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Case Study
Proc IMechE Part B:
J Engineering Manufacture
115IMechE 2015
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DOI: 10.1177/0954405415599927
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Analysis and optimization of cuttergeometric parameters for surface
integrity in milling titanium alloy usinga modified greyTaguchi method
Junxue Ren1, Jinhua Zhou1 and Jingwen Zeng2
Abstract
Surface integrity determines the performance and quality of the end product. It often needs to change the input para-
meters, such as cutting parameter, cutting tool geometry and material, and tool coating, to obtain the best machiningsurface integrity. This article presents and demonstrates the effectiveness for the multi-objective optimization of cuttergeometric parameters for surface integrity of milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium alloy via grey relational analysiscoupled with the Taguchi method, entropy weight method, and analytic hierarchy process. The main influence factorsare chosen as radial rake angle, primary radial relief angle, and helix angle, while surface roughness and residual stressare taken as performance characteristics. Based on the Taguchi method, an L16 (4 3) orthogonal array is chosen for theexperiments. The effect of cutter geometric parameters on surface roughness and residual stress is analysed by signal-to-noise ratio. Then, the multiple objectives optimization problem is successfully converted to a single-objective optimi-zation of grey relational grade with the grey relational analysis. The weight coefficient for grey relational grade is deter-mined by entropy weight method integrated with analytic hierarchy process. The results show that the order ofimportance for controllable factor to the milling surface integrity, in sequence, is radial rake angle, primary radial relief
angle, and helix angle. The validation experiment verifies that the proposed optimization method has the ability to findout the optimal geometric parameters in terms of milling surface integrity.
Keywords
Surface integrity, Ti-5Al-5Mo-5V-1Cr-1Fe, cutter geometric parameter, multi-objective optimization, greyTaguchi,weight coefficient
Date received: 1 August 2014; accepted: 13 July 2015
Introduction
The use of titanium and its alloys has increased recentlydue to their superior properties and improvements in
machinability. Manufacturing of titanium alloys that
are critical structural components of the aerospace
industry is also a point of emphasis. Their poor machin-
ability often results in unfavourable accuracy of the
machined product dimensions or end product quality
issues such as surface integrity and lower lifetime prone-
ness. The final manufacturing process is decisive about
the product surface quality, so it should be controlled
and optimized. This study investigates the surface integ-
rity of finish milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium
alloy and explores possible way to adjust cutter geo-
metric parameters to achieve better surface integrity.Many researches focused on the machined surface
integrity in recent years. The research of Sun showed
that the machined surface exhibits an anisotropic
nature for end milling Ti-6Al-4V, and the b phase
experiences more deformation and volume shrinkage inthe near surface. The compressive residual stress in cut-
ting direction is about 30% larger than that in feed
direction and both increased with cutting speed.1 Hioki
et al.2 evaluated the influence of the cutting parameters
1The Key Laboratory of Contemporary Design and Integrated
Manufacturing Technology, Ministry of Education, School of Mechanical
Engineering, Northwestern Polytechnical University, Xian, China2Xian Microelectronics Technology Institute, Xian, China
Corresponding author:
Junxue Ren, The Key Laboratory of Contemporary Design and Integrated
Manufacturing Technology, Ministry of Education, School of MechanicalEngineering, Northwestern Polytechnical University, P.O. Box 552, Xian
710072, China.
Email: [email protected]
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of high-speed milling on the surface integrity properties
of hardened AISI H13 steel. They found that the
engagement, feed per tooth, and cutting speed show
strong influence on the machined surface integrity.
Pawade et al.3 studied the effect of machining para-
meters and cutting edge geometry on surface integrity
of high-speed turned Inconel 718. They observed thatthe highest cutting speed, lowest feed rate, and moder-
ate depth of cut coupled with the use of honed cutting
edge can induce compressive surface residual stress.
Ezilarasan et al.4 found that a combination of 190m/
min cutting speed and 0.102 mm/rev feed rate is the crit-
ical condition for turning Nimonic C-263 alloy based
on surface roughness using whisker-reinforced ceramic
insert. Many researchers reported that cutting speed,
feed, and depth of cut are effective to some degree
increasing surface roughness, and increase in depth of
cut and feed has some effect on making the residual
stress more tensile at the surface and more compressivein the peak compressive depth.5 Additionally, appropri-
ate lubrication and inclination angles produce substan-
tial benefit in terms of milling surface integrity.6,7
These literature surveys focus on the surface integ-
rity and provide practical approach to obtain the opti-
mal process parameters. But it also has heighted the
dearth of information available to understand the effect
of cutter geometric parameters on surface integrity. In
addition, it is challenging to obtain excellent surface
integrity without losing production efficiency for the
cutting parameter optimization. A large material
removal rate improves production efficiency, but maybe
generates unfavourable surface integrity properties.8
Hence, it is the advantage to obtain a better surface
integrity by optimizing the cutter geometric parameters.
