Grey relation analysis (Deng, 1989a, b; Deng, 1987) is an important
part of grey system theory. The Grey theory can provide a solution of
a system in which the model is unsure or the information is incomplete.
It also provides an efficient solution to the uncertainty, multi-input
and discrete data problem. The relation between machining parameters and
performance can be found out with the Grey relational analysis (Deng,
1989a). It has been widely used in many applications (Chen et al.,
2000; Bin et al., 2002; Edwards et al., 1997; Baoqing et
al., 2004). In recent years, It also provides an efficient solution
to the uncertainty, multi-input and discrete data problems.
The relation between machining parameters and performance can be found
out with the Grey relational analysis. Grey relation analysis has even
been applied in facing recognition combining with other statistical methods
(Chen et al., 2000). It is important to select machining parameters
in Electrical Discharge Machining (EDM) for achieving optimal machining
performance (Tarng et al., 1995). The Taguchi method (Taguchi,
1990; Ghani et al., 2004), is a systematic application of design
and analysis of experiments for the purpose of designing and improving
product quality. Dr. Taguchi proposes the plan of quality project (called
the Taguchi method) with the robust design based on design of experiment,
to simplify a great quantity of fully factor experimentation. It had extensively
applied in engineering design and the analysis of optimal manufacturing.
To deliberate multiple performance characteristics by the Taguchi method
it requires further research effect. For the Electrical Discharge Machining
(EDM) process, material removal rate is a higher-the-better performance
characteristic. However, surface roughness and electrode wear ratio are
a lower-the-better performance characteristic. As a result, an improvement
of one performance characteristic may require a degradation of another
one. Hence, optimization of the multiple performance characteristics is
much more complicated than optimization of a single performance characteristic.
The grey relational analysis based on grey system theory can be used to
solve the complicated interrelationships among the multiple performance
characteristics effectively. In the grey relational analysis, a grey relational
grade is obtained to evaluate the multiple performance characteristics.
In this way, optimization of the complicated multiple performance characteristics
can be converted into optimization of a single grey relational grade.
It is shown by this study that the use of the orthogonal array with the
grey relational analysis can greatly simplify the optimization procedure
for determining the optimal machining parameters with the multiple performance
characteristics in the EDM process. As a result, the method developed
in this study is very suitable for practical use in a machine shop.
MATERIALS AND METHODS
The application of Design-of-Experiments (DOE) requires careful
planning, prudent layout of the experiments and expert analysis of the
results (Montgomery, 1997; Logothetis and Haigh, 1988). Taguchi has standardized
methods for each of these DOE application steps.
A statistical Analysis of Variance (ANOVA) is performed to identify
the process parameters that are statistically significant. Based on ANOVA
the optimal combinations of the process parameters are predicted.
In the grey relational analysis, the experimental results of electrode
wear ratio, material removal rate and surface roughness are first normalized
in the range between zero and one, which is also called the grey relational
generating. Next, the grey relational coefficient is calculated from the
normalized experimental results to express the relationship between the
desired and actual experimental results. Then, the grey relational grade
is computed by averaging the grey relational coefficient corresponding
to each performance characteristic. As a result, optimization of the complicated
multiple performance characteristics can be converted into optimization
of a single grey relational grade. Optimal level of the process parameters
is the level with the highest grey relational grade. Furthermore, a statistical
Analysis of Variance (ANOVA) is performed to see which process parameters
||Experimental layout using L27 orthogonal array
significant. With the grey relational analysis and statistical analysis
of variance, the optimal combination of the process parameters can be
predicted. Based on the above discussion, the orthogonal array with the
grey relational analysis to optimize the process is investigated.
The Taguchi method is devised for process optimization and identification
of optimal combinations of factors for given responses (Cox and Reid,
2000; Logothetis and Haigh, 1988). In the present analysis, an L27 orthogonal
array with four columns and twenty seven rows is used. This array can
handle four-three level process parameters and has twenty six degrees
of freedom. Therefore only twenty seven experiments are required to study
the entire machining parameters using the L27 orthogonal array.
