Multi-Response Optimization of Process Parameters using Weight ...
57
MULTI-RESPONSE OPTIMIZATION OF PROCESS PARAMETERS
USING WEIGHT BASED GREY ANALYSIS AND WEIGHT BASED
DESIRABILITY FUNCTION IN THE TAGUCHI METHOD
V. Jai Ganesh1 and R. Raju2
Department of Industrial Engineering, Anna University, Chennai-600025, Tamilnadu,
India, jaiganeshannauniv@gmail.com
1
Department of Industrial Engineering, Anna University, Chennai-600025, Tamilnadu,
India, krrajuin@yahoo.co.in
2
Abstract: The objective of this paper is to propose a weight based Desirability
function and weight based Grey Relational Analysis (GRA) by assigning a
weight to each factor in optimizing the Wire-cut Electrical Discharge
Machining process parameters. This method will provide a direct solution
and there is no external intervention to search for an optimal solution. In
this paper, the new weight based step-by-step procedure for applying the
grey relation analysis called Weight Based Grey Relation Analysis (WBGRA)
and Weight Based Desirability Method (WBDM) is compared with classical
techniques like factor analysis and grey relation analysis for solving the
multi-response problem. Eigen value based weight assigning method is used
to demonstrate the work.
Keywords: Desirability, Factor Analysis, Grey Relation Analysis, Eigen value,
Taguchi Techniques; Wire-cut EDM.
1. INTRODUCTION
Multi-response optimization receives more attention because today
industries are facing real and complex problems in there production shops.
Because of its nature and complexity in their manufacturing processes
their outputs get affected. But a common problem in product or process
design is the selection of parameter levels for optimizing multiple
responses simultaneously, which is called a multiresponse problem. Multiresponse optimization is an interesting area in which more than one factors
are taking into account simultaneously to obtain an optimal solutions for
their problems. Unfortunately, it is difficult to optimize a multiresponse
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
problem by the taguchi method. In order to solve the multiresponse
problem the basic steps was to convert the multiple values into single
response value then optimal solutions were obtained. Until now,
engineering judgement has been primarily used to solve such a complicated
multiresponse problem. But the method increases the uncertainty of the
decision making process. So weight assigning must be scientific to avoid
uncertainty.
But how to determine a definite weight for each response in a real
case still remains difficult. Pignatiello (1993), Reddy et al. (1997), and
Logothetis et al (1998) have applied regression technique to optimize the
multi-response problem. However, there approach increases more
complexity to the computational process, and the possible correlation
among the responses may still not be considered. Su et al. (1997) and
Antony (2000) proposed the multi-response methods which are based on
a principal component analysis (PCA) method to optimize the multiresponse problem. In this method to transform the normalized multiresponse value into uncorrelated linear combinations. If n linear
combinations are obtained, then n principal components will also be
formed. Su et al. (1997) also used the Kaiser’s study (1960) to choose the
components whose eigenvalue is bigger than 1 to replace the original multiresponse values. In his study the when the number of components is bigger
than 1, and he had suggested that “trade-off might be necessary to select
a feasible solution”.
Antony (2000) indicated that “rule of thumb is to choose those
components with an eigenvalue greater than or equal to one”. There is no
further discussion about how to deal with situations where the number of
components grater than 1. Hung – Chung Liao (2006) proposed a weighted
principal component (WPC) method. The WPC method uses the explained
variation as the weight to combine all principal components in order to
form a multi-response performance index. Noorul Haq, et al., (2007) used
grey relation analysis to solve the drilling parameters. However the average
values have been directly taken for calculating the grey grade. However
the average value brings uncertainty to the problem, so there is a need for
finding some scientific method to get the optimal solution. In this proposed
method the grey relational grade was calculated based on the new
weight which is multiplied with the existing grey relation coefficient for
obtaining the grey grade which is called Weight Based Grey Relational
Analysis (WBGRA).
Multi-Response Optimization of Process Parameters using Weight ...
59
And the weight multiplied with the individual desirability (d1, d2, d3
….) to find out the overall desirability (D) is called Weight Based
Desirability method (WBDM).
2. LITERATURE REVIEW
Tara man (1974) investigates multi machining output, multi independent
variable turning research by response surface methodology. The purpose
of this research was to develop a methodology which would allow
determination of the cutting conditions (cutting speed, feed rate and depth
of cut) such that specified criterion for each of the several machining
dependent parameters such as (surface finish, tool force and tool life – TL)
could be achieved simultaneously. Derringer and Suich (1980) made use
of modified desirability functions, which measure the designer's
desirability over a range of response values. They have utilized the
modified desirability function approach for the development of a tyre tread
compound, which involves four responses (abrasion index, modulus,
elongation at break and hardness) and three independent variables.
Byrne and Taguchi (1987) illustrate a case of the optimization of two
quality characteristics: the force required to insert the tube into the
connector and the pull off force. The selected quality characteristics were
independently optimized using Taguchi approach and then the results
were compared subjectively to select the best levels in terms of the quality
characteristics of interest.
Kacker (1985) and Leon etal, (1987) in this approach the process
parameters are divided into two groups: Those which affect mean and
variability, and those which affect only mean. The PerMIA approach looks
for a transformation of the response variables such that the standard
deviation of the transformed output becomes independent of the mean.
