SBLMD–ANN–MOPSO-based hybrid approach for determining optimum parameter in CNC milling

Chatter is a type of self-excited vibration that expresses variations in frequency and energy dispersion during the milling process and invariably results in poor part quality and a lower material removal rate. An efficacious chatter detection approach is necessary to anticipate the chatter's nascent stage. Feature extraction is an important step in identifying chatter. In this paper, an efficient PF-based multi-mode signal processing method, i.e., Spline-Based Local Mean Decomposition (SBLMD), has been used to decompose the experimentally acquired sound signals into a series of PFs and then, selected PFs have been used to reconstruct the new chatter signal, which is rich in information. Further, two three-layer ANN-based prediction models are established to predict the CI and MRR. Three process-related variables, such as tool speed, feed speed, and cut depth, have been used to develop ANN models. Statistical comparisons have been conducted to obtain the optimal training algorithm and have found that the LM-based training algorithm was the best among the five other training algorithms. For the estimates of CI and MRR, the neurons in the hidden layer are optimized at 5 and 7 with the least mean squared errors (MSE), respectively. After applying MOPSO to ANN-based prediction models, the MOPSO-optimized ANNs are very sensitive to optimizing input layers. The optimal ranges for the Cut Depth (CD), Feed Speed (FS) and Tool Speed (TS) are 1.56–1.77 mm, 77–97 mm/min, and 2380–2880 rpm, respectively.


