1. Department of Industrial Engineering, Anna University, Chennai, 600025, India,
e mail: nishmech@gmail.com
Abstract—The Taguchi method aims at reducing response
variation from target so as to increase yields and product lifetime,
reduce defects, and improve performance. However, the Taguchi
method is only efficient for optimizing a single quality response.
This research, therefore, proposes a Grey relational analysis –
Simulated Annealing approach for solving the multiresponse
problem in the Taguchi method. The proposed approach includes
two phases. The first phase utilizes grey relational analysis to
calculate grey relational co-efficient for the normalized S/N ratio
values for all combinations of factor levels. Whereas, the second
phase employs simulated annealing to decide optimal combination
of factor levels for multiresponse problem. Here the multi
response is converted in to single response by sum of each
responses multiplying with its optimal weight (optimal weights are
determined using Simulated Annealing), then solved by Taguchi
method. A case study is provided for illustration. In conclusion,
the G-SA approach may provide a great assistance to practitioners
for solving the multiresponse problem in real life applications on
the Taguchi method.
Keywords— Grey Relational Analysis, Simulated Annealing,
Taguchi method, multiresponse problem.
I. INTRODUCTION
In practice, a great deal of engineering time is spent
generating information bout how different design
parameters affect performance under different usage
conditions. The Taguchi [1] method is a widely accepted
approach for robust design. The overall goal of robust
design is to find settings of the controllable factors so that
the response is least sensitive to variations in the noise
variables, while still yielding an acceptable mean level of
the response. Generally, a process’s or a product’s quality
response can be divided into three main types: the smaller-
the-better (STB); the nominal-the-best (NTB); and the
larger-the-better (LTB) type responses. To optimize a
quality response by the Taguchi method, an orthogonal
array (OA) is utilized to reduce the number of experiments
under permissive reliability. Then, signal-to-ratio (S/N)
ratio is employed as a quality measure to decide the optimal
combination of factor levels.
In today’s highly competitive markets, however,
customers are concerned about more than one quality
response. The Taguchi method has been extensively
employed in manufacturing to robustly design a product or
process with only one quality response [2-3]. Recently, the
multiresponse problem in the Taguchi method has received
an increasing research attention. For example, Phadke [4]
employed pure engineering judgment for optimizing
concurrently three quality responses in a very-large-scale
integrated circuit-manufacturing process. However, human
judgment increases uncertainty in decision-making process.
Pignatello [5] discussed a manufacturing process with five
responses. They used data-driven transformations for each
response variable in a multiple-univariate or one-at-a-time
manner. Then, the regression technique based approaches
were utilized to determine tentative optimal factor levels.
Also, Reddy et al. [6] employed regression techniques
based approaches during unifying goal programming to
optimize simultaneously several responses. Regression
approaches, however, increase the complexity of
computational process. Thereafter, Antony [7] utilized
principal component analysis (PCA) to transform the
multiresponses in few uncorrelated ones, which were then
employed for solving the multiresponse problem. However,
PCA is based on some rigid assumptions, such as the error
terms are multivariate normally distributed random
variables, which may limit its use in practical applications.
Jeyapaul et al. [8] utilized genetic algorithm for determining
a weight for the S/N ratio of each response. Then, the
weighted sum of S/N ratios was used to decide optimal
factor levels. However, the genetic algorithm is a search
heuristic that provides near optimal solutions for complex
search spaces, such as scheduling and transportation
problems.
Grey relational analysis and Simulated Annealing (SA)
have been broadly used for optimizing many a
manufacturing process or a product. This research,
therefore, provides a combined approach of Grey relational
analysis and Simulated Annealing for solving the
multiresponse problem in the Taguchi method. Relevant
background of Grey relational analysis and Simulated
Annealing are introduced in Section II. The proposed
Grey-Simulated Annealing Approach for Solving the
Multiresponse Problem in Taguchi Method
Nishanth.G, Rajmohan.M