Then a two step optimization is performed by initially working with the
first group and afterwards working with the second group. Bryne and
Taguchi, (1986); Phadke, (1989), The S/N ratio developed by Dr. Taguchi
is a performance measure to choose control levels that best cope with noise.
Hence noise factors are difficult to control in real time applications. The
S/N ratio takes both the mean and the variability into account. In its
simplest form, the S/N ratio is the ratio of the mean (signal) to the standard
deviation (noise). The S/N equation depends on the criterion for the quality
characteristic to be optimized. Phadke (1989) presents a case of products
with multiple characteristics such as surface defects and thickness in his
example of polysilicon deposition. In order to estimate the loss caused by
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
quality characteristics, he assigned a weight from experience to each quality
characteristic. Roy (1990) proposed a simple and pure engineering
methodology for the optimization of multiple responses. The methodology
is called Overall Evaluation Criteria (OEC) and the approach involves a
high degree of subjectivity as the relative weights of responses vary from
one company to another. Hatzis and Larntz (1992) constructed locally Doptimal designs for a nonlinear multiresponse model describing the
behavior of a biological system. Delcastillo and Montgomery (1993) A more
general approach that can be used is to formulate a multiple response
problem as a constrained optimization problem. Young-Jou Lai and Shing
I Chang, (1994) Multiresponse optimization techniques are used to identity
settings of process parameters that make the product's performance close
to target values in the presence of multiple quality characteristics. Sunk.
H. Park (1996) has used graphical superimposition for optimizing multiresponse problem is difficult to apply when the number of input
variables exceeds three.
Tong et al. (1997) propose a procedure on the basis of the quality loss
of each response so as to achieve the optimization on multi-response
problems in the taguchi method. Rajkumar & etal, (2000) in his study the
S/N ratio is measured in unit decibels (db). Higher S/N ratio is preferred
because a high value of S/N implies that the signal is much higher than
the uncontrollable noise factors such temperature, humidity and
consistency in measurement of data.
Antony (2001) has developed a simple and practical step-by-step
approach for tackling multiple response or quality characteristic problems
in Taguchi’s parameter design experiments. The methodology uses the
Taguchi's quality loss function for identifying the significant factor/
interaction effects and also for determining the optimal condition of the
process. In-Jun Jeong (2003), A common problem encountered in product
or process design is the selection of optimal parameters that involves
simultaneous consideration of multi-response characteristics, called a
multi-response surface (MRS) problem. The existing MRO approaches
require that all the preference information of a decision maker be
articulated prior to solving the problem.
J.A. Ghani et al (2004) used an orthogonal array, S/N ratio and pareto
ANOVA to analyze the effect of these milling parameters. Hung-Chang
Liao (2004) proposes an effective procedure on the basis of the neural
network (NN) and the data envelopment analysis (DEA) to optimize the
multi-response problems. Kun-Lin Hsieh et al., (2005), This study proposes
Multi-Response Optimization of Process Parameters using Weight ...
61
a procedure utilizing the statistic regression analysis and desirability
function to optimize the multi-response problem with Taguchi’s dynamic
system consideration. Firstly, the regression analysis is employed to screen
out the control factors significantly affecting the quality variation, and the
adjustment factors significantly affecting the sensitivity of a Taguchi’s
dynamic system.
Hari Singh et al.,(2006) Optimizing multi-machining characteristics
through Taguchi’s approach and utility concept. The multi-machining
characteristics have been optimized simultaneously using Taguchi’s
parameter design approach and the utility concept. The paper used a single
performance index, utility value, as a combined response indicator of
several responses. Ramakrishnan, et al (2006) in this study optimization
of WEDM operations using Taguchi’s robust design methodology with
multiple performance characteristics is proposed. In order to optimize the
multiple performance characteristics, Taguchi parametric design approach
was not applied directly. Since each performance characteristic may not
have the same measurement unit and of the same category in the S/N
ratio analysis. However they used average weighting method for
calculating the multi-response optimization.
3. DESCRIPTION OF RESEARCH WORK
3.1 Introduction
WEDM is a widely accepted non-traditional, non-conventional, thermal
machining process capable of machining components with intricate shapes
and profiles. It is used for making simpler profiles like tools, dies, microscale parts to aerospace, surgical components with excellent accuracy and
surface finish. In WEDM, the conductive material is eroded by means of
discrete sparks produced between material and wire separated by a film
of dielectric fluid. Dielectric fluid is fed continuously into the machining
zone [1]. The temperature at machining zone ranges from 8000-20000°C
[2] or as high as 20000°C [3] which are sufficient for melting any kind of
material. During the time of pulse-OFF in AC voltage, the reaction of
dielectric fluid causes material to cool suddenly results in microscopic
debris [4].
Tarang et.al [5] founded the factors that affect the machining
performance as Pulse duration, pulse interval, Peak current, Open circuit
voltage, Servo voltage, Electric capacitance and table speed. Lin et.al [6]
used grey analysis based on Taguchi method to solve multi-optimization
problem. Chin et.al [7] used Principal component analysis (PCA) to solve
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
multi-response optimization problem. Jeyapaul et.al [8] used factor analysis
to solve multi-response problem and compared the results with several
methods. Govaerts et.al used desirability index to multi-response problem.
