Abstract—This paper considers a machining process in which its surface roughness is influenced by radial rake angle, speed and feed rate cutting condition simultaneously. In previous works, researchers have been used a response surface methodology (RSM), soft-computing techniques based regression model and integration of linear/nonlinear regression and soft-computing techniques to estimate the proper cutting conditions in this machining process. It is noticed that the combined linear regression and genetic algorithm (GA) had limitation regarding to the issue of multicollinearity. While the hybrid of nonlinear regression and GA faced the issue of complexity of the nonlinear regression’s structure since more complex structure of regression can increase the chance of overfitting, In this paper, an alternative techniques based on support vector regression (SVR) and GA is proposed to address the above issuess. SVR employs relatively a new technique, call it structural risk minimization (SRM), to obtain a nonlinearity relationship between the cutting conditions and its surface roughness. The alternative framework gave more accurate prediction model compared to RSM and soft-computing techniques based regression model in a machining dataset, and has less complex structure of regression compared to KPCR and GA.
Index Terms— Surface roughness, support vector regression, nonlinear regression, genetic algorithms.
I. INTRODUCTION
REGRESSION models and soft-computing techniques including Genetics Algorithm (GA), Simulated Annealing (SA), Tabu Search, Particle Swarm Optimization (PSO) and Neural Networks (NN) have proposed to estimate the highest quality of machined surface [1-2]. Previously, one may used the conventional techniques such as response surface methodology (RSM) technique to estimate the proper parameters in machining process [1]. Recently, some researchers integrated regression models and soft-computing techniques to estimate the parameters of
Antoni Wibowo is with Computer Science Department, Binus Graduate Program - Master in Computer Science, Bina Nusantara University , Anggrek
Campus Jl. Kebon Jeruk Raya No. 27, Kebon Jeruk, West Jakarta 11480 Indonesia (phone: +62-21-53696969 ext 1803; e-mail:[email protected]).
interest in machining process. In machining process, a surface roughness is one of the most common performance measurements and an effective parameter to measure the quality of machined surface. A machined surface has the higher quality than another machined surface when it has lower surface roughness than the other one. Therefore, the proper values of the parameters of interest are selected to have as low as possible of the surface roughness
This paper consider to estimate the optimum cutting conditions of surface roughness in machining process, which is influenced by radial rake angle, speed and feed rate cutting condition, and its corresponding roughness surface. The ideal mathematical model of this machining is given by [2-3]:
(1)
m f cv Ra k lm
where
R
ais the experiment (measured) surface roughness (m), is the cutting speed (m/min), f is the feed rate (mm/tooth), is the radial rake angle (˚), is the experimental error and c, m, k and l are the model parameters to be estimated, respectively.It is noticed that [2] integrated the ordinary linear regression and genetic algorithm (GA ) to estimate the optimum values of the cutting conditions and the quality of surface roughness for the experiment data conducted by [4]. [2]’s technique was conducted by constructing the ordinary linear regression models from these data, and followed by model selection among the regression models. Then, they developed a linear programming problem in which the objective function was one of their regression models and the constraints were developed based on the lower and upper bounds of radial rake angle, speed and feed rate cutting condition data. However, it is noticed that the severe multicollinearity exists on the linear model [3]. When a severe multicollinearity exists in a regression model, it can givethe negative effects such as its prediction model can be inappropriate to be used [5]. Beside, it is obvious that model (1) is, itself, nonlinear which implies that a linear regression model can be inappropriate model to approximate it.
Due to the above limitations, [3] presented the integration
of kernel principal component regression (KPCR) and GA to
estimate the optimum of the cutting conditions and its
Optimization for Cutting Conditions of Surface
Roughness in Machining Process Using Support
Vector Regression and Genetic Algorithm
corresponding surface roughness. In this technique, KPCR is used to overcome the issues of linearity and multicollinearity on [4]’s data and GA is conducted to obtain the optimum values of the cutting conditions such that the estimated surface roughness is as low as possible. KPCR is developed based on linear regression model, kernel principal
component analysis (KPCA) and empirical risk minimization
(ERM). The ERM is a conventional method in regression analysis to estimate regression parameters. KPCA has been used for nonlinear systems by mapping an original input space into a higher-dimensional feature space [6-9]. While KPCR was studied by [10-15] and can be used to perform a nonlinear prediction and to dispose the effects of multicollinearity in regression model. However, KPCR can face the issues of complexity structure of nonlinear regression since more complex structure of regression can increase the chance of overfitting.