Introduction
Chatter is a serious obstacle to high part surface quality and productivity. Damage to the part quality and MRR can be prevented if the chatter is discovered in its nascent stage and reduced as soon as possible. In recent years, lots of process-based approaches have been proposed by various researchers for the detection, suppression, and recognition of the chatter in its early stage (Al-Hadithi et al. 2016;Xu et al. 2020). In this process-based approach, chatter recognition is the first key technology to avoid chatter. In this regard, various signals have been used by specialists and scholars, such as acceleration signals (Wang et al. 2021a), sound signals (Zhou et al. 2021;Mishra and Singh 2022a), torque signals (Tansel et al. 2012), motor currents (Liu et al. 2016), cutting forces (Wang et al. 2022) and instantaneous angular speeds (Lamraoui et al. 2014). Researchers have compared these signals and sensors and found that acquiring sound signals is easy and cost-efficient. The major disadvantage of a microphone is that it may receive some noisy signals due to environmental conditions along with the machining signal (Delio et al. 1992).
In order to segregate the chatter data from the acquired signal, the signal needs to be decomposed and again reconstructed using a threshold value. In this regard, signal processing methods play an important role in selecting the chatter spectrum bands to define the threshold. At present, time-frequency (TF) domain-based methods have been widely used by researchers. These TF methods include the Short-Time Fourier Transform (STFT) (Daldal et al. 2020), Continuous Wavelet Transform (CWT) (Tran et al. 2020), and Synchronous Compression Wavelet (SCWT) (Yoon and Chin 2005). However, due to the Heisenberg uncertainty principle, these T-F methods are not able to perform for chatter signal (Qi et al. 2007;Liu et al. 2016).
In order to deal with the nonlinearity and non-stationarity of the signal, various self-adaptive signal analysis methods were presented by the researchers. Researchers (Huang et al. 1998) proposed a novel signal processing method, i.e., empirical mode decomposition (EMD). Liu et al. ) used the EMD technique along with wavelet packet decomposition (WPD) in their work and proposed a chatter identification method. However, EMD has a major shortcoming in mode mixing, which may lead to extreme aliasing of frequency bands and loss of dominant IMF (Chen et al. 2019;Hu et al. 2019). Researchers (Wu and Huang 2009) proposed an updated EMD known as ensemble empirical mode composition (EEMD) to overcome the drawbacks of EMD. Ji et al. (Ji et al. 2017) proposed an EEMD-based chatter recognition system using FD and PSE. A comparative study has been done between EMD and EEMD by Yogesh et al. (Shrivastava and Singh 2019) and concluded that EEMD can deal with mode mixing. Still, it takes too much time for signal segregation which ultimately hampers the real-time monitoring condition. In 2014, Dragomiretskiy (Dragomiretskiy and Zosso 2014) proposed Variational Mode decomposition (VMD). In the literature, it has been shown that VMD can disintegrate a multi-component signal into a number of quasiorthogonal IMFs. VMD is very much advanced as compared to EMD, EEMD and EWT in terms of mode aliasing, time consumption and many more (Wang et al. 2015). However, to use VMD, one must optimize some parameters such as K and a which play an important role in VMD (Li et al. 2017). In this regard, Liu et al.  used Fast Local Mean Empirical Mode Decomposition (FLMEMD) to optimize VMD hyperparameters. Wang et al. (Wang et al. 2021b) proposed k-Map-based method to optimize the hyperparameters of the VMD-SVM model. In 2005, Smith proposed Local Mean Decomposition (LMD), a novel signal processing method, which was motivated by EMD (Smith 2005). However, there are certain flaws in LMD itself (Deng and Zhao 2014). Recently, Rohit et al. used the PF's-based multi-mode-based novel technique to identify tool chatter in the milling process (Mishra and Singh 2022b).
After chatter feature extraction, the chatter prediction model has been developed by using ANN (Mishra and Singh 2022c), fuzzy logic (Devillez and Dudzinski 2007), SVM (Chen and Zheng 2018), RSM (Mishra and Singh 2022d) and other classifications models (Shaul Hameed et al. 2021). However, as per the knowledge of the authors, no work has been reported considering chatter with MRR. Previous researchers have not considered the effect of input milling parameters on MRR during chatter investigation. In the present scenario, modern manufacturing industries are seeking means to enhance productivity as well as better surface quality for the machined product. Productivity is related to MRR, and the intensity of chatter adversely affects the surface finish. So, these two features cannot be overlooked. This motivated the present research work. Therefore, two three-layer models were developed using TANSIG as an activation function between the first layer and hidden layer and PURELIN as an activation function between the hidden layer and output layer for chatter and MRR, respectively. The main problem with ANN modeling is determining to optimize the training algorithm and optimized neurons in the hidden layer. However, various scholars have proposed several methods to optimize neurons (Zhu et al. 2020). Still, an efficient methodology is missing. After developing the prediction model of chatter and MRR, the Multi-objective optimization technique (MOPSO) has been used to obtain the optimal milling process variable, which results in less chatter and Higher MRR.
The structure of the manuscript is as follows: an outline of the presented methodology is presented in Sect. 2. The milling dynamic mathematical model is introduced in Sect. 3. Section 4 describes the analysis between conventional LMD (C-LMD) and SB-LMD for the developed simulated signal. Signal acquisition, processing and feature extraction are depicted in Sect. 5. The prediction model of CI and MRR using optimal ANN is presented in Sect. 6. In Sect. 7, MOPSO has been utilized to optimize the developed prediction models of CI and MRR. Finally, the presented study has been concluded with future scope in Sect. 8.

Outline of the presented methodology
The outline of the presented paper is shown in Fig. 1. The current work is distributed into three parts for a clear understanding of the proposed framework. In the first part, the CNC milling machine, types of milling cutter and workpiece and range of milling process variables are selected. Signals are acquired using the microphone. In the second part, an effective signal processing method, i.e., SBLMD, has been employed to sieve the unwanted noisy data and gather dominant PFs. Thereafter, chatter signals have been again reconstructed to extract chatter and MRR features.
In the third part, these features (CI and MRR) are used as an output for predicting the chatter and MRR using the ANN model. In this part, the developed ANN model has been optimized using the suitable algorithm and optimal neurons in hidden layers. Thereafter, these ANN-based prediction models are used for getting the optimal milling process variable using the MOPSO algorithm.

