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Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9208-9216 9208
www.etasr.com Le: Multi-Criteria Decision Making in the Milling Process Using the PARIS Method
Multi-Criteria Decision Making in the Milling
Process Using the PARIS Method
Hong Ky Le
Vinh Long University of Technology Education
Vinh Long, Vietnam
kylh@vlute.edu.vn
Received: 8 July 2022 | Revised: 25 July 2022 | Accepted: 25 July 2022
Abstract-The Multi-Criteria Decision-Making (MCDM) process
of milling SNCM439 steel is presented in this study. In this
experimental study, 3 cutting tool parameters, namely the
number of pieces, cutting piece material, and tip radius were
considered and 3 cutting mode parameters, i.e. cutting speed,
feed rate, and depth of cut changed in each experiment. SR and
MRR are selected as the output parameters of the milling
process. The PARIS method was used for MCDM, in which, the
weights of SR and MRR were determined by 3 methods, namely
AW, EW, and MW. Twenty-seven sets of ranking results for 27
alternatives (experiments) are presented. The GINI index was
used to evaluate the stability of ranking alternatives. The results
have determined the value of 6 input parameters to ensure the
minimum SR and the maximum MRR simultaneously.
Keywords-MCDM; PARIS; average weight; entropy weight;
Merec weight; GINI index; milling
I. INTRODUCTION
Milling the plane with a face milling cutter is a machining
method that gives the highest productivity of all cutting
machining methods, because it has many teeth simultaneously
involved in cutting and a tool with a large diameter can be
chosen [1-3]. Therefore, this method is increasingly used in
machine manufacturing. Thanks to the development of
machine tools as well as cutting tool manufacturing
technology, the accuracy of this method is increasingly
improved. This method is even used instead of grinding when it
is necessary to machine surfaces that require high precision. In
addition, the residual stress on the surface layer of the part
during milling is usually compressive residual stress, whereas
the residual stress on the surface layer during grinding is
usually tensile residual stress. This is also the advantage of
milling over the grinding method. Among many criteria to
evaluate the machining process, such as MRR, surface
hardness, cutting force, cutting heat, etc., SR and MRR are the
two most used parameters in published documents. This can be
easily understood because MRR is an important parameter to
evaluate cutting productivity, while SR is a parameter that has
a great influence on the workability as well as the life of the
product. On the other hand, determining the value of SR and
MRR is also quite simple, specifically an SR measuring device
is quite more popular than a force measuring device or a heat
measuring device, and MRR can be calculated from simple
math formulas. As with most machining processes, it is
desirable to have minimum SR and maximum MRR when
milling the plane with a face mill. However, these requirements
cannot be achieved simultaneously with each specific
machining condition. Some examples to support this statement
follow. In [4], when performing 27 tests of milling SCM440
steel, the one with the smallest SR was also the one with the
smallest MRR. When performing 9 tests for milling 060A4
steel, the one with the smallest SR is had a very small MRR
[5]. Among 27 SKD11 steel milling experiments, the one with
the smallest SR almost had the smallest MRR [6], etc. Thus, in
cases like these it is necessary to define an experiment where
SR is considered smallest and MRR is considered maximum,
i.e. an MCDM [4-6] problem. There are various mathematical
methods that support MCDM such as: SAW, WASPAS,
TOPSIS, VIKOR, MOORA, COPRAS, PIV, PSI, EDAS,
MARCOS, CODAS, WASPAS, WPAS, etc. These methods
have been widely applied to MCDM in many different fields.
The common feature of these methods is that each method
gives only one set of ranking results for the alternatives.
The PARIS method was first proposed in 2020 by Ardil [7].
PARIS is also an MCDM method, but, unlike the above
techniques, it performs threefold data normalization in 3
different ways. In addition, for each data normalization
method, the ranking of the alternatives is performed in 3 stages.
Thus, when applying this method, 9 different ratings will be
conducted for the alternatives. In the next section of this paper,
this method will be presented in detail. This method has been
applied to make multi-criteria decisions in some specific cases
such as: When deciding in choosing 1 of 6 aircraft types, each
aircraft is evaluated through 7 criteria. The PARIS method was
applied to accomplish this in [7]. In this study, two methods,
i.e. AW and EW were used to determine the weights for 7
criteria. The TOPSIS method was also used and compared with
the PARIS method. When using the PARIS method with two
different weighting methods (AW and EW) it gave 18 ranking
options. When using the TOPSIS method with two different
weighting methods two ranking options were proposed. An
interesting result was obtained that all 20 ranking options
identified the best aircraft according to the 7 given criteria.