Under the above circumstance, this paper presents another nonlinear framework based on support vector regression (SVR) and GA to estimate the optimum values of the cutting conditions. SVR employs the structural risk minimization
(SRM) principle which is relatively new approach for classification and regression [16-19]. This method can be used as an alternative technique for determining the optimum values of parameters of interest in machining process. As KPCR did, SVR transforms the coresspond data into a higher dimensional space as well, called it the feature space, and construct linear SVR model in this feature space. Afterward, a kernel trick is conducted to have a nonlinear SVR and followed by the Cross Validation (CV) method to select the ‘best’ regression model. The final stage of this framework is developing nonlinear programming problems using the best regression models and the constraints are lower and upper bound of [4]’s experiment design. Then, GA is used to solve the nonlinear optimization problems.
The rest of the manuscript is organized as follows. In Section II, review theories and methods of SVR and GA is presented. In Section III, case study using data from [4], model selection, constructing nonlinear programming problems of the surface roughness and discussion are described. Finding the minimum of the surface roughness and its corresponding variables is given to close this section. Finally, conclusions are given in Section IV.
II.THEORY AND METHOD
A.Structural Risk Minimization
Given training data (xTi ,yi)p1 generated from an independent and identically distributed (iid) probability distribution P(x,y)where i = 1, 2, , N, N is the number
of data, is the set of real numbers and
pip i
i
i x1 x2 x
x andyi is the corresponding
output value of xi, respectively. Then, a class of function
) , (xα
f can be found to approximate the relationship between the input vector and the output variable where α is the parameter vector of the function. The goal is to select a function from the class of function f(x,α) which minimize
( , ( , )) ( , ) (2) )
( L y f dP y
Rα x α x
where L(y,f(x,α)) denotes the loss between the actual output y and the function output f(x,α) based on the training data. One popular measure of loss in function approximations is the squared loss which is given as follows:
(3) )) , ( ( )) , ( ,
(y f xα y f xα 2
L
It is evident that from (2) and (3), one can obtain
( ( , )) ( , ) (4) )
( y f 2dP y
R α x α x
Suppose that the distribution of P(x,y) is unknown which implying a difficulty arises to find R(α).
To overcome this difficulty, one can only use the given data (xTi,yi)for estimating a function f that minimizes R(α) and conduct an induction principle is required to minimized the empirical risk Remp(α) which is given by
(5) )) , ( ( 1 ) (
1
2
N
i
emp y f
N
R α xα
instead of the expected risk R(α) This procedure to obtain a function f via Eq.(5) is called the empirical risk minimization
(6) ) ( ) ( h N RR α emp α
where h is called the VC dimension. The VC dimension can be interpreted as the model capacity or the complexity of the learning machine in which the VC dimension h controls the confidence interval Ω for the learning machine. If the chosen structure of the learning machine is too complex such that the empirical risk can be minimized then the confidence interval Ω will be large. In the case, even the empirical risk can be minimized to zero, however, the confidence interval Ω may be large. Therefore, the SRM principle suggests a tradeoff between the quality of approximation and the complexity of the approximating function [20]. Hence, a best selected function that minimizes the expected risk is controlled by the above two terms.
B.Support Vector Regression
SVR is constructed bases SRM paradigm. To construct SVR, it is assumed that there is a function, say:p F, and data xi is mapped to a feature space F through(xi). Then, a linear regression function is constructed in the feature space F as follows:
(7) ) ( , ) ,
( b b
f w w x
where f :p, w and b are the parameters vectors of the function.. The objective of SVR is to find the best linear function f which provides the best approximation of the actual output y with an error tolerance ϵ and is concurrently as flat as possible. It indicates that the parameter vector
w w
w , is needed to as small as possible with the notation zi,zj denotes the inner product of zi and
z
j.