2. approach is outlined in Section III. A case study is provided
for illustration in SectionIV. Finally, conclusions are made
in SectionV.
II. RELEVANT BACKGROUND
A- Grey Relational Analysis
Grey relational analysis, proposed by Deng [9], is a
method of measuring degree of approximation among
sequences according to the grey relational grade. Grey
relational analysis is part of grey system theory, which is
suitable for solving the complicated interrelationships
between multiple factors and variables. The major
advantage of Grey theory is that it can handle both
incomplete information and unclear problems very
precisely. The grey relational analysis has been widely
employed for solving the multiresponse problem in many
manufacturing applications [10-13].
Grey analysis uses a specific concept of information. It
defines situations with no information as black, and those
with perfect information as white. However, neither of these
idealized situations ever occurs in real world problems. In
fact, situations between these extremes are described as
being grey, hazy or fuzzy. Therefore, a grey system means
that a system in which part of information is known and part
of information is unknown. With this definition, information
quantity and quality form a continuum from a total lack of
information to complete information – from black through
grey to white. Since uncertainty always exists, one is always
somewhere in the middle, somewhere between the extremes,
somewhere in the grey area.
Grey analysis then comes to a clear set of statements
about system solutions. At one extreme, no solution can be
defined for a system with no information. At the other
extreme, a system with perfect information has a unique
solution. In the middle, grey systems will give a variety of
available solutions. Grey analysis does not attempt to find
the best solution, but does provide techniques for
determining a good solution, an appropriate solution for real
world problems.
Based on the above introduction, this research proposes a
combined approach using grey relational analysis and
Simulated Annealing for solving the multiresponse problem
in the Taguchi method.
B- Simulated Annealing
Simulated annealing was developed in the mid 1970's by
Scott Kirkpatric, along with a few other researchers.
Simulated annealing was original developed to better
optimize the design of integrated circuit (IC) chips.
Simulated annealing simulates the actual process of
annealing. Annealing is the metallurgical process of heating
up a solid and then cooling slowly until it crystallizes. The
atoms of this material have high energies at very high
temperatures. This gives the atoms a great deal of freedom
in their ability to restructure themselves. As the temperature
is reduced the energy of these atoms decreases. If this
cooling process is carried out too quickly many
irregularities and defects will be seen in the crystal
structure. The process of too rapid of cooling is known as
rapid quenching. Ideally the temperature should be
deceased at a slower rate. A slower fall to the lower energy
rates will allow a more consistent crystal structure to form.
This more stable crystal form will allow the metal to be
much more durable.
Simulated annealing seeks to emulate this process.
Simulated annealing begins at a very high temperature
where the input values are allowed to assume a great range
of random values. As the training progresses the
temperature is allowed to fall. This restricts the degree to
which the inputs are allowed to vary. This often leads the
simulated annealing algorithm to a better solution, just as a
metal achieves a better crystal structure through the actual
annealing process.
The process of annealing is one in which a solid, usually
metal, is first heated, and then allowed to cool slowly. As
the solid cools, a change of state takes place in which
individual atoms arrange themselves into a regular array
corresponding to a minimum energy arrangement. Such an
arrangement cannot easily propagate throughout the solid if
the cooling occurs quickly, and boundaries between
different ``domains of regularity'' occur. Such boundaries
introduce potential “fault-lines” along which a fracture is
most likely to occur when the material is stressed. To avoid
such potential failures, metal is often cooled slowly, in a
process known as annealing to permit re-arrangements at
these boundaries so the same local minimum energy
arrangement occurs throughout the material.
This process is imitated in numerical optimization. The
idea originated with metropolis et al 1953 when trying to
simulate such thermodynamic systems. Given a potential
state change from one with energy E1 to one with energy
E2, they chose to accept it with a probability
( )











 −−
=
kT
EE
acceptprob 12
exp,1min)(
Where T is the ``temperature'' and k is a constant in this
application it is Boltzmann's constant. In words this always
accepts a change if it moves to a state of lower energy, but
sometimes accepts the change even though the system
moves to a state with a higher energy. Note that for small T
there is a very small probability of accepting an unfavorable
move, while for large T, the probability of acceptance can
be quite high. With this in mind we now describe the
requirements in order to apply the same ideas to a more
general minimization problem.
We need
1. A coding of the possible system states;

3. ),.....,2,1,min(),.....,2,1,max(
),.....,2,1,max(
niyniy
yniy
Z
ijij
ijij
ij
=−=
−=
=
2. An objective function which we are trying to
minimize;
3. A mechanism for proposing random changes to the
state of the system; and
4. A control parameter T, analogous to the
temperature above, which governs the probability of
acceptance of the proposed change, together with an
annealing schedule specifying how the temperature is to be
lowered.
III. METHODOLOGY
The procedure for the application of Grey relational
analysis and Simulated Annealing (SA) to solve multi-
response problems is given below.
Step 1: Choose the appropriate orthogonal array for the
problem and design the experiment layout. The selection of
OA depends on the number of factors (f) and the number of
interactions (if any). The number of degrees of freedom
associated with the experiment is always greater than or
equal to the number of degrees of freedom required for
studying the main and interaction effects.
Step 2: Conduct the experiment by setting levels as per the
selected orthogonal array and obtain the responses ( yij ).
Step 3: Calculate the SN ratio for a given response using
one of the formulae depending upon the type of quality
characteristic.
i Larger-the-better
S/N Ratio (η) = ∑=