Milling dynamic model
In this section, a simulation of the milling chatter based on the two degree of freedom was developed, as presented in Fig. 2. Simulating chatter signals in real time was developed considering an end mill that has 'N' number of teeth with zero helix angles. The forces applied to these teeth excite the mechanism in the directions of feed (X) and normal (Y). Later, these excitation forces in the lateral and longitudinal axes lead to dynamic tool displacements in the x and y directions, respectively. However, before developing the force matrix in the time domain, the typical equation of motion in the lateral and longitudinal axis can be written as Eq. 1 (Altintas 2012); For developing this model, some notion has been used, which are as follows:k = arbitrary tooth on the mill arbor X = angular speed of the cutter T ¼ time period for one revolution per number of teeth ¼ 2p NX Then, the time-varying chip thickness in the radial direction is generated by the regenerative chatter at the present and previous tooth periods Dxðt; TÞ ¼ x tool ðtÞ À x workÀpiece ðtÞ Â Ã À x tool ðT À tÞ À x workÀpiece ðT À tÞ Â Ã ; Dyðt; TÞ ¼ y tool ðtÞ À y workÀpiece ðtÞ Â Ã À y tool ðT À tÞ À y workÀpiece ðT À tÞ Â Ã 2 6 6 6 4 3 7 7 7 5 can be expressed as Eq. 2; The tangential and radial cutting forces acting on the random tooth k can be expressed as Eq. 3 (Altintaş and Budak 1995); where; 'b' represents the cut depth, 0 sðh k Þ 0 represents the time-varying chip thickness, 0 K 0 t and 0 K 0 r are the cutting coefficients are constant. Components of forces in the x and y directions can be expressed as Eq. 4.
Finally, total time-varying milling forces in x and y directions can be found by summing up all the components in the respective direction for all tooth and can be written and expressed as Eq. 5; where; h k ¼ h þ kh p and the cutter pitch angle is h p ¼ 2p N :.
By reorganizing Eq. 5, Net force in the matrices form in the x and y directions expressed as Eq. 6 (Budak et al. 1996); where, dynamic directional dynamic milling force coefficients can be expressed as Eq. 7.
Àu k ½sin 2h k þ K r ð1 À cos 2h k Þ; Net force in time domain is expressed by Eq. 8; A simulated signal has been developed using an aforementioned mathematical model. During acquisition of machining signals via microphone, there is a high chance that it may acquire noisy data which is not related to machining information. Sources of noisy data are termed as 'environmental noise' in the literature. As per available literature, these environmental noisy data can be presented in the form of white Gaussian noise (Du et al. 1992). Therefore, in this study, in order to make it more realistic, Gaussian white noise has been introduced. A simulated signal that has been presented without white Gaussian noise and with white Gaussian noise is presented in Fig. 3a and b, respectively.

Formulation of SBLMD
Confirmation and recognition of the exact nature of chatter is hampered because of unwanted noises. So, in order to segregate these unwanted signals from the acquired milling signals, SBLMD technique has been used. The main difference between the conventional LMD and the proposed LMD is that in the proposed technique, cubic spline-based interpolation is used in place of the moving averaging method, as follows: 1. Initially, all local maxima and minima extremes of the signal have been found. Then, collected local upper extremes are connected with one type of cubic spline line function, and local minima extremes are associated with another cubic spline line function. As a consequence, the top and bottom layers will form, i.e., P tl ðtÞ and P bl ðtÞ.
2. The local mean and local envelope function, i.e., m 11 (t) and a 11 (t), respectively, are calculated using Eq. 9; 3. The other steps will be the same as conventional LMD. Figure 4 depicts the flowchart of the SBLMD.

Processing and result by conventional-LMD
In this section, conventional LMD has been applied to develop a simulated chatter signal (in Sect. 3). The simulated signal has been decomposed, and generated PFs are shown in Fig. 5a. Moreover, in order to realize the frequency peaks of the developed simulated chatter signal, FFT has been utilized. FFT has been applied to the first three PFs to gather the frequency peaks, as presented in Fig. 5b. It is evident from Fig. 5b that the spectra of the first PFs are not clear and are not the same as the original frequency peaks. Therefore, it can be concluded that conventional LMD is able to extract the original frequency peaks.