Many researches indicated that cutter geometry has
significant influence on the machining process and
quality characteristics. Different combinations of cutter
geometric parameters might produce large variations in
the final product quality. For instance, a variable pitch
or helix milling tool can be used to reduce the cutting
force and improve the machined surface quality by sup-
pressing the machining chatter.912 Zain et al.13 applied
the genetic algorithm and regression model to find the
optimal solution of the cutting conditions (radial rakeangle, cutting speed, and feed rate) that yielded the
minimum value surface roughness. Wang et al.14 built
an analysis model of parameters affecting performance
in high-speed milling of AISI H13 tool steel considering
cutter geometric parameters and cutting parameters.
Their experimental results indicated that the contribu-
tions of tool grinding precision, geometric angle, and
cutting conditions to the performance characteristics
are 11.8%, 9.8%, and 73.1%, respectively.
Arunachalam et al.15 studied the effect of insert shape,
cutting edge preparation, type and nose radius on both
residual stresses, and surface finish. They suggestedthat coated carbide cutting tool inserts of round shape,
chamfered cutting edge preparation, negative type and
small nose radius (0.8 mm), and coolant would generate
primarily compressive residual stress.
From the above analyses, it can be seen that cutter
geometric parameters influence the machining process
and performance significantly. Therefore, this study
attempts to determine the influence weight of these fac-
tors on multiple surface integrity properties.Considering the structure complexity and variety of
mill, the current techniques challenge is to design the
various cutter geometric parameters that yield opti-
mum surface integrity, which is a multi-objective opti-
mization problem.
The grey relational analysis (GRA) with the Taguchi
method (greyTaguchi) is an effective approach to
solve the multi-objective optimization problem. This
method has been widely applied in recent years for
optimal process parameter design of multiple perfor-
mance characteristics.1623 In the traditional machining,
Kopac and Krajnik24 applied the greyTaguchi method
to the robust design of flank milling parameters dealing
with the optimization of the cutting loads, milled sur-
face roughness, and the material removal rate. They
obtained optimal parameter combination of coolant
employment, number of end mill flutes, cutting speed,
feed, axial depth of cut, and radial depth of cut. Tsao25
adopted the greyTaguchi method to optimize the
milling parameters on A6061P-T651 aluminium alloy
with multiple performance characteristics. Haq et al.26
optimized drilling parameters with the considerations
of multiple responses such as surface roughness, cutting
force, and torque for drilling Al/SiC metal matrix com-
posite with the GRA in the Taguchi method. Ko klu27focused on the optimization of the continuous and
interrupted cylindrical grinding of AISI 4140 steel con-
sidering the effect of workpiece speed, depth of cut,
and the number of slots on the surface roughness and
roundness error using the grey-based Taguchi method.
To sum up, even though the effect of process para-
meters on the machined surface integrity has been accu-
mulated a lot, results are mainly dependent on
complicated factors. Furthermore, the factors that
these studies focused on are very dispersed and the sur-
face integrity is not integrately evaluated. The multi-
objective optimization of radial rake angle, primaryradial relief angle, and helix angle for milling surface
integrity is not yet available. More empirical studies
should be carried out to test the impact and optimiza-
tion processes of different combination of cutter geo-
metric parameters. Therefore, this research focuses on
the mill geometry optimization and machined surface
integrity for milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium
alloy and introduces the greyTaguchi method to
search the optimal cutter geometric parameters. The
design factors are selected as radial rake angle, primary
radial relief angle, and helix angle, while the surface
integrity are evaluated by surface roughness and sur-
face residual stress. Additionally, the influences of cut-ter geometric parameters on surface integrity are
analysed with the Taguchi method. Then, the
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correlations between the factors and surface integrity
are studied using the GRA method. Finally, a valida-
tion experiment verifies the effectiveness of this
approach. The multi-objective optimization flow dia-
gram of cutter geometric parameters using a modified
greyTaguchi method is illustrated in Figure 1.
Experimental procedureMachining setup
A set of milling experiments is conducted in a three-
coordinate vertical computer numerical control (CNC)
machining centreJOHNFORD VMC-850 with maxi-
mum spindle speed of 8000 rpm, maximum feed rate of
12 m/min, and spindle power of 10 hp. The workpiece
material used in all experiments is Ti-5Al-5Mo-5V-1Cr-
1Fe titanium alloy. The chemical composition is Al:
4.4-5.9wt%, Mo: 4.0-5.5wt%, V: 4.0-5.5wt%, Cr: 0.5-
1.5wt%, Fe: 0.5-1.5wt%, C: 0.1wt%, Si: 0.15wt%, Zr:
0.3wt%, N: 0.05wt%, H: 0.015wt%, O: 0.2wt%, and
the rest of Ti. The shapes of workpieces are oblong
blocks with the size of 71mm 3 55mm 3 49 mm.
The cutters are four-flute toroidal end mills with the
carbide body K40, diameter of 12 mm, circular arc
radius of 2 mm, and uncoated edge. To reduce the
influence of tool wear, a fresh cutter is used in each
experiment. The milling parameters in each experi-
ment are fixed at the level with spindle speed
s = 500 r/min, axial milling depth ap = 5 mm, radial
milling depth ae = 1 mm, and feed rate fz = 0.035
mm/z. The overhang length of toroidal end mill is
fixed as 44 mm. All cutting experiments are per-
formed in down milling using emulsified liquid.Figure 2 shows the milling process.
