# Multi-objective optimization in WEDM of D3 tool steel using integrated approach of Taguchi method & Grey relational analysis

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## Abstract

In this paper, wire electrical discharge machining of D3 tool steel is studied. Influence of pulse-on time, pulse-off time, peak current and wire speed are investigated for MRR, dimensional deviation, gap current and machining time, during intricate machining of D3 tool steel. Taguchi method is used for single characteristics optimization and to optimize all four process parameters simultaneously, Grey relational analysis (GRA) is employed along with Taguchi method. Through GRA, grey relational grade is used as a performance index to determine the optimal setting of process parameters for multi-objective characteristics. Analysis of variance (ANOVA) shows that the peak current is the most significant parameters affecting on multi-objective characteristics. Confirmatory results, proves the potential of GRA to optimize process parameters successfully for multi-objective characteristics.

## Keywords

ANOVA D3 tool steel Grey relational analysis Multi-objective optimization Taguchi method Wire electrical discharge machining (WEDM)## Introduction

## Literature review

Every machining process deals with Single characteristics and/or Multi-objective characteristics, which may be equally important and affect the product quality and economy. Numbers of different techniques such as Taguchi robust design method, Factorial design method, Statistical regression analysis, GRA, principle component analysis, Response surface methodology, utility theory, etc., are available to optimize the Single and multi-objective characteristics optimization. Several authors have successfully employed different optimization techniques to improve the process performance.

Tosun (2003) modelled the variation of response variables with the machining parameters of a WEDM process using regression analysis method, and then applied simulated annealing based searching for determination of the machining parameters that can simultaneously optimize all the performance measures. Liao et al. (1997) established mathematical models relating the machining performances with various machining parameters, and then determined the optimal parametric settings for the WEDM processes using feasible-direction method of non-linear programining. Ahmad et al. (2001) studied machinability of aluminum matrix composite (AMC). The full factorial design of experimental approach with two levels were used to determine the combination of machining parameter based on pulse-off time, servo voltage and wire tension. They concluded that the servo voltages have significant influence on the MRR. Mahapatra and Patnaik (2006) established the relationship between various process parameters and responses using non-linear regression analysis and then employed genetic algorithm to optimize the WEDM process. Singh and Garg (2009) machined hot die steel (H-11) on WEDM. They studied the effects of process parameters on MRR using one variable at a time approach. Finally conclude that by increasing the pulse on time the MRR also increases and vice versa. Jangra et al. (2011) investigated the Influence of taper angle, peak current, pulse-on time, pulse-off time, wire tension and dielectric flow rate for MRR and surface roughness during machining of WC–Co composite. In order to optimize MRR and surface roughness GRA is used along with Taguchi method. Durairaj et al. (2013) examined the effect of process parameters on surface roughness and kerf width on WEDM using SS304. For experimentation Taguchi L16 OA has been used. By using multi-objective optimization technique GRA theory, the optimal value is obtained for surface roughness and kerf width and by using Taguchi optimization technique, optimized value is obtained separately. Shivade and Shinde (2013) proposed different approaches which may also be quite useful for multi-objective optimization of WEDM processes. Chiang and Chang (2006) optimized the surface roughness and MRR of a WEDM process for Al_{2}O_{3} particle reinforced material based on GRA method. Ramakrishna and Karunamoorthy (2006) used multi-objective optimization method using Taguchi’s robust design approach for WEDM. Each experiment had been performed under different cutting conditions of pulse on time, wire tension, delay time, wire feed speed and ignition current intensity. Three responses, namely MRR, surface roughness and wire wear ratio had been considered. It was observed that the Taguchi’s parameter design is a simple, systematic, reliable and more efficient tool for optimization of the machining parameters. It was identified that the pulse on time and ignition current had influenced more than the other parameters. Sarkar et al. (2005) presented an approach to select the optimum cutting condition with an appropriate wire offset setting in order to get the desired surface roughness and dimensional accuracy for machining of γ-titanium aluminide alloy. The process has been modeled using additive model in order to predict the response parameters, i.e., cutting speed, surface roughness and dimensional deviation. Muthu et al. (2011) investigated experimentally the influence of the machining parameter on the kerf width, metal removal rate and the surface roughness of the machined workpiece surface using Taguchi method. Jangra et al. (2012) proposed integrated approach for multi-objective optimization for WC-5.3 %Co composite on WEDM using Taguchi, GRA and entropy method. Hewidy et al. (2005) developed a mathematical model based on response surface methodology for correlating the relationships of various WEDM parameters of Inconel 601 material such as peak current, duty factor, wire tension and water pressure on the MRR, wear ratio and surface roughness. Rao and Pawar (2009) highlighted the development of mathematical models using response surface modeling for correlating the relationships of various WEDM parameters such as pulse on-time, pulse off-time, peak current, and servo feed setting on the machining speed and surface roughness. ABC algorithm was then applied to find the optimal combination of process parameters with an objective of achieving maximum machining speed for a desired value of surface finish. Datta and Mahapatra (2010) derived a quadratic mathematical model to represent the process behavior of WEDM operation. Experiments have been conducted with six process parameters: discharge current, pulse duration, pulse frequency, wire speed, wire tension and dielectric flow rate; to be varied in three different levels. Process responses such as Material Removal Rate, surface finish and kerf have been measured. These data have been utilized to fit a quadratic mathematical model (response surface model) for each of the responses.GRA has been adopted to convert this multi-objective criterion into an equivalent single objective function; Optimal setting has been verified through confirmatory test; showed good agreement to the predicted value. Kumar et al. (2013) investigated effect of various WEDM parameters on four response variables, i.e., machining rate, surface roughness, dimensional deviation and wire wear ratio on pure titanium (Grade-2) using RSM. The experimental plan is based on Box–Behnken design. The six parameters, i.e., pulse on time, pulse off time, peak current, spark gap voltage, wire feed and wire tension have been varied to investigate their effect on output responses. The ANOVA has been applied to identify the significance of developed model. Shandilya et al. (2011) used a RSM and artificial neural network based mathematical modeling for average cutting speed of SiC_{p}/6061 Al metal matrix composite during WEDM. Four WEDM parameters namely servo voltage, pulse-on time, pulse-off time and wire feed rate were chosen as machining process parameters. They developed a back propagation neural network to establish the process model. The performance of the developed artificial neural network models was compared with the RSM mathematical models of average cutting speed. Sharma et al. (2013) investigated the effect of parameters on metal removal rate for WEDM using high strength low alloy as work-piece and brass wire as electrode. They observed that material removal rate and surface roughness increase with increase in pulse on time and peak current. RSM is used to optimize the process parameter for Material removal rate and surface roughness. They developed a mathematical model which correlates the independent process parameters with the desired metal removal rate and Surface Roughness. The central composite rotatable design has been used to conduct the experiments.