Itcan be interpreted that if one has a smaller w then she/he have a smoother and less complex approximating function. It can be written this problem as a convex optimization problem as follows [17-18]:
. , . 2 , 1 , 0 , ) ( , (8) ) ( , . . 2 1 * * 1 * 2 * , , ,min
N i y b b y t s C i i i i i i i i N i i i b
x w x w w wwhere iand i* are slack variables that specify the upper
and the lower training errors subject to an error tolerance and C>0 that determine the trade-off between the flatness of f
and the amount up to which deviations larger than ϵ are tolerated. In the optimization problem (8), one expects the most data examples are to be in the -tube. If an item of data
) ,
(xTi yi is outside the tube, then an error iori* exist,
which is to be minimized in the objective function. The tolerated errors within the extent of the -tube, as well as the penalized losses when data concern the outside of the tube, are defined by the so-called Vapnik’s -insentive loss function as [17-18]:
(9) for for 0
It is evident that the function f is influenced by the
regularization term 2
2 1
w and the training error term
N i i i C 1 * . SVR avoids underfitting and overfitting
problem with respect to the training data by minimizing the two terms. It is noticed that minimizing of the first term is equivalent to minimizing the confidence interval, and minimizing of the second term corresponds to minimizing the empirical risk. The optimization problem (8), in most cases, can be solved more easily in its dual formulation. The dual formulation provides an important key to develop nonlinear SVR. The dual problem of (8) can be described in Eq. (10) [17-18].
. , , 2 , 1 , 0 , , 2 , 1 , 0 0 . . (10) ) ( ), ( 2 1 1 1 1 1 , * , , ,min
N i C N i C t s y i N i i i N i N i i i i i i j i j j N j i i i b
x x wAfterward, the kernel trick is conducted using Mercer Theorem. According to Mercer Theorem [6-7,14], if one choose a continuous, symmetric and positive semidefinite kernel :pp then there exists :pF such that (xi,xj) (xi),(xj) . Hence, Instead of choosing
(11)
) , ( 2
1
1 1
1 , * , , ,
min
N
i
N
i
i i i i
i
j i j j N
j i
i i b
y
x x
w
.
,
,
2
,
1
,
0
,
,
2
,
1
,
0
0
.
.
1
N
i
C
N
i
C
t
s
i N
i
i i
It is evident that the optimization problem (11) is a quadratic optimization problem which can be solved using available software such as Matlab. When data xi is in inside of the ϵ -insentive tube, the value (ii) is zero. Therefore, the best function f involves only the remaining nonzero coefficients (ii). The data that have nonzero Lagrange multipliers are called the support vectors. In another word, it can be said that support vectors are those data that “support” the definition of function f, whereas other data can be regarded as redundant [21]. Furthermore, after the Lagrange multipliers i and i are obtained, then the best parameter vectors w and b, say w**and, b**, can be estimated and are
given as follows [17]:
( ) (12)1
*
N
i
i i
i x
w
and
( , ) ( , )
(13) 21
) ( ) ( , 2 1
N
1 i
* * *
r i s i i i
r s
b
x x x x x x w
Furthermore, the kernel the best nonlinear SVR with respect to a kernel
, say f*, can be expressed as follows:
( , ) (14) )( *
1
* b
f i
N
i
i
i
x x
x
Furthermore, the suitable parameter C, ϵ and kernel are selected using the cross validation (CV) method.
C.Genetic Algorithm
This section briefly introduce GA which was developed by Holland in 1975. His theory has been further developed and now GA stand up as a powerful tool for solving search and optimization problems. The modified of GA and the hybrid of GA with another meta-heuristic techniques have been proposed to improve the performance of pure GA [22-24].
Given the following optimization problem
D z
z
. .
) G( min
t s
where D is the feasible space of (20). The basic of GA to solve the above optimization problem can be summarized as follows [25-26]:
1. Initialization: In this stage, an initial random population Po containing chromosome
z
k is constructed where k = 1, 2, , Npop and Npop is the total number of chromosome and set index i = 0. 2. Fitness Evaluation: Calculate the value of the fitnessG(z) of each chromosome
z
k in the current population.3. New Population: Set i = i+1 and create a new population Piby repeating the following steps:
Selection: Select two parent chromosomes from a population based on their fitness values.
Crossover: Make crossover the parents to form new offspring (children) with a certain crossover probability. If no crossover was performed, offspring is the exact copy of parents.
Mutation: Make mutate new offspring at some positions in chromosome with a certain mutation probability.
Accepting: Place new offspring in the new population.
Stopping Criteria: If the end condition is satisfied, stop, and return the best solution in the current population. Otherwise, go to Step 2 for fitness evaluation for the population Pi.
