Relevance vector machine is found to be one of the best predictive models in the area of pattern recognition and machine learning. The important performance parameters such as the material removal rate (MRR) and surface roughness (SR) are influenced by various machining parameters, namely, discharge current (
Electrodischarge machining (EDM) has tremendous potentials on account of its versatile application in industry. The applications are like high precision machining of all types of electrical conductors and hard material such as metal and metal alloys like tool steel and steels used for die making in metal forming processes and manufacturing of molds, automotive, aerospace, and surgical components. It has enormous advantages without any physical contact between tool and work piece. This nonconventional machining process uses the thermoelectric energy for machining and removes the material by thermal erosion process. The EDM process involves finite discrete periodic electric sparks created by the electric pulse generator at short intervals between the tool electrode (anode) and work electrode (cathode) separated by a thin film of dielectric liquid that causes the material removal in motion and vaporize form, and these tiny molten vaporize particles are flushed away from the gap by continuous flushing of dielectric liquid. This machining process provides productive with increasing strength of the work material and maintains desire shape, accuracy, and surface integrity requirements [
A number of studies have been investigated for material removal rate and surface roughness in EDM process and effect of machining parameters on machining performance. The modelling of material removal rate through RSM, Rahman et al. [
Those above mentioned investigations were restricted to studying the effect of various discharge machining parameters on the MRR and SR in EDM with EN19 tool steel and other machining parameters called duty cycle. Hence the present study attempts to investigate the machining performance using EN19 as work piece material. EN19 has tremendous application towards the components of mediums and large cross section, requiring high tensile strength and toughness for automatic engineering and gear and engine construction such as crane shafts steering knocking connecting rods.
Although the response surface methodology and other prediction tools are a powerful approach for the investigation of material removal rate and the surface roughness during the EDM process, it is timeconsuming. During recent decades, the RVM was been a popular machine learning tool for solving complex problems. In this model a general Bayesian framework is used for obtaining the sparse solutions to regression (function estimation) and classification tasks by utilizing linear models in the parameters [
It was concluded from the above study that limited work has been done on EDM of EN19 (alloy steel) and use of RVM model to predict EDM responses namely MRR and SR. Therefore it is needed to work out the present research for EN19 as work piece material and RVM model as the machine learning algorithim. The present study was initiated to develop a multiinputmultioutput RVM model to predict the values of MRR and SR resulting from an EDM process. The three process parameters, namely, discharge current, pulse on time, and duty cycles, were varied to investigate their effect on response parameters such as material removal rate and surface roughness. The process optimizations of these parameters were done by response surface methodology. In this paper we use the application of large scale multikernel RVM for the prediction of MRR and SR for EDM process.
The experiments are carried out utilizing LEADER1 ZNC electrical discharge machine. The EDM has provision of movement in three axes,
Chemical composition of work piece (EN19).
Elements  Percentage of weight 

C  0.38–0.43 
Mn  0.75–1.00 
P  0.035 
S  0.04 
Si  0.15–0.3 
Cr  0.8–1.10 
Mo  0.15–0.25 
Mechanical properties of work piece (EN19).
Density (Kg/m^{3}) 

Poisson’s ratio  0.27–0.3 
Elastic modulus (GPA)  190–210 
Hardness (HB)  197 
Experimental setup (electrode, work piece, and flushing arrangement).
The machining was carried out for a fixed depth of machining of 0.2 mm and for each depth of machining the time is measured by using a digital stopwatch. The experimental schedules are shown in Table
The design of experiments for exploring the effect of various predominant EDM process parameters (e.g., Pulse on time, discharge current, and duty cycle) on the machining characteristics (e.g., the material removal rate and surface roughness) was modelled. The main objective of experimental design is to study the relations between the response as a dependent variable and the various process parameter levels. It provides an opportunity to study the individual effects of each factor.
In the present work experiments were designed on the basis of experimental design technique using RSM that is a mathematical and statistical techniques which are useful for the modelling and analysis of an experiment in which a response of interest is influenced by several variables and the objective is to optimize the response. The coded levels for all process parameters used are displayed in Table
EDM process parameters and their levels assigned for experiment.
Process parameters  Unit  Levels  

−2  −1  0  1  2  
Discharge current ( 
Ampere  1  2  4  7  8 
Pulse on time ( 
Ms  50  200  500  750  1000 
Duty cycle (tau)  %  45  55  65  80  85 
CCD for process parameters using RSM and experimental results.
Sl. no. 