Besides, Ardil also used the PARIS method [8] to decide which
one of the 3 types of military aircraft to choose. In this case,
each aircraft is evaluated on 7 criteria. AW and EW were again
Corresponding author: Hong Ky Le
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used to determine the weights of the criteria and TOPSIS was
also used and compared with the PARIS method. The
calculated results showed that all 20 ranking options (including
18 options of PARIS and 2 options of TOPSIS) identified the
same best aircraft. In [9], the PARIS method was also used to
select 1 out of 5 software candidates, considering 8 criteria. In
total, there were 31 sub-criteria out of which 8 were applied. In
addition, 3 different sets of weights that are random numbers
(RN) have been assigned to the criteria. The calculated results
showed 27 rating options. In particular, the different best
alternatives depended on the random weights assigned to the
criteria.
The above results show that the use of PARIS method is
quite effective. However, the PARIS method has only been
used in the mentioned studies. Up to now, and to the best of our
knowledge, there have been no studies using this method in
MCDM for the milling process and mechanical processing in
general. AW, EW, and RN methods were used to determine the
weights for the criteria in the above studies when applying the
PARIS method. This is understandable because AW is the
simplest method to determine the weights for criteria where the
weights of the criteria are equal, and EW is a method with high
accuracy that has been widely used. Its use is recommended in
MCDM [10]. The RN method was used because out of the 31
criteria for software evaluation, there are both qualitative and
quantitative criteria. However, for quantitative criteria it is not
necessary to use this method. The MW weighting method was
first proposed in 2021 [11]. It has been used in several studies
to determine weights in MCDM [12, 13]. However, this
method has not been used to determine weights for the criteria
of any milling process. Therefore, the use of the MW method
for determining the weights for the criteria of the milling
process along with the two methods already used (AW and
EW) ensures the novelty of the current study. The use of 3
methods of determining weights in a study is the basis for
assessing the stability in determining the best solution.
SNCM439 steel (according to JIS standard - Japan) is a
high-alloy steel and products like gears, wood cutters, dies, etc.
are usually made from it. This steel is equivalent to some steels
according to other standards such as, AISI - 4340, EN -
36CrNiMo4, BS - EN24, JIS - SNCM439, DIN - 150Cr14,
GOST - 9CrSi (or 9XC or 9HS or 9KHS). There have been a
few studies regarding the milling of this steel (or equivalent
steels). In [14], the authors investigated the influence of the
parameters of the MQL-type cooling lubrication on SR when
hard milling of 9CrSi steel. In [15], the Response Surface
Method (RSM) and an Artificial Neural Network (ANN) model
were combined to predict the value of tool wear and cutting
force for the dry milling of EN24 steel. In [16], the shear force
and friction force were investigated when milling EN24 steel
with different cooling lubrication conditions. In [17], the
optimal values of cutting speed, feed rate, and depth of cut
were determined to ensure minimum SR when milling En24
steel. The influence of cutting speed, feed rate, and depth of cut
on SR on cutting temperature when hard milling AISI-4340
steel were investigated in [18]. In [19], it was determined that
the value of cutting speed, feed rate, and depth of cut improve
the surface hardness when milling AISI-4340 steel. Research
on milling steel SNCM439 (or equivalent steels) has been
conducted in a number of studies as described above. However,
to date, there has not been any research done in MCDM when
machining this steel. This is the reason that this steel was
chosen as the research object in this paper.
Like the SR parameter, the MRR is a very common
parameter used to evaluate the milling process. This parameter
reflects the processing capacity. Considering both SR and
MRR in a study makes both economic and technical
implications. That is why this study has selected both SR and
MRR as the criteria to evaluate the milling process. Besides,
the three cutting parameters (cutting speed, feed rate, and depth
of cut) that have been investigated in many studies, the
parameters of the cutting tool (number of inserts, cutting tool
material, and nose radius) are also parameters that have a great
influence on the surface roughness during milling [20-22].
However, until now, no study has been found that considers all
6 of these parameters during the milling process. Therefore, the
consideration of all these 6 parameters is also a novelty of this
work.
This study will conduct SNCM439 steel milling
experiments. In each experiment, 6 parameters will be changed
including cutting speed, feed rate, depth of cut, number of
inserts, insert material, and nose radius. SR and MRR were
selected as the two output criteria. PARIS method is used for
MCDM with 3 weighting methods including AW, EW, and
MW.
II. THE PARIS METHOD
The PARIS method is performed according to the following
steps [7].
Step 1: Building of the decision matrix.
� = �
��� ��� ��� �� ��� ��� ��� �� �
� �
� �
� �
��� ��� ��� �� � (1)
In which: m is the number of options, i = 1, 2, …, m and n
is the number of criteria, j= 1, 2, …, n.
If xij is negative then do the calculation x’ij = xij = min(xij),
then x’ij is used to calculate the next steps.
Step 2: Normalizing the decision matrix.