III. CASE STUDY
A.Experiment Data
In this case study, this paper also considers the machining experiment conducted by [4]. In these data, the surface roughness value in end milling which was performed using an annealed alpha–beta titanium alloy Ti–6Al–4V (Ti–64) in which the properties of mechanical and chemical composition of Ti–6Al–4V are given in Table 1. Furthermore, a CNC MAHO 700S machine centre for the milling experiments and Taylor Hobson Surftronic+ instrument were conducted to measure the surface roughness value of the machined workpiece. Before that, the instrument was calibrated using a standard specimen roughness delivered to ensure the consistency and accuracy of surface roughness values before the experiment was conducted.
coating in which the composition and properties of these cutting tools are illustrated in Table 2.
For conducting the experiments, the geometry of the end mills was prepared according to the experimental design, that is, radial rake angle and helix angle. In addition, [4] conducted design of experiments for three independent variables to obtain sample data. The coding variables for the end milling of titanium Ti-6Al–4V is illustrated in Table 3. Then, the whole experiments were carried out with a constant axial depth of cut (aa) = 5 mm and radial depth of cut (ae) = 2 mm and under flood conditions (6% concentration of water base coolants).
TABLEI
CHEMICAL COMPOSITION AND MECHANICAL PROPERTIES OF TI-6A1-4V
Chemical
Compo-sitions Mechanical properties Value
Al 6.37 Tensile strength (MPa) 960–1270
V 3.89 Yield strength (MPa) 820
Fe 0.16 Elongation 5D (%) ≥8
C 0.002 Reduction in area (%) ≥25
Mo <0.01 Density (g/cm3) 4.42
Mn <0.01 Modulus of elasticity tension (GPa) 100–130
Si <0.01 Hardness (Hv) 330–370
Ti Balance Thermal conductivity (W/mK) 7
TABLEII
PROPERTIES OF THE CUTTING TOOL USED IN EXPERIMENTS
Tool type WC–Co TiAlN TiAlN coated Supernitride Coated Substrate
(wt%) WC 94 94 94
Co 6 6 6
Properties Grade K30 K30 K30
Grain Size
(µm) 0.5 0.5 0.5
Coating Process - PVD–HIS PVD–HIS Coating
Thickness - Monolayer Multilayer (3–4 µm) (1–8 µm)
Film composition (mol-%AIN)
- Approx. 54 Approx. 65– 67
In this experiment, twenty four experimental trials were executed that were based on eight data of two levels of DOE 2k full factorial analysis with four centre data and twelve axial data. All the data were tested in real machining for three different cutting tools, which are uncoated, TiA1N coated and SNTR coated cutting tools to show the actual value (experiment result) of Ra. Then, five measurements were conducted at the location of the length of cut on the workpiece and the average surface roughness (Ra) value was
recorded. The R a values of each type of cutting tool are given in Table 4.
B.Model Selection and Estimating the Surface Roughness
B.1. Support Vector Regression Models
TABLEIII
CHEMICAL COMPOSITION AND MECHANICAL PROPERTIES OF TI-6A1-4V Independent
Var. Units -1.4142 -1 0 1 1.4142 Cutting
speed (v) m/min 124.53 130 144.2 160 167.03 Feed rate (f) mm/tooth 0.025 0.03 0.046 0.0
7 0.083 Radial rake
( ) angle ˚ 6.2 7 9.5 13 14.8
It is well known that the regression model is invariant due to transforming data from the original scale to another scale. Therefore, transform the original data into another scale is needed in order to have the suitable prediction model.
In this paper, transformation the original data in Table 4 using the relation: ~vv102, f~ f10 and ~102Moreover, the function kernels in SVR in this experiment are as follows:
1. Linear kernel: (xi,xj) xi,xj
2. Polynomial kernel: ( , ) ( , 1)p1
j i j
i x x x
x
3. Sigmoid kernel: )
) ( , tanh( ) , ( 2 1 p lenght p i j i j
i
x x x x x
4. Radial basis function (RBF) kernel:
) 2 exp( ) , ( 2 2 1 p j i j i x x x
x
to obtain SVR functions for estimating of high quality of this machining process. LetRˆauncoated, RˆaTiAIN and RˆaSNTR
be the prediction of SVR for Rauncoated, RaTiAIN and
SNTR a
R in the new scale, respectively. Then, one can has
where bj for j1,2,3 is the bias term and cij is the
Lagrange multipliers jiji corresponding to the three
cutting tools used.
