The use of the orthogonal array with grey relational ...

International Journal of Machine Tools & Manufacture 42 (2002) 237–244

The use of the orthogonal array with grey relational analysis to optimize the electrical discharge machining process with multiple performance characteristics J.L. Lin *, C.L. Lin Department of Automatic Engineering, Fushin Institute of Technology, Toucheng, I-Lan 261, Taiwan, R.O.C. Received 19 November 2000; received in revised form 2 July 2001; accepted 12 July 2001

Abstract In this paper a new approach for the optimization of the electrical discharge machining (EDM) process with multiple performance characteristics based on the orthogonal array with the grey relational analysis has been studied. A grey relational grade obtained from the grey relational analysis is used to solve the EDM process with the multiple performance characteristics. Optimal machining parameters can then be determined by the grey relational grade as the performance index. In this study, the machining parameters, namely workpiece polarity, pulse on time, duty factor, open discharge voltage, discharge current, and dielectric fluid are optimized with considerations of multiple performance characteristics including material removal rate, surface roughness, and electrode wear ratio. Experimental results have shown that machining performance in the EDM process can be improved effectively through this approach.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Electrical discharge machining; Orthogonal array; Grey relational analysis; Optimization

1. Introduction The Taguchi method [1–3] is a systematic application of design and analysis of experiments for the purpose of designing and improving product quality. In recent years, the Taguchi method has become a powerful tool for improving productivity during research and development so that high quality products can be produced quickly and at low cost [4]. However, the original Taguchi method has been designed to optimize a single performance characteristic. Handling multiple performance characteristics by the Taguchi method requires further research effect [5,6]. This is because optimization of the multiple performance characteristics is concerned with optimizing a vector of objectives. 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

* Corresponding author. Tel.: +886-3-9771997x288; fax: +886-39774852. E-mail address: [email protected] (J.L. Lin).

lower-the-better performance characteristic. As a result, an improvement of one performance characteristic may require a degradation of another performance characteristic. Hence, optimization of the multiple performance characteristics is much more complicated than optimization of a single performance characteristic. In this paper, the orthogonal array with the grey relational analysis [7] is used to investigate the multiple performance characteristics in the EDM process. The grey system theory initiated by Deng [8] in 1982 has been proven to be useful for dealing with poor, incomplete, and uncertain information. The grey relational analysis based on the grey system theory can be used to solve the complicated interrelationships among the multiple performance characteristics effectively [9–11]. Through the grey relational analysis, a grey relational grade is obtained to evaluate the multiple performance characteristics. As a result, 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

0890-6955/02/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 1 ) 0 0 1 0 7 - 9

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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. The paper is organized in the following manner. An overview of the optimization of the multiple performance characteristics by the orthogonal array with the grey relational analysis is given first. Then, the selection of machining parameters and the evaluation of machining performance in the EDM process are discussed. Optimization of the EDM process based on the orthogonal array with the grey relational analysis is described in detailed. Finally, the paper concludes with a summary of this study.

2. Optimization of multiple performance characteristics Optimization of process parameters is the key step in the Taguchi method to achieving high quality without increasing cost. This is because optimization of process parameters can improve performance characteristics and the optimal process parameters obtained from the Taguchi method are insensitive to the variation of environmental conditions and other noise factors. Basically, classical process parameter design [12] is complex and not easy to use. A large number of experiments have to be carried out when the number of the process parameters increases. To solve this task, the Taguchi method uses a special design of orthogonal arrays to study the entire process parameter space with a small number of experiments only. A loss function is then defined to calculate the deviation between the experimental value and the desired value. Taguchi recommends the use of the loss function to measure the performance characteristic deviating from the desired value. The value of the loss function is further transformed into a signal-to-noise (S/N) ratio. Usually, there are three categories of the performance characteristic in the analysis of the S/N ratio, that is, the lower-the-better, higher-the-better, and nominal-the-better. The S/N ratio for each level of process parameters is computed based on the S/N analysis. Regardless of the category of the performance characteristic, the larger S/N ratio corresponds to the better performance characteristic. Therefore, optimal level of the process parameters is the level with the highest S/N ratio. This is true for the optimization of a single performance characteristic. However, optimization of the multiple performance characteristics can not be straightforward like optimization of a single performance characteristic. The higher S/N ratio for one performance characteristic may correspond to the lower S/N ratio for another performance characteristic. As a result, the overall evaluation of the S/N ratio is required for the optimization of the multiple performance characteristics. To