Most of the researchers used response surface method for multi-objective optimization. In modeling and optimization of manufacturing processes using RSM, the sufficient data is collected through designed experimentation. In general, a second-order regression model is developed because first-order models often give lack-of-fit, on the other hand, the major drawback of RSM is to fit the data to a second-order polynomial. It cannot be said that all systems containing curvature are well accommodated by the second-order polynomial (Rao 2011). To overcome this, the data can be converted into another form that can be explained by the second-order model; this disadvantage is overcome in other multi-objective method like GRA. This is an impacting measurement method in grey system theory that analyzes uncertain relations between one main factor and all the other factors in a given system. In the case when experiments are ambiguous or when the experimental method cannot be carried out exactly, grey analysis helps to compensate for the shortcomings in statistical regression. GRA is actually a measurement of the absolute value of the data difference between sequences, and it could be used to measure the approximate correlation between sequences (Tosun 2006).

## Experimental works

In this study, Taguchi method is used for single characteristics optimization and GRA has been used to establish correlation between the independent variables and the multi-objective characteristics; therefore, the experiments were performed according to a Taguchi design of experiments.

### Work material and cutting tool (electrode)

The work material selected in this investigation was AISI D3 tool steel. The chemical composition of the D3 tool steel includes: C,2.25;Si,0.60,Mn,0.60;Cr,12;Ni,0.30;W,1;V,1;Cu0.25;P,0.03;S,0.03. A Commercially available D3 plate with 30 mm thickness was used to prepare 20 × 20 mm Square shaped specimens for performing WEDM experiments. A commercially available Molybdenum wire of diameter 0.18 is used as electrodes material.