Scott et.al [9] used factorial design to find optimal combination of control
parameters in WEDM by measuring performance measures as MRR and
SF. Mahaputra et.al [10] used genetic algorithm to solve multi-response
problem by measuring performance measures as MRR, SF and Kerf.
It was founded from past experiments that the most important
performance measures in WEDM are MRR, SF and Kerf. Here MRR gives
importance to rate of production and economics of machining whereas
kerf determines precision of machining. Machining parameters such as
Input peak voltage, Pulse-ON time, Pulse-OFF time, Dielectric fluid
pressure, Wire feed, Wire speed and Servo voltage has much influence
over those responses. By setting the inputs such that MRR and SF are
maximized while kerf should be minimized. Here Taguchi method
is employed to achieve optimal machining performance. Advantage over
this method is that it reduces experimental cost and time. Comparison
between desirability index and factor analysis is made to find
optimal combination.
Fig. 1: Details of WEDM Cutting Gap
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Multi-Response Optimization of Process Parameters using Weight ...
3.2 Experimental Methods
3.2.1 Machine Description
The experiments were performed using typical four-axes Supercut-734
CNC-wire cut EDM, which is manufactured by Electronica machine tools
ltd. It consists of wire, servo control, work table, power supply, dielectric
supply and monitor. The inputs like as Input peak voltage, Pulse-ON time,
Pulse-OFF time, Dielectric fluid pressure, Wire feed, Wire speed and Servo
voltage can be easily fed either in manual or auto mode. Here brass wire
of diameter 0.25mm is employed as a cutting tool and distilled water is
used as dielectric.
3.2.2 Material
The material chosen for our experimental study was Oil Hardened NonShrinkage Steel, OHNS (0.85%C, 0.27%Si, 0.043%S, 0.04%P, 0.25%Cu and
1.24%Cr) of size 150 × 150 × 9 mm.
3.2.3 Test Condition
According to the Taguchi method-based robust design [11], a L 18 (21 × 37)
mixed orthogonal array is employed for the experimentation. Here the
first control factor A (Dielectric fluid pressure) is kept at two levels and
the other six control factors, such as factor B (pulse on-time), factor C(pulse
off-time), factor D(peak voltage), factor E(wire feed), factor F(wire tension)
and factor G(servo feed) are kept at three levels for the experiment. Each
time an experiment is conducted with a particular set of input parameters
shown in table 1.
Table 1
Factors and Levels
Factors/
Levels
Control Parameters
Level 1
Level 2
Level 3
A
Dielectric fluid
pressure kg/cm2
High
Low
-
B
C
D
E
F
G
Pulse on time µs
Pulse off time µs
Peak voltage Volts
Wire feed m/min
Wire tension gms
Servo feed mm/min
4
15
120
2
3
90
5
16
130
3
4
100
6
17
140
4
5
110
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
3.2.4: Performance Measures
The formulae for measuring preferred performance measures are listed
below,
3.2.4.1 Metal Removal Rate (MRR)
The metal cutting speed data is directly available from the monitor of
supercut-734 CNC machine for various experimental setting. From which
MRR can be calculated as,
MRR = k. t. vc. ρ mm3/min
(1)
Here, k is the kerf, t is the thickness of work piece (9 mm), vc is cutting
speed and ρ is the density of work piece material (7.6 g/cm3).
3.2.4.2 Surface Roughness
Surface roughness is calculated by measuring mean deviation (Ra) using
TIME TR100 surface roughness tester. A total of six readings were taken
on each work piece at different points and the average of these values is
surface roughness.
3.2.4.3 Kerf
It is average three measurement of wire-work piece gap along the length
of cut. It is measured using Tool Maker’s microscope.
3.3 Determination of S/N Ratio by Taguchi Method
To evaluate optimal combination of machining parameters, a specially
designed experimental procedure is required. In this case, Taguchi method
is used for maximizing MRR and SF and minimizing Kerf. For our factors
and its levels, mixed level L18 (21 × 37) orthogonal array is selected.
For every experiment, corresponding three responses are measured.
Then these measured responses are transferred into S/N (signal to noise)
ratio. For different type of responses, different S/N ratios are available.
The selection of suitable S/N type depends on the nature of the response.
According to quality engineering [12], the response that higher the value
represents better machining performance like MRR and SF are called “HB”,
higher the better. Similarly the response that lower value represents better
machining performance like Kerf is called “LB”, lower the better. The S/
N ratios for HB and LB are shown below,
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Multi-Response Optimization of Process Parameters using Weight ...
S / N HB = −10.log 1/n ∑ i = 1, 2 , .....n 1/Yi2 


(2)
S / N LB = −10.log 1/n ∑ i = 1, 2 , .....n Yi2 


(3)
Where ‘n’ denotes the number of experiment and ‘Yi’ denotes the
corresponding responses.
The S/N values are then normalized to avoid the effect of adopting
different units and to reduce the variability. Nagahanumaiah et.al. [13]
details the normalization formula.