The experimental layout for the machining parameters using the L27 orthogonal
array is shown in Table 1.
ELECTRICAL DISCHARGE MACHINING PROCESS
Electrical Discharge Machining is one of the most accurate manufacturing
processes available for creating complex or simple shapes and geometries
within parts and assemblies. EDM works by eroding material in the path
of electrical discharges that form an arc between an electrode tool and
the work piece. EDM manufacturing is quite affordable and a very desirable
manufacturing process when low counts or high accuracy is required.
The EDM system consists of a shaped tool or wire electrode and the part.
The part is connected to a power supply. Sometimes to create a potential
difference between the work piece and tool, the work piece is immersed
in a dielectric (electrically nonconducting) fluid which is circulated
to flush away debris.
EDM removes metal by discharging an electric current across a narrow
dielectric filled gap between the tool and the workpiece. It uses heat
to produce a tiny crater by melting and vaporization. Common methods of
evaluating machining performance in the EDM operation are based on the
following performance characteristics: material removal rate, surface
roughness and electrode wear ratio. Basically, material removal rate,
surface roughness and electrode wear ratio are correlated with the machining
parameters such as workpiece polarity, pulse on time, duty factor, open
discharge voltage, discharge current and dielectric fluid. Proper selection
of the machining parameters can obtain higher material removal rate, better
surface roughness and lower electrode wear ratio.
Machining Parameter Selection
The essential steps include identifying the factors that are to be
included in the study and determining the factor levels. It was decided
to study the effect of the parameters viz., pulse on-time (A) in the range
20-300 μs, Discharge current (B) in the range 1.5-6 A, Discharge
voltage (C) in the range 100-150 V and Duty factor (D) in
the range 0.3-0.7 on
The responses viz., Material Removal Rate (MRR), Electrode Wear Ratio
(EWR) and Surface Roughness (SR). The range of the input parameters was
fixed as shown in Table 2.
In this study, L27 orthogonal array was chosen that include
13 columns that can be used to assign test factors and their interactions.
For 3 factor with 3 level setup the total number of experiments to be
conducted is given by 33 = 27.
||Machining parameters and their levels
Machining Performance Assessment
Material removal rate, surface roughness and electrode wear ratio
are used to evaluate machining performance. The Material Removal Rate
(MRR) is expressed as the Work Piece Removal Weight (WRW) over a period
of machining time (T) in min, that is:
The Electrode Wear Ratio (EWR) is defined as the ratio of the Electrode
Wear Weight (EWW) to the Work Piece Removal Weight (WRW) and is usually
expressed as a percentage, that is:
The higher the material removal rate in the EDM process, the better
is the machining performance. However, the smaller the electrode wears
ratio and surface roughness in the EDM process, the better is the machining
performance. Therefore, the material removal rate is the higher-the-better
performance characteristic and the electrode wear ratio and surface roughness
are the lower-the-better performance characteristics.
The experimental runs conducted on Aristech LS-350 model EDM machine.
a pure copper with diameter 6.00 mm is used as a electrode. The work piece
has been made by tool steel SKD11. lonoPlus 3000 is used as a dielectric.
After removing material from work piece, the result surface roughness
is measured using profile meter (3D-Hommelewerk). The experimental results
are shown in Table 3.
||Experimental results for machining parameters
RESULTS AND DISCUSSION
Normalization of the Experimental Results and Computing the Grey
Relational Analysis for the Experimental Results
A linear normalization of the experimental results for the responses
viz. MRR, EWR and SR is performed in the range between 0 and 1, which
is called as the Grey relational generating. In the study, a linear data
preprocessing method (Hsia and Wu, 1998) for the material removal rate
can be expressed as:
And for electrode wear ratio and surface roughness can be expressed
where, yij is the ith experimental results in the jth experiment.
According to Deng (1989b), larger normalized results correspond to the
better performance and the best-normalized result should be equal to 1.
