INTRODUCTION
Manufacturing cost is always very important issue either manufacturing assembly
or machining process. Optimizations are one of the methods to minimize the operations
hours and reduce the process time itself. Response Surface Method (RSM) is a
collection of statistical and mathematical methods that are useful for the modelling
and optimization of the engineering problems. In this technique, the main objective
is to optimize the responses that are influencing by various parameters (Montgomery,
1984). RSM also quantifies the relationship between the controllable parameters
and the obtained response. In modelling of the manufacturing processes using
RSM, the sufficient data is collected through designed experimentation. In general,
a second order regression model is developed because of first order models often
give lack off fit. The study uses the BoxBehnken design in the optimization
of experiments using RSM to understand the effect of important parameters. BoxBehnken
design is normally used when performing nonsequential experiments. That is,
performing the experiment only once. These designs allow efficient estimation
of the first and secondorder coefficient. Because BoxBehnken design has fewer
design points, they are less expensive to run than central composite designs
with the same number of factors. The RSM is practical, economical and relatively
easy for use and it was used by lot of researchers for modeling machining processes
(ElBaradie, 1993; Hasegawa et
al., 1976; Sundaram and Lambert, 1981). Mead
and Pike (1975) and Hill and Hunter (1966) reviewed
the earliest study on response surface methodology. Response surface methodology
is a combination of experimental and regression analysis and statistical inferences.
The concept of a response surface involves a dependent variable y called the
response variable and several independent variables x_{1},x_{2},.
. .,x_{k} (Hicks and Turner, 1999). Consequentially,
the RSM is utilized to describe and identify, with a great accuracy, the influence
of the interactions of different independent variables on the response when
they are varied simultaneously. In addition, it is one of the most widely used
methods to solve the optimization problem in the manufacturing environment as
studied by Grum and Slab (2004), Puri
and Bhattacharyya (2003) and Kansal et al. (2005).
Therefore, the quadratic model of RSM associated with the Sequential Approximation
Optimization (SAO) method was used in an optimal setting of machining parameters.
The main aim of the study is to develop the first and second order model by
using response surface methodology. From this model, the relationship between
the factors and the response can be investigated. The objective of this study
is to develop the mathematical model and predicted the torque in endmilling.
Yalcinkaya and Bayhana (2009) presents a modelling and
solution approach based on discreteevent simulation and response surface methodology
for dealing with average passenger travel time optimization problem inherent
to the metro planning process. The objective is to find the headways optimizing
passenger average travel time with a satisfactory rate of carriage fullness.
Armarego and Wright (1984) developed a model which can be used to estimate
thrust and torque for three different drill flank configurations. This model
uses the findings of their analysis of cutting mechanisms and fundamental machining
data such as shear stress and chip length. Design of experiments is a powerful
analysis tool for modeling and analyzing the influence of process variables
over some specific variable, which is an unknown function of these process variables
(Paulo, 2001). The most important stage in the design
of experiment lies in the selection of the control factors. As many as possible
should be included, so that it would be possible to identify nonsignificant
variables at the earliest opportunity (Nian et al.,
1999). In general, the thrust and torque parameters will mainly depend on
the manufacturing conditions employed, such as: feed, cutting speed, tool geometry,
machine tool and cutting tool rigidity, etc.
Horng et al. (2008) have attempt to model the
machinability evaluation through the Response Surface Methodology (RSM) in machining
Hadfield steel. The combined effects of four machining parameters, including
cutting speed, feed rate, depth of cut and tool corner radius, on the basis
of two performance characteristicsflank wear (VBmax) and surface roughness
(Ra), were investigated and the centered Central Composite Design (CCD) and
the analysis of variance (ANOVA) were employed. The quadratic model of RSM associated
with the Sequential Approximation Optimization (SAO) method was used to find
optimum values of machining parameters.
MATHEMATICAL MODELING
The proposed relationship between the responses (torque and torque) and machining
independent variables can be represented by the following:
where, τ is the torque in Nm, V, F, A_{x} and A_{r }are
the cutting speed (m sec^{1}), feed rate (mm rev^{1}), axial
depth (mm) and radial depth (mm). C, m, n, y and z are the constants.
Equation 1 can be written in the following logarithmic form
as in Eq. 2:
Equation 2 can be written as a linear form:
where, y = lnτ is the torque, x_{0} = 1 (dummy variables), x_{1}=
lnV, x_{2 }= lnF, x_{3 }= lnA_{x} , x_{4} =
lnA_{r} and ε = ln ε’, where, ε is assumed to be
normallydistributed uncorrelated random error with zero mean and constant variance,
β_{0} = lnC and β_{1}, β_{2}, β_{3}
and β_{4} are the model parameters.
The second model can be expressed as:
The values of β_{1}, β_{2}, β_{3} and
β_{4} are to be estimated using the method of least squares. The
basic formula can be expressed as in Eq. 5:
where, x^{T} is the transpose of the matrix x and (x^{T}x)^{1}
is the inverse of the matrix (x^{T}x) and y is the value from experiment.
The details solution of this matrix approach is explained by Montgomery
(1984). The parameters have been estimated by the method of leastsquare
using a Matlab computer codes.
ENGINEERING DESIGN
To develop the firstorder, a design consisting 27 experiments were conducted.
BoxBehnken design method is normally used when performing the nonsequential
experiments. These designs allow the efficient estimation of the first and secondorder
coefficients because of the BoxBehnken design has fewer design points; they
are less expensive to run than central composite designs with the same number
of factors. BoxBehnken design no axial points, thus all design points fall
within the safe operating. BoxBehnken design also ensures that all factors
are never set at their high levels simultaneously (Box and
Draper, 1986; Khuri and Cornell, 1987). Preliminary
tests were carried out to determine the suitable cutting speed, federate, axial
and radial depth of cutting as shown in Table 1.
Table 1: 
Levels of independent variables 