solve this problem, the grey relational analysis is adopted in this study. 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. The overall evaluation of the multiple performance characteristics is based on the grey relational grade. 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 are statistically significant. With the grey relational analysis and statistical analysis of variance, the optimal combination of the process parameters can be predicted. Finally, a confirmation experiment is conducted to verify the optimal process parameters obtained from the process parameter design. Based on the above discussion, the use of the orthogonal array with the grey relational analysis to optimize the process with multiple performance characteristics includes the following steps: 1. identify the performance characteristics and process parameters to be evaluated, 2. determine the number of levels for the process parameters, 3. select the appropriate orthogonal array and assign the process parameters to the orthogonal array, 4. conduct the experiments based on the arrangement of the orthogonal array, 5. normalized the experimental results of electrode wear ratio, material removal rate and surface roughness, 6. perform the grey relational generating and calculate the grey relational coefficient, 7. calculate the grey relational grade by averaging the grey relational coefficient, 8. analyze the experimental results using the grey relational grade and statistical analysis of variance, 9. select the optimal levels of process parameters, 10. verify the optimal process parameters through the confirmation experiment.

3. Electrical discharge machining process The EDM removes workpiece by an electrical spark erosion process. Common methods of evaluating mach-

J.L. Lin, C.L. Lin / International Journal of Machine Tools & Manufacture 42 (2002) 237–244

ining 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. 3.1. Machining parameter selection An EDM machine (KT-200) was used as the experimental machine in this study. Cylindrical pure copper with a diameter of 8 mm was used as an electrode to erode a workpiece of SKD11 alloy steel with a diameter of 12 mm. The workpiece and electrode was separated by a moving dielectric fluid. In the experiment, kerosene with aluminum oxide power was used as the dielectric fluid to improve machining performance. The schematic diagram of the experimental setup is shown in Fig. 1. Machining experiments for determining the optimal machining parameters were carried out by setting negative polarity or positive polarity of workpiece, pulse on time in the range of 20–300 ␮s, duty factor in the range of 0.3–0.7, open discharge voltage in the range of 100– 150 V, discharge current in the range of 1.5–6.0 A and the concentration of the aluminum oxide power in the

Fig. 1.

239

dielectric fluid in the range of 2–8 g/l. Machining time for each workpiece in the experiments was 30 minutes and each experiment is repeated three times. To perform the experimental design, the levels of the machining parameters were selected and listed in Table 1. 3.2. Machining performance evaluation Material removal rate, surface roughness, and electrode wear ratio are used to evaluate machining performance. The material removal rate (MRR) is expressed as the workpiece removal weight (WRW) under a period of machining time in minute (T), that is: WRW . MRR (g/min)⫽ T

(1)

A profilemeter (3D-Hommelewerk) measured the machined surface roughness. The average surface roughness Ra that is most widely used surface roughness parameter in industry is selected in this study. The electrode wear ratio (EWR) is defined by the ratio of the electrode wear weight (EWW) to the workpiece removal weight (WRW) and usually expressed as a percentage, that is: EWR (%)⫽

EWW ⫻100. WRW

(2)

Basically, the higher the material removal rate in the EDM process, the better is the machining performance. However, the smaller the electrode wear ratio and sur-

Schematic diagram of the EDM process.

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Table 1 Machining parameters and their levels Symbol

Machining parameter

A B C D

Workpiece polarity Pulse on time Duty factor Open discharge voltage Discharge current Dielectric fluid (kerosene+Al2O3 power)

E F

Unit

Level 1

Level 2

Level 3

V

Negative 20 0.3 100

Positive 150 0.5 120

300 0.7 150

A g/l

1.5 2

4.0 4

␮s

face 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 characteristic.

