48

CHAPTER 3

WEIGHT-BASED GREY RELATION ANALYSIS

TO SOLVE MULTI-RESPONSE PROBLEMS

3.1 INTRODUCTION

The application of parameter design methodology has been

considerable in recent years to make system performance robust over a wide

range of input conditions. This notion has been referred to as a robust design

with dynamic characteristics. Due to product complexity, multiple correlated

characteristics must be simultaneously evaluated for improving product

quality. Multi-response optimization is becoming an important issue to

contemporary industry. The methodology that utilizes both Taguchi method

and principal component analysis (PCA) is a quite practical and effective

procedure for tackling multi-response problems. By using PCA, a set of

original responses is transformed into a set of uncorrelated components so

that the optimal factor/level combination can be found. However, there are

several limitations that may come across when we deal with real applications.

Those cases studied in the published literature have had one Eigen value

larger than one; and thus, the first principal component has been the only

component extracted. This does not occur in the majority of cases in today’s

complex manufacturing processes we have.

In particular, it is doubtful whether the factor/level combination,

determined by the first principal component only, is optimal or not. Therefore,

the optimizing Eigen value larger than one and more than one principal

49

component is extracted. This study has developed a novel procedure of

optimizing dynamic multi-responses using principal component analysis

(PCA) and multiple response evaluation of the grey relation model. PCA can

consider the correlations among multiple quality characteristics to obtain

uncorrelated components. These components are then substituted into

multiple response evaluation of the grey relation model to determine the

optimal factor level combination. A case study demonstrates the effectiveness

of the proposed procedure for optimizing multi-response processes.

3.2 GREY RELATION ANALYSIS

Deng (1989) proposed a Grey relational analysis. The Grey

relational analysis is a method for measuring the degree of approximation

among the sequences using a Grey relational grade. Theories of the Grey

relational analysis have attracted considerable interest among researchers.

Some other researchers have also examined the optimization of process

parameters. For example, Huang and Lin (2002) applied the Grey relational

analysis to design the die-sinking EDM machining parameters. Fung et al.

(2003) studied the Grey relational analysis to obtain the optimal parameters of

the injection molding process for mechanical properties of yield stress and

elongation in polycarbonate/ acrylonitrile butadiene-styrene (PC/ABS)

composites. Shen et al (2004) studied different polymers (such as PP, PC, PS,

POM) with various process parameters of the micro gear. The simulation used

the Taguchi method and the Grey relational analyses were provided. Lin

(2004) employed the Taguchi method and the Grey relational analysis to

optimize the turning operations with multiple performance characteristics.

Chiang and Chang (2006) used the Grey relational analysis to

optimize the wire electric discharge machining process of particle-reinforced

material with multiple performance characteristics. Yang et al (2006) also

applied the Taguchi method and the Grey relational analysis to optimize the

50

dry machining parameters for high-purity graphite in end milling process.

Planning the experiments through the Taguchi orthogonal array has been used

quite successfully in process optimization by Chen and Chen (2007); Fung

and Kang (2005); Tang et al (2007); Vijian and Arunachalam (2006); Yang

(2007); Zhang et al (2007), etc. Grey relational analysis compares quantitative

analysis to the development between every factor in the grey system

dynamically, and describes the relation degree among main factor and other

factors in the grey system.

3.3 OPTIMIZATION STEPS USING WEIGHT-BASED GREY

RELATION ANALYSIS

The steps used for determining the optimal combinations in the

existing method for both desirability index and grey relation are replicated in

the proposed method except the weight assignment for the process parameters

for converting the multi-responses into single response.

3.3.1 Weight-Based Grey Relation Analysis

Step1: Calculate S/N ratio using the following formula.

i. Larger–the–Best

l

n1

10 2n i 1 ij

1S / N ratio( ) 10 logy

(3.1)

where n = number of replications yij = Observed response value

where i =1,2,…n; j = 1,2,…..k.

51

ii. Smaller–the–Better

n

2ijs 10

i 1

1S / Natio( ) 10 log yn

(3.2)

This is termed the smaller-the-better type problem where

minimization of the characteristic is intended.

iii. Nominal-the-Best

n

2

10 2μS/N ratio(η ) =10logσ

(3.3)

where 1 2 3 n(y y y y ) / n ; 2

2 i(y y)n 1

Step 2: Normalize the S/N values

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.

ij ij,

ijij, ij

y min (y i 1,2,...,n)z

max (y i 1,2,...,n)) min (y ,i 1,2,..., n)

(3.4)

(To be used for S/N ratio with larger the better manner)

ij ijij

ij ij

max (y ,i 1,2,..., n) yz

max (y ,i 1,2,...,n) min (y ,i 1,2,..., n)

(3.5)

(To be used for S/N ratio with smaller the better manner)

ij ijij

ij ij

(y t arget) min ( y t arg et ,i 1,2,..., n)z

max( y t arg et ,i 1,2,...n) min( y t arg et ,i 1,2,...,n)

(3.6) (To be used for S/N ratio with nominal the best manner).