Processing and result by SB-LMD
The aforementioned analysis makes it clear that the traditional LMD method is not always suitable for analyzing a time-and frequency-varying signal. In this section, SB-LMD is employed to alleviate the drawback of standard LMD. The decomposed PFs using SBLMD of the simulated chatter signal are presented in Fig. 6a, and the ensuing frequency domain of the first three PFs are given in Fig. 6b. It is evident from Fig. 6b that SBLMD is able to extract the original frequencies of the simulated signals. Thus, it can be said that The SB-LMD approach is recommended for handling time-and frequency-varying signals.

Signal acquisition and processing
In this section, SBLMD has been used to process the realtime experimental-based machining signal. Milling is one of the prime machining processes that have been used by the manufacturing industries to produce the product. Therefore, authors have decided to use a CNC milling machine to conduct experiments. Al-6061 T6 has been selected as a workpiece material because of its wide applicability such as the aviation, automotive and construction industries. The reason for selecting HSS 4 fluted milling cutter as a tool is because it allows higher feed and better surface finish for non-ferrous alloys. A photograph of the signal acquisition set-up is shown in Fig. 7. After selecting the machine and tool-workpiece, experiments have been performed on MTAB XL-MILL (CNC Mill). Ranges of variables are selected from pilot experiments. The available CNC machine has the specification to perform machining for a maximum tool speed (TS) of up to 3000 rpm, cut depth (CD) of up to 2 mm, and feed speed (FS) can go up to a maximum of 100 mm/min. In the present work, based on full factorial design, a total of 27 experiments are performed and listed in Table 1. A unidirectional microphone has been selected as a sensor for acquiring the real-time experimental signal.

Evaluation of responses
In this study, MRR has been taken as a first response. MRR is evaluated using the following relation (Eq. 10) and is presented in Fig. 11:

Signal reconstruction using SBLMD
In the experimental section, signals were acquired using the microphone. Microphones are very good sensors as compared to other sensors in terms of acquiring information about the overall process and high sampling rate, but the acquired signal using a microphone is corrupted with ambient noises (Cuka and Kim 2017). The signal-to-noise ratio (SNR) of the useful and constructive sound data is calculated by subtracting the unwanted noise from the recorded signal. Therefore, it is pertinent to refine the acquired machining signal using a suitable signal processing technique. In this section, the proposed multi-PFbased signal processing technique, i.e., SBLMD is used to remove useful information from the acquired signal in terms of PFs as shown in Fig. 8. Thereafter, for signal reconstruction, which is having useful machining information, two factors are utilized named as Pearson correlation coefficient (CC) and energy ratio (NER). Their values are also shown in Fig. 8. The importance of the first three PFs is evident from Fig. 8 due to their higher NER and CC. Therefore, using these important PFs, the signal is reconstructed for all 27 experiments. Figure 9 presents the FFT of the SBLMD-based newly reassembled signal. It is clear from Fig. 9 that there are clear frequency peaks. Therefore, it can be stated that SBMD is very much efficient to sieve out the signal feature from the sound signal.

Calculation of response for the prediction model
Although feed speed rate is another crucial input element that affects tool chatter, researchers have hitherto ignored this study. The primary cause of this indifference is that the Stability Lobe Diagram (SLD) makes it challenging to understand how this phenomenon affects tool chatter. Therefore, the current study's inspiration came from this research gap.

Chatter indicator
The first response which is taken in this study is Chatter Indicator (CI), which represents the chatter value for all 27 experimental runs. CI value has been calculated using the coefficient of variance, as expressed in Eq. 11.
The Chatter Indicator has been calculated for each of the 27 experimental runs and displayed in Fig. 10. Upper threshold, which is shown in red color, and the lower threshold, which is shown in green color, are evaluated using the 3r criterion to classify the three domains of chatter intensity as shown in Fig. 10. The CI values which are below the green line are satisfactory and vice versa.

MRR
MRR has been considered another response for developing the prediction model, and their value is measured using Eq. 10 and presented in Fig. 11.