Table 2
Results of Performance Measures and Its S/N Values
Exp
No
MRR
(mm3/min)
S/N
ratio
SF
(µm)
S/N
(ratio)
Kerf
(mm)
S/N
ratio
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
7.8788
6.9311
6.3053
7.1413
7.1428
7.9344
7.1557
8.4233
8.0854
7.9344
6.4152
7.4675
7.4236
7.8496
6.8122
8.398
7.9311
7.7463
17.9292
16.8160
15.9941
17.0756
17.0774
17.9903
17.0930
18.5097
18.1540
17.9903
16.1442
17.4635
17.4123
17.8970
16.6658
18.4835
17.9867
17.7819
4.025
3.125
3.760
2.890
3.505
3.900
3.540
3.780
2.905
2.830
2.975
3.575
3.580
4.135
3.580
3.030
4.565
3.700
87.9047
90.1030
88.4962
90.7820
89.1062
88.1787
89.0199
88.4502
90.7371
90.9643
90.5303
88.9345
88.9223
87.6705
88.9223
90.3712
86.8112
88.6360
0.2895
0.2775
0.2975
0.2750
0.2300
0.2900
0.2525
0.2845
0.2500
0.2910
0.2910
0.2850
0.2635
0.2925
0.2880
0.2790
0.2705
0.2025
10.7670
11.1347
10.5303
11.2134
12.7654
10.7520
11.9548
10.9184
12.0412
10.7221
10.7221
10.9031
11.5844
10.6775
10.8122
1.0879
11.3567
13.8715
4. DESIRABILITY INDEX
The balance between different responses is usually measured by
desirability index, a concept introduced by Harrington in 1980’s.
Harrington suggest to associate a value belonging to [0, 1], D(x) to each
combination of factor levels x, representing the desirability to resulting
product quality. The desirability index has to be maximized. It allows to
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
transform multi-response into single objective optimization. D(x) is usually
defined as the combination of desirability functions di (Yi), i = 1, 2, … p
representing the confinity of each individual response to its specifications.
In this research, the Derringer’s desirability function approach is
transformed between the intervals [0, 1], where individual value ‘1’
represents most acceptable response and individual value ‘0’ represents
unacceptable response ‘r’ is assigned value between ‘0’ and ‘1’ (0 < r < 1).
di = [(Yi – a)/(b – a)]r
(4)
Where, ‘di’ represents individual desirability, ‘a’ is the minimum
S/N value along a column and ‘b’ is the maximum S/N value of the
same column.
D = Σ di . wi / Σ wi
(5)
Where, D is the overall desirability index and wi represents weight
assigned to each response. Here weights are taken from eigen values
of responses.
Table 3
Desirability Index for the Experiments
Exp No d1
d2
d3
D
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
0.963918
0.905033
0.513125
0.890291
1
0.977815
0.74338
0.573828
0.729269
0.940132
0.972263
1
0.946307
0.715022
0.712975
0.454871
0.712975
0.925843
0.867565
0.161211
0.141109
0.877072
0.571613
0
0.655669
0.656222
0.890807
0.660952
1
0.926624
0.890807
0.244264
0.764282
0.750845
0.869733
0.516715
0.994793
0.89
0.843026
0.891941
0.575354
0.966242
0.757402
0.740129
0.970862
0.970862
0.942556
0.827351
0.977721
0.956887
0.912743
0
0.176459
0.091117
0.180005
0.119477
0.628205
0.21962
0.057793
0.174387
0.149329
0.147052
0.110845
0.284042
0
0.662857
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Multi-Response Optimization of Process Parameters using Weight ...
From the value of desirability index shown in table 4, the average
value of control parameters under each level is tabulated on table 5 and
factor effects are plotted on graph 1.
Table 4
Mean Desirability Index
Factors/Levels
Level 1
Level 2
Level 3
A
B
C
D
E
F
G
0.145001
0.136068
0.195405
0.120988
0.110923
0.146824
0.179571
0.133929
0.142468
0.108847
0.11257
0.183809
0.161025
0.138891
0.139858
0.114143
0.184837
0.123663
0.110546
0.099932
Graph 1: Factor Effects on Desirability Index
Larger the value of desirability index implies the better quality.
Therefore from table 4 and graph 1, the optimal parameter combination is
A1 B2 C1 D3 E2F2 G1.
5. FACTOR ANALYSIS (PCA) METHOD
The basic concept behind factor analysis is grouping the original input
variables into factors. Each factor is accounted for one or more input
variables. Principle component method (PCM) of factor analysis is used
as extraction techniques for grouping factors. In PCM, information in
original variables is converted in uncorrelated combinations with
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
minimum loss of information. The analysis combines the variables but
accounts for largest amount of variance to form first principle component
and second corresponds to next largest amount of variance and so on.
Here the components that are having eigen values more than one are
selected for analysis. The components are derived using SPSS software.
Table 5
Component Matrix
Components
1
2
MRR
SF
KERF
-0.782
0.630
0.428
-0.0187
-0.5800
0.8210
Extraction Method: Principal Component Analysis
For calculating the Multi response performance index (MRPI), the
normalized S/N ratio of a response is multiplied with sum of its
components. MRPI can calculated as,
MRPI i 1 = −0.8007 Zi 1 + 1.249 Zi 2 + 0.05 Zi 3
(6)
Where Zi1, Zi2 and Zi3 represents the normalized S/N values for the
responses MRR, kerf and SF at ith trial respectively. MRPI can be treated
as the overall evaluation of experimental data for multi response process.
