly and Engineering Metrology, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 70800, Ostrava, Czech Republic
Robert Čep
Department of Mechanical Engineering, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, 600 062, India
Kanak Kalita
Jadara University Research Center, Jadara University, Irbid, 21110, Jordan
Kanak Kalita
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Author ContributionsConceptualization: Baneswar Sarker, Shankar Chakraborty, Robert Čep, Kanak Kalita; Formal analysis: Baneswar Sarker; Investigation: Baneswar Sarker; Methodology: Baneswar Sarker, Shankar Chakraborty, Robert Čep, Kanak Kalita; Writing – original draft: Baneswar Sarker, Shankar Chakraborty; Writing – review & editing: Baneswar Sarker, Shankar Chakraborty, Robert Čep, Kanak Kalita; All authors have read and agreed to the published version of the manuscript.
Correspondence to Shankar Chakraborty or Kanak Kalita.
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Sarker, B., Chakraborty, S., Čep, R. et al. Development of optimized ensemble machine learning-based prediction models for wire electrical discharge machining processes. Sci Rep 14, 23299 (2024). https://doi.org/10.1038/s41598-024-74291-x
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Received: 23 July 2024
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DOI: https://doi.org/10.1038/s41598-024-74291-x
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