tau  MRR (mm^{3}/min)  SR ( 

1  2.00  750.00  55.00  0.31107  3.72 
2  7.00  750.00  80.00  4.62736  7.8 
3  2.00  200.00  80.00  1.77819  4.17 
4  4.00  500.00  65.00  2.68492  5.13 
5  4.00  500.00  65.00  2.59718  4.79 
6  7.00  200.00  55.00  3.97481  11.27 
7  7.00  750.00  55.00  4.57803  9.12 
8  4.00  500.00  65.00  2.69626  5.2 
9  4.00  500.00  65.00  2.65843  5.21 
10  2.00  750.00  80.00  0.39685  3.8 
11  7.00  200.00  80.00  4.06222  10.33 
12  2.00  200.00  55.00  1.8393  3.53 
13  4.00  500.00  65.00  2.78798  4.7 
14  8.00  500.00  65.00  5.404  11.43 
15  4.00  50.00  65.00  2.43872  5.83 
16  4.00  1000.00  65.00  0.76172  4.13 
17  4.00  500.00  65.00  2.79229  4.87 
18  1.00  500.00  65.00  0.6589  3.47 
19  4.00  500.00  85.00  2.69359  4.73 
Relevance vector machine (RVM) is a machine learning technique based on a Bayesian formulation of a linear model with Proper selection of prior that result in a sparse representation RVM is a special type of a sparse linear model, in which the basis functions are formed by using kernel functions centered at the different training points [
Shows the block diagram of RVM.
The output function
Let us assume the sparse Bayesian regression model and associated inference procedures to predict both MRR and SR for EDM process. The experimental data set of inputoutput pairs obtained from RSM analysis are given in form of
Due to the assumption of independence of the likelihood function, the complete data set can be written as follows:
To modify this approach we should follow the Bayesian prior probability distribution. At first we encode a preference for smoother functions by making the popular choice of a zeromean Gaussian prior distribution over
The Bayesian inference obtained from Bay’s rule, which is given by the following
Relevance vector machine method is a learning procedure to search for the best hyper parameters posterior mode that is, the maximization of
For
During the convergence of the hyper parameter estimation procedure, we have to make predictions based on the posterior distribution over the weights, in which the conditions of the maximizing values are
ANOVA and main effect plots for MRR and SR are done by response surface methodology using MINTAB 16. And with the help of 40 sets of experimental inputoutput patterns, the proposed RVM modelling are carried out. The software programs for RVM model are implemented using MATLAB version 10.1.
The influences of various machining parameters on MRR are shown in Figure
Analysis of variance (ANOVA) for MRR and SR.
Source  MRR (mm^{3}/min)  SR (  



% of contribution 


% of contribution  

10377.67  0.000  84.34  2145.29  0.000  85.84 

377.47  0.000  4.13  120.39  0.000  3.84 
tau  Insignificant  14.87  0.002  0.224  

199.47  0.000  6.77  

653.71  0.000  5.8  Insignificant  

662.65  0.000  5.58  60.75  0.000  2.04 

Insignificant  24.67 

0.84  


Main effects of machining parameters on MRR.
The effects of various input parameters on SR are shown in Figure
Main effects of machining parameters on SR.
The proposed modelling is carried out by relevance vector machine (RVM) with the help of 40 sets of experimental inputoutput patterns obtained from RSM (response surface methodologies) in EDM process. As lesser amount of data available from the experimental design we again sampled the data and further used those data for both training and testing of RVM model. The different machining parameters such as discharge current
Regression test error for different kernel functions.
Noise factor = 0.1, number of iterations = 100  

Kernel functions  Regression test error for MRR  Regression test error for SR 
Gaussian kernel  0.083636  0.079863 
Laplace kernel  0.077997  0.074544 
Linear spline kernel  0.088783  0.078388 
Cubic kernel  0.070611  0.066244 
Distance kernel  0.129122  0.103465 
Thinplate spline  0.086694  0.075625 
Neighborhood indication  0.257006  0.220205 
Regression test error for different noise factors.
For cubic kernel function, number of iterations = 100  

Noise factor  Regression test error for MRR  Regression test error for SR 
0.05  0.037754  0.035251 
0.01  0.012026  0.018637 
Regression test error for number of iterations.
For cubic kernel function, noise factor = 0.01  

No. of iterations  Regression test error for MRR  Regression test error for SR 
100  0.012026  0.018637 
200  0.010694  0.015266 
250  0.01293  0.015358 
From Table
Experimental and predicted values of MRR and SR.


tau  MRR (mm^{3}/min)  SR ( 
MRR (predicted using RVM) in mm^{3}/min  SR (predicted using RVM) in 

2.0  750.00  55.00  0.311073  3.72  0.33439221  3.710796635 
7.00  750.00  80.00  4.627355  7.8  4.602598191  7.90183655 
2.00  200.00  80.00  1.77819  4.17  1.775926522  4.169651963 
4.00  500.00  65.00  2.684917  5.13  2.700207547  4.943501339 
4.00  500.00  65.00  2.59718  4.79  2.700207547  4.943501339 
7.00  200.00  55.00  3.974812  11.27  4.014896139  11.20501356 
7.00  750.00  55.00  4.578032  9.12  4.540240762  8.886500507 
4.00  500.00  65.00  2.69626  5.2  2.700207547  4.943501339 
4.00  500.00  65.00  2.658429  5.21  2.700207547  4.943501339 
2.00  750.00  80.00  0.39685  3.8  0.379393774  3.585126803 
7.00  200.00  80.00  4.062215  10.33  4.005057172  10.32266034 
2.00  200.00  55.00  1.839296  3.53  1.868379133  3.56482374 
4.00  500.00  65.00  2.78798  4.7  2.700207547  4.943501339 
8.00  500.00  65.00  5.404  11.43  5.343913023  11.43558399 
4.00  50.00  65.00  2.43872  5.83  2.426300736  5.825788079 
4.00  1000.00  65.00  0.761718  4.13  0.881463669  4.105567142 
4.00  500.00  65.00  2.792288  4.87  2.700207547  4.943501339 
1.00  500.00  65.00  0.6589  3.47  0.672458618  3.782897979 
4.00  500.00  85.00  2.69359  4.73  2.745193125  4.986478772 
4.00  500.00  45.00  2.6557  5.43  2.62671966  5.343644844 
The comparison between predicted MRR and experimental MRR.
The comparison between predicted SR and experimental SR.
The experimental design of EDM was successfully implemented using CCD based on RSM. From the RSM analysis, the most significant parameter towards the response was found to be the discharging current. The proper selection of pulse time can save the machining time as well as cost of component in manufacturing industry. High value of pulse duration decreases the MRR. Roughing operation “
The authors would like to thank Department of Mechanical Engineering, National Institute of Technology, Rourkela, India, for providing the facility to conduct the experiment. Also, they are thankful to anonymous reviewers and their suggestions for helping them to improve the quality of the paper.