The optimal level of the process parameters is the level associated with
highest MRPI. MRPI value for all experiments is calculated and shown
in Table 7.
Table 6
MRPI Values for the Experiments
Exp No Normalized
SN for MRR
Normalized
SN for SF
Normalized
SN for KERF
MRPI
1
2
3
4
5
6
7
8
0.929138
0.819085
1
0.795559
0.331032
0.933624
0.573658
0.883847
0.263298
0.792619
0.405736
0.956122
0.552614
0.329279
0.531833
0.394641
0.676199
1.157725
1.451868
1.127492
0.344963
0.69535
0.632624
0.500546
0.769256
0.326742
0
0.429902
0.430627
0.793537
0.436857
1
Table Cont’d
Multi-Response Optimization of Process Parameters using Weight ...
69
Table Cont’d
9
10
11
12
13
14
15
16
17
18
0.858632
0.793537
0.059665
0.584127
0.563768
0.756435
0.266994
0.989613
0.792101
0.710694
0.547791
0.942572
0.942572
0.888412
0.68451
0.955938
0.915633
0.833099
0.752668
0
0.945295
1
0.895497
0.511257
0.508334
0.206908
0.508334
0.857185
0
0.439379
0.469331
1.041888
1.577248
0.897544
0.657711
0.691743
1.18401
0.67675
0.305848
0.34936
From the value of MRPI shown in table 7, the average value of control
parameters under each level is tabulated on table 8 and factor effects are
plotted on graph 2.
Table 7
Mean MRPI
Factors/Levels
Level 1
Level 2
Level 3
A
B
C
D
E
F
G
0.7840
1.1337
0.8021
0.8900
0.6539
0.6222
0.6260
0.7425
0.7835
0.7630
0.5568
0.9240
0.6898
0.8362
0.3726
0.7247
0.8430
0.7119
0.9779
0.8275
Graph 2: Factor Effects on MRPI
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
The main effects are tabulated in table-8 and larger the value of MRPI
implies the better quality. Therefore the optimal conditions are A1 B1 C1
D1 E2F3 G2.
6. GREY RELATION ANALYSIS
Grey theory, proposed by Deng in 1982, is an effective mathematical means
to deal with systems analysis characterized by incomplete information.
Grey system theory presents a grey relation space, and a series of
nonfunctional type models are established in this space so as to overcome
the obstacles of needing a massive amount of samples in general statistical
methods, or the typical distribution and large amount of calculation work.
Grey relation refers to the uncertain relations among things, among
elements of systems, or among elements and behaviors. Grey theory is
widely applied in fields such as systems analysis, data processing,
modeling and prediction, as well as decision-making and control. The grey
relation grades were calculated based on the formula (8). The optimal
combination of values obtained from the table is shown in Table (4)
Table 8
Mean Parameter Values
Parameters
1
2
3
MINI-MAX
A
B
C
D
E
F
G
0.6353
0.67595
0.68614
0.65622
0.61089
0.63750
0.66377
0.65164
0.60738
0.62845
0.54309
0.68717
0.67726
0.63834
0.64725
0.61592
0.73121
0.63245
0.61575
0.62840
0.01624
0.06862
0.07021
0.18812
0.07628
0.06150
0.03537
Graph 3: Grey Relation Grade
71
Multi-Response Optimization of Process Parameters using Weight ...
There fore the optimal combination of parameter value can be chosen
as per the quality characteristics.
OPTIMIAL COMBINATION A2 B1 C1 D3 E2 F2 G1
7. STEPS USED FOR PROPOSED OPTIMIZATION
Step 1 Calculation S/N ratio for the corresponding responses using
the following formula
i. Larger –the –better
S / N ratio ( η) = −10 log 10
n
∑
i=1
1
y ij2
(1)
Where n = number of replications yij = observed response value where
i = 1, 1, … n; j = 1,2,….. k.
This is applied for problem where maximization of the quality
characteristics of interest is sought. This is referred as the large-the -better
type problem.
ii. Smaller – the – better
S / N ratio ( η) = −10 log 10
n
∑
i=1
1
y ij2
(2)
This is termed as the smaller-the-better type problem where
minimization of the characteristic is intended.
iii. Nominal-the-best
 µ2 
S / N ratio ( η) = −10 log 10  2 
σ 
where
µ=
(3)
yi + y 2 + y 3 + ... + yn
n
Σ ( yi − y )2
σ2 =
n−1
This is called nominal-the-best type of problem where one tries to
minimize the mean squared error around a specific target value. Adjusting
the mean on target by any means renders the problem to a constrained
optimization problem. Normalization is a transformation performed on a
single data input to distribute the data evenly and scale it into an acceptable
range for further analysis.
72
Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
Step 2 Normalization
The value yij is normalized as zij (0 ≤ zij ≤ 1) by the following formula to
avoid the effect of adopting different units and to reduce the variability.