### Design of experiments

In present work, four different input process parameters and four performance measure selected which are discussed in following section.

#### Input process parameters

##### Pulse on time

The pulse on time is referred as *T*_{on} and it represents the duration of time in micro seconds (μs), for which the current is flowing in each cycle. During this time, the gap voltage is applied across the electrodes. The single pulse discharge energy increases with increasing *T*_{on} period, resulting in higher cutting rate. With higher values of *T*_{on}, however, surface roughness tends to be higher. The higher value of discharge energy may also cause wire breakage.

##### Pulse off time

The pulse off time is referred as T_{off} and it represents the duration of time in micro seconds (μs), between the two simultaneous sparks. The voltage is absent during this part of the cycle. With a lower value of *T*_{off}, there is more number of discharges in a given time, resulting in increase in the sparking efficiency. As a result, the cutting rate also increases. Using very low values of *T*_{off} period, however, may cause wire breakage which in turn reduces the cutting efficiency. As and when the discharge conditions become unstable, one can increase the *T*_{off} period. This will allow lower pulse duty factor and will reduce the average gap current.

##### Peak current

The peak current (*I*_{p}) is the maximum value of the current passing through the electrodes for the given pulse. Increase in the I_{P} value will increase the pulse discharge energy which in turns improves the cutting rate further. For higher value of *I*_{p}, gap conditions may become unstable with improper combination of *T*_{on}, *T*_{off} or other parameter settings. As and when the discharge conditions become unstable one must reduce the *I*_{p} value.

##### Wire speed

Wire speed is the speed at which the wire electrode travels along the wire guide path and is fed continuously for sparking. In WEDM, wire electrode contributes 70 % of the machining cost. Therefore, it is desirable to set proper wire feed rate for stable machining with no or less wire breakage occurs.

#### Output process parameters

##### Material removal rate (MRR)

^{2}/min. Cutting speed (Cs) was observed from machine tool monitor screen which gives linear cutting speed in terms of mm/min, therefore, MRR for the WEDM operation can be calculated using Eq. 1.

MRR = material removal rate.

Cs = cutting speed in mm*/*min.

L = thickness of the material in mm.

##### Dimensional deviation

*ζ*) must be equal to dimensional deviation (

*D*). The orientation of this wire offset (i.e., left or right with respect to the programmed path) depends upon the direction (clockwise or counter clockwise) of cutting and type of job (i.e., die or punch). It may be noted that though the magnitude of dimensional deviation and wire offset are equal but their usage is kept different. The term “dimensional deviation” has been used as response parameter during cutting experiment in WEDM with zero wire offset. But, “wire offset” is a control setting in WEDM part programming to eliminate or minimize dimensional inaccuracy during actual machining.

##### Gap current

In wire cut electric discharge machine, specimen is mounted on the machine and during the process of cutting, a small amount of gap is maintained between the job and the electrode wire as shown in Fig. 2. To initiate the cutting a pulse of current is generated by the pulse generator in order to start the cutting process and the current that passes through the material is measured and named as gap current. The gap current is read on an ammeter, which is an integral part of the machine, in ampere.

##### Machining time

It is the actual time required for cutting complete path on work piece in wire cut electric discharge machine and generally expressed in minutes or in seconds. Machining time is an important parameter for any process for deciding cost of product.

Level values of input parameters

Sr. no. | Parameters | Unit | Level 1 | Level 2 | Level 3 |
---|---|---|---|---|---|

1 | Pulse on time ( | µs | 3 | 6 | 9 |

2 | Pulse off time ( | µs | 2 | 4 | 6 |

3 | Peak current ( | Amp | 1 | 2 | 3 |

4 | Wire speed (Ws) | m/min | 3 | 5 | 7 |

^{4}) OA is selected (Table 2). Each machining parameter is assigned to a column of OA and 9 machining parameter combinations are designed. The response variables chosen for the present investigation are: MRR, Dimensional deviation, gap current and machining time. The “higher-the-better” quality characteristic has been used for calculating the signal to noise (S/N) ratio of MRR and Gap current, whereas “lower-the-better” quality characteristic for dimensional deviation and machining time. See Eq. (3 and 4) (Taguchi et al. 2005).