52

Step 3: Calculate the grey relation co-efficient for the

normalized S/N ratio values

0 ioj

min max(y (k), y (k))(k) max

(3.7)

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. oj o jy (k) y (k) = the absolute value of the difference

between yo (k) and yj (k)

4. j i k o jmin min min y (k) y (k) is the smallest value of

yj(k)

5. o jj i kmax max max y (k) y (k)

is the largest value of yj(k)

6. is the distinguishing coefficient, which is defined in the

range 0 1 (the value may be adjusted based on the

practical needs of the system).

Step 4: Generate the grey relational grade.

m

j iji 1

1k

(3.8)

where j is the grey relational grade for the jth experiment and k is

the number of performance characteristics.

53

The grade values for the effect of factor i can be calculated by

multiplying with the corresponding weights. For example, the grey

grade for first value in table (3.5) can be calculated by

Grey grade = GV1ij* weight1 + GV2ij*Weight 2 (3.9)

Step 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 determined.

For example, the normal procedure for estimating 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) (3.10)

If the factor i is controllable, the best level j*, is determined by

J* = max j (AGVij) (3.11)

Step 6: Perform ANOVA for identifying the significant factors

The main purpose of the analysis of variance (ANOVA) is the

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, 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

54

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.

3.4 CASE-1: OPTIMIZATION OF WIRE ELECTRICAL

DISCHARGE MACHINING OPERATION WITH

MULTIPLE QUALITY CHARACTERISTICS

3.4.1 Introduction

Wire-EDM 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,

micro-scale 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.

The temperature at machining zone ranges from 8000-200000C or as high as

200000C 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, resulting in microscopic debris.

3.4.2 Wire Cutting

EDM wire cutting uses a metallic wire to cut a programmed

contour in a work piece. Extrusion dies and blanking punches are very often

machined by wire cutting. Cutting is always through the entire work piece. To

start machining it is necessary to drill a hole in the work piece or start from

the edge. On the machining area, each discharge creates a crater in the work

55

piece and an impact on the tool. The wire can be inclined, thus making it

possible to make parts with taper or with different profiles at the top and

bottom. Therefore, there is no mechanical contact between the electrode and

the work piece. The wire is usually made of brass or stratified copper and

diameter in the range of 0.1 to 0.3 mm diameter.

Depending on the accuracy and surface finish a part will either be

cut or roughed and skimmed. On cutting, the wire ideally passes through a

solid part and drops a slug or scrap piece when it is done. This will bring

adequate accuracy for the job and skimming is also necessary in most of the

times. A skim cut is where the wire is passed back over the roughed surface

again with a lowered power setting and low pressure flush. Therefore, one to

nine skim passes is required depending upon the accuracy and surface finish

needed. In practice there are just two skim passes which remove 0.002” of

material or even less tan 0.0001”. During roughing (i.e. the first cut) the water

is forced at high pressure in order to provide plenty of cooling and eliminate

eroded particles rapidly. In Skimming (accuracy/finish cuts) water gently

flows over in order to prevent the wire deflection.

3.4.3 Features of Wire EDM Process

Low work holding force

Low cutting force

High accurate process tolerances up to ± 0.0001”.

Complex profile capability

No tool wear

Eco-friendly

Hardened materials can be easily machined.

56

3.4.4 Machine Specifications

The specification of Supercut-734 CNC-wire cut EDM is detailed

below:

Travel range:

Longitudinal : Y-axis = 400 mm

V-axis = ± 40 mm

Lateral : X-axis = 300 mm

U-axis = ± 40 mm

Vertical : Z-axis = 225 mm

Working Size:

Table Size (w*D) : 110*450 mm

Maximum Work piece Size : 400*500*250 mm

Maximum Work piece Weight : 400 kg

Feed:

Main Table Feed Rate : 170 to 900mm/min.

Resolution : 0.001 – 0.0005 mm

Wire feed rate : 0-15 m/min

Wire Tension : 1.5 – 2.5 kgf

Wire Guide:

Type : Diamond closed type.