Artificial neural network for CI and MRR predictions
In this section, a typical feed-forward with backpropagation has been utilized in order to predict CI and MRR. In order to make an analytical model for CI and MRR, at least three layers-the input layer, the hidden layer, and the output layer-should be used. The machining input parameters (tool speed, feed speed, and cut depth) are used for the input layer. Two responses, i.e., CI and MRR, are used for the output layer. However, for the hidden layer, the number of neurons and training function should be selected as per requirements. The selection of the wrong number of neurons and training function may lead to overfitting or underfitting the model. Therefore, in this study, two main objectives, i.e., the numbers of neurons in the hidden layer and optimal training function, have been investigated. In order to make this more illustrative and conclusive this section, a flowchart is presented in Fig. 12.
In this section, a methodology has been proposed to get the optimal number of neurons in the hidden layer based on MSE. Moreover, six Back Propagation (BP) based training algorithms have been taken, and their effect on the developed prediction model has been studied and presented in the subsequent section. The list of six BP training algorithms is as follows: 1. Resilient Propagation (RP), 2. Conjugate Gradient by Polak-Ribeiro (CGP), 3. Scaled Conjugate Gradient (SCG), 4. Quasi-Newton method by Bryden, Fletcher, Goldfarb, and Shanno (BFGS), 5. One Step Secant (OSS), and 6. Levenberg-Marquardt (LM).

Optimal neurons in the hidden layer
In this section, the first objective regarding the ascertainment of optimal neurons in the hidden layer has been presented. The optimal neurons for all six training algorithms have been obtained using the MSE criterion. The neuron for which the difference between the validation MSE value and training MSE is lowest is referred to as the optimal neuron for the respective training functions. MSE values for training and validation data for CI and MRR data sets are shown in Figs. 13 and 14, respectively. Based on the minimal difference, the optimal neurons for CGP, RP, SCG, BFGS, OSS, and LM are 9, 8, 11, 5, 5, and 5 neurons, respectively, for CI data. Similarly, the optimal neurons for CGP, RP, SCG, BFGS, OSS, and LM are 7, 8, 8, 7, 5 and 7 neurons, respectively, for MRR data. These optimal  Tool speed (TS, rpm) 1 9 10 3 2 9 10 3 3 9 10 3

Comparison of six training functions
In this section, six training algorithms (RP, CGP, SCG, BFG, OSS, and LM) have been compared, and the best training function has been selected based on statistical parameters such as Average Absolute Percentage Deviation (AAPD), Mean Square Error (MSE), number of iteration and R-value. Here, two three-layer ANN models have been developed for predicting CI and MRR. Each ANN model has been trained using optimal neurons in the hidden layer, considering all six training algorithms separately. For the input layer, input machining parameters, i.e., Tool Speed (TS), Feed Speed (FS), and Cut Depth (CD), have been normalized so that the effect of a large number can be minimized on the developed prediction model. The existing data set is first separated into three parts: training, validation and testing sets in a 70:15:15 ratio, respectively. The gradients or Jacobians are then computed with the data from the training set, and adjustments to the weights are made after each iteration. During the training phase, the network is monitored in the validation set, and changes in In the CI case, firstly, the RP algorithm is invoked to check the gradient error and the number of epochs. At 12 epochs, a gradient error of 0.0013424 has been found. At the epoch of 14, the obtained gradient error for the CGP is 0.005212. For the SCG condition, the gradient error is roughly 0.003214 at 13 epochs. The gradient error of 0.0023154 is obtained for the BFG case at an epoch of 21. At 17 epochs, a gradient error of 0.0018524 is observed for the OSS training algorithm. While for LM-based algorithm, the gradient error was 0.000056432 at eight epochs. The training was stopped when a minimum gradient error was obtained for each training algorithm. As compared to RP, CGP, SCG, BFG, and OSS algorithms, the gradient error using the LM method is 2278. 79, 9135.895, 5595.35, 4002.99, and 3182.535% lower, respectively. In addition, when compared to the RP, CGP, SCG, BFG, and OSS algorithms, the number of epochs utilized by the LM method to obtain the least gradient error is 50, 42.85, 62.5, 162.5, and 112.5% lower, respectively.
While in the MRR case, using the RP technique, the gradient error and epochs scenario are 0.005498 and 16, respectively. At the epoch of 18, the observed gradient  Along with this, AAPD (Average absolute percentage deviation) is also studied, and results are presented in Table 2. The Levenberg-Marquardt (LM) training method (trainlm) was determined to be the best-performing training  Table 2. When compared to other training algorithms like RP, CGP, SCG, BFG and OSS, LM (trainlm) provided lower MSE values as well as lower AAPD and Iteration numbers. Furthermore, when compared to other training algorithms, the total R (coefficient of correlation) value of the LM training function is the greatest.