Experimental design
This study discusses the relationship between cutter
geometric parameters and the surface integrity of
milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium alloy in order
to obtain the optimal geometric parameter combina-
tion. First, the objective performance characteristics
are focused on two aspects: surface roughness and sur-
face residual stress. The performance characteristics for
surface roughness contain the surface roughness of
machined bottom surface and side surface, denoted as
SRb and SRs respectively. The residual stresses are the
surface residual stress of machined bottom surface and
side surface, denoted as RSband RSs, respectively. The
machined bottom surface is formed by the end edges,
while the machined side surface is formed by the side
edges. Next, the control process parameters havinginfluence on the surface integrity properties are radial
Figure 1. Multi-objective optimization flow diagram of cutter geometric parameters.
Figure 2. Milling process.
Ren et al. 3
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rake angle (A), primary radial relief angle (B), and helix
angle (C). Table 1 lists the cutter geometric parameters
and their levels. Figure 3 diagrammatizes these three
kind of angles. Radial rake angle indicates the angle of
the flute face with respect to a line drawn from the cut-
ting edge at the outer diameter to the centre of the tool.
Most toroidal end mills are ground with positive rake
angles. In application, larger positive rake angles are
used on softer materials, and smaller positive rake
angles are used on harder materials. Primary radial
relief angle is ground for the length of cut to provideclearance behind the cutting edge. In general, larger
relief angle is favoured for softer materials, and smaller
relief angle are used for harder materials. Helix angle is
defined as the inclination of the cutting edges with
respect to the axis of the tool. Then, these experiments
are conducted with a three controllable four-level fac-
tors and four response variables. Therefore, the
Taguchi orthogonal array L16 (43) is used to reduce
the number of experiments, as shown in Table 2.
Measurement procedure
Surface integrity includes the mechanical properties,such as residual stress, hardness, and microstructural
changes, and topological parameters such as surface
Table 1. Cutter geometric parameters and their levels.
Symbol Geometric parameter Level 1 Level 2 Level 3 Level 4
A Radial rake angle () 4 8 12 16B Primary radial relief angle () 10 12 14 16C Helix angle () 30 40 50 60
Table 2. Taguchi L16 (43) orthogonal array and experimental results.
Experiment no. A B C SRb(mm) SRs(mm) RSb(Mpa) RSs(Mpa)
1 1 1 1 0.199 0.169 2335.4 2185.92 1 2 2 0.115 0.160 2201.4 2117.33 1 3 3 0.148 0.177 2276.6 2130.24 1 4 4 0.140 0.209 2217.7 2121.25 2 1 2 0.232 0.175 2267.7 297.26 2 2 1 0.246 0.147 2170.9 2133.37 2 3 4 0.225 0.196 2162.9 2137.58 2 4 3 0.140 0.123 2241.9 2140.19 3 1 3 0.262 0.201 2179.4 2142.310 3 2 4 0.280 0.199 2222.2 2167.411 3 3 1 0.226 0.132 2208.6 2149.812 3 4 2 0.362 0.117 2183 2130.813 4 1 4 0.119 0.199 2209.1 2122.914 4 2 3 0.159 0.167 2158.9 211615 4 3 2 0.248 0.160 2212.3 2120.316 4 4 4 0.347 0.173 2171.7 2147.5
Note: Values in bold face are idea values .
Figure 3. Geometrical parameters of a toroidal end mill.
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roughness. In this study, the surface roughness and resi-
dual stress are taken as assessment criteria for surface
integrity of milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium
alloy. The expressed surface roughness in this article is
the arithmetic mean deviation of the surface roughness
profile Ra. The surface roughness of machined surface
is measured in feed direction by surface roughness tester
MarSurf M 300 C, made by MAHR Co. Ltd, as shown
in Figure 4. An average value of five measurements of
surface roughness is used to evaluate geometric accu-
racy of machined surface. Measuring residual stress is
very difficult. X-ray diffraction (XRD) method stands
out as the reliable, nondestructive, and easily accessible
technique. In this method, the surface residual stress is
measured in PROTO LXRD MG2000, made by
PROTO Co. Ltd, with Gu-Ka radiation using XRD
method. Figure 5 illustrates the measurement process of
surface residual stress. The surface residual stress in
feed direction is used to evaluate the physical property
of machined surface. An average of two measurementsof surface residual stress is taken as the results.
Experimental results and discussion
Analysis of signal-to-noise ratio for single
performance characteristic
Table 2 shows the measurement results of surface
roughness and residual stress for the 16 experiments.
Negative sign of residual stress only represents that theresidual stress is compressive and is not taken in the cal-
culation. In order to study the effect of cutter geometry
on the surface integrity, the Taguchi method is used to
seek the optimal level combination of cutter geometric
parameters for single surface integrity property.
The Taguchi method is a simple and effective solu-
tion for parameter design and experiment planning.28 In
this method, Taguchi recommended analysing the per-
formance of process response using signal-to-noise (S/
N) ratio, in which the largest value of S/N is required.