TABLEIV a
R VALUES FOR REAL MACHINING EXPERIMENT
i
Data Setting values of experimental
Experimental surface roughness value (µm)
Source
v f Ra_uncoat ed
Ra_TiA1N Ra_SNTR
1 DOE 2k 130 0.03 7 0.365 0.32 0.284
2 160 0.03 7 0.256 0.266 0.196
3 130 0.07 7 0.498 0.606 0.668
4 160 0.07 7 0.464 0.476 0.624
5 130 0.03 13 0.428 0.26 0.28
6 160 0.03 13 0.252 0.232 0.19
7 130 0.07 13 0.561 0.412 0.612 8 160 0.07 13 0.512 0.392 0.576
9 Centre 144.2 0.046 9.5 0.464 0.324 0.329
10 144.2 0.046 9.5 0.444 0.38 0.416 11 144.2 0.046 9.5 0.448 0.46 0.352 12 144.2 0.046 9.5 0.424 0.304 0.4
13 Axial 124.5 0.046 9.5 0.328 0.36 0.344
14 124.5 0.046 9.5 0.324 0.308 0.32 15 167 0.046 9.5 0.236 0.34 0.272 16 167 0.046 9.5 0.24 0.356 0.288 17 144.2 0.025 9.5 0.252 0.308 0.23 18 144.2 0.025 9.5 0.262 0.328 0.234 19 144.2 0.083 9.5 0.584 0.656 0.64 20 144.2 0.083 9.5 0.656 0.584 0.696 21 144.2 0.046 6.2 0.304 0.3 0.361 22 144.2 0.046 6.2 0.288 0.316 0.36 23 144.2 0.046 14.8 0.316 0.324 0.368 24 144.2 0.046 14.8 0.348 0.396 0.36
Minimum 0.236 0.232 0.19
B.2 Model Selection
The leave one out Cross Validation (CV) method, which is widely used to select the best model, was conducted to select the best model for estimating the surface roughness of uncoated, TiA1N coated and SNTR [29]. In this paper, the CV method works based on the sum of absolute errors (SAE)
to obtain the appropriate kernel function
, C, and ϵ,respectively, in which the best model is chosen if it has the minimum SAE. In this experiment, the modified of SVR code, which was developed in Matlab by [17], to select the suitable , C, and ϵ. For the sake of comparisons, this
paper just used ‘einsensitive’ loss function in the Matlab code. Furthermore, the procedure of SAE is given in the following manners. First, a certain kernel function
and certain values of C and ϵ were employed, respectively. Then, deleting the ith data (i=1, 2, , 24) and obtain the estimate of parameter vectors w and b from the remaining data. In this stage, the prediction of Ra at the ith data, say Rˆai wasobtained. Then, SAE of SVR with respect to
, C, and ϵ is given by(16) ˆ
) , , (
24
1
i
ai ai R
R C
SAE
It should be noticed thatbj is equal to zero. Furthermore,
from Table 5-7 that it can be seen that the minimum SAE ofRˆauncoated , RˆaTiAIN and RˆaSNTR are achieved when
selecting RBF kernel and ϵ=0.01, respectively, with parameters C and p 1 are infinity (Inf) and 1.01, 5 and 2; and infinityand 2.6, respectively.
B.3 Estimating The Surface Roughness And Its Optimum Variables
The next stage of the proposed frameworks is constructing the three nonlinear programming problems to estimate the optimum values of the radial rake angle, speed, feed rate cutting condition that give minimum surface roughness as follows:
1.48 ~ 62 . 0
8.3 ~ 5 2
6703 . 1 ~ 2453 1
(17.a) ) ~
~ ~ , ~ ~ ~ ( ˆ 24
1 * 1 *
f .
v .
s.t.
f v f v c R
i i i
i i uncoated
a
1.48 ~ 62 . 0
8.3 ~ 5 2
6703 . 1 ~ 2453 1
(17.b) ) ~
~ ~ , ~ ~ ~ ( ˆ 24
1 * 2 2 *
1
f .
v .
s.t.