4. Determination of optimal machining parameters In this section, the use of the orthogonal array with the grey relational analysis for determining the optimal machining parameters is reported step by step. Optimal machining parameters with considerations of the multiple performance characteristics are obtained and verified. 4.1. Orthogonal array experiment To select an appropriate orthogonal array for experiments, the total degrees of freedom need to be computed. The degrees of freedom are defined as the number of comparisons between machining parameters that need to be made to determine which level is better and specifically how much better it is. For example, a three-level machining parameter counts for two degrees of freedom. The degrees of freedom associated with interaction between two machining parameters are given by the product of the degrees of freedom for the two machining parameters. In the present study, the interaction between the machining parameters is neglected. Therefore, there are 11 degrees of freedom owing to one two-level machining parameter and five three-level machining parameters in the EDM process. Once the degrees of freedom are known, the next step is select an appropriate orthogonal array to fit the specific task. The degrees of freedom for the orthogonal array should be greater than or at least equal to those for the machining parameters. In this study, an L18 orthogonal array is used because it has 17 degrees of freedom greater than 11 degrees of freedom in the selected machining parameters. This array has eight columns and 18 rows and it can handle one two-level machining para-

6.0 8

meter and seven three-level machining parameters at most. Each machining parameter is assigned to a column and 18 machining parameter combinations are required. Therefore, only 18 experiments are needed to study the entire machining parameter space using the L18 orthogonal array. The experimental layout for the machining parameters using the L18 orthogonal array is shown in Table 2. 4.2. Grey relational analysis for the experimental results In the grey relational analysis, data preprocessing is first performed in order to normalize the raw data for analysis. In this study, a linear normalization of the experimental results for material removal rate, surface roughness and electrode wear ratio shown in Table 3 were performed in the range between zero and one, which is also called the grey relational generating. The normalized experimental results xij can be expressed as: yij −min yij j xij ⫽ max yij −min yij j

(3)

j

yij for the ith experimental results in the jth experiment. Table 4 shows the normalized results for material removal rate, surface roughness and electrode wear ratio. Basically, the larger normalized results corresponds to the better performance and the best-normalized results should be equal to one. Next, the grey relational coefficient is calculated to express the relationship between the ideal (best) and actual normalized experimental results. The grey relational coefficient xij can be expressed as: min min |xoi −xij |+z max max |xoi −xij | xij ⫽

i

j

i

j

|xoi −xij |+z max max |xoi −xij | i o i

(4)

j

where x is the ideal normalized results for the ith performance characteristic and z is the distinguishing coefficient which is defined in the range 0ⱕzⱕ1.

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Table 2 Experimental layout using an L18 orthogonal array No.

Machining parameter

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

A B Workpiece polarity Pulse on time

C Duty factor

D Discharge voltage

E Discharge current

F Dielectric fluid

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2

1 2 3 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1

1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

Table 3 Experimental results for electrode wear ratio, material removal rate and surface roughness No.

Electrode wear ratio (%)

Material removal rate (g/min)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

32.21 23.96 29.19 7.16 3.55 33.06 44.95 10.79 5.80 24.71 179.17 28.23 17.31 99.21 39.62 47.89 30.00 94.66

0.00149 0.00362 0.00324 0.00452 0.00451 0.00323 0.00125 0.00321 0.00316 0.00028 0.00016 0.00041 0.00017 0.00042 0.00018 0.00024 0.00023 0.00044

38.57 25.02 25.19 8.72 3.62 57.18 61.10 3.74 2.49 7.92 5.02 37.80 20.97 76.35 66.67 21.54 57.32 99.36

25.24 27.29 23.66 6.31 3.04 53.62 38.39 5.57 8.40 21.28 52.54 30.56 16.18 74.34 21.74 101.72 10.53 111.36

0.00117 0.00408 0.00356 0.00413 0.00461 0.00128 0.00128 0.00303 0.00348 0.00034 0.00166 0.00027 0.00021 0.00049 0.00011 0.00043 0.00027 0.00052

Then, the grey relational grade that 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, that is:

冘 m

gj ⫽

1 x m i⫽1 ij

(5)

where gj is the grey relational grade for the jth experi-

Surface roughness (␮m) 0.00139 0.00276 0.00361 0.00322 0.00384 0.00134 0.00141 0.00323 0.00345 0.00031 0.00039 0.00024 0.00023 0.00051 0.00023 0.00019 0.00032 0.00044

3.02 2.67 5.61 3.62 3.50 2.50 4.87 3.20 2.23 1.78 1.65 5.62 2.65 2.36 1.63 4.45 3.43 1.89

3.17 3.46 3.99 3.73 4.57 3.43 2.58 2.91 2.45 2.00 1.71 3.98 1.53 1.50 2.33 4.06 2.08 1.82

2.07 3.47 4.35 3.65 4.27 3.32 3.15 2.18 2.80 2.06 1.74 5.96 3.50 2.32 2.30 4.43 3.67 2.17

ment and m is the number of performance characteristics. Table 5 shows the grey relational grade for each experiment using the L18 orthogonal array. The higher grey relational grade represents that the corresponding experimental result is closer to the ideally normalized value. Experiment 9 has the best multiple performance characteristics among 18 experiments because it has the highest grey relational grade shown in Table 5. In other words, optimization of the complicated multiple per-

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Table 4 Data preprocessing of the experimental result for each performance characteristic No.