L9 design matrix

Expt. no. | Para. 1 | Para. 2 | Para. 3 | Para. 4 |
---|---|---|---|---|

E 1 | 3 | 2 | 1 | 3 |

E 2 | 3 | 4 | 2 | 5 |

E 3 | 3 | 6 | 3 | 7 |

E 4 | 6 | 2 | 2 | 7 |

E 5 | 6 | 4 | 3 | 3 |

E 6 | 6 | 6 | 1 | 5 |

E 7 | 9 | 2 | 3 | 5 |

E 8 | 9 | 4 | 1 | 7 |

E 9 | 9 | 6 | 2 | 3 |

*y*

_{ j }is the response value for

*i*th experiment

### Experimental planning

## Experimental results and discussion

### Effect of process parameters on MRR

Mean values and S/N ratios of observed results

Ortho. array | Raw data | Signal to noise ratios (DB) | ||||||
---|---|---|---|---|---|---|---|---|

MRR | Dim. deviation | Gap current | M/C time | MRR | Dim. deviation | Gap current | M/C time | |

1 | 17.7 | 1.655 | 0.5 | 135.62 | 24.96 | −4.38 | −6.021 | −42.65 |

2 | 28.6 | 1.667 | 1.3 | 83.98 | 29.13 | −4.437 | 2.279 | −38.48 |

3 | 29.5 | 1.825 | 1.4 | 81.37 | 29.40 | −5.225 | 2.923 | −38.21 |

4 | 38.4 | 1.742 | 1.7 | 62.51 | 31.69 | −4.819 | 4.609 | −35.92 |

5 | 40.1 | 1.697 | 2.0 | 59.86 | 32.06 | −4.592 | 6.021 | −35.54 |

6 | 21.1 | 1.595 | 0.6 | 113.84 | 26.49 | −4.055 | −4.437 | −41.13 |

7 | 59.5 | 1.708 | 2.2 | 40.34 | 35.49 | −4.651 | 6.848 | −32.12 |

8 | 22.2 | 1.672 | 0.7 | 108.12 | 26.93 | −4.463 | −3.098 | −40.68 |

9 | 36.4 | 1.702 | 1.7 | 65.94 | 31.22 | −4.617 | 4.609 | −36.38 |

Response table for MRR (S/N Data) and raw data

Levels | Pulse on time ( | Pulse off time ( | Peak current ( | Wire speed (Ws) | Pulse on time ( | Pulse off time ( | Peak current ( | Wire speed (Ws) |
---|---|---|---|---|---|---|---|---|

1 | 27.83 | 30.71 | 30.71 | 29.41 | 25.27 | 38.53 | 20.33 | 31.40 |

2 | 30.08 | 29.37 | 29.37 | 30.37 | 33.20 | 30.30 | 34.47 | 36.40 |

3 | 31.21 | 29.03 | 29.03 | 29.34 | 39.37 | 29.00 | 43.03 | 30.05 |

Delta | 3.39 | 1.68 | 1.68 | 1.03 | 14.10 | 9.53 | 22.70 | 6.37 |

Rank | 2 | 3 | 3 | 4 | 2 | 3 | 1 | 4 |

### Selection of optimal levels

In order to study the significance of the process variables towards MRR, analysis of variance was performed (Appendix: A). From these tables, it is clear that pulse on time, pulse off time, peak current and wire speed significantly affect both the MRR values. The ranks and the delta values show that current has the greatest effect on MRR and is followed by pulse on time, peak current, pulse off time and wire speed in that order. As MRR is the “higher the better” type quality characteristic, it can be seen from Fig. 5 that the third level of pulse on time (A3), first level of pulse off time (B1),third level of peak current (C3) and second level of wire speed (D2) provide maximum value of MRR. The S/N data analysis suggests the same levels of the variables (A3, B1, C3, and D2) as the best levels for maximum MRR in WEDM process.