There are three types of S/N ratiothe larger-the-better
model (LBM), the smaller-the-better model (SBM), andthe nominal-the-better model (NBM).29
1. LBM
Maximum response characteristic means that the
target extreme value is infinity. The S/N ratio is as
below
S=N = 10 log 1
N
XNi= 1
1
y2i
! 1
where yiis the response value of the ith test and N is
the number of measurements in each test.
2. SBM
Minimum response characteristic means that the
target extreme value will be 0. The S/N ratio with
a smaller-the-better characteristic is defined as fol-
lows
S=N = 10 log 1
N
XNi= 1
(yi)2
" # 2
3. NBM
Targeted response characteristic means that theresponse result is the target value. The S/N ratio
can be expressed as below
S=N=10log u2
s2
3
where
u = 1
NXN
i= 1
yi
and
Figure 4. Surface roughness measurement.
Figure 5. Surface residual stress measurement.
Ren et al. 5
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s2 = 1
N1
XNi= 1
(yiu)2
Obviously, smaller values of surface roughness are
desirable. Thus, the data sequences have a smaller-the-
better characteristic and the SBM, and equation (2) is
used to calculate the S/N ratio. Higher tensile residual
stress tends to present potential risk in terms of crack
initiation and propagation, and fatigue failure of end
products, but the compressive residual stress has the
opposite effect. Now that all these measurement results
present compressive surface residual stresses, and the
LBM of S/N ratio can be used to calculate the S/N ratio
for surface residual stress.
Table 3 shows the results of S/N ratio. A higher S/N
ratio value represents that the response value is closer
to the expected performance characteristic. According
to this criterion, it is obviously observed that experi-
ment no. 2 has the maximum S/N ratio for bottom sur-
face roughness. It means the optimum combination of
cutter geometric parameters is A1B2C2 among the
experiment arrays. Side surface roughness has a higher
S/N ratio in experiment no. 12 than that in the others.
Therefore, the design factors A3B4C2should be selectedif only considering the side surface roughness for
milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium alloy. As to
the surface residual stress, the largest value can beobtained from experiment no. 1 at levels A1B1C1among the 16 experiments.
The response table for the Taguchi method is used
to calculate the mean S/N ratios for each factor level.
First, group the S/N ratios by factor level for each col-
umn in the orthogonal array. Next, take their average.
For example, the S/N ratio for A at level 1 can be cal-
culated as follows
MA1=1
4(14:023+18:786+16:595+17:077)=16:620
4
The mean S/N ratios for each cutter geometric para-
meter level are calculated using the same process
method.
Based on the data presented in Table 4, the optimal
combination of parameters is A1B2C3 for the bottom
surface roughness, namely, radial rake angle is 4, pri-
mary radial relief angle is 12, and helix angle is 50.
Figure 6 shows the fluctuation of mean S/N ratio of
bottom surface roughness with the change in cutter
geometric parameters. The bottom surface roughness
increases with the radial rake angle and primary radial
relief angle on the overall trend. The reason could bethat larger rake angle and relief angle weaken the cutter
which leads to stronger cutter wear and increases
Table 3. The S/N ratio of the experimental results.
Experiment no. S/N (dB)
SRb SRs RSb RSs
1 14.023 15.442 50.511 45.3862 18.786 15.918 46.081 41.386
3 16.595 15.041 48.837 42.2924 17.077 13.597 46.757 41.6705 12.690 15.139 48.553 39.7536 12.181 16.654 44.655 42.4977 12.956 14.155 44.238 42.7668 17.077 18.202 47.673 42.9299 11.634 13.936 45.076 43.06410 11.057 14.023 46.935 44.47511 12.918 17.589 46.386 43.51012 8.826 18.636 45.249 42.33213 18.489 14.023 46.407 41.79114 15.972 15.546 44.023 41.28915 12.111 15.918 46.539 41.60516 9.193 15.239 44.695 43.376
Note: Values in bold face are idea values .
Table 4. Mean S/N ratio for bottom surface roughness.
Factors Level (S/N) Maximumminimum
1 2 3 4
A 16.620 13.726 11.109 13.941 5.511B 14.209 14.499 13.645 13.043 1.456C 12.079 13.103 15.320 14.895 3.241
Note: Values in bold face are idea values .
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vibration in axial direction. Simultaneously, a radial
rake angle also provides a better bottom surface finish
since it aids the chip to flow out from the workpiece.30
Therefore, the bottom surface roughness has obvious
decrease when radial rake angle reaches 16. From thefigure, it can be observed that the bottom surface
roughness decreases as the helix angle increases.
From Table 5, the optimum cutter geometric para-
meters for side surface roughness are as follows: radial
rake angle of 12, primary radial relief angle of 16, and
helix angle of 40. The mean S/N ratio plot of side sur-
face roughness with respect to radial rake angle, pri-mary radial relief angle, and helix angle is shown in
Figure 7. According to parallel shear zone theory,
Figure 6. S/N response graph for bottom surface roughness.
Figure 7. S/N response graph for side surface roughness.
Table 5. Mean S/N ratio for side surface roughness.
Factors Level (S/N) Maximumminimum
1 2 3 4
A 15.000 16.038 16.046 15.182 1.046B 14.635 15.535 15.676 16.419 1.784C 16.231 16.403 15.681 13.950 2.453
Note: Values in bold face are idea values .