Optimized learnt weight in prediction models
After deciding the best training algorithm, prediction models of CI and MRR are established using three neurons in the input layer, one neuron in the output layer and 5 and 7 neurons in the hidden layer for CI and MRR, respectively. Learnt weights values for the developed ANN model are presented in Figs. 15 and 16 for CI and MRR, respectively. Different color coding has been used to visualize the connection between the input and hidden layer. Learnt weights are real numbers associated with each character and indicate the significance of that feature in predicting the final output.

Mathematical formulation for CI and MRR
In this sub-section, analytical expressions are devised for two output responses, i.e., CI and MRR. In order to devise the analytical expression, some notations have been used and are presented as follows: f is representing the weight between layers, a is the ascertained optimal number of neurons in the hidden layer, b is representing the bias and w is representing the final output.
The mathematical relation between input and hidden layers can be written and presented as Eq. 12 (Hagan et al. 2014); where; f superscript is representing the relation between the input layer and hidden layer, and the first and second numbers in f subscript are indicating the position of the neuron in the second layer (hidden layer) and the first layer (input layer). Thereafter, the output of Eq. 12 has been used as input to get the optimal weight at the hidden layer. These optimal weights have been ascertained using activation function (TANSIG) and can be expressed as Eq. 13 (Hagan et al. 2014); Since the output layer has only one neuron, either CI or MRR, output at the output layer can be expressed as Eq. 14.
Now, the final output at the last layer can be ascertained using the same TANSIG function again; it can be expressed as Eq. 15.
Using Eq. 15, prediction models for the output responses, i.e., CI and MRR have been developed and presented in Eq. 16 and 17, respectively.
Thereafter, these developed mathematical models have been utilized to develop a final multi-objective optimization model and explored to study the effect of individual parameters.

Optimization model and constraints
The multi-objective optimization model of process parameters for a CNC milling process is established as follows:

Overview of MOPSO
Coello and Lechuga's MOPSO (Coello Coello and Lechuga 2002) was put out in 2002 to help with multiobjective problem optimization. For calculating the particle's flight direction, they applied a Pareto dominancebased notion. In order to retain the diversity of solutions along the space solution, they recommended adding a global repository (an external memory) to store the nondominated solution and to keep it updated depending on particle experience (iteration per iteration).

Problem initialization
(1) Set up the particles in a search space at random to begin the population.
(2) Set the speed of each particle to zero at the start.
(3) Analyze the cost of each particle in the swarm using the objective function. (4) Establish the repository and register the coordinates of each particle that characterize the nondominated values. (5) Create grids of the search space and use them as a reference system to find particles. The positions of each particle are determined by their costs (objective function values). (6) The particle's best position (P B ) has been evaluated and stored in the repository.

MOPSO main loop
For each iteration and each particle in the swarm, do the following: 1) In order to select the leader among nondominated particles, stochastic universal sampling has been utilized. The position of the leader has been designated as global best (P GB ).
2) Velocity of the particle (k) has been calculated for iteration (T) and for decision variable using Eq. 18: Where, C 1 and C 2 are tuned using Taguchi method. d 1 and d 2 are random between [0,1] 3) Particle's new position has been calculated using the position of the particle in the previous iteration and presented as Eq. 19: y ðk;l;Tþ1Þ ¼ y ðk;l;TÞ þ v ðk;l;Tþ1Þ ð19Þ 4) Hold particles inside the search area in case they travel outside the limits. 5) Updating the repository's data in accordance with where the particles are located inside the grids. 6) To determine whether the particle's present situation is superior to the P B from the previous iteration, Pareto dominance has been utilized. If the new assignment dominates the P B from the previous cycle, the P B is modified using Eq. 20 and saved in the repository. If not, the repository retains the previous value.
Steps used in MOPSO are represented in Fig. 17.