Normalizing way 1 (Vector normalization):
� = �
��∑ �
���
�� If j is the criterion, the bigger the better (2)
� = 1 − �
��∑ �
���
�� If j is the criterion, the smaller the better (3)
Normalizing way 2 (Linear normalization):
� = �
������ If j is the criterion, the bigger the better (4)
Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9208-9216 9210
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� = ���
�
� If j is the criterion, the smaller the better (5)
Normalizing way 3 (Max - Min linear normalization):
� = �
� − ���
����� − ���
If j is the criterion, the bigger the better (6)
� = ����� − �
������ − ���
If j is the criterion, the smaller the better (7)
Step 3: Calculating of the weighted normalized value: �
� = �� ∙
� (8)
Step 4: Summarizing the weighted normalized values as: �
� = ∑ �� ∙
� ��� (9)
where i = 1, 2,… m; j = 1, 2,..., n.
Step 5: Rank the alternatives. The solution with the largest
value of �
� is the best solution.
Step 6: Identify the elements of the reference ideal solution:
��∗ = ���∗, … , ��∗� = �� !�
�
� �" ∈ $%, � &'
�
��" ∈ (%� (10)
where B represents a criterion as large as possible, C represents
a criterion as small as possible.
Step 7: Calculate the distance from the reference ideal
solution: �
∗ = ∑ ���∗ − �
� % ��� (11)
Step 8: Rank the alternatives according to the principle that
the one with the smallest value of πi is the best one.
Step 9: Calculate the distance from the alternatives to the
ideal solution:
)
= ���
� − �
�,��� %� + ��
∗ − �
∗,�
%� (12)
Step 10: Rank the alternatives according to the principle
that the one with the smallest Ri value is the best one.
III. METHODS OF DETERMINING WEIGHTS
A. The Average Weight Method
AW is determined according to the following formula [23]:
+� = � (13)
where n is the number of criteria.
B. The Entropy Weighted Method
EW is determined according to the following steps [10].
Step 1: Determine the normalized values for the indicators:
,ij = �ij�-. �ij/0123 (14)
where xij is the value of criterion j corresponding to option i
and m is the number of alternatives.
Step 2: Calculate the value of the Entropy measure for each
indicator with:
( ) ( )
j ij ij1
ij ij1 1
ln(p )
1 ln 1
m
i
m m
i i
me p
p p
=
= =
= − ⋅ −
− ⋅ −
(15)
Step 3: Calculate the weight for each indicator:
+� = �4�56∑ ��4�56%0623 (16)
C. The MEREC Weight Method
MW is determined according to the following steps [11]:
Step 1: Is similar to step 1 of the PARIS method.
Step 2: Calculate the normalized values according to:
ℎ
� = �
�16�16 If j is the criterion, the bigger the better. (17)
ℎ
� = �16����16 If j is the criterion, the smaller the better (18)
Step 3: Calculate the overall efficiency of the alternatives
according to:
8
= 9' :1 + ;� ∑ <='�ℎ
� %< � >? (19)
Step 4: Calculate the performance of the alternatives
according to:
8
�@ = 9' :1 + ;� ∑ <='�ℎ
� %< A,AB� >? (20)
Step 5: Calculate the absolute value of the deviations
according to: C� = ∑ <8
�@ − 8
<�
(21)
Step 6: Calculate the weight for the criteria according to:
+� = D6∑ DE0E (22)
IV. MILLING EXPERIMENT
A. Experimental Design
Experiments were carried out on a 3-axis machining center.
The values of the 6 input parameters are presented in Table I
[1, 24]. As the number of inserts varies in each experiment, 3
different types of inserts have been used. In addition, each type
of tool head has several insert positions of 2, 3, and 4. All 3
types of cutters have a diameter of 40mm. The basic
geometrical parameters of the selected insert types are the
same. Specifically, the main cutting angle is 90
0
, the main
cutting-edge length is 10mm, and the blade width is 6.8mm
[24]. The Taguchi method was used for the experimental
design due to its advantages [25, 26]. In this experiment, an
experimental matrix of 27 experiments was designed (Table II).
Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9208-9216 9211
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TABLE I. INPUT PARAMETERS
Parameter Symbol Unit
Value at level
1 2 3
Number of insert N - 2 3 4
Insert material M - TiN TiCN TiAlN
Nose radius r mm 0.3 0.5 0.8
Cutting speed vc m/min 120 150 180
Feed rate vf mm/min 300 400 500
Depth of cut ap mm 0.2 0.35 0.5
TABLE II. EXPERIMENTAL MATRIX AND RESULTS
Trial N M
r
(mm)
vc
(m/min)
vf
(mm/min)
ap
(mm)
MRR
(mm
3
/min)
SR
(µµµµm)
1 2 TiN 0.3 120 300 0.2 2400 2.287
2 2 TiN 0.3 120 400 0.35 5600 3.152
3 2 TiN 0.3 120 500 0.5 10000 4.017
4 2 TiCN 0.5 150 300 0.2 2400 1.377
5 2 TiCN 0.5 150 400 0.35 5600 2.242
6 2 TiCN 0.5 150 500 0.5 10000 3.107
7 2 TiAlN 0.8 180 300 0.2 2400 0.490
8 2 TiAlN 0.8 180 400 0.35 5600 1.240
9 2 TiAlN 0.8 180 500 0.5 10000 2.105
10 3 TiN 0.5 180 300 0.35 4200 0.245
11 3 TiN 0.5 180 400 0.5 8000 0.793
12 3 TiN 0.5 180 500 0.2 4000 1.163
13 3 TiCN 0.8 120 300 0.35 4200 1.104
14 3 TiCN 0.8 120 400 0.5 8000 1.969
15 3 TiCN 0.8 120 500 0.2 4000 2.339
16 3 TiAlN 0.3 150 300 0.35 4200 0.838
17 3 TiAlN 0.3 150 400 0.5 8000 1.703
18 3 TiAlN 0.3 150 500 0.2 4000 2.073
19 4 TiN 0.8 150 300 0.5 6000 0.345
20 4 TiN 0.8 150 400 0.2 3200 0.456
21 4 TiN 0.8 150 500 0.35 7000 0.890
22 4 TiCN 0.3 180 300 0.5 6000 0.611
23 4 TiCN 0.3 180 400 0.2 3200 0.241
24 4 TiCN 0.3 180 500 0.35 7000 0.624
25 4 TiAlN 0.5 120 300 0.5 6000 0.657
26 4 TiAlN 0.5 120 400 0.2 3200 1.027
27 4 TiAlN 0.5 120 500 0.35 7000 1.892
B. Results and Discussion
The SR for each experiment can be seen presented in Table
II. These values are the mean values of at least 3 consecutive
measurements. In addition, the MRR at each experiment is also
been summarized in Table II. The values are calculated by (23),
where vf, ap and bw are the feed rate, the depth of cut, and the
wide cut respectively. F)) = GH ∙ !I ∙ JK (23)
The extent and influence of the parameters on SR are
shown in Figure 1. From this graph, it is shown that:
• The number of inserts, cutting speed, and feed rate have a
great influence on SR. The nose radius and depth of cut also
affect the SR, but to a lesser extent than the 3 mentioned
above parameters. The insert material has no significant
effect on SR.
• When the number of insert increases, the SR decreases.
This can be explained by the fact that as the number of
inserts increases, each area of the machined surface is cut
more than once. That means that after a insert cuts off a
layer of the material, the metal layer on the surface of the
part will be elastically deformed. This metal is then
removed by other cuttings, which causes the SR to
decrease.
• As the cutting speed increases, the cutting tool rotates at a
faster speed. Then a point on the surface of the workpiece
will be repeatedly cut by the cutting edges, even the
undulations caused by plastic deformation will be
eliminated, which leads to a reduced SR. The case is similar
with the increasing of the number of inserts discussed
above.
• As the feed rate increases, the time the cutting tool is in
contact between an area of the part surface and the cutting
edge decreases, causing plastic layers of metal on the
surface to not be removed, leading to an increase in SR.
Increasing the nose radius will cause the SR to decrease. It
can be understood that the height of the surface undulation
is inversely proportional to the nose radius, as discussed in
[27]. Furthermore, the large SR at large feed rates and small
nose radius are also consistent with the SR calculation
formula used in some studies [27]: )� = 1000 ∙ 0.0321 ∙ GH�/
Fig. 1. Main effects plot for surface roughness.
From Figure 1, if the SR has a small value, choose a large
cutting speed, a small feed rate, and a small depth of cut.
However, according to (23), when the feed rate and cutting
depth are small, the MRR will also be small, which is
undesirable. Therefore, choosing the value of the cutting
parameters to ensure that the SR is small and the MRR is large,
it is necessary to make the right decisions. This right decision
can be made with a MCDM method. In this problem, the
PARIS method will be used.
V. MULTI-CRITERIA DECISION MAKING
A. Determining the Weights of the Criteria
Equation (1) is used to build the decision matrix. The last
two columns in Table II are the decision matrix. Equation (13)
was applied to determine the weights of the criteria according
to the AW method. In addition, (14)-(16) are used to determine
the weights for the criteria according to the EW method.
Equations (17)-(22) determine the weights of the criteria of the
MW method. The results are shown in Table III.
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TABLE III. CRITERIA WEIGHTS
Method Ra MRR
AW 0.5 0.5
EW 0.6859 0.3141
MW 0.3663 0.6337
B. Applying the PARIS Method
Equations (2) and (3) are used to normalize the matrix in
way 1. Equations (4) and (5) are applied to normalize the
matrix in way 2. Equations (6) and (7) are used to normalize
the matrix in way 3. The results are presented in Table V.