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larger positive radial rake angle provides higher shear
angle.31 It also produces sharper cutting edges on the
periphery, which leads to diminishing peripheral milling
force or lateral cutting force. But the excessive rake
angle weakens the cutter and possibly increases accel-
eration amplitude in feed direction.32 Therefore, the
side surface roughness first decreases approaching a
value at a radial rake angle of 12 and then increases.
The side surface roughness decreases with primary
radial relief angle due to the fact that larger relief angle
weakens the friction effect between radial relief surface
and side surface of the workpiece by shortening the
contact length. It can be observed from Figure 7 that as
the helix angle increases, the side surface roughness alsoincreases. A possible reason is that higher helix angle
leads to more roughness heterogeneity bands due to the
grinding errors and high eccentricity of the cutters used
in experiment.33
As to the residual stress, Tables 6 and 7 illustrate the
results of mean S/N ratio for the machined bottom and
side surface. According to the Taguchi method, the
maximum compressive residual stress can be obtained
for bottom surface and side surface at the parameter
levels A1B1C2and A3B4C1, respectively. The mean S/N
ratios at each level are plotted as a response graph
shown in Figures 8 and 9. It can be seen that the resi-
dual compressive stress of the bottom surface monoto-
nically decreases with the radial rake angle. But the
other residual stress curves show no obvious regularity.
Generally, cutting residual stress can be affected bymechanical loading and thermal effects.34 In cutting
process, mechanical load caused by cutting force
Table 6. Mean S/N ratio for residual stress of bottom surface.
Factors Level (S/N) Maximumminimum
1 2 3 4
A 48.047 46.280 45.912 45.416 2.631B 47.637 45.424 46.500 46.094 2.213C 46.562 46.606 46.402 46.084 0.522
Note: Values in bold face are idea values .
Table 7. Mean S/N ratio for residual stress of side surface.
Factors Level (S/N) Maximumminimum
1 2 3 4
A 42.684 41.986 43.345 42.015 1.359B 42.499 42.412 42.543 42.577 0.165C 43.692 41.269 42.394 42.676 2.423
Note: Values in bold face are idea values .
Figure 8. S/N response graph for residual stress of bottom surface.
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induces residual compressive stress, while thermal load
caused by cutting temperature induces residual tensile
stress. Larger cutting force leads to higher cutting tem-
perature. Therefore, residual compressive stress may
increase or decrease with the change in cutting force
caused by varying these cutter geometric angles.
Multi-objective optimization of cutter geometric
parametersAnalysis of S/N ratios is available for single-objective
optimization problem, but ineffective for multi-
response characteristics. It often exists in multi-
objective optimization problem that the higher S/N
ratio for one performance characteristic may corre-
spond to a lower S/N ratio for another. So, it is essen-
tial to evaluate overall S/N ratios in multi-objective
optimization problem. In this study, the multiple per-
formance characteristics are evaluated using the GRA,
which converts a multiple response process optimiza-
tion into a single-objective optimization of the grey
relational grade (GRG).In the GRA, the performance characteristics are first
normalized, ranging from 0 to 1. This experiment data
process is called grey relational generation. The second
step is to calculate the grey relational coefficient (GRC)
based on the normalized experimental data, which rep-
resents the correlation between the desired data
sequence and the actual experimental data sequence.
Finally, the GRG sequence can be obtained from the
weighted average of the GRC. The surface integrity of
milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium alloy is eval-
uated by the GRG.
Grey relational generation. In the GRA, raw data prepro-
cessing is the first step, which is known as grey
relational generation. If the purpose is the larger-the-
better, then the normalized results can be expressed as
xi(k) =
x(0)i (k)minfx
(0)i (k)g
maxfx(0)i (k)g minfx(0)i (k)g
,
i= 1, . . . , m, k = 1, . . . , n
5
where xi(k) is the normalized value of the kth perfor-
mance characteristic in the ith experiment, while x(0)i (k)
is the original result of the kth performance characteris-
tic in the ith experiment;m is the total number of tests;
and n is equal to the number of performance
characteristics.
If the target value of the original sequence is the
smaller-the-better performance characteristic, then the
original sequence is normalized as follows
xi(k) =
maxfx(0)i (k)g x(0)i (k)
maxfx(0)i (k)g minfx(0)i (k)g
,
i= 1, . . . , m, k = 1, . . . , n
6
As mentioned above, a larger S/N ratio is desirable
and the larger-the-better is adopted. Consequently,
equation (5), a linear normalization, is used to prepro-
cess the origin response characteristic sequences. The
values of the surface roughness and surface residual
stress are set to be the origin sequence x(0)i (k), where k
is less than or equal to 4 corresponding to the number
of performance characteristics and i is not more than
16 corresponding to the number of experiments. Then,
the S/N ratios obtained by the Taguchi method are nor-
malized in the range of 01. The origin matrix O is the
composition of origin sequence x(0)i (k). After grey rela-
tional generation, the matrix S shows the normalizedresults for surface roughness and surface residual stress.
Basically, the larger normalized results correspond to
Figure 9. S/N response graph for residual stress of side surface.