Electrode wear ratio

Material removal rate

Surface roughness

Ideal

1

1

1

1

1

1

1

1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.8368 0.8838 0.8540 0.9795 1.0000 0.8320 0.7643 0.9588 0.9872 0.8795 0.0000 0.8595 0.9217 0.4553 0.7946 0.7475 0.8494 0.4812

0.6275 0.7674 0.7657 0.9357 0.9884 0.4354 0.3950 0.9871 1.0000 0.9440 0.9739 0.6355 0.8093 0.2375 0.3375 0.8034 0.4340 0.0000

0.7950 0.7761 0.8096 0.9697 1.0000 0.5331 0.6736 0.9766 0.9505 0.8316 0.5430 0.7460 0.8787 0.3418 0.8273 0.0890 0.9309 0.0000

0.3053 0.7934 0.7077 1.0000 0.9977 0.7039 0.2510 0.7008 0.6894 0.0283 0.0000 0.0581 0.0031 0.0604 0.0038 0.0176 0.0168 0.0635

0.2348 0.8815 0.7667 0.8933 1.0000 0.2593 0.2593 0.6489 0.7481 0.0504 0.3444 0.0363 0.0215 0.0852 0.0000 0.0719 0.0363 0.0911

0.3269 0.7032 0.9352 0.8292 1.0000 0.3132 0.3324 0.8320 0.8932 0.0329 0.0548 0.0128 0.0091 0.0858 0.0100 0.0000 0.0338 0.0676

0.6516 0.7393 0.0025 0.5013 0.5313 0.7820 0.1880 0.6065 0.8496 0.9624 0.9950 0.0000 0.7444 0.8170 1.0000 0.2932 0.5489 0.9348

0.4560 0.3616 0.1889 0.2736 0.0000 0.3713 0.6482 0.5407 0.6906 0.8371 0.9316 0.1922 0.9902 1.0000 0.7296 0.1661 0.8111 0.8958

0.9218 0.5900 0.3815 0.5474 0.4005 0.6256 0.6659 0.8957 0.7488 0.9242 1.0000 0.0000 0.5829 0.8626 0.8673 0.3626 0.5427 0.8981

Table 5 Grey relational grade for each experimental No.

Grey relational grade

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.7007 0.7824 0.7150 0.8130 0.8121 0.6847 0.6511 0.8293 0.8619 0.7194 0.6840 0.5822 0.6902 0.6404 0.6702 0.5826 0.6524 0.6178

formance characteristics can be converted into optimization of a single grey relational grade. Since the experimental design is orthogonal, it is then possible to separate out the effect of each machining parameter on the grey relational grade at different levels. For example, the mean of the grey relational grade for the workpiece polarity at levels 1 and 2 can be calculated by averaging the grey relational grade for the experiments 1 to 9 and 10 to 18, respectively (Table 2). The mean of the grey relational grade for each level of the other machining parameters can be computed in the

similar manner. The mean of the grey relational grade for each level of the machining parameters is summarized and shown in Table 6. In addition, the total mean of the grey relational grade for the 18 experiments is also calculated and listed in Table 6. Fig. 2 shows the grey relational grade graph and the dash line indicated in Fig. 2 is the value of the total mean of grey relational grade. Basically, the larger the grey relational grade, the better is the multiple performance characteristics. However, the relative importance among the machining parameters for the multiple performance characteristics still needs to be known so that the optimal combinations of the machining parameter levels can be determined more accurately. 4.3. Analysis of variance The purpose of the analysis of variance (ANOVA) is to investigate which machining parameters significantly affect the performance characteristic. This is accomplished by separating the total variability of the grey relational grades, which is measured by the sum of the squared deviations from the total mean of the grey relational grade, into contributions by each machining parameter and the error. First, the total sum of the squared deviations SST from the total mean of the grey relational grade gm can be calculated as:

冘 p

SST⫽

(gj ⫺gm)2

(6)

j⫽1

where p is the number of experiments in the orthogonal array and gj is the mean of the grey relational grade for the jth experiment.