### Effect of process parameters on dimensional deviation

Response table for dimensional deviation (S/N Data) and raw data

Levels | Pulse on time ( | Pulse off Time ( | Peak current ( | Wire speed (Ws) | Pulse on time ( | Pulse off time ( | Peak current ( | Wire speed (Ws) |
---|---|---|---|---|---|---|---|---|

1 | −4.679 | −4.616 | −4.298 | −4.537 | 1.716 | 1.702 | 1.641 | 1.686 |

2 | −4.489 | −4.497 | −4.633 | −4.381 | 1.678 | 1.678 | 1.705 | 1.657 |

3 | −4.586 | −4.641 | −4.823 | −4.836 | 1.696 | 1.709 | 1.743 | 1.746 |

Delta | 0.191 | 0.144 | 0.525 | 0.455 | 0.038 | 0.031 | 0.103 | 0.089 |

Rank | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 |

### Selection of optimal levels

In order to study the significance of the process variables towards dimensional deviation, ANOVA was performed. It was found that the wire feed are most significant process parameters for dimensional deviation as compared to pulse on and pulse off time. The ranks and the delta values for various parameters show that wire speed has the greatest effect on dimensional deviation and is followed by current, pulse on time and pulse off time in that order. As dimensional deviation is the “lower the better” type quality characteristic, from Fig. 6, highest S/N ratio corresponds to, second level of pulse on time (A2), second level of pulse off time (B2), first level of current (C1) and second level of wire speed (D2) provide minimum value of dimensional deviation. The S/N data analysis (Fig. 6) also suggests the same levels of the variables (A2, B2, C1, and D2) as the best levels for minimum dimensional deviation in WEDM process.

### Effect of process parameters on gap current

Response table for gap current (S/N Data) and raw data

Levels | Pulse on time ( | Pulse off time ( | Peak current ( | Wire speed (Ws) | Pulse on time ( | Pulse off time ( | Peak current | Wire speed (Ws) |
---|---|---|---|---|---|---|---|---|

1 | −0.2731 | 1.8123 | −4.5185 | 1.5363 | 1.0667 | 1.4667 | 0.6000 | 1.4000 |

2 | 2.0642 | 1.7338 | 3.8323 | 1.5634 | 1.4333 | 1.3333 | 1.5667 | 1.3667 |

3 | 2.7865 | 1.0315 | 5.2639 | 1.4778 | 1.5333 | 1.2333 | 1.8667 | 1.2667 |

Delta | 3.0595 | 0.7808 | 9.7824 | 0.0856 | 0.4667 | 0.2333 | 1.2667 | 0.1333 |

Rank | 2 | 3 | 1 | 4 | 2 | 3 | 1 | 4 |

### Selection of optimal levels

To study the significance of the process variables towards gap current, ANOVA was performed (Appendix: 1). It was found that wire feed are non significant process parameter for gap current and it is observed that pulse on time, pulse off time and peak current significantly affect both the mean and the variation in the gap current values. The ranks and the delta values for various show that pulse on time has the greatest effect on gap current and is followed by pulse off time and peak current in that order. As gap current is the “larger the better” type quality characteristic, from Fig. 7, it can be seen that the third level of pulse on time (A3), first level of pulse off time (B1) and third level of peak current (C3), provide maximum value of gap current. The S/N data analysis (Fig. 5) also suggests the same levels of the variables (A3, B1, and C3) as the best levels for maximum gap current in WEDM process.

### Effect of process parameters on machining time

Response table for machining time (S/N Data) and raw data

Levels | Pulse on time ( | Pulse off time ( | Peak current ( | Wire speed (Ws) | Pulse on time ( | Pulse off time ( | Peak current ( | Wire speed (Ws) |
---|---|---|---|---|---|---|---|---|