−
r
i ijyr 1
210
11
log10 (1)
Where r = number of replications; yij=observed response
value, where i=1, 2...n; j=1, 2...k;
This is applied for problems where maximization of the
quality characteristic of interest is sought. This is referred to
as the larger-the-better type problem.
ii Smaller-the-better
S/N Ratio (η) = 





⋅− ∑=
r
i
ijy
r 1
2
10
1
log10 (2)
This is termed a smaller-the-better type problem where
minimization of the characteristic is intended.
iii Nominal-the-best
S/N Ratio (η) = 





2
2
10log10
σ
µ
(3)
Where
r
yyy r+⋅⋅⋅++
= 21
µ
( )
∑= −
−
=
r
i
i
r
yy
1
2
1
σ
This is called a nominal-the-best type of problem where
one tries to minimize the mean squared error around a
specific target value. Adjusting mean to the target by any
method renders the problem to a constrained optimization
problem.
Step 4: yij is normalized as Zij (0 ≤Zij ≤ 1) by the following
formula to set right the effect of adopting different units:
),.....,2,1,min(),.....,2,1,max(
),.....,2,1,max(
niyniy
niyy
Z
ijij
ijij
ij
=−=
=−
= (4)
(To be used for SN ratios with larger-the-better manner)
(5)
(To be used for SN ratios with smaller-the-better manner)
(6)
(To be used for SN ratios with nominal-the-best manner)
Where j = 1, 2… k;
Step 5: Grey Relational Analysis: Calculate the grey
relational co-efficient for the normalized S/N ratio values.
max)(
maxmin
))(),((
∆+∆
∆+∆
=
ξ
ξ
γ
k
kyky
oj
io (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.
3. =−=∆ ||)()(|| kyky jooj The absolute value of
the difference between yo(k) and yj(k).
( )
minmax
minarg
DVDV
DVetTy
Z
ij
ij
−
−−
=
( )nietTyDV ij ,,2,1,argmaxmax ⋅⋅⋅⋅=−=
( )nietTyDV ij ,,2,1,argminmin ⋅⋅⋅⋅=−=

4. 4. ||)()(||minminmin kyky jo
kij
−=∆
∀∈∀
is the
smallest value of yj(k).
5. ||)()(||maxmaxmax kyky jo
kij
−=∆
∀∈∀
is the
largest value of yj(k).
6. K is the distinguishing coefficient, which is defined
in the range 0≤K≤1 (the value may adjusted based
on the
practical needs of the system)
Step 6: Simulated Annealing: The procedure for the
application of SA is presented below
Step 6.1: Generation of Initial Seed: In this algorithm
initialization is often carried out randomly. The initial seed
is generated randomly. But the sum of weight of all the
response should be equal to one. The objective of this
algorithm is to get the optimal weights so as to maximize
the grey grade.
Step 6.2: Evaluation: Calculate the value of the objective
function for the initial seed. The objective function
quantifies the performance level of the seed. The objective
function value for the problem is given
f(x) = ∑∑= =
k
j
n
i
ijj ZW
1 1
(8)
Where f(x) is the total weighted grey grade (WGG) ratio to
be maximized,
Wj= Weights for each response,
Zij= Grey coefficient values,
n = Number of Experiments under each response, and
k = Number of responses.
Step 6.3: Neighborhood generation: Generate some of
the neighborhoods to the initial seed and Evaluate in the
objective function. The neighborhood weights are generated
by pair wise transfer some less percentage of value from
one weight to another one within the seed. This percentage
which is to be transferred is determined randomly. The best
neighborhood which has maximum WGG among the
neighborhoods is taken and compared with the initial seed
objective function value to check whether it is acceptable or
not.
Step 6.4: Probability for acceptance: When the
neighborhood seed gives the inferior solution, we go to
check for the probability of acceptance.
Probability (Pacc) =
( )





