**Table 2.**

MRR and TWR can be found out. For MRR the range is between 1045.3295 cm<sup>3</sup> /min to 1437.789 cm3 /min and for TWR, it varies between 149.402 gm/min to 1216.031 gm/min respectively. Therefore, in order to arrive at an optimal or near optimal parametric combination which will consecutively satisfy the contradictory nature of the responses Fuzzy Gray Relational Analysis is conducted.

**7. Multi-criteria decision making (MCDM) analysis**

rather than the lower TWR.

material in **Table 2**.

*Weight criteria for deference responses.*

**MRR (cm3 /min)**

*Grey relation co-efficient along with grades and ranks.*

**TWR (gm/min)**

**Table 3.**

**Table 4.**

**147**

**Exp. No**

set as Maximum MRR and minimum TWR.

*DOI: http://dx.doi.org/10.5772/intechopen.96725*

**7.1 Optimization of the parameters**

In this chapter the two contradictory responses i.e. MRR and TWR have got a dissimilar level of rank. In this case the maximized MRR is the primary objective

The specified weights for MRR and TWR are 84.4% and 15.6% respectively as calculated by using Entropy method [17]. **Table 4** represents the grey relation coefficient and grades corresponding to parametric settings and responses for the

**Criteria Linguistic terms Fuzzy number BNP** MRR VH 0.9,1.0,1.0 0.844 TWR ML 0.1,0.3,0.5 0.156

**Responses Grey Co-efficient Grey**

**MRR (cm<sup>3</sup> /min)**

 1045.330 1216.031 0.458 0.135 0.296 16 1436.914 149.402 0.997 1.000 0.999 2 1100.345 961.288 0.495 0.170 0.333 10 1437.789 149.464 1.000 1.000 1.000 1 1167.082 185.945 0.550 0.820 0.685 3 1074.937 1057.663 0.477 0.155 0.316 11 1201.488 827.616 0.584 0.197 0.390 8 1145.313 683.333 0.531 0.238 0.384 9 1233.717 531.626 0.619 0.303 0.461 4 10 1070.928 1178.077 0.474 0.139 0.307 14 1154.800 633.317 0.539 0.256 0.398 7 1305.472 875.273 0.715 0.186 0.451 5 1072.248 1070.531 0.475 0.153 0.314 13 14 1049.991 1138.689 0.461 0.144 0.302 15 1111.972 461.582 0.504 0.348 0.426 6 1074.937 1057.913 0.477 0.155 0.316 12

**Grade**

**TWR (gm/min)** **Rank**

**Table 3** is tabulated by using the weightage from the fuzzy set theory. The closed value to the 1 gives the ideal solution between the comparative sequences. In this case, 16 simulated data from the genetic algorithm have been used for further evaluation. In this case, the criteria for decision making have been

*Nature Inspired Metaheuristic Approach for Best Tool Work Combination for EDM Process*

The different parametric combinations with respective responses as obtained through GA are shown in **Table 2** below.

#### **5. Multi objective solution using Grey Relation Analysis (GRA)**

GRA can be employed to simultaneously find out the optimized solution for several contradictory responses [13]. This theory has been proposed by Deng in 1982's [14]. In modern research work, this theory is a very essential tool to design the model for the unknown of partially known or unspecified data.

GRA form the link between