Wire electrode diameter : 0.25 standard and 0.15, 0.2, 0.3

Taper cutting : (optional)

Maximum taper angle : ± 15/100 mm.

57

3.4.5 Input Parameters

3.4.5.1 Wire feed

Wire feed is the rate at which the wire electrode travels along the

wire guide path. Due to spark erosion, the traveling wire-electrode becomes

thin and brittle. It is always desirable to set the wire feed to maximum. This

will result in less wire breakage; better machining stability and higher cutting

speed with wire feed at 8 m/min, on an average of 0.25mm diameter brass

wire spool of 5 kg will last for 24 hours.

Setting wire feed at 15 will correspond to 15 m/min (approx).

3.4.5.2 Peak current

Increase in the IP value will increase the pulse discharge energy

which in turn can improve the cutting rate further. For higher value of IP, gap

conditions may become unstable with improper combination of Ton and Toff

setting. As and when the discharge conditions become unstable one must

reduce the IP value (and or increase the Toff period). Selection of pulse current

in this study Range is 10- 230 (in step of 10) volts.

3.4.5.3 Spark gap voltage

Reference voltage for the actual gap voltage Range is 0-99 (in step

of 1) volts.

3.4.5.4 Servo feed (with constant feed option)

Servo feed decides the servo speed, at the set value of servo feed,

can vary in proportion with the gap voltage (Normal feed mode) or can be

held constant while machining (with constant feed mode).

58

Range: 0000 – 0990 for normal feed

1000 – 1999 for constant feed.

In general, normal feed is set with machining conditions as those

applied in rough cuts, depending upon the machining stability. However,

while operating with finishing cuts, selection of constant feed option can give

better results. In constant feed mode, the least 3 significant digits of servo

feed define the feed rate in tenths of mm/min.

3.4.5.5 Flushing pressure of water dielectric

Flushing pressure of water dielectric is used for selection of

flushing input pressure.

Range: 0 or 1 (0 – low pressure, 1 – high pressure).

High input pressure of water dielectric is necessary for cutting with

higher values of pulse power and also while cutting the jobs of higher

thicknesses. Low input pressure is used for thin and trim cuts.

3.4.5.6 Wire tension

A brass wire of 0.2 mm in diameter can be applied with a

maximum tension of 1200 gm (1600 gm in case of a wire with 0.25 mm

diameter). Wire tension determines how much the wire is to be stretched

between upper and lower wire guides. Higher the thickness in job, higher the

tension required. Improper setting of tension may result in the job

inaccuracies as well as wire breakage. Setting wire tension to 15 will

correspond to 2300 – 2500 gm (approx).

59

3.4.5.7 Pulse ON time

During pulse ON time the voltage is applied across the electrodes.

Range: 001 – 031 (in step of 1) with ISO pulse OFF.

100 – 131 (in step of 1) with ISO pulse ON.

Higher the Ton setting larger is the pulse on period. The single pulse

discharge energy increases with increasing Ton period, resulting in higher

cutting rate with higher values of Ton. However, surface roughness tends to be

higher. The higher value of discharge energy may also cause wire breakage.

ISO pulse – ISO pulse in EDM is a term for constant energy pulse.

To set ISO pulse mode ON, set \first digit of Ton = 1.

3.4.5.8 Pulse OFF time

Voltage for the gap is absent during this period. Range: 1-31

(in step of 1). Higher the Toff setting larger is the pulse off period. With a

lower value of Toff, there is more number of discharges in a given time,

resulting in increase in the spark efficiency. As a result, the cutting rate also

increases.

Using very low values of Toff period, however, may cause breakage

which in turn reduces the cutting efficiency. As and when the discharge

condition becomes unstable, one can increase the Toff period. This will lower

pulse duty factor and will reduce the average gap current.

This study presents an investigation on the effect of machining

parameters on multiple performance characteristics (Material removal rate,

Surface finish and Kerf) pertaining to a Wire-EDM operation in an

automobile industry. The machining process is controlled by Input peak

60

voltage, Pulse-ON time, Pulse-OFF time, Dielectric fluid pressure, Wire feed,

Wire speed and Servo voltage.

Tarng et al (1995) found the factors that affect the machining

performance as pulse duration, pulse interval, peak current, open circuit

voltage, servo voltage, electric capacitance and table speed. It was found from

the 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, by setting the inputs such that MRR and SF are maximized while

Kerf should be minimized.