f v f v c b R
i i i
i i N
1.48 ~ 62 . 0
8.3 ~ 5 2
6703 . 1 ~ 2453 1
(17.c) ) ~ ~ ~ , ~ ~ ~ ( ˆ 24
1 * 3 3 *
f .
v .
s.t.
f v f v c b R
i i i
i i SNTR
a
TABLEV SAE OF Rˆauncoated
Kernel function
ϵ=0.01 ϵ=0.1
C C
5 10 20 30 inf 5 10 20 30 inf
Linear 1.6358 1.6246 1.6187 1.6177 1.4786 1.4436 1.418 1.4062 1.4102 1.414
Polynomial 1.5504 1.5836 1.6162 1.6355 1.4788 1.523 1.4983 1.4207 1.4188 1.4073
(Best p1) (1) (1) (1) (1) (1) (2) (1) (1) (1) (2)
RBF 0.8758 0.8878 0.7461 0.6934 0.6719 1.4137 1.3007 1.2789 1.1967 1.107 (Best p1) (0.5) (0.5) (0.75) (0.75) (1.01) (4.25) (0.75) (0.75) (1) (2.5)
Sigmoid 2.6679 2.3137 2.555 2.5605 2.5825 2.1643 2.3621 2.6303 2.5541 2.5622 (Best p1, p2=0.5) (1.25) (1.75) (-1.5) (1.5) (5.25) (1) (1) (2) (2) (5.25)
Sigmoid 2.7 2.4097 2.3881 2.6399 2.5824 2.51 2.1226 2.6195 2.5594 2.5621 (Best p1, p2=1) (0.75) (1.75) (1.25) (1.5) (5.25) (2) (1) (1) (2.5) (5.25)
Sigmoid 2.4646 2.3218 2.727 2.6458 2.5824 2.145 2.0464 2.6235 2.5615 2.5622 (Best p1, p2=1.5) (0.75) (1.5) (1.5) (1.25) (5.25) (0.75) (0.75) (1.75) (1.5) (5.25)
TABLEVI
SAE OF RˆaTiAIN
Kernel function
ϵ=0.01 ϵ=0.1
C C
5 10 20 30 Inf 5 10 20 30 inf
Linear 1.3686 1.3623 1.354 1.3529 1.2938 1.3797 1.3689 1.3816 1.3857 1.2697
Polynomial 1.3711 (3)
1.3710 (3)
1.3711 (3)
1.3685 (1)
1.3049 (1)
1.2465 (3)
1.2471 (3)
1.2470 (3)
1.2465 (3)
1.2697 (1) (Best p1)
RBF 1.1386
(2.0)
1.1615 (2.25)
1.1625 (0.5)
1.1625 (0.5)
1.1523 (0.5)
1.1370 (0.75)
1.1371 (0.75)
1.1370 (0.75)
1.1369 (0.75)
1.1560 (0.75) (Best p1)
Sigmoid 2.4865 (1)
2.7297 (1.25)
2.7892 (1)
2.0862 (0.5)
2.0649 (5.25)
1.7263 (1.5)
1.8821 (3.25)
2.2333 (3.25)
2.0641 (3.25)
2.1883 (4) (Best p1, p2=0.5)
Sigmoid 2.6652 (1)
2.6638 (1)
2.6837 (1)
2.5602 (1.25)
2.0649 (5.25)
1.7263 (2)
1.8135 (1.5)
2.2333 (3.25)
2.0668 (3.5)
2.1847 (3.75) (Best p1, p2=1)
Sigmoid
2.7307 (0.75)
2.9691 (0.75)
2.5631 (1)
3.1098 (1)
2.0648 (5.25)
1.7263 (1.75)
1.8821 (2.75)
2.1021 (1)
2.0641 (3)
TABLEVII
SAE OF RˆaSNTR
Kernel function
ϵ=0.01 ϵ=0.1
C C
5 10 20 30 Inf 5 10 20 30 inf
Linear 0.9086 0.9012 0.8936 0.8911 0.9863 1.0312 1.0333 1.0309 1.0323 0.9937
Polynomial 0.8941 (1)
0.8996 (1)
0.9120 (1)
0.8965 (1)
0.9862 (1)
1.0307 (1)
1.0255 (1)
1.0326 (1)
1.0326 (1)
0.9937 (1) (Best p1)
RBF 0.5963
(1)
0.5855 (1)
0.5830 (1.25)
0.5708 0.5541 0.7869 (3.5)
0.7936 (3.5)
0.7927 (3.5)
0.7918 (3.5)
0.7569 (3.75)
(Best p1) (1.25) (2.6)
Sigmoid 2.2937 (1)
3.0927 (1)
3.6514 (1.25)
3.1891 (0.5)
3.1512 (5.25)
2.5225 (2)
3.4675 (1.5)
3.5165 (2)
3.2768 (0.5)
3.3012 (5.25) (Best p1, p2=0.5)
Sigmoid 2.3422 (1)
3.1270 (1)
3.5348 (0.75)
3.4853 (1)
3.1512 (5.25)
2.5185 (2)
3.4833 (1.25)
3.5107 (1.25)
3.3641 (5.25)
3.3012 (5.25) (Best p1, p2=1)
Sigmoid
2.2910 (0.75)
3.1704 (0.75)
3.5289 (1)
3.3868 (1)
3.1512 (5.25)
2.5607 (1.75)
3.4638 (1.25)
3.4657 (1.25)
3.3641 (5.25)
3.3012 (5.25) (Best p1, p2=1.5)