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the better performance and the best normalized results
should be equal to 1
O1634=
14:023 15:442 50:511 45:386
18:786 15:918 46:081 41:386
16:595 15:041 48:837 42:292
17:077 13:597 46:757 41:670
12:690 15:139 48:553 39:753
12:181 16:654 44:655 42:497
12:956 14:155 44:238 42:766
17:077 18:202 47:673 42:929
11:634 13:936 45:076 43:064
11:057 14:023 46:935 44:475
12:918 17:589 46:386 43:510
8:826 18:636 45:249 42:332
18:489 14:023 46:407 41:791
15:972 15:546 44:023 41:289
12:111 15:918 46:539 41:605
9:193 15:239 44:695 43:376
2
6666666666666666666666666666666664
3
7777777777777777777777777777777775
S1634=
0:522 0:366 1:000 1:000
1:000 0:461 0:317 0:290
0:780 0:287 0:742 0:451
0:828 0:000 0:421 0:340
0:388 0:306 0:698 0:000
0:337 0:607 0:097 0:487
0:415 0:111 0:033 0:535
0:828 0:914 0:563 0:564
0:282 0:067 0:162 0:588
0:224 0:085 0:449 0:838
0:411 0:792 0:364 0:667
0:000 1:000 0:189 0:458
0:970 0:085 0:367 0:362
0:717 0:387 0:000 0:273
0:330 0:461 0:388 0:329
0:037 0:326 0:104 0:643
26666666666666666666666666666666664
37777777777777777777777777777777775
GRC. After obtaining the normalized sequence, the
next step is to calculate the GRC. In the GRA, a higher
value of the GRC, ranging from 0 to 1, corresponds to
intense relational degree between the desired perfor-
mance characteristics and the actual performance char-
acteristics. The GRC is defined as follows
gi(k) =g(x0(k), x
i(k)) =
min8i
min8k
D0i(k) + zmax8i
max8k
D0i(k)
D0i(k) + zmax8i
max8k
D0i(k) ,
i= 1m, k = 1n
7
where
D0i(k) = jxi(k) x
0(k)j 8
0\g(x0(k), xi(k))\ 1
where x0(k) is the reference sequence, xi(k) is the com-
parability sequence, D0i(k) is the deviation sequence of
xi(k) and x
0(k), and z is the distinguishing coefficient
between 0 and 1.
The reference sequence indicates the expected
sequence. According to the normalized results, the ref-
erence sequence should be taken the maximum as
follows
x0= 1,1,1,1 9
The comparability sequencexi(k) has been obtained
from the previous step. Then, according to equation
(8), the maximum and minimum of deviation sequences
are calculated as follows
max8i
max8k
D0i(k) = j01j = 1 , 14i416,14k44 10
min8i
min8k
D0i(k) = j11j = 0 , 14i416,14k44 11
The value ofz is smaller and the identification abil-
ity is larger. In this study, it is set as 0.5.17 With equa-
tion (7), it is easy to obtain the GRC matrix R from the
normalized matrix S
R163 4=
0:511 0:441 1:000 1:0001:000 0:481 0:423 0:4130:694 0:412 0:660 0:477
0:744 0:333 0:463 0:4310:450 0:419 0:623 0:3330:430 0:560 0:356 0:4940:461 0:360 0:341 0:5180:744 0:853 0:534 0:5340:411 0:349 0:374 0:5480:392 0:353 0:476 0:7550:459 0:706 0:440 0:6000:333 1:000 0:381 0:4800:943 0:353 0:441 0:4390:639 0:449 0:333 0:4070:427 0:481 0:450 0:4270:342 0:426 0:358 0:583
2
666666666666666666666666664
3
777777777777777777777777775
Weight coefficient. In order to obtain the GRG sequence,
the weight of multi-performance characteristics needs
to be determined. It is also the weight of the column
vector of matrix R. The previous researchers prefer to
use the same weight, which ignores the difference of
multi-performance characteristics. The weight of vari-
ous performance is different from each other for the
current engineering problem due to variety of surface
integrity properties. In this section, a weight determina-
tion method is proposed to calculate the weight of
milling surface integrity by entropy weight method(EWM) combined with analytic hierarchy process
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(AHP). The amalgamative method reflects the objective
competition level of multi-performance characteristics
by EWM and the specialistic evaluation for these
indexes by AHP.
The process of determining weight of multi-
performance characteristics is as follows:
Step 1. Calculate the objective weight by EWM.