Tuning of MOPSO parameters:
Due to their direct impact on the effectiveness of the outcomes, tuning the MOPSO parameters is essential. These parameters are problem dependent and, therefore, must be tuned as the requirement. As per the available literature, the number of iterations, repository size, particle size, inertia weight, global and personal learning coefficients are the important MOPSO parameters. The settings of these parameters may perform satisfactorily for some optimization problems and may fail despondently for others. In this study, the aforementioned parameters have been taken into consideration and tuned using a well-known one of statistical method, i.e., Taguchi. Taguchi helps in reducing the number of experiments in order to tune the MOPSO parameters. In this study, for all six factors, three levels have been taken. After constructing the DOE, objective function values have been evaluated as per the designed DOE. Thereafter, the effect of each factor is analyzed and presented in Fig. 18.
The Taguchi method, as a strong tool for the DOE, is used for tuning the MOPSO parameters. For this, a threelevel Taguchi experiment is implemented to examine the impact of the main MOPSO parameters. The best level for all these factors (MOPSO parameters) is presented in Table 3.
Thereafter, MOPSO code was developed in MATLAB using open-source code (Heris 2015). In this study, the following parameters have been considered; 'w' is an inertia weight parameter and is taken as 0.7; Population size or the number of particles (N p ) is taken as 100; the maximum number of iterations is taken as 100, repository size is equal to 50 and local and global learning coefficient, i.e., 'C p ' and 'C G ' are taken as 1.25 and 1, respectively. Other parameters such as mutation rate, inertia weight damping rate, inflation rate, leader and deletion selection pressure have been taken as 0.1, 0.99, 0.01, 2 and 2, respectively. Obtained parent solutions are presented in Fig. 19.