It is necessary to normalize the original data before analyzing them with
the grey relation theory or other methodologies. An appropriate value is
deducted from values in the same array to make the value of this array
approximate to 1. Since the process of normalization affects the rank, we
also analyzed the sensitivity of the normalization process on the sequencing
results. Thus we recommend that the s/n ratio value be adopted when
normalizing data in grey relation analysis.
zij =
yij − min ( yij , i = 1, .....,n )
max( yij , i = 1, 2 , ...., n ) ) − min ( yij ,i = 1, 2 ,..., n )
(4)
(To be used for S/N ratio with larger the better manner)
zij =
max ( yij , i = 1, 2 , ..., n ) − yij
max( yij , i = 1, 2 , ...., n ) ) − min ( yij ,i = 1, 2 ,..., n )
(To be used for S/N ratio with smaller the better manner)
zij =
(
( y ij − target) − min(| y ij − target|, i = 1, 2, ...n)
)
max | y ij − target|, i = 1, 2, ...n − min (| y ij − target|, i = 1, 2, ...n)
(6)
(To be used for S/N ratio with nominal the best manner).
Step 3: Calculate the grey relation co-efficient for the normalized s/
n ratio values.
η ( y o ( k ), y i ( k )) =
∆ min + ξ∆ max
∆ oj ( k ) + ξ∆ max
(7)
where
1. j = 1, 2, ... n; k=1, 2,….m, n is the number of experimental data items
and m is the number of responses.
2. Yo (k) is the reference sequence (yo (k) = 1, k = 1, 2…m); yj (k) is the
specific comparison sequence.
73
Multi-Response Optimization of Process Parameters using Weight ...
3. ∆ oj = y o ( k ) − y j ( k ) = the absolute value of the difference between yo
(k) and yj (k)
min y o ( k ) − y j ( k ) is the smallest value of y (k)
4. ∆ min = min
j
∀j ∈i
∀k
max y o ( k ) − y j ( k ) is the largest value of y (k)
5. ∆ max = max
j
∀ j ∈i
∀k
6. ζ is the distinguishing coefficient, which is defined in the range 0 ≤ ζ ≤ 1
(The value may adjusted based on the practical needs of the system)
Step 4: Generate the grey relational grade.
γj =
1 m
∑ γ ij
k i =1
(8)
where γ j is the grey relational grade for the jth experiment and k is the
number of performance characteristics.
Steps 4: The grade values the effect of factor i, can be calculated by
multiplying with the weights and then the new grey grade can be obtained.
For Example the grey grade for first value in table (3) can be calculated by
Grey grade = GV1ij* weight1 + GV2ij*Weight2
Steps 5: Determine the optimal factor and its level combination.
The higher grey relation grade implies the better product quality;
therefore, on the basis of the grey relational grade, the factor effect can be
estimated and the optimal level for each controllable factor can also
be determined.
For example, The normal procedure for estimate the effect of factor i,
we calculate the average of grade values (AGV) for each level j, denoted
as AGVij, then the effect, Ei is denoted as:
Ei = max (AGVij) – min (AGVij)
(9)
If the factor i is controllable, the best level j*, is determined by
J* = maxj (AGVij)
(10)
Step 6: Perform ANOVA for identifying the significant factors.
The main purpose of the analysis of variance (ANOVA) is the
74
Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
application of a statistical method to identify the effect of individual factors.
Results from ANOVA can determine very clearly the impact of each factor
on the process results. The Taguchi experimental method could not judge
the effect of individual parameters on the entire process; thus, the
percentage of contribution using ANOVA is used to compensate for this
effect. The total sum of the squared deviations SST is decomposed into
two sources: The sum of the squared deviations due to each process
parameter and the sum of the squared error. The percentage contribution
by each of the process parameter in the total sum of the squared deviations
SST can be used to evaluate the importance of the process-parameter
change on the performance characteristics. Usually, the change of the
process parameter has a significant effect on the performance characteristic
when the F value is large.
Step 7: Calculate the predicted optimum condition.
Once the optimal level of the design parameters has been selected, the
final step is to predict and verify the quality characteristic using the optimal
level of the design parameters. Here we have used the factor levels obtained
by using Eqs. (9) and (10). The estimated S/N ratio using the optimal level
of the design parameters can be calculated as the following:
q
η = ηm + ∑ ( ηl − ηm )
i=1
ηm = Average S / N ratio
η = average S/N ratio corresponding to ith
(11)
significant factor on jth level
q = number of significant factors
9. Application of Weight Based Grey Relation Analysis for Wire
Electrical Discharge Machining Process Parameters
In the proposed method the Steps (1-3) are common for calculating the
grey coefficient. In Step 4 the weights have been multiplied with the grey
co-efficient to avoid the engineering judgement. In the proposed method
the weight are assigned based on Eigen values and also individual weight
have been determined for each coefficient.
75
Multi-Response Optimization of Process Parameters using Weight ...