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Table 6 Response table for the grey relational grade Symbol

Machining parameter

Grey relational grade Level 1

A B C D

Workpiece polarity 0.7611 Pulse on time 0.6973 Duty factor 0.6928 Open discharge 0.7303 voltage E Discharge current 0.6631 F Dielectric fluid 0.6889 (kerosene+Al2O3 power) Total Mean Value of the Grey Relational Grade=0.7050

Level 2

Level 3

Max–Min

0.6488 0.7184 0.7334 0.6893

0.6992 0.6886 0.6952

0.1123 0.0448 0.0410

0.7099 0.7391

0.7418 0.6858

0.0787 0.0434

4.4. Confirmation tests Once the optimal level of the machining parameters is selected, the final step is to predict and verify the improvement of the performance characteristic using the optimal level of the machining parameters. The estimated grey relational grade gˆ using the optimal level of the machining parameters can be calculated as:

冘 q

gˆ ⫽gm⫹

(g¯ i⫺gm)

(7)

i⫽1

Fig. 2.

Grey relational grade graph.

The total sum of the squared deviations SST is decomposed into two sources: the sum of the squared deviations SSd due to each machining parameter and the sum of the squared error SSe. The percentage contribution by each of the machining parameter in the total sum of the squared deviations SST can be used to evaluate the importance of the machining parameter change on the performance characteristic. In addition, the Fisher’s Ftest [12] can also be used to determine which machining parameters have a significant effect on the performance characteristic. Usually, the change of the machining parameter has a significant effect on the performance characteristic when F is large. Results of analysis of variance (Table 7) indicate that workpiece polarity is the most significant machining parameter for affecting the multiple performance characteristics. Based on the above discussion, the optimal machining parameters are the workpiece polarity at level 1, pulse on time at level 2, duty factor at level 2, open discharge voltage at level 1, discharge current at level 3 and dielectric fluid at level 2.

where gm is the total mean of the grey relational grade, g¯ i is the mean of the grey relational grade at the optimal level, and q is the number of the machining parameters that significantly affects the multiple performance characteristics. Based on Eq. (7), the estimated grey relational grade using the optimal machining parameters can then be obtained. Table 8 shows the results of the confirmation experiment using the optimal machining parameters. As shown in Table 8, material removal rate is accelerated from 0.00350 to 0.00466 g/min, surface roughness is improved from 3.20 to 1.81 ␮m, and electrode wear ratio is greatly reduced from 22.5% to 1.9%. It is clearly shown that the multiple performance characteristics in the EDM process are greatly improved through this study.

5. Conclusion The use of the orthogonal array with grey relational analysis to optimize the edm process with the multiple performance characteristics has been reported in this paper. A grey relational analysis of the experimental results of electrode wear ratio, material removal rate and surface roughness can convert optimization of the multiple performance characteristics into optimization of a

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Table 7 Results of the analysis of variance Symbol

Machining parameter

A B C D E F

Workpiece polarity Pulse on time Duty factor Open discharge voltage Discharge current Dielectric fluid (kerosene+Al2O3 power) Error Total

Mean square

F

0.0568 0.0016 0.0073 0.0059 0.0188 0.0106

0.0568 0.0008 0.0037 0.0030 0.0094 0.0053

6.2179 0.0900 0.4021 0.3235 1.0296 0.5784

0.0183 0.1193

0.0091

Table 8 Results of machining performance using the initial and optimal machining parameters

References

Initial machining parameters

Setting level Material removal rate (g/min) Surface roughness (␮m) Electrode wear ratio (%) Grey relational grade Improvement of

Degree of freedom Sum of square 1 2 2 2 2 2 6 17

Optimal machining parameters

Prediction

Experiment

A1B1C2D2E2F2 0.00350

A1B2C2D1E3F2 0.00466

A1B2C2D1E3F2

3.20

1.81

22.5

1.90

0.7824

0.7979

0.8947

the grey relational grade=0.1123

single performance characteristic called the grey relational grade. As a result, optimization of the complicated multiple performance characteristics can be greatly simplified through this approach. It is shown that the performance characteristics of the edm process such as material removal rate, surface roughness, and electrode wear ratio are improved together by using the method proposed by this study.

Contribution (%) 47.5910 1.3783 6.1560 4.9519 15.7606 8.8543 15.3079 100

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