First, calculate the weight of ith experiment for each
performance characteristic by equation (12). In other
words, calculate the weight of ith row by column. The
matrix P can be obtained from the matrix R
Pij= RijPm
i= 1
Rij
(i=1,2, . . . , m;j=1,2, . . . , n) 12
where the Pijis the weight ofith experiment forjth per-
formance characteristic, m is the total number of test,
andn is the number of performance characteristics
P1634=
0:057 0:055 0:131 0:1180:111 0:060 0:055 0:0490:077 0:052 0:086 0:0570:083 0:042 0:060 0:0510:050 0:053 0:081 0:0390:048 0:070 0:047 0:0590:051 0:045 0:045 0:0610:083 0:107 0:070 0:0630:046 0:044 0:049 0:0650:044 0:044 0:062 0:0890:051 0:089 0:057 0:071
0:037 0:125 0:050 0:0570:105 0:044 0:058 0:0520:071 0:056 0:044 0:0480:048 0:060 0:059 0:0510:038 0:053 0:047 0:069
2666666666666666666666666664
3777777777777777777777777775
Second, calculate entropy value of the jth perfor-
mance characteristic. The entropy row vector e can be
obtained from the matrix P by equation (13)
ej= 1
ln m
Xmi= 1
PijlnPij (j=1,2, . . . , n) 13
where e
j is the entropy of the jth performancecharacteristic
e = 0:9782 0:9778 0:9821 0:9861
Finally, determine the entropy weight vector a for
the multi-performance characteristics by equation (14)
aj= 1 ej
nPnj= 1
ej
(j=1,2, . . . , n) 14
where aj is the entropy weight vector for the multi-
performance characteristics
a= 0:288 0:292 0:235 0:183
Step 2. Calculate the subjective weight by AHP.
The subjective weight depends on the specialistic eva-
luation on the specific engineering problems. Table 8
shows the relative importance degree by pairwise com-
parison of the multi-performance characteristics.
Therefore, the judgment matrix J is as follows
J434=
1 1 1
3
1
3
1 1 1
3
1
33 3 1 1
3 3 1 1
2666664
3777775
The weight vector b for multi-performance charac-
teristics can be easily calculated by asymptotic normali-
zation coefficient method
b= 0:125 0:125 0:375 0:375
Step 3. Determine the amalgamative weight vector.
The amalgamative weight of each performance charac-
teristic can expressed as follows
vj=ajbj
Pn1 a
jbj
15
where vj is the weight of the jth performance
characteristic.
The amalgamative weight vector v is calculated by
equation (15) as follows
v= 0:157 0:159 0:384 0:299
GRG. The GRG expresses the correlation between the
comparability sequence and the reference sequence. A
higher GRG presents that the corresponding multi-
performance characteristics are closer to the ideal value.The GRG ranges from 0 to 1 and equals to 1 when the
two sequences agree with each other completely. The
GRG can be expressed as follows
gi(x0, x
i) =
Xnk = 1
vkgi(k), i= 1m, k = 1n
or
g=R3vT 16
where v is the weight factor of the kth performance
characteristic.
The GRG is used to evaluate the overall surfaceintegrity. The parametric combination with highest
GRG implies that the corresponding experimental run
Table 8. Comparison of importance degree.
SRb SRs RSb RSs
SRb 1 1 1/3 1/3SRs 1 1 1/3 1/3RSb 3 3 1 1
RSs 3 3 1 1
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is closest to the optimal value of the desired multiple
performance characteristics.23 The GRG vector g can
be determined by equation (16) as follows. It is clearly
observed that g1 has the largest value among the com-
ponents of vector g. It also suggests that the experi-
ment no. 1 has the optimal surface integrity among the
16 experiments
g=
0:8330:5190:571
0:4760:4760:4410:4150:6170:4270:5260:5330:5010:5050:4210:444
0:433
2
666666666666666666666666664
3
777777777777777777777777775
The average GRGs for each factor level have been
calculated using the process approach similar to that of
the mean S/N ratios, as shown in Table 9. The average
of GRG at each level is plotted as a response shown in
Figure 10. Since the GRG reflects the level of correla-
tion between the comparability and the reference
sequences, a larger GRG represents the comparability
sequence exhibiting a stronger correlation with the ref-
erence sequence [1, 1, 1, 1]. Based on this basic criterion
of grey system theory, one can select a combination of
the design factor levels that provide the largest average
performance characteristics. As listed in Table 9, the
combination of A1, B1, and C1 exhibits the largest
value of the GRG for the design factors A, B, and C,
respectively. Consequently, A1B1C1 with a radial rake
angle of 4, primary radial relief angle of 10, and helix
angle of 30is the optimum cutter geometric parameter
combination.
From Table 9, the difference between the maximum
and minimum values of the GRG of the cutter geo-
metric parameters is 0.149 for A, 0.084 forB, and 0.080
for C. These difference values reflect the level of effect
of cutter geometric parameters on the performance
characteristics. In other words, the comparison among
the difference values will qualitatively give the level of
significance of the control factors over the milling sur-face integrity. It can be easily observed that the maxi-
mum value among 0.149, 0.084, and 0.080 is 0.149,
Table 9. Response table of the average grey relational grade.
Factors GRG Maximumminimum Rank
1 2 3 4
A 0.600 0.487 0.497 0.451 0.149 1B 0.560 0.477 0.491 0.507 0.084 2C 0.560 0.485 0.509 0.481 0.080 3
Note: Values in bold face are idea values
GRG: grey relational grade.
Total mean value of the GRG was 0.509.
Figure 10. Response graph of average grey relational grade.
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which means radial rake angle has the most remarkable
effect on the multiple performance characteristics
among the cutter geometric parameters. That is to say
the order of optimization for cutter geometric para-
meters should be A (radial rake angle) . B (primary
radial relief angle) . C(helix angle) in this study.