TABLE IV. VALUE OF Z*J
Normalization SR MRR
Way 1 0.255 0.200
Way 2 0.030 0.500
Way 3 0.000 0.500
To rank the alternatives for weight calculation according to
the AW method we apply (8) to calculate the weighted
normalized value (zij) and (9) to calculate the weighted
normalized sum (π
ω
i). The results are presented in Table VI.
The results of ranking the alternatives according to the value of
π
ω
i are also presented in this Table. Equation (10) was applied
to determine the factors of the ideal solution (z
*
j) and the results
are presented in Table IV. Equation (11) is used to determine
the distance from the reference ideal solution (πi). The results
are presented in Table VII. The results of ranking the
alternatives according to the value of π
*
i are also presented in
this Table. The distance from the alternatives to the ideal
solution (Ri) is calculated by (12). The results are presented in
Table VIII. The results of ranking options according to the
value of Ri are also shown in this Table.
TABLE V. NORMALIZED MATRICES
Trial.
Way 1 Way 2 Way 3
SR MRR SR MRR SR MRR
1 0.721 0.096 0.105 0.240 0.458 0.000
2 0.616 0.224 0.076 0.560 0.229 0.421
3 0.511 0.400 0.060 1.000 0.000 1.000
4 0.832 0.096 0.175 0.240 0.699 0.000
5 0.727 0.224 0.107 0.560 0.470 0.421
6 0.622 0.400 0.078 1.000 0.241 1.000
7 0.940 0.096 0.492 0.240 0.934 0.000
8 0.849 0.224 0.194 0.560 0.735 0.421
9 0.744 0.400 0.114 1.000 0.506 1.000
10 0.970 0.168 0.984 0.420 0.999 0.237
11 0.903 0.320 0.304 0.800 0.854 0.737
12 0.858 0.160 0.207 0.400 0.756 0.211
13 0.866 0.168 0.218 0.420 0.771 0.237
14 0.760 0.320 0.122 0.800 0.542 0.737
15 0.715 0.160 0.103 0.400 0.444 0.211
16 0.898 0.168 0.288 0.420 0.842 0.237
17 0.793 0.320 0.142 0.800 0.613 0.737
18 0.747 0.160 0.116 0.400 0.515 0.211
19 0.958 0.240 0.699 0.600 0.972 0.474
20 0.944 0.128 0.529 0.320 0.943 0.105
21 0.892 0.280 0.271 0.700 0.828 0.605
22 0.926 0.240 0.394 0.600 0.902 0.474
23 0.971 0.128 1.000 0.320 1.000 0.105
24 0.924 0.280 0.386 0.700 0.899 0.605
25 0.920 0.240 0.367 0.600 0.890 0.474
26 0.875 0.128 0.235 0.320 0.792 0.105
27 0.770 0.280 0.127 0.700 0.563 0.605
TABLE VI. SOME PARAMETERS IN PARIS RANKED BY THE VALUE OF πωi
Trial
Way 1 Way 2 Way 3
zij
ππππ
ωωωω
i Rank
zij
ππππ
ωωωω
i Rank
zij
ππππ
ωωωω
i Rank
SR MRR SR MRR SR MRR
1 0.361 0.048 0.409 27 0.053 0.120 0.173 27 0.229 0.000 0.229 27
2 0.308 0.112 0.420 26 0.038 0.280 0.318 21 0.115 0.211 0.325 26
3 0.255 0.200 0.455 23 0.030 0.500 0.530 8 0.000 0.500 0.500 18
4 0.416 0.048 0.464 22 0.088 0.120 0.208 26 0.350 0.000 0.350 24
5 0.363 0.112 0.476 21 0.054 0.280 0.334 19 0.235 0.211 0.446 22
6 0.311 0.200 0.511 18 0.039 0.500 0.539 7 0.120 0.500 0.620 10
7 0.470 0.048 0.518 16 0.246 0.120 0.366 17 0.467 0.000 0.467 20
8 0.424 0.112 0.537 12 0.097 0.280 0.377 16 0.368 0.211 0.578 13
9 0.372 0.200 0.572 7 0.057 0.500 0.557 4 0.253 0.500 0.753 2
10 0.485 0.084 0.569 8 0.492 0.210 0.702 1 0.499 0.118 0.618 11
11 0.452 0.160 0.612 1 0.152 0.400 0.552 5 0.427 0.368 0.795 1
12 0.429 0.080 0.509 19 0.104 0.200 0.304 22 0.378 0.105 0.483 19
13 0.433 0.084 0.517 17 0.109 0.210 0.319 20 0.386 0.118 0.504 17
14 0.380 0.160 0.540 11 0.061 0.400 0.461 13 0.271 0.368 0.640 9