The optimization problems (17a), (17b) and (23c) are used to find the optimum solutions and the corresponding surface roughness of uncoated, TiA1N coated and SNTR cutting tools, respectively. The coefficients of the objective functions of those models are given in Table 8. Then, GA tool in Matlab is used to obtain the optimum solutions and the corresponding minimum surface roughness by solving the optimization problems (23a), (23b) and (23c). Table 9 illustrates the optimum solutions and the corresponding minimum surface roughness. From this table, it can be seen that the minimum prediction of the surface roughness is achieved using SNTR coated cutting tools and choose the cutting conditions v~1.6703, ~f 2.500 and~1.4789, and the corresponding minimum surface roughness is 0.115. In the original scale, those values are equivalent tov167.03, f 0.025 and ~14.789,respectively.
C.Discussion
This paper used combined SVR and GA to estimate the optimum values of cutting conditions and its roughness surface in the machining experiment data where the surface roughness is influenced by radial rake angle, speed and feed rate cutting condition of the machine used. It is noticed that the previous works in this machining process used softcomputing based regression models and the combination of the statistical based regression model and metaheuristic techniques. It is also noticed that [2] proposed a nonlinear framework based on kernel principal component regression
(KPCR), which was developed using conventional regression method and kernel trick, and GA to find optimum values of the cutting conditions
To compare the results of the previous works and the proposed technique, the value of cutting conditions level in Table 3 are classified as the lowest, lower, medium, high, and highest. Using this classification, the comparison of the results of the previous works and proposed technique as stated in Table 11. It is noticed that hybrid linear regression and GA involved limitation due multicollinearity issue in regression model [3]. Furthermore, it can be seen that the proposed technique gave better results compared to [2, 4, 30] and provided almost similar reault as similar result compared to hybrid of KPCR and GA. KPCR-GA and SVR-GA gave better results in terms of minimum surface roughness compared to the other taechniques. The cutting speed and the feed for the two techniques are the same values, while the radial rake angle is a little bit difference. It can be estimated the optimum values of the radial rake angle is in the range interval of 12.635 and 14.789, while the minimum surface roughness is in the range of 0.104 m and 0.115m.
TABLEVIII
OBJECTIVE FUNCTION’S COEFFICIENTS OF (23A),(23B) AND (23C)
i c1*i c*2i c3*i
1 0 4.630834 0
2 0 -4.999582 0
3 0 5 0
4 0 -3.434254 0
5 0 -2.79613 0
6 0 -1.263625 0
7 0 -2.577803 0
8 0 0 0
9 1.000001 -4.229638 -3.524991
10 0 5 3.175007
11 0 5 -1.224994
12 -0.999999 -5 1.575008
13 0 5 0.199991
14 -0.000001 -5 -0.200007 15 -0.000001 0.686761 -0.000013
16 0 5 0
17 0 -0.463854 0
18 0 4.098524 0
19 -2.599999 5 -1.8
20 2.599999 -3.837288 1.8
21 0 -5 0
22 0 0 0
23 -0.599999 -4.816409 0
24 0.599999 5 0
IV. CONCLUSION
This paper presented the combined of SVR and GA to estimate the optimum value of cutting conditions and its corresponding surface roughness. Simply Utilization of SVR to construct nonlinear prediction of machining data and followed by model selection using CV method. Then, nonlinear optimization problems were constructed where their objective functions were the best models found by CV method and their constraints were lower and upper bounds of experiment data.