The experiments were performed using typical four-axis 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 (Figure 3.1). The inputs like 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. The material chosen for our experimental study was Oil

Hardened Nitride Steel (OHNS) (0.85%C, 0.27%Si, 0.043%S, 0.04%P,

0.25%Cu and 1.24%Cr) of size 150×150×9mm.

Mixed orthogonal array L18 (21×37) 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 as shown in Table 3.1.

61

Figure 3.1 Details of WEDM cutting gap

Table 3.1 Levels for various control factors for Case-1

Factors/Levels Level Level Level

1 2 3

A Dielectric fluid pressure (kg/cm2)

Low High -

B Pulse on time (µs) 4 5 6

C Pulse off time (µs) 15 16 17

D Peak voltage (Volts) 120 130 140

E Wire feed (m/min) 2 3 4

F Wire tension (gms) 3 4 5

G Servo feed (mm/min) 90 100 110

62

3.5 PERFORMANCE MEASURES

3.5.1 Metal Removal Rate (MRR)

The metal cutting speed data is directly available from the monitor

of Supercut-734 CNC machine for various experimental settings. MRR can be

calculated as,

MRR= k. t. vc. ρ mm3/min (3.12)

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).

3.5.2 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.

3.5.3 Kerf

It is an average of three measurements of wire-work piece gap

along the length of cut. It is measured using Tool Maker’s microscope.

3.5.4 Determination of S/N Ratio by Taguchi Method

To estimate S/N ratio, Taguchi method is used. For our factors and

their levels, mixed level L18 (21×37) orthogonal array is selected. This array is

shown in Table 3.2.

63

Table 3.2 Standard OA for L18 (21×37) Taguchi design

Exp.No A B C D E F G Error 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 3 1 1 3 3 3 3 3 3 4 1 2 1 1 2 2 3 2 5 1 2 2 2 3 3 1 3 6 1 2 3 3 1 1 2 1 7 1 3 1 2 1 3 2 3 8 1 3 2 3 2 1 3 1 9 1 3 3 1 3 2 1 2

10 2 1 1 3 3 2 2 2 11 2 1 2 1 1 3 3 3 12 2 1 3 2 2 1 1 1 13 2 2 1 2 3 1 3 1 14 2 2 2 3 1 2 1 2 15 2 2 3 1 2 3 2 3 16 2 3 1 3 2 3 1 3 17 2 3 2 1 3 1 2 1 18 2 3 3 2 1 2 3 2

For every experiment, corresponding three responses are measured.

Then these measured responses are transferred into S/N ratio. For different

types of responses, different S/N ratios are available. The selection of suitable

S/N type depends on the nature of the response. The S/N ratios for higher the

better and lower the better are given below:

S/N HB = -10.log 1/n [∑i=1, 2, ….n1/Yi2] (3.13)

S/N LB = -10.log 1/n [∑i=1, 2, ….nYi2] (3.14)

Where n denotes the number of experiments and Yi denotes the corresponding

responses. S/N ratio for the three responses is given in Table 3.3. S/N values

are then normalized Vijayakumar et al (2006) as given in Table 3.4 to avoid

the effect of adopting different units and to reduce the variability.

64

3.6 DETERMINING OPTIMAL COMBINATION FOR WEDM OPERATIONS USING THE EXISTING GREY RELATION METHOD

Using equations (3.13) and (3.14) given above, S/N ratios for MRR, SF and Kerf are calculated as given in Table 3.3. These S/N ratios are

normalized using equations (3.4) and (3.5) and the results are given in Table 3.4.

The grey relation coefficients for each output variable were found

out using the equation (3.7) given above and grey relation grades were

calculated based on the formula (3.8). The results are given in Table 3.5. The

optimal combinations of values are obtained from the mean parameter values given in Table 3.6 and the factor effects are shown in Figure 3.2.

Table 3.3 Results of performance measures and their S/N values

Exp. No.