The AISI 618 stainless steel workpieces were provided in fully annealed condition
in sizes of 65x170 mm. The tools used in this study are carbide inserts PVD
coated with one layer of TiN. The inserts are manufactured by Kennametal with
ISO designation of KC 735 M. They are specially developed for milling applications
where stainless steel is the major machined material. The endmilling tests
were conducted on Okuma CNC machining centre MX45VA. Every one passes (one
pass is equal to 85 mm), the cutting test was stopped. The same experiment has
been repeated for 3 times to get more accurate results.
RESULTS AND DISCUSSION
Firstorder torque model: The first order torque model can be expressed
as in Eq. 6:
The transforming equations for each of the independent variables are:
The torque model can be expressed as Eq. 8:
Table 2 shows the 95% confidence interval for the experiments
and analysis of variance. For the linear model, the pvalue for lack of fit
is 0.196 and the Fstatistics is 5.1033. Therefore, the model is adequate. The
experimental and predicted torque results for the first order model are given
in Table 3.
Equation 8 shows that the torque increases with decreases
of the cutting speed while increases of the feed rate, axial and radial depth
of cut. It also indicates that the feed rate has the most significant effect
on the torque, follow by radial and axial depth of cut and cutting speed. Equation
8 is utilized to develop torque contour at selected cutting speed and feed
rate. Figure 1ac show that the torque
contours in the axialradial depth plane for different cutting speed and feed
rate.
Table 2: 
Analysis of variance (ANOVA) for firstorder torque model
with 95% confidence interval 

Table 3: 
The predicted result for first order torque model 



Fig. 1: 
Torque contours in the axialradial depth plane for different cutting
speed and feed rate; (a) V_{c} = 100 m min^{1} and f =
0.1 mm rev^{1}; (b) V_{c} = 140 m min^{1} and
f = 0.15 mm rev^{1} and (c) V_{c} = 180 m min^{1}
and f = 0.2 mm rev^{1} 
It is helpful to predict the torque at any experimental zone. It is clearly
shown that the cutting speed, feed rate, axial depth of cut, radial depth of
cut and feed rate are strongly related with the torque in endmilling. It can
be seen that the increases of torque with increases of cutting speed and feed
rate. The torque obtained the highest value about 25 N at cutting speed 180
m min^{1}.
Secondorder torque model: The secondorder model was postulated in
obtaining the relationship between the cutting force and the machine independent
variables. The second order model equation can be expressed as in Eq.
9:
Table 4 is given the 95% confidence interval for the experiments
and analysis of variance. For the secondorder model, the pvalue for lack of
fit is 0.221 and the Fstatistics is 4.5249. Therefore, the model is adequate.
The secondorder model is more adequate because of the predicted result is much
more accurate than the first model. The pvalue higher than the first order
predicted value. The predicted torque results for second order model are given
Table 5.
Thirdorder torque model: The thirdorder model as shown below was use
is obtained to investigate the 3way interaction between the variables:
Table 4: 
Analysis of variance (ANOVA) for secondorder model with 95%
confidence interval 

Table 5: 
The predicted result for second order torque model 

From this model the most important points are the main effects, 2way interaction
and 3way interaction. The third order torque model can be presented as in Eq.
11:
The thirdorder model parameters can be solved using leastsquares method.
β’s are the model parameters, x_{1} = cutting speed, x_{2}
= feedrate, x_{3} = axial depth and x_{4} = radial depth. The
third order model for torque can be rewrite in Eq. 12:
The model adequate and significant of 3way interaction can be seen form Table
6. From the ANOVA both model not significant to the 3way interaction since
the p>0.05. The thirdorder model adequate for torque since the pvalue for
lack of fit for torque is 0.818 and the Fstatistics is 0.52. It indicates that
this model is not suitable as much as secondorder torque model.
Fourthorder torque model: The fourthorder model as shown below is
obtained to investigate the 4way interaction between the variables.
From this model the most important points are the main effects, 2way, 3way
and 4way interactions. So, the fourth order model can be reduced as in Eq.
14:
Table 6: 
Analysis of Variance (ANOVA) for thirdorder torque model 

Table 7: 
Analysis of Variance (ANOVA) for fourthorder torque model 

This model parameters can be solved using least squares method. β’s
are the model parameters, x_{1} = cutting speed, x_{2} = feed
rate, x_{3} = axial depth and x_{4} = radial depth. The fourth
order torque model can be presented as in Eq. 15:
The model adequate and significant of 4way interaction for both model are
also presented in Table 7. It can be seen that from the ANOVA
analysis both model not significant to 4way interaction since the p>0.05.
The thirdorder model adequate for torque since the pvalue for lack of fit
for torque is 0.818 and the Fstatistics is 0.52. It indicates that this model
is not suitable as much as secondorder torque model.
CONCLUSION
Reliable torque model have been developed and utilized to enhance the efficiency of the milling 618 stainless steel. The torque equation shows that feed rate, cutting speed, axial depth and radial depth plays the major role to produce the torque. The higher the feed rate, axial depth and radial depth, the torque generates very high compare with low value of feed rate, axial depth and radial depth. Contours of the torque outputs were constructed in planes containing two of the independent variables. These contours were further developed to select the proper combination of cutting speed, feed, axial depth and radial depth to produce the optimum torque. The third order model and fourth order model are very important to investigate the 3way interaction and 4way interaction. The third order model and fourth order model, shows that the 3way interaction and 4way interaction not significant.
ACKNOWLEDGMENTS
The authors would like to thanks the Universiti Tenaga National for provided laboratory facilities and Universiti Malaysia Pahang for providing financial support for provided research grant (0399030011EA 0041).