15 0.358 0.080 0.438 25 0.052 0.200 0.252 25 0.222 0.105 0.327 25
16 0.449 0.084 0.533 14 0.144 0.210 0.354 18 0.421 0.118 0.539 15
17 0.396 0.160 0.556 9 0.071 0.400 0.471 12 0.306 0.368 0.675 8
18 0.374 0.080 0.454 24 0.058 0.200 0.258 24 0.257 0.105 0.363 23
19 0.479 0.120 0.599 3 0.349 0.300 0.649 3 0.486 0.237 0.723 4
20 0.472 0.064 0.536 13 0.264 0.160 0.424 14 0.472 0.053 0.524 16
21 0.446 0.140 0.586 4 0.135 0.350 0.485 10 0.414 0.303 0.717 5
22 0.463 0.120 0.583 5 0.197 0.300 0.497 9 0.451 0.237 0.688 6
23 0.485 0.064 0.549 10 0.500 0.160 0.660 2 0.500 0.053 0.553 14
24 0.462 0.140 0.602 2 0.193 0.350 0.543 6 0.449 0.303 0.752 3
25 0.460 0.120 0.580 6 0.183 0.300 0.483 11 0.445 0.237 0.682 7
26 0.437 0.064 0.501 20 0.117 0.160 0.277 23 0.396 0.053 0.449 21
27 0.385 0.140 0.525 15 0.064 0.350 0.414 15 0.281 0.303 0.584 12
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TABLE VII. π*i VALUES AND RATINGS
Trial
Way 1 Way 2 Way 3
ππππ
*
i Rank ππππ
*
i Rank ππππ
*
i Rank
1 0.047 27 0.357 27 0.271 27
2 0.035 26 0.212 21 0.175 26
3 0.000 23 0.000 8 0.000 18
4 -0.009 22 0.322 26 0.150 24
5 -0.020 21 0.196 19 0.054 22
6 -0.055 18 -0.009 7 -0.120 11
7 -0.063 16 0.164 17 0.033 20
8 -0.081 12 0.153 16 -0.078 13
9 -0.116 7 -0.027 4 -0.253 2
10 -0.114 8 -0.172 1 -0.118 6
11 -0.156 1 -0.022 5 -0.295 1
12 -0.054 19 0.226 22 0.017 19
13 -0.061 17 0.211 20 -0.004 17
14 -0.085 11 0.069 13 -0.140 10
15 0.018 25 0.278 25 0.173 25
16 -0.078 14 0.176 18 -0.039 15
17 -0.101 9 0.059 12 -0.175 9
18 0.002 24 0.272 24 0.137 23
19 -0.144 3 -0.119 3 -0.223 4
20 -0.081 13 0.106 14 -0.024 16
21 -0.130 4 0.045 10 -0.217 5
22 -0.127 5 0.033 9 -0.188 7
23 -0.094 10 -0.130 2 -0.053 14
24 -0.147 2 -0.013 6 -0.252 3
25 -0.125 6 0.047 11 -0.182 8
26 -0.046 20 0.253 23 0.051 21
27 -0.069 15 0.116 15 -0.084 12
TABLE VIII. Ri VALUES AND RATINGS
Trial
Way 1 Way 2 Way 3
Ri Rank Ri Rank Ri Rank
1 0.287 27 0.748 27 0.976 27
2 0.271 26 0.543 21 0.750 18
3 0.221 23 0.243 8 0.418 3
4 0.209 22 0.699 26 0.912 26
5 0.193 21 0.521 19 0.681 14
6 0.143 18 0.231 7 0.343 2
7 0.132 16 0.475 17 0.860 25
8 0.106 12 0.459 16 0.624 13
9 0.056 7 0.204 4 0.298 1
10 0.060 8 0.000 1 0.700 15
11 0.000 1 0.212 5 0.427 4
12 0.145 19 0.563 22 0.757 19
13 0.134 17 0.541 20 0.737 17
14 0.101 11 0.340 13 0.454 6
15 0.246 25 0.637 25 0.834 24
16 0.111 14 0.492 18 0.724 16
17 0.078 9 0.327 12 0.444 5
18 0.223 24 0.627 24 0.814 22
19 0.018 3 0.074 3 0.563 10
20 0.107 13 0.393 14 0.791 21
21 0.037 4 0.306 10 0.499 8
22 0.041 5 0.289 9 0.569 11
23 0.088 10 0.059 2 0.781 20
24 0.014 2 0.224 6 0.495 7
25 0.045 6 0.309 11 0.570 12
26 0.156 20 0.600 23 0.820 23
27 0.123 15 0.408 15 0.536 9
TABLE IX. RANKING WHEN THE WEIGHTS ARE DETERMINED WITH THE EW METHOD
Trial
Ranking by value of ππππ
ωωωω
i Ranking by value of ππππ
*
i Ranking by value of Ri
Way 1 Way 2 Way 3 Way 1 Way 2 Way 3 Way 1 Way 2 Way 3
1 25 27 26 25 27 25 25 27 27
2 26 23 27 26 23 27 26 23 24
3 27 13 25 27 13 26 27 13 13
4 20 26 20 20 2 20 20 26 26
5 21 22 22 21 22 22 21 22 17
6 23 12 21 23 12 21 23 12 3
7 10 9 13 10 9 13 10 9 22
8 12 17 14 12 17 14 12 17 12
9 17 11 10 17 11 10 17 11 1
10 4 1 5 2 1 5 2 1 14
11 3 6 1 4 6 1 4 6 2