MRR (mm3/min)

S/N ratio (db)

SF (µm)

S/N ratio (db) Kerf (mm) S/N ratio

(db) 1 7.8788 17.9292 4.025 87.9047 0.2895 10.7670 2 6.9311 16.816 3.125 90.103 0.2775 11.1347 3 6.3053 15.9941 3.76 88.4962 0.2975 10.5303 4 7.1413 17.0756 2.89 90.782 0.275 11.2134 5 7.1428 17.0774 3.505 89.1062 0.23 12.7654 6 7.9344 17.9903 3.9 88.1787 0.29 10.752 7 7.1557 17.093 3.54 89.0199 0.2525 11.9548 8 8.4233 18.5097 3.78 88.4502 0.2845 10.9184 9 8.0854 18.154 2.905 90.7371 0.25 12.0412

10 7.9344 17.9903 2.83 90.9643 0.291 10.7221 11 6.4152 16.1442 2.975 90.5303 0.291 10.7221 12 7.4675 17.4635 3.575 88.9345 0.285 10.9031 13 7.4236 17.4123 3.58 88.9223 0.2635 11.5844 14 7.8496 17.897 4.135 87.6705 0.2925 10.6775 15 6.8122 16.6658 3.58 88.9223 0.288 10.8122 16 8.398 18.4835 3.03 90.3712 0.279 11.0879 17 7.9311 17.9867 4.565 86.8112 0.2705 11.3567 18 7.7463 17.7819 3.7 88.636 0.2025 13.8715

65

Table 3.4 Normalized S/N ratio values for Case-1

Exp. No. Normalized Value for

MRR Normalized value

for SF Normalized Value

for Kerf

1 0.769256 0.929138 0.263298

2 0.326742 0.819085 0.792619

3 0 1 0.405736

4 0.429902 0.795559 0.956122

5 0.430627 0.331032 0.552614

6 0.793537 0.933624 0.329279

7 0.436857 0.573658 0.531833

8 1 0.883847 0.394641

9 0.858632 0.547791 0.945295

10 0.793537 0.942572 1

11 0.059665 0.942572 0.895497

12 0.584127 0.888412 0.511257

13 0.563768 0.68451 0.508334

14 0.756435 0.955938 0.206908

15 0.266994 0.915633 0.508334

16 0.989613 0.833099 0.857185

17 0.792101 0.752668 0

18 0.710694 0 0.439379

66

Table 3.5 Grey relational grade values for Case-1

Sl. No.

Grey Relation Coefficient for

MRR

Grey Relation Coefficient for

SF

Grey Relation Coefficient for

KERF Grey Grade

1 0.68423415 0.875868424 0.404301117 0.65480123

2 0.426163725 0.734306044 0.706832669 0.622434146

3 0.333333333 1 0.456928127 0.59675382

4 0.467246925 0.70978265 0.919323819 0.698784465

5 0.467563703 0.427727705 0.527767985 0.474353131

6 0.707751149 0.882805769 0.427087239 0.672548052

7 0.470303619 0.539757455 0.516439829 0.508833634

8 1 0.811486757 0.452341728 0.754609495

9 0.77958364 0.525094806 0.901380013 0.73535282

10 0.707751149 0.896976829 1 0.868242659

11 0.347141464 0.896976829 0.827125755 0.690414682

12 0.545927219 0.817543837 0.505692581 0.623054546

13 0.534055661 0.613128303 0.50420202 0.550461995

14 0.672436169 0.919012907 0.38667009 0.659373055

15 0.405513031 0.855626687 0.50420202 0.588447246

16 0.979648776 0.749736468 0.777828769 0.835738004

17 0.706315449 0.669046689 0.333333333 0.569565157

18 0.633467882 0.333333333 0.471421931 0.479407715

67

Table 3.6 Mean parameter values of GRA for Case-1

Parameters Levels

Max-Min 1 2 3

A 0.6353 0.65164 - 0.01634

B 0.67595 0.60738 0.64725 0.06857

C 0.68614 0.62845 0.61592 0.07022

D 0.65622 0.54309 0.73121 0.18812

E 0.61089 0.68717 0.63245 0.07628

F 0.6375 0.67726 0.61575 0.06151

G 0.66377 0.63834 0.6284 0.03537

0.5

0.55

0.6

0.65

0.7

0.75

1 2 3

Levels

Mea

n G

rey

Rel

atio

n G

rade A

BCDEFG

Figure 3.2 Factor effects on GRA for Case-1

Therefore the optimal combination of parameter values are: A2 B1

C1 D3 E2 F2 G1.

68

The result of pooled ANOVA given in Table 3.7 shows that

factor D (Peak voltage) is the most affecting critical factor with a contribution

of 51.61%.

Table 3.7 Results of the pooled ANOVA using GRA for Case-1

Factors SS Dof MS F %

Contribution

C 0.016825 2 0.0084125 1.410568114 8.06954436

D 0.10762 2 0.05381 9.022605674 51.616307

E 0.018552 2 0.009276 1.5553557 8.89784173

Error 0.065603 11 0.00596391 31.4642686

Total 0.2085 17

3.7 DETERMINING OPTIMAL COMBINATION FOR WEDM

OPERATIONS BY USING THE PROPOSED WEIGHT–

BASED GREY RELATION METHOD FOR CASE-1

In the proposed method, the weights are assigned based on PCA

Eigen values. The Eigen value-based weights are derived through PCA

method. By applying PCA on WBGRA, the optimal weights are obtained.