1 | −39.78 | −36.89 | −41.48 | −38.19 | 100.32 | 79.49 | 119.19 | 87.14 |

2 | −37.53 | −38.23 | −36.93 | −37.24 | 78.74 | 83.99 | 70.81 | 79.39 |

3 | −36.39 | −38.57 | −35.29 | −38.27 | 71.47 | 87.05 | 60.52 | 84.00 |

Delta | 3.39 | 1.68 | 6.19 | 1.03 | 28.85 | 7.56 | 58.67 | 7.75 |

Rank | 2 | 3 | 1 | 4 | 2 | 3 | 1 | 4 |

### Selection of optimal levels

In order to study the significance of the process variables towards machining time, ANOVA was performed. From these tables (Appendix: 1), it is clear that pulse on time, pulse off time, peak current and wire speed significantly affect both the machining time values. The ranks and the delta values show that current has the greatest effect on machining time and is followed by pulse on time, peak current, pulse off time and wire speed in that order. As machining time is the “Smaller is better” type quality characteristic, it can be seen from Fig. 8 that the third level of pulse on time (A3), first level of pulse off time (B1),third level of peak current (C3) and second level of wire speed (D2) provide less value of machining time. The S/N data analysis (Fig. 6) also suggests the same levels of the variables (A3, B1, C3, and D2) as the best levels for minimizing machining time in WEDM process.

### Estimation of optimum response characteristics

*η*_{opt} is the total mean of the machining characteristic under consideration,

*η*_{m} is the mean value at the optimum level (from response Tables) and *q* is the number of process that significantly affects on machining characteristics.

Optimal value of individual machining characteristics

Sr. no. | Machining characteristics | Optimal combination | Predicted optimal value | Experimental value |
---|---|---|---|---|

1 | MRR | A3B1C3D2 | 59.5 mm | 59.5 mm |

2 | Dimensional deviation | A2B2C1D2 | 1.56 % | 1.69 % |

3 | Gap current | A3B1C3 | 1.17 A | 1.2 A |

4 | Machining time | A3B1C3D2 | 40.34 min | 40.34 min |

Sequence after data pre-processing

Ex. no. | MRR | Dimensional deviation | Gap current | Machining time |
---|---|---|---|---|

Ref. sequence | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Comparability sequence | ||||

1 | 0.0000 | 0.2741 | 0.0000 | 1.0000 |

2 | 0.3958 | 0.3263 | 0.6449 | 0.6047 |

3 | 0.4213 | 1.0000 | 0.6949 | 0.5787 |

4 | 0.6388 | 0.6530 | 0.8260 | 0.3612 |

5 | 0.6745 | 0.4587 | 0.9357 | 0.3254 |

6 | 0.1449 | 0.0000 | 0.1231 | 0.8556 |

7 | 1.0000 | 0.5096 | 1.0000 | 0.0000 |

8 | 0.1868 | 0.3485 | 0.2271 | 0.8131 |

9 | 0.5947 | 0.4806 | 0.8260 | 0.4053 |

### Multi-objective characteristics optimization

In order to optimize the MRR, Dimensional deviation, gap current and machining time simultaneously GRA is used. The following stepwise procedure of GRA optimization is used to solve the current formulation (Deng 1989).

#### Normalization of S/N ratio

- 1.
Larger is better

- 2.
smaller-the-better

#### Determination of deviation sequence

#### Calculation of grey relational coefficient (GRC)

${\mathrm{\Delta}}_{0i}\left(k\right)$ is the deviation sequence and *ξ* = distinguishing coefficient, *ξ* ∈ (0,1) and *ξ* is set as 0.5 in this study (Dabade 2013).

*k*) and ${\mathrm{\Delta}}_{\mathrm{min}}$ (

*k*) are as follows:

The deviation sequences, Grey relational coefficients and grade values

Deviation sequences | Grey relational coefficients (GRC) and grade values | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Deviation sequence | ∆ | ∆ | ∆ | ∆ | Compa sq. | MRR | Dim. deviation | Gap current | M/C time | Grade value | Rank |

No.1, i = 1 | 1.0000 | 0.7259 | 1.0000 | 0.0000 | 1 | 0.33333 | 0.4079 | 0.3333 | 1.0000 | 0.5186 | 6 |

No.2, i = 2 | 0.6042 | 0.6737 | 0.3551 | 0.3953 | 2 | 0.45281 | 0.4260 | 0.5847 | 0.5585 | 0.5055 | 7 |

No.3, i = 3 | 0.5787 | 0.0000 | 0.3051 | 0.4213 | 3 | 0.46353 | 1.0000 | 0.6211 | 0.5427 | 0.6568 | 2 |

No.4, i = 4 | 0.3612 | 0.3470 | 0.1740 | 0.6388 | 4 | 0.58059 | 0.5903 | 0.7418 | 0.4391 | 0.5880 | 4 |