 − =
x
xx
T
ff 1
exp (9)
If the probability of acceptance is within the
generated random number, the inferior seed is accepted.
Step 6.5: Termination condition: When the program
meets the final temperature or number of continuous
unacceptable inferior neighborhood generation (freezer
count) is equal to pre-specified value, the program will stop.
Step 7: Get the final best seed which maximizes the
objective function from the simulated annealing algorithm
and using this weights (W1, W2...Wj), calculated weighted
grey grade
WGG = W1Z11 + W2Z12 + …..+ WjZij (10)
Step 8: Determine the optimal level combination for the
factors. Maximization of weighted grey grade consequences
the better product quality; therefore, on the basis of
weighted grey grade, the main effects of the control factor is
calculated and the optimal level for each controllable factor
is determined. For example, to calculate the main effect
of factor i on Weighted grey grade, we calculate the average
of weighted grey grade values (WGG) for each level j,
denoted as WGGij, then the difference in the main effect, εi,
is defined as:
( ) ( )ijiji WGGWGG minmax −=ε (11)
The best level j* of the controllable factor ‘i’ is selected by
j* = max (WGGij) (12)
Step 9: Perform ANOVA to identify the significant factors
and percentage of contribution of the factors.
Step 10: Calculate the Predicted value of SN ratio for the
selected optimal levels and calculate the improvement in SN
ratio and overall improvement percentage as the ratio
between sum of the improvement values of all responses
and sum of SN ratios of initial conditions of all responses.
The predicted SN ratio using the selected optimal level is
can be calculated as:
( )∑=
−+=
f
i
mim
1
ηηηη (13)
Where,

5. mη = Average SN ratio.
η = Average SN ratio corresponding to ith
factor on fth
level.
f = Number of factors.
IV. IMPLEMENTATION OF THE SOLUTION METHODOLOGY -
CASE STUDY
In this work, LM 25-based aluminium alloy (Cu:
7.15%,Mg: 0.49%, Mn: 0.11%, Fe: 0.47%, Ni: 0.002%, Ti:
0.064%, Zn: 0.017%, Pb: 0.003%, Sn: 0.005%) reinforced
with green bonded silicon carbide particle of size 25 µm
with 10% volume fractions manufactured through stir
casting route is used for experimentation. The drilling tests
are carried out on radial drilling machine under dry
condition. In order to conduct experiments, the work
materials are cut into plates of 150×50×20 mm and faced in
a lathe to obtain flat surface. Then the plate is fastened to
the rigid fixture attached to the strain gauge dynamometer
which is mounted on the table. Equal spacing is maintained
between successive drilled holes in the plate. The cutting
point of a standard HSS twist drills of 10 mm diameter with
various cutting point angles (90, 115 and 140 degrees),
coated by TiN are used throughout the experimental work.
The average surface roughness (Ra), cutting force (Fc) and
torque (T) are considered as responses for this study. The
surface roughness is measured at three positions spaced at
120° intervals around the hole circumference. The surface
roughness of each hole is taken as the mean of three
circumferential readings. The cutting force and torque for
each trial is measured by using strain gauge dynamometer.
Plan of investigation
The factors and their levels considered in this study are
shown in Table 1. Experiments are conducted with three
factors each at three levels and hence a three level
orthogonal array (OA) is chosen. Degrees of freedom (Dof)
required for the design are six. The OA, which satisfies the
required Dof is L9.The experiments are conducted using L9
OA and the response values obtained are given in Table 2.
Step 1: Calculate the S/N ratios for a given response and
predicted S/N ratios of the starting conditions using one of
the Eqs. (1), (2) and (3) depending upon the type of quality
characteristics. The computed S/N ratios for each quality
characteristic are shown in Table 3.
Table 1. Factors and levels
Levels
Parameters Unit
1 2 3
Cutting
speed (V)
m/min 35.18 56.54 87.96
Feed (F) mm/rev 0.050 0.125 0.20
Point angle
(PA)
Degree 90 115 140
Table 2. L9 Orthogonal array with factors and responses
Responses
Trial
No.
V F PA
Ra in
µm
Fc in N
T in
Nm
1 1 1 1 7.83 107.87 0.88
2 1 2 2 4.01 254.96 2.06
3 1 3 3 2.22 470.67 2.26
4 2 1 2 6.70 186.31 1.96
5 2 2 3 5.80 539.331 1.28
6 2 3 1 6.09 1186.53 3.24
7 3 1 3 6.01 274.57 0.69
8 3 2 1 8.27 1078.66 2.55
9 3 3 2 6.20 1274.78 2.16
Step 2: Normalize the S/N ratio values by Eqs. (4), (5) and
(6). The results are given in Table 3.
Step 3: Perform the grey relational analysis. From the data
in Table 3, calculate the grey relational co-efficient for the
normalized S/N ratio values by using Eq. (7). The value for
ξ∆max is taken as 0.5 in Eq. (7). Since all the process
parameters are of equal weighting. The results are given in
Table 4.
Table 4. Grey relational co-efficient
Grey relational co-efficient
Trial No.
Ra Fc T
1 0.923 0.333 0.372
2 0.476 0.434 0.631
3 0.333 0.553 0.682
4 0.757 0.391 0.606
5 0.650 0.589 0.454
6 0.682 0.945 1.000
7 0.673 0.446 0.333
8 1.000 0.881 0.764
9 0.695 1.000 0.656
Table 3. S/N ratio values and normalized S/N ratio values
S/N ratios Normalized values of S/N ratios Zij
Trial No.
Ra Fc T Ra Fc T
1 −17.875 −40.658 1.110 0.958 0.000 0.157