12 16 20 17 15 20 17 15 20 18
13 13 19 15 13 19 15 13 19 16
14 18 15 16 18 15 16 18 15 5
15 24 25 24 24 25 24 24 25 25
16 11 16 11 11 16 11 11 16 15
17 14 14 12 16 14 12 16 14 4
18 22 24 23 22 24 23 22 24 23
19 1 3 2 1 3 2 1 3 8
20 9 5 9 9 5 9 9 5 20
21 8 10 7 8 10 7 8 10 7
22 5 7 4 6 7 4 6 7 9
23 7 2 8 5 2 8 5 2 19
24 2 4 3 3 4 3 3 4 6
25 6 8 6 7 8 6 7 8 10
26 15 21 18 14 21 19 14 21 21
27 19 18 19 19 18 18 19 18 11
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TABLE X. RANKING WHEN THE WEIGHTS ARE DETERMINED WITH THE MW METHOD
Trial
Ranking by value of ππππ
ωωωω
i Ranking by value of ππππ
*
i Ranking by value of Ri
Way 1 Way 2 Way 3 Way 1 Way 2 Way 3 Way 1 Way 2 Way 3
1 27 27 27 27 27 27 27 27 27
2 24 18 22 24 18 22 24 18 15
3 14 3 9 14 3 10 14 3 3
4 25 26 26 25 26 26 25 26 26
5 20 17 16 20 17 17 20 17 14
6 10 2 3 9 2 3 10 2 2
7 21 21 23 21 21 23 21 21 25
8 13 15 13 13 15 14 13 15 13
9 2 1 1 2 1 1 2 1 1
10 11 5 14 11 5 15 11 5 16
11 1 6 2 1 6 2 1 6 4
12 19 22 20 19 22 8 19 22 19
13 18 20 18 18 20 19 18 20 18
14 9 10 7 10 10 7 9 10 6
15 26 24 25 26 25 25 26 24 23
16 16 19 15 16 19 16 16 19 17
17 6 9 5 6 9 5 6 9 5
18 23 23 24 23 23 24 23 23 21
19 5 4 8 5 4 9 5 4 10
20 17 16 19 17 16 20 17 16 22
21 4 11 6 4 11 6 4 11 8
22 7 12 10 7 12 11 7 12 11
23 15 8 17 15 8 18 15 8 20
24 3 7 4 3 7 4 3 7 7
25 8 13 11 8 13 12 8 13 12
26 22 25 21 22 24 21 22 25 24
27 12 14 12 12 14 13 12 14 9
The ranking results in Tables V-X show 27 different
ranking options. From these results it is shown that:
• 22/27 times experiment #1 was determined to be the worst.
In this experiment, MRR = 2400mm
3
/min was one of the 3
smallest values in Table II (equal to the MRR in
experiments #4 and #7). In addition, Ra = 2.287µm is very
large compared to the surface texture in other experiments
(only smaller than the surface texture in 4 experiments: #2,
#3, #5, and #15). That allows the claim that the experiment
#1 is the worst to be entirely reasonable.
• 8/27 times determined experiment #9, 6/27 times
determined experiment #10, 10/27 times determined
experiment #11, and 3/27 times determined experiment #19
as the best. Thus, determining which experiment is the best
would not be achieved if the work stopped here. To
determine the best experiment, in addition to the ranking
results, it is also necessary to add the stability of the ratings.
In this study, the GINI index value will be used to
determine the stability in ranking the alternatives [29]. The
GINI index value is determined by [29, 30]:
QR)S = TR�4�S;U/4VWXY RU∙Z/SV> ∑ ∑ |)\ − )] |U]�\-�U4�\�� (24)
where m is the number of options, z is the number of MCDM
methods used, Rh and Rl are the ranking values of the
alternatives of the decision method h and l, and D(R) ∈[0,1].
When D(R) = 0, the rank of an alternative is the same when
ranking by different methods. In contrast, when D(R) = 1, the
ranking of the alternatives is most different when using
different ranking methods. When comparing two alternatives,
the one with the smaller GINI index value is the better one.
Equation (24) has been applied to calculate the GINI index
value for the data in Tables V-X. The results are presented in
Table XI.