Table 9
Normalized SN Ratio, Grey Relational Co-efficient and Grey Grade Values
Exp Normalised
No S/N Value
Grey Relational Grey
Co-efficient
Grade
Exp
No
Normalised
S/N Value
Grey Relational Coefficient
Grey
Grade
1
0.684234
0.875868
0.404301
0.612921
0.801113
0.563648
0.6555941
2
0.426164
0.734306
0.706833
0.46562
0.653002
0.483382
0.5313127
3
0.333333
1
0.45693
0.428571
1
0.466667
0.6233234
4
0.467247
0.709783
0.919324
0.484143
0.632737
0.492195
0.5341424
5
0.467564
0.427728
0.527768
0.484291
0.466299
0.492267
0.4813454
6
0.707751
0.882806
0.427087
0.631115
0.810118
0.57545
0.6686495
7
0.470304
0.539757
0.51644
0.48558
0.520702
0.492893
0.4992762
8
1
0.811487
0.452342
1
0.726202
1
0.9130236
9
0.779584
0.525095
0.90138
0.694043
0.51287
0.620381
0.6109124
10
0.707751
0.896977
1
0.631115
0.829156
0.57545
0.6746972
11
0.347141
0.896977
0.827126
0.433705
0.829156
0.468913
0.5715519
12
0.545927
0.817544
0.505693
0.524069
0.732648
0.512331
0.5862519
13
0.534056
0.613128
0.504202
0.517628
0.563779
0.508972
0.5292833
14
0.672436
0.919013
0.38667
0.604183
0.860604
0.55815
0.6696553
15
0.405513
0.855627
0.504202
0.456835
0.775948
0.479311
0.5660109
16
0.979649
0.749736
0.777829
0.960889
0.666432
0.927454
0.8557401
17
0.706315
0.669047
0.333333
0.629973
0.601719
0.574695
0.6018032
18
0.633468
0.333333
0.471422
0.577013
0.428571
0.541719
0.5176027
Table 10
Main Effects on Grade
Factors/Levels
Level 1
Level 2
Level 3
Max-Min
A
0.613064
0.6191774
B
0.607122
0.5748478
0.666393
0.091545
C
0.624789
0.6281153
0.595458
0.032657
D
0.590002
0.5241787
0.734182
0.210003
E
0.597055
0.6644136
0.586894
0.077519
F
0.627884
0.5897205
0.617912
0.038163
G
0.632216
0.5951477
0.614821
0.037068
0.006113
There fore the optimal combination of parameter value can be chosen
as per the quality characteristics.
76
Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
Graph 4: Optimal Graph Chart for WBGRA
The optimal combinations are A2B3C2D3E2F1G1.
8. COMPARISONS OF RESULTS
The initial settings for the WEDM process were A 2B1C1D2E2F2G1. The
optimal factor settings based on the previous research comparisons with
proposed methodology are A2B1C1D3E2F2G1, A1 B1 C1 D1 E2 F3 G2 and for the
proposed combinations are A2 B1 C1 D3 E2 F2 G1 and A2 B3 C2 D3 E2 F1 G1. To
find the improvements under the optimum condition, the SN ratio for all
the responses are determined using the additive model. The overall
improvement percentage is calculated as the ratio between sum of the
improvement values of all the responses and sum of SN ratios of initial
conditions of all responses. Table 11 presents the comparison of results.
From the table, it has seen that the results from equal weight GRA have
shown an improvement of 1.84 % from the initial condition. And the
proposed desirability weight method and weight based Grey relation
analysis has the improvement of 1.846 % and 0.58 %.
Table 11
Comparison of Solutions for Wire EDM Process Parameters
Responses
Initial
Grey
Relation
Analysis
MRR
17.02362
18.201233
SF
89.00231
90.939333
KERF
11.11263
10.160816
Optimal
A2B1C1D2
A2B1C1D3
Setting
E 2 F 2 G1
E 2 F 2 G1
Improvement of SN Ratio Value
MRR
1.1776133
SF
1.9370233
KERF
–0.9518133
Overall Improvement (%) 1.84
Factor
Analysis
Desirability
Method
(WBDM)
Weight Based
Grey Relation
Analysis
(WBGRA)
16.464683
90.782972
10.019744
A1B1C1D1
E 2 F 3 G2
18.20123333
90.93933333
10.16081667
A1B2C1D3
E 2 F 2 G1
19.1504833
87.9573833
10.71375
A2B3C2D3
E 2 F 1 G1
–0.5589367
1.7806622
-1.0928856
0.109
1.177613333
1.937023333
-0.95181333
1.846
2.12686333
-1.04492667
-0.39888
0.58
77
Multi-Response Optimization of Process Parameters using Weight ...
9. MULTI-RESPONSE OPTIMIZATION ON INJECTION
MOULDING PROCESS DISCUSSION CASE STUDY PROBLEMS
The company manufactures polypropylene components through
injection moulding process. The raw material is fed to the machine and
then the injection moulding process starts. The components are ejected by
automatic mechanisms which are inspected by visual inspection. Then
the components are sent to packing section.
The common defects occurring in the components are tearing along
the hinge and poor surface finish. The tensile strength and surface
roughness of the mould have been identified as the causes for the
above defects.
The factors and their levels considered in this study are shown in table
1. An experiment was conducted with three factors each at three levels
and hence a three level orthogonal array is chosen. Degrees of freedom
required for the design are 7. The OA, which satisfies the required degrees
of freedom, is L9. Experiments were conducted using L9 orthogonal array
and the response values obtained are given in table 13.