6. 2 -12.063 -48.129 -6.277 0.450 0.348 0.707
3 -6.927 -53.454 -7.082 0.000 0.597 0.767
4 -16.521 -45.405 -5.845 0.840 0.221 0.675
5 -15.269 -54.637 -2.144 0.730 0.652 0.400
6 -15.692 -61.486 -10.211 0.767 0.971 1.000
7 -15.577 -48.773 3.223 0.757 0.378 0.000
8 -18.350 -60.658 -8.131 1.000 0.932 0.845
9 -15.848 -62.109 -6.689 0.781 1.000 0.738
Step 4: By applying SA, the optimal weights corresponding
to each response are obtained. The optimal weights are [
0.997172, 0.000932, 0.001896 ]. So the WGG equation is
WGGi1 = 0.994582 Zi1 + 0.00352 Zi2 + 0.001898 Zi3
where Zi1, Zi2 and Zi3 represent the grey grade values for the
responses Ra, Fc and T at ith
trial respectively. The WGG
values are computed and listed in Table 5.
Table 5. Grey relational co-efficient and weighted grey grade
Grey relational co-efficient
Trial No.
Ra Fc T
WGG
1 0.923 0.333 0.372 0.9531075
2 0.476 0.434 0.631 0.4511233
3 0.333 0.553 0.682 0.0035572
4 0.757 0.391 0.606 0.8385025
5 0.650 0.589 0.454 0.7300956
6 0.682 0.945 1.000 0.7681603
7 0.673 0.446 0.333 0.7542291
8 1.000 0.881 0.764 0.9994665
9 0.695 1.000 0.656 0.7816893
Step 5: From the Table 6, the effect on WGG and Fig.1
plots their corresponding factor effects. The controllable
factors on WGG value in order of significance are V, F, PA.
The larger the WGG value implies the better the quality.
Consequently, the optimal condition can be set in the order
V3F1 PA1.
Table 6, Main effects on weighted grey grades
Factors Level 1 Level 2 Level 3 Max-Min
Cutting
speed (v)
0.4693 0.7789 0.8451 0.375866
Feed (F) 0.8486 0.7269 0.5178 0.330811
Point angle
(PA) 0.9069 0.6904 0.496 0.410951
Fig. 1 Factor Effects on WGG Values
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V1 V2 V3 F1 F2 F3 PA1 PA2 PA3
V. ANALYSIS OF VARIANCE
The purpose of the ANOVA is to examine the process
parameters which significantly affect the performance
characteristics. This is accomplished by separating the total
variability of the multi-response weighted grey grade, which is
measured by the sum of the squared deviations from the total
mean of the WGG, into contributions by each of the process
parameters and the error. The Results of the Pooled ANOVA is
shown in Table 7.
Table 7, Results of pooled ANOVA

7. VI. CONCLUSION
Taguchi method can optimize the single response problem,
but cannot optimize the multiple responses problem and
currently receive little attention to multi response problem. In
many cases, pure engineering judgments are used to optimize
multiple responses, which often bring a certain degree of
uncertainty to the decision-making process. This paper has
presented the use of Grey relational analysis – Simulated
annealing algorithm (G-Sa) and ANOVA to the Taguchi
method for the optimization of the face milling process with
multiple performances characteristic.
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Factor SS Dof MS F % Contribution
V 0.24155164 2 0.1207758 3.35811141 32.8645264
F 0.16792635 2 0.0839632 2.33455417 22.847371
PA 0.25358314 2 0.1267916 3.52537628 34.5014825
Error 0.07193087 2 0.0359654 9.78662013
Total 0.734992 8 100