Validation tests
The estimated GRG gusing the optimum cutter geo-
metric parameters can be expressed as
g=gm+Xni= 1
(gi gm) 17
wheregm is the total mean of the GRG, gis the mean
of the GRG at the optimal level, and n is the number
of control factors that significantly affects the multiple
performance characteristics.
Generally, the GRG under optimal parameters can
be calculated by equation (17) in greyTaguchi method.
As mentioned above, the GRG reaches its maximum
value at A1B1C1 which coincidentally corresponds to
experiment no. 1 in the Taguchi orthogonal array L16
(43). Therefore, one more validation test is superfluous.
The effectiveness of the modified greyTaguchi method
can be verified by comparing the response results of the
parameter combination A1B1C1 and initial parameter
combination. The initial cutter geometric parameters
are selected as A2B1C2 with a radial rake angle of 8,primary radial relief angle of 10, and helix angle of 40
according to engineering experience.
Table 10 illustrates the comparison of the experi-
mental results using the initial and optimal cutter geo-
metric parameters. Under the condition with the levels
A1B1C1of the optimum parameters, the GRG has been
improved by 0.357; the bottom and side surface rough-
ness are decreased to 0.199 (an improvement of 14.2
%) and 0.169 mm (an improvement of 3.40 %), respec-
tively; and the compressive residual stress of bottom
and side surface is improved from 267.7 and 97.2MPa
to 335.4 and 185.9 MPa, respectively. In summary, it isclearly shown that the surface integrity of milling Ti-
5Al-5Mo-5V-1Cr-1Fe titanium alloy can be
significantly improved by optimization of cutter geo-
metric parameters.
Conclusion
This study applies the GRA integrated with the
Taguchi method, EWM, and AHP to optimize the cut-
ter geometric parameters in terms of surface integrity
for milling Ti-5Al-5Mo-5V-1Cr-1Fe titanium alloy.
Conclusions are summarized as follows:
1. The validation experiment indicates that the grey
Taguchi method is an effective approach of multi-
objective optimization to the cutter geometry for
machined surface integrity. With this method, the
GRG of the multiple performance characteristics
is significantly improved by 0.357.
2. With the analysis of S/N ratio, the optimum radialrake angle, primary radial relief angle, and helix
angle for bottom surface roughness are 4, 12, and
50, respectively; the optimal controllable factors for
side surface roughness are radial rake angle of 12,
primary radial relief angle of 16, and helix angle of
40; the optimum radial rake angle, primary radial
relief angle, and helix angle for residual stress of bot-
tom surface are 4, 10, and 40, respectively; the
optimal controllable factors for residual stress of
side surface are radial rake angle of 12, primary
radial relief angle of 16, and helix angle of 30.
3. The radial rake angle is the most significant con-trol factor for the milling surface integrity among
the three cutter geometric parameters. The largest
value of GRG is obtained at the combination of
cutter geometric parameters with a radial rake
angle of 4, primary radial relief angle of 10, and
helix angle of 30 It is the recommended levels of
cutter geometric parameters in terms of surface
roughness and residual stress for milling Ti-5Al-
5Mo-5V-1Cr-1Fe titanium alloy.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interestwith respect to the research, authorship, and/or publi-
cation of this article.
Table 10. Comparison results of the initial and optimal cutter geometric parameters.
Initial cutter geometric parameters Optimal cutter geometric parameters Improvement rate (%)
Prediction Validation tests
Level A2B1C2 A1B1C1SRb(mm) 0.232 0.199 14.2SRs(mm) 0.175 0.169 3.40RSb(MPa) 2267.7 2335.4 25.3RSs(MPa) 297.2 2185.9 91.3GRG 0.476 0.702 0.833 75.0
GRG: grey relational grade.
Improvement in the GRG is 0.357.
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Funding
The author(s) disclosed receipt of the following finan-
cial support for the research, authorship, and/or publi-
cation of this article: This work was supported by the
National Science and Technology Major Project of
China (no. 2013ZX04001081) and the Doctorate
Foundation of Northwestern Polytechnical University(no. CX201514).
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Appendix 1
Notationap axial milling depth
ae radial milling depth
A radial rake angle
B primary radial relief angle
C helix angle
e entropy row vector
fz feed rate
J judgment matrix
m total number of tests
n number of performance characteristics
N number of measurements in each test
O matrix composed of origin resultsx(0)
i (k)
Pij weight of theith experiment for thejth
performance characteristic
R matrix composed of the grey relational
coefficients
RSb residual stress of bottom surface
RSs residual stress of side surface
s spindle speedS matrix composed of normalized results of
performance characteristicsxi(k)
S/N signal-to-noise ratio
SRb bottom surface roughness
SRs side surface roughness
x(0)i (k) original result of thekth performance
characteristic in theith experiment
xi(k) normalized value of thekth performance
characteristic in theith experiment
xi(k) comparability sequence
x0(k) reference sequence
yi response value of theith test
a entropy weight vector calculated by EWM
b weight vector calculated by AHP
gi grey relational grade of theith experiment
gi(k) grey relational coefficient of theith
performance characteristic in theith
experiment
g estimated grey relational grade
D0i(k) deviation sequence ofxi(k) andx
0(k)
z distinguishing coefficient
v amalgamative weight vector
Ren et al. 15