Table 12
Factor and Levels
Factor
Level 1
Melt Temperature (C )
Injection time (sec)
Injection Pressure (kg/cm2)
Level 2
250
3
30
Level 3
265
6
55
280
9
80
Table 13
Experimental Results and SN Ratio Values
No Control Factors
1
2
3
4
5
6
7
8
9
A
B
C Error
1
1
1
2
2
2
3
3
3
1
2
3
1
2
3
1
2
3
1
2
3
2
3
1
3
1
2
1
2
3
3
1
2
2
3
1
Tensile
Strength (TS)
1
2
1075
1044
1062
1036
988
985
926
968
957
1077
1042
1062
1032
990
983
926
966
959
Surface
Roughness (SR)
S/N Ratio Values
1
2
TS
SR
0.3892
0.3397
0.6127
0.964
0.4511
0.3736
1.2712
1.291
0.1557
0.3896
0.3399
0.6127
0.962
0.4515
0.3736
1.271
1.2908
0.1557
0.63624543
60.36568617
60.52249033
60.29041078
59.90392583
59.85990197
59.33221973
59.70852948
59.62731018
8.192081
9.375533
4.255042
0.327474
6.910963
8.551863
-2.08359
-2.21785
16.15423
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Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
10. DATA ANALYSIS
The data are analyzed using proposed weight based Grey Relation Analysis
(WBGRA) and Weight based Desirability method (WBDM).
10.1 Weight Based Grey Relation Analysis (WBGRA)
The experimental data are pre processed in order to normalize the SN
ratios. The SN ratio values are presented in Table 13. Table 14 shows the
normalized SN rations, with grey relational co-efficient and grey relational
grade for each experiment.
The main effects are tabulated in Table (14) and the factor effects are
plotted in graph 4. The optimal factor levels are obtained as A 1 B1 C1. The
factors on grade value in the order of significance B, A and C.
Table 14
Normalized SN Ratio, Grey Relational Co-efficient and Grey Grade Values
Exp.
No
Normalized SN
Ratio Values
TS
SR
Grey Relational
Co-efficient
TS
SR
Grey Grade
Value
1
1
0.433383
1
0.468772
0.390354505
2
0.79252
0.368967
0.706734
0.442074
0.298741054
3
0.912766
0.647678
0.851449
0.586633
0.371066525
4
0.734795
0.861457
0.65342
0.783033
0.353461949
5
0.438416
0.50313
0.470994
0.50157
0.241807953
6
0.404656
0.4138
0.456478
0.46032
0.22903195
7
0
0.992692
0.333333
0.985595
0.301293382
8
0.288575
1
0.412737
1
0.327579603
9
0.226292
0
0.392555
0.333333
0.184054046
Table 15
Table Main Effects on Grade
Symbol
Factors
A
B
C
1
2
3
Max-Min
Melt Temperature 0.353387
0.322112
0.270976
0.082412
Injection time
0.34837
0.289376
0.261384
0.086986
Injection pressure
0.315655
0.278752
0.304723
0.036903
79
Multi-Response Optimization of Process Parameters using Weight ...
Graph 4: Factor Effects on Grade
From ANOVA Table 16, it is shown that the controllable factor B
contributes 66.38% and A contributes 5.8 % respectively.
Table 16
Result of the Pooled ANOVA on Grade
Factor
A
SS
0.000498
Dof
2
MS
0.000249
F
0.41519
% Contribution
5.775922
B
0.005724
2
0.002862
4.771184
66.38831
Error
0.002399
4
0.0006
Total
0.008622
8
27.82417
10.2 Comparison of Results
The initial settings for the injection moulding process were A 3 B2 C3. The
optimal factor settings based on the previous research comparisons with
proposed methodology are A 3 B 3 C3, A 1 B 3 C2, and for the proposed
combinations are A1 B2 C1 A1B1C1. To find the improvements under the
optimum condition, the SN ratio for all the responses are determined using
the additive model. The overall improvement percentage is calculated as
the ratio between sum of the improvement values of all the responses and
sum of SN ratios of initial conditions of all responses. Table 17 presents
the comparison of results. From the table, it has seen that the results from
factor analysis have shown an improvement of 24.99% from the initial
condition. And the proposed desirability weight method and weight based
Grey relation analysis has the improvement of 10.0 % and 6.3 %.
80
Journal of Microwave Science and Technology, Volume 1, Nos. 1-2, January-December 2011
Table 17
Comparative Analysis Among Methodologies for Injection Moulding
Responses
Initial
Grey
Factor
Relation
Analysis
Analysis
(Jeyapaul’s Method)
TS
59.41345
59.42397
SR
0.675439
5.639692
Optimal
A 3 B2 C 3
A 3 B3 C 3
Setting
Improvement of SN Ratio Value
TS
0.01052
SR
4.964253
Overall Improvement (%) 8.279023
Desirability
Method
(WBDM)
Weight Based
Grey Relation
Analysis
(WBGRA)
60.55102
14.55468
A 1 B3 C 2
60.088889
6.238772
A 1 B2 C 1
60.60783161
3.269183667
A 1 B1 C 1
1.137565
13.87924
24.99099
0.675439
5.563333
10.38
1.194381606
2.593744667
6.3
11. CONCLUSIONS
This study discusses the feasibility for solving the multi-response problems
in a simplest way by assigning a new weight assigning method. Vijay
Kumar et al., (2006) the grey relation grade can be calculated by averaging
the grey relation coefficient corresponding to each performance
characteristic. Highest grey relational grade corresponds to the optimal
level of experiments. The comparison made between the average method
and the proposed weight assigned grey grade values and weight assigned
desirability method for obtaining the optimal solutions. The above
drawbacks have been identified and based on that the new weight
assigning method was proposed. The results have improved with different
optimal level combinations and also in overall improvement
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