TABLE XI. GINI INDEX VALUE OF THE ALTERNATIVES
Experiment GINI index Experiment GINI index
1 0.002536 15 0.003381
2 0.018808 16 0.015427
3 0.048605 17 0.017751
4 0.018174 18 0.004861
5 0.012468 19 0.013314
6 0.040152 20 0.019231
7 0.01754 21 0.019019
8 0.014159 22 0.015216
9 0.019442 23 0.040997
10 0.032967 24 0.012046
11 0.016272 25 0.015216
12 0.018597 26 0.016061
13 0.014582 27 0.011834
14 0.017117
The results in Table XI show that:
• In experiment #1, the minimum GINI index value is
0.002536. This proves that experiment #1 has the highest
stability when ranking in different times. Up to 22/27
options confirmed this experiment as the worst (ranked
27th), 4/27 options indicate that this experiment is the
second worst (ranked 26), and 1/27 indicates that this
experiment is the third worst (ranked 25). On the other
hand, 27th or 26th or 25th ranking is very close. That
proves that experiment #1 has the highest stability when
ranking according to different options.
Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9208-9216 9215
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• In experiment #3, the largest GINI index value was
0.048605, proving that this experiment has the lowest rank
stability when ranking according to the alternatives.
According to the data in Tables VI, VIII-XI, experiment #3
came 5 times at the 3rd place, 3 times at the 8th, 1 time at
the 9th, 1 time 10th, 4 times came at the 13th, 3 times at the
14th place, 2 at the 18th, 3 at the 23rd place, and 1 at the
25
th
, 26th time, and 27th place. So, the stability in the
ranking of experiment #3 is very weak. This experiment
ranked in a variety of categories, with both good (3) and
bad (27) ranks.
• Among the 4 experiments #9, #10, #10 and #19, experiment
#19 has the smallest GINI index value. That proves that
experiment #19 has a higher stability rating than the other 3
experiments. Thus experiment #19 is the best of these 4
experiments and it is also the best of the 27 experiments
performed. The best values of the input parameters to
ensure minimum SR and maximum MRR at the same time
are: 4 as the number of inserts, TiN as the insert material,
0.8mm nose radius, 150m/min cutting speed, 30mm/min
feed rate, and 0.5 mm depth of cut.
• The use of different weighting methods leads to different
ranking orders. Responding to different data normalization
ways will result in different ranking orders for the
alternatives. However, the simultaneous use of multiple
weighting and multiple data normalization methods to give
different ranking results, and then the use of the GINI index
to choose the best solution will form the basis for
determining which option is the best.
VI. CONCLUSIONS
In this study, 27 SNCM439 steel milling experiments were
performed. At each experiment, 6 parameters were considered:
number of inserts, cutting material, nose radius, cutting speed,
feed rate, and depth of cut. SR and MRR were determined in
each experiment. The PARIS method was used to rank the
alternatives, and the stability in ranking was evaluated by the
GINI index. Some drawn conclusions are:
• The number of inserts, cutting speed, and feed rate have a
great influence on surface roughness. Increasing the number
of inserts or cutting speed reduces the surface roughness,
while increasing the feed rate increases it. The nose radius
and depth of cut also affect the surface roughness. Surface
roughness is reduced if the tip radius is increased, or the
depth of cut is decreased. On the other hand, the insert
material does not significantly affect surface roughness.
• The use of 3 data normalization methods is what
distinguishes the PARIS method from other methods. For
each data normalization method, the PARIS method also
gives 3 ranking results for the alternatives. This is also its
difference from the other MCDM methods.
• The combination of the PARIS method and 3 different
weighting methods (AW, EW, and MW) resulted in 27
different ranking options. The combination of the PARIS
method and the GINI index to determine the best solution
has higher reliability instead of using just one method that
only gives a ranking solution for the alternatives.
• To ensure minimum SR and maximum MRR
simultaneously, it is recommended to use the TiN insert
with parameter values of the number of inserts, tool radius,
cutting speed, feed rate, and depth of cut respectively as 4
pieces, 0.8mm, 150m/min, 30mm/min, and 0.5mm.
NOMENCLATURE
PARIS Preference Analysis for Reference Ideal Solution
SAW Simple Additive Weighting
WASPAS Weighted Aggregates Sum Product ASsessment
TOPSIS
Technique for Order of Preference by Similarity to Ideal
Solution
VIKOR
Vlsekriterijumska optimizacija i kompromisno resenje in
Serbian
MOORA
Multiobjective Optimization On the basis of Ratio
Analysis
COPRAS COmplex Proportional ASsessment
PIV Proximity Indexed Value
PSI Preference Selection Index
EDAS Evaluation based on Distance from Average Solution
MARCOS
Measurement Alternatives and Ranking according to
COmpromise Solution
CODAS COmbinative Distance based Assessment
WASPAS Weighted Aggregated Sum Product Assessment
WPAS Weighted Product Assessment
MCDM Multi-Criteria Decision-Making
MEREC Method based on the Removal Effects of Criteria
AW Average Weight
EW Entropy Weight
MW Merec Weight
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