No.5, i = 5 | 0.3255 | 0.5413 | 0.0643 | 0.6746 | 5 | 0.60572 | 0.4802 | 0.8860 | 0.4257 | 0.5994 | 3 |

No.6, i = 6 | 0.8551 | 1.0000 | 0.8769 | 0.1444 | 6 | 0.36898 | 0.3333 | 0.3631 | 0.7759 | 0.4603 | 9 |

No.7, i = 7 | 0.0000 | 0.4904 | 0.0000 | 1.0000 | 7 | 1.00000 | 0.5048 | 1.0000 | 0.3333 | 0.7095 | 1 |

No.8, i = 8 | 0.8132 | 0.6515 | 0.7729 | 0.1869 | 8 | 0.38076 | 0.4342 | 0.3928 | 0.7279 | 0.4839 | 8 |

No.9, i = 9 | 0.4053 | 0.5194 | 0.1740 | 0.5947 | 9 | 0.55229 | 0.4905 | 0.7418 | 0.4567 | 0.5603 | 5 |

#### Determination of grey relational grade (GRG)

*γ(x*

_{ 0 }

*, x*

_{ i }

*)*is the grey relational grade for the

*j*th experiment and

*m*is the number of performance characteristics.

#### Analysis of GRG and selection of optimal level of parameters

ANOVA has been performed using statistical software MINITAB 16 of Grey relational grade values to evaluate the influence of process parameters on multi-objective characteristics. ANOVA for grade values (Appendix: 1) shows that all four parameters, pulse on time (A), pulse off time (B), current (C) and wire speed (D) significantly affect the multi-objective characteristics under 95 % confidence levels. It is clearly observed from Table 10 for grey relational grade that the process parameter “setting of experiment No. 7” has the highest grey relational grade (0.7095) thus the seventh number experiment gives the best multiple performance characteristics among the nine experiments.

Response table for grey relational grade (GRG)

Levels | Pulse on time ( | Pulse off time ( | Peak current ( | Wire speed (Ws) |
---|---|---|---|---|

1 | 0.5603 | 0.6054 | 0.4876 | 0.5595 |

2 | 0.5492 | 0.5296 | 0.5513 | 0.5585 |

3 | 0.5846 | 0.5592 | 0.6553 | 0.5762 |

Max–Min | 0.0354 | 0.0758 | 0.1676 | 0.0175 |

Ranking | 3 | 2 | 1 | 4 |

Total mean value of GRG is 0.5647 |

#### Prediction of grey relational grade under optimum parameters

*η*

_{opt}) is predicted using Eq. 6 as described below:

Predicted and experimental values

Sr. no. | Machining characteristics | Initial setting | Predicted value | Experimental value |
---|---|---|---|---|

1 | Optimal parameter | A3B1C3D2 | A3B1C3D3 | A3B1C3D3 |

2 | MRR (mm | 59.5 | 57.9 | |

3 | Dimensional deviation % | 1.70 | 1.73 | |

4 | Gap current (A) | 2.2 | 2.2 | |

5 | Machining time (min) | 40.34 | 41.45 | |

6 | Grey relational grade | 0.7095 | 0.7274 | 0.7218 |

7 | Improvement in grey relational grade = 1.22 % |

## Conclusions

- 1.
Experimental results of WEDM of D3 tool steel indicate current and pulse on time have significant effect on MRR and Gap current. Pulse on time and current have direct relation with MRR and Gap current but there is an inverse relation Machining time and them. Apart from these parameters, Current and wire speed have major contribution on dimensional deviation value because wire speed increases, fresh wire comes in contact with cutting zone which rapidly increases material erosion which causes dimensional deviation. Optimum combinations and corresponding value are show in Table 8.

- 2.
Using GRA, Initial setting (A3B1C3D2) grade i.e., 0.7095 increases by using new optimum combination A3B1C3D3 up to 0.7218 means there is increment in grade of 1.22 %, Therefore, using present approach process parameters have been successfully optimized for better machining characteristics.

## Notes

### Acknowledgments

The author would like to thank Dr. Kamal Jangra (Assistant Professor, PEC University of Technology, Chandigarh) and all those who contributed direct or indirectly.

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