Computing the Grey Relational Coefficients
In the grey relational analysis, experimental data (electrode wear
ratio, material wear rate and surface roughness) are first normalized
in the range between 0 and 1, which is also called the grey relational
generating. Next, the grey relational coefficient is calculated from the
normalized experimental data to express the relationship between the desired
and actual experimental data. The Grey relational coefficient δij
can be expressed as:
||The gray relational coefficient
the ideal normalized results for the ith performance characteristics and
ξ is the distinguishing coefficient which is defined in the range
Table 4 shows the grey relational coefficient for each
experiment using the L27 orthogonal array.
COMPUTING THE GREY RELATIONAL GRADES
The grey relational grade is computed by averaging the grey relational
coefficient corresponding to each performance characteristic. The overall
evaluation of the multiple performance characteristics is based on the
grey relational grade.
where, γj is the Grey relational grade for the jth experiment
and m is the number of performance characteristics. The higher the Grey
relational grade represents that the experimental result is closer to
the ideally normalized value. in here, experiment 20 has the best multi
response characteristics among the 27 experiments conducted (Table
5). The mean of the Grey relational grade for each level of the machining
parameter can be calculated by averaging the Grey relational grade for
pulse on time for experiment number 1-9 as level 1, experiment number
10-18 as level 2 and experiment number 19-27 as level 3. Similarly, it
is calculated for the respective levels for discharge current, discharge
voltage and duty factor and in addition, the total mean of the grey relational
grade for the 27 experiments is also calculated and is shown in Table
6. The larger the value of the Grey relational grade, the better is
the multi response characteristics.
||Gray relational grade for each experiment
||Response table for the gray relational grade
|Mean value of the Gray relational Grade = 0.6156
||The analysis of variance table for process parameters.
Analysis of variance table [classical sum of squares-type II]
Analysis of Variance
Since the experimental design is orthogonal, it is then possible to
separate out the effect of each process parameter at different levels.
The mean of the gray relational grade for each process parameter is calculated
(Table 6). The optimal level of the process is the level
with the greatest gray relational grade. According to Ghani et al.
(2004), from figure of the gray relational grade for various levels (Table
6), it can be visually understood about to be significance of each
parameter. Where there is a large slope for the graph lines, it can be
understood that the parameter has significant effect on process. In this
study, it can be found that there is no line with a significant slope
for a pulse on time; however the slope of line for discharge voltage and
duty factor is fairly high. May be it can visually understood that discharge
current has a significant effect too. But judge about that is relatively
hard. However, visually judgment about the process is a strait forward
method and there is no need for complex analysis of variance and everybody
without more knowledge can get useful information about the process. However,
for more precision it has been came ANOVA analysis too to do confirmation
on conceptual approach (Table 1).
From statistical analysis it known that p-value less than 0.05 shows
that specific parameter has significant effect on process and also p-value
between 0.05 and 0.1 shows the low significant parameter (Table
7). If the p-value is more than 0.1 it understood that the specific
parameter has not significant effect on process. the ANOVA result for
this study indicates that pulse on time is low significant, while three
other process parameters means, Discharge current, discharge voltage and
duty factor are most significant affecting the multiple process response.
As it can be seen (Table 6), the conceptual approach
for grey relational analysis is confirmed by the ANOVA analysis.
Electric Discharge Machining (EDM) is a relatively new method to
product a precision parts. The optimization of multiple performance characteristics
of EDM process using Grey relational analysis based on Taguchi orthogonal
array is investigated. The response table and response graph for each
level of the machining parameters is obtained. The study of grey relational
graph for machining parameters shows that pulse on time is low significant,
while three other process parameters means, discharge current, discharge
voltage and duty factor are most significant affecting the multiple process
response. While the greater grey relational grade is the best, so level
2 for Pulse on-time, Discharge current as well as Duty factor and level
3 for Discharge voltage is proposed.
It means that by selecting these levels for parameters, it result the
maximum material removal rate, minimal electrode wear ratio and low Surface
roughness. These optimal points are for condition that three responses
are important simultaneously and where there is only one criterion for
example surface roughness; maybe different optimal point will be achieved
by analysis. It seems that Gray relational analysis is a strait forward
method for optimizing multi-criteria problems in EDM process where two
or more response can be study simultaneously.