Performance Metrics
Performance metrics are used to investigate the effectiveness of meta-heuristics optimization techniques for multiobjective problems. Performance metrics help in understanding the accuracy and variability of the Pareto solution obtained by meta-heuristic methods. In this study, MOPSO performance has been evaluated using three metrics, i.e., Generational Distance (GD), Error Ratio (ER) and Spacing (S), are as follows: a) Generational Distance (GD): GD metric has been used to measure the averaged distance between the Pareto optimal front and the true Pareto optimal front. A low GD number denotes good performance. GD can be represented as Eq. 21: where N presents the number of solutions, d i is the minimum Euclidean distance. b) Error ratio (ER): The number of outcomes in the Pareto optimum front that are not its members may be counted with the help of ER. Good performance is indicated by a low ER value. ER is presented using Eq. 22: c) Spacing Metric (SM): The Spacing metric, as presented in Eq. 23, helps in finding the dispersion of Pareto optimal solutions. SM is a good indicator to measure the variance between the Pareto optimal front and its closest neighbors. Low value of SM is recommended to ensure the performance of the algorithm.
where SM is the Spacing metric, N is the number of Pareto optimal solutions produced by the algorithm, X is the mean of all X i , X i is the minimum distance between the i th solution and all other solutions in the Pareto set. The result of the performance metrics is presented in Table 4. It is evident from Table 4 that results are promising in terms of identifying the optimal solution of milling parameters for improved MRR and minimal Chatter.  Max 9.62 9 10 -3 SM Mean 7.14 9 10 -2 Std 7.87 9 10 -3 Min 6.27 9 10 -2 Max 8.00 9 10 -2 7.5 Analysis of pareto solutions As shown in Figs. 20, 21 and 22, 2D plots of responses (CI and MRR) were created using two inputs at a time while maintaining the third unchanged. The green color represents an excellent CI and MRR zone, while the red color represents the poor condition. The 2D plot for CI and MRR, as illustrated in Fig. 20, was generated with unchanged 'FS' and varied 'TS' and 'CD'. The range of 'TS' and 'CD', as shown in Fig. 20, is 1085-2880 rpm and 1.04-1.96 mm, respectively. It is clear from this diagram that there is no confluence set of process variables for both green and mint colors. The major aim of the study is to achieve the lowest CI and highest MRR. As a result, milling input variables were chosen from the lesser CI (up to the blue color) and higher MRR (blue to green zone) ranges, as shown in Table 5.
In the same way, 2D plots for CI and MRR are included in Fig. 21. At a fixed 'CD,' these charts indicate the effect of 'TS' and 'FS' on responses. 'TS' and 'FS' can be adjusted between 1085 and 2880 rpm and 77 and 97 mm/ min, respectively. It is clear that for both green and dark green colors, there is no overlap zone of process variables. As a result, the sets of process variables have been chosen using the blue color as a barrier, as seen in Table 6.
In Fig. 22, the influences of 'FS' and 'CD' at a consistent 'TS' (2000 rpm) on responses are presented. When evaluating the 2D plots of CI and MRR at the same time, it is clear that the set of variables does not overlap until the dark green color region. As a result, as indicated in Table 7, a variety of milling process variables were chosen with the blue color zone in focus.
After determining the correct limits of processing variables ('TS', 'CD' and 'FS') from Figs. 20, 21 and 22, which are shown in Tables 5, 6 and 7, an ideal set of milling process variables are retrieved. Finally, by collecting common ranges of values from Tables 5, 6 and 7, ideal parameters corresponding to desirable outcomes are determined and are shown in Table 8.
The stated optimal range is further confirmed by 20 supplementary experiments. These tests were conducted using a variety of milling setting combinations, utilizing Table 8's established optimal range as a benchmark. A number of validation trials are shown in Table 9.
From experiments 1-3, it can be concluded that 'TS' plays an important role in the chatter excitation. As 'TS' increases, chatter also increases. When MRR values are observed for these first three validation experiments, it can be deduced that 'TS' does not have much influence on MRR while 'FS' is an important parameter. When experiments 4-12 are studied, it can be said that as the value of the 'CD' is increasing, chatter and MRR also enhance. However, it can be observed that, while studying the experiments from 1 to 12, MRR is increasing along with chatter value. This study aims to find the optimal range which ensures the minimum chatter along with higher MRR. This objective is not fulfilled in the experiments from 1-12.
Thereafter, 3 more experiments (13, 14 and 15) have been performed considering the optimal range which is given in Table 8. It is obtained that the outcomes of experiment No. 13 are in the desirable range, where CI is 1.39 and MRR is 2.95. The same results were obtained for experiments No. 14 and 15. If the calculated CI is compared with that in Fig. 10

Conclusion and future scope
This study's major objective is to find a set of machining settings that provide consistent performance, increased productivity, and increased quality. To arrive at this outcome, a novel improved LMD approach relying on cubic     B-splines has been used. Furthermore, ANN modeling was used to investigate the impact of input machining characteristics on chatter quality and MRR. In order to train these neural networks, 6 different backpropagation training algorithms: RP, CGP, SCG, BFG, OSS and LM were used. Each one was designed to train the ANN via optimal sets of neurons in the hidden layer. Three layers ANN-based model showed précised and effective predictions of CI and MRR for testing sets (R = 0.99119 and 0.98638) having three milling process variables. The optimal neurons in the hidden layer were obtained as 5 and 7 with the corresponding minimum MSE values of 0.08473 and 0.094743, respectively. Statistical measure index (R, MSE, AAPD and IN) shows that proposed LM-based ANN models have an effective and efficient prediction ability as compared to RP, CGP, SCG, BFG and OSS. Finally, LM-ANN-based prediction models were refined for analyzing a consistent set of milling process variables using a MOPSO technique. Cut depth, feed speed, and tool speed was found to be best at 1.56-1.77 mm, 77-97 mm/min, and 2380-2880 rpm, respectively. The proposed methodology can have importance in numerous applications such as fault diagnosis in rotating machines and other machining operations like turning, grinding and boring. The results reveal that the proposed algorithm can predict optimal machining parameters that result in better productivity at minimal chatter.
Funding The present work has received no funds.
Availability of data and material Not Applicable.
Code availability Not Applicable.

Declarations
Conflicts of interest The authors declare that they have no known competing financial interests.
Ethical approval In this manuscript, the authors declare that no investigations have been carried out using either human volunteers or animals.