Based on the above experiment, the proposed technique provided better solution compared to RSM, hybrid of a linear regression model and GA, and Neural Network. While, it gave almost similar result with KPCR-GA which was the previous work as well. However, SVR had more less complex structure regression model compared to KPCR. The proposed method can be used as alternative in prediction of machining process. It can be conducted both KPCR-GA and SVR-GA to estimate the range of optimum values of parameters in the machining and its corresponding surface roughness.
ACKNOWLEDGMENT
The author would like to express a sincere gratitude to the anymous reviewers for their valuable comments and suggestions to improve the quality of this manuscript. In addition, the authors would also like to thank Bina Nusantara University for supporting this research project. This work is supported by the Directorate General of Strengthening for Research and Development, Ministry of Research, Technology, and Higher Education, Republic of Indonesia as a part of Applied Research entitled “Regresi Non-Linier Berganda Berbasis Kernel Principal Component Analysis dan Regresi Ridge untuk Penemuan Pengetahuan Pada Himpunan Data Besar” with contract number: 12/AKM/PNT/ 2019 and contract date: 27 March 2019.
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Antoni Wibowo has received the first degree of Applied Mathematics in 1995
and master degree of Computer Science in 2000. In 2003, He awarded a Japanese Government Scholarship (Monbukagakusho) to attend Master and PhD programs at Systems and Information Engineering in University of Tsukuba-Japan. He completed the second master degree in 2006 and PhD degree in 2009, respectively. His PhD research focused on machine learning, operations research, multivariate statistical analysis and mathematical programming, especially in developing nonlinear robust regressions using statistical learning theory. He has worked from 1997 to 2010 as a researcher in the Agency for the Assessment and Application of Technology – Indonesia. From April 2010 – September 2014, he worked as a senior lecturer in the Department of Computer Science - Faculty of Computing, and a researcher in the Operation Business Intelligence (OBI) Research Group, Universiti Teknologi Malaysia (UTM) – Malaysia. From October 2014 – October 2016, he was an Associate Professor at Department of Decision Sciences, School of Quantitative Sciences in Universiti Utara Malaysia (UUM). Dr. Eng. Wibowo is currently working at Binus Graduate Program (Master in Computer Science) in Bina Nusantara University-Indonesia as a Specialist Lecturer and continues his research activities in machine learning, optimization, operations research, multivariate data analysis, data mining, computational intelligence and artificial intelligence.
TABLEIX
OPTIMAL VALUES AND THE ASSOCIATED BEST PREDICTION
v~ ~f ~ The best prediction of Ra
uncoated a
Rˆ 1.6703 2.5 0.62 0.129 TiAIN
a
Rˆ 1.6703 2.893 1.4799 0.217 SNTR
a
Rˆ 1.6703 2.5 1.4789 0.115
TABLEX
CLASSIFICATION OF CUTTING CONDITION TOOLS
Decision variables Unit
Classification of cutting conditions scale
Lowest Lower Medium High Highest
Cutting speed m/min 124.43 130 144.22 160 167.03
Feed rate (f) Mm/tooth 0.025 0.03 0.046 0.07 0.083
Radial rake angle ˚ 6.2 7 9.5 13 14.8
TABLEXI
COMPARISONS OF THE OPTIMAL CONDITIONS AND BEST PREDICTION WITH THE OTHER TECHNIQUES
Technique
The best prediction of Ra Optimal Level Optimal Level Optimal Level
Experimental (Mohruni, 2008) 130 Lower 0.03 Lower 7 Lower 0.19
RSM (Mohruni, 2008) 160 High 0.044 Medium 7 Lowest 0.277
Neural Network (Zain et al., 2012) 144.22 Medium 0.025 Lowest 9.5 Medium 0.188 Linear Regression and GA 167.029 Highest 0.025 Lowest 14.769 Highest 0.138
KPCR and GA (Wibowo et al, 2012) 167.03 Highest 0.025 Lowest 12.635 High 0.104