The optimal weights are 0.3351, 0.317667, and 0.347233. These weights are

multiplied with the corresponding grey co-efficient and grey grades are

obtained. The results are given in Table 3.8.

The main effects on the grade are given in Table 3.9. The optimal

combination of parameters based on the main effects is: A2B3C1D3 E2F1G1.

The factor effects are shown in Figure 3.3.

69

The results of pooled ANOVA given in Table 3.10 show that

factor D (Peak voltage) is the most affecting critical factor with a contribution

of 57.44%.

Table 3.8 Normalized S/N ratio values, grey relation co-efficient

values and grey grade for WBGRA Case-1

Exp. No.

Normalized S/N Ratio Grey Relation Coefficient Grey

Grade

1 0.769256 0.929138 0.263298 0.684234 0.875868 0.612921 0.7203477

2 0.326742 0.819085 0.792619 0.426164 0.734306 0.46562 0.537751

3 0 1 0.405736 0.333333 1 0.428571 0.5781811

4 0.429902 0.795559 0.956122 0.467247 0.709783 0.484143 0.5501593

5 0.430627 0.331032 0.552614 0.467564 0.427728 0.484291 0.4607175

6 0.793537 0.933624 0.329279 0.707751 0.882806 0.631115 0.7367496

7 0.436857 0.573658 0.531833 0.470304 0.539757 0.48558 0.4976713

8 1 0.883847 0.394641 1 0.811487 1 0.9401156

9 0.858632 0.547791 0.945295 0.779584 0.525095 0.694043 0.6690384

10 0.793537 0.942572 1 0.707751 0.896977 0.631115 0.7412512

11 0.059665 0.942572 0.895497 0.347141 0.896977 0.433705 0.5518636

12 0.584127 0.888412 0.511257 0.545927 0.817544 0.524069 0.624621

13 0.563768 0.68451 0.508334 0.534056 0.613128 0.517628 0.5534703

14 0.756435 0.955938 0.206908 0.672436 0.919013 0.604183 0.7270657

15 0.266994 0.915633 0.508334 0.405513 0.855627 0.456835 0.56632

16 0.989613 0.833099 0.857185 0.979649 0.749736 0.960889 0.9000994

17 0.792101 0.752668 0 0.706315 0.669047 0.629973 0.6679678

18 0.710694 0 0.439379 0.633468 0.333333 0.577013 0.5185219

70

Table 3.9 Main effects on grey grade values WBGRA for Case -1

Parameters Levels

Max-Min 1 2 3

A 0.632304 0.6501312 - 0.017827

B 0.625669 0.5990804 0.698902 0.099822

C 0.6605 0.6475802 0.615572 0.044928

D 0.620949 0.5321255 0.770577 0.238452

E 0.62537 0.686511 0.611771 0.07474

F 0.661207 0.6239646 0.638135 0.037242

G 0.666887 0.6362782 0.615385 0.051502

0.5

0.55

0.6

0.65

0.7

0.75

0.8

1 2 3

Levels

Mea

n G

rey

Rel

atio

n G

rade A

BCDEFG

Figure 3.3 Factor effects for WBGRA Case -1

71

Table 3.10 Results of the pooled ANOVA for WBGRA Case -1

Factors SS Dof MS F %

Contribution

B 0.032069 2 0.0160345 2.01517204 10.5701186

C 0.00642 2 0.00321 0.40342401 2.11606728

D 0.174274 2 0.087137 10.9511395 57.4416681

E 0.019018 2 0.009509 1.19506507 6.26843731

Error 0.071612 9 0.00795689 23.6037087

Total 0.303393 17

3.8 SUMMARY

This chapter has presented a new methodology for solving the

multi-response optimization problems. In this chapter Grey Relation Analysis

is used by assigning a new weight based on Eigen value to obtain an optimal

combination. The existing GRA with the weight assigned or weight based

GRA has improved in the results with different optimal combinations

discussed in chapter 4. The results in Table 3.10 show that factor D (Peak

voltage) is the most affecting critical factor with a contribution of 57.44%.

The author has contributed the first case and the other three case studies from

literature are solved using this proposed methodology and the results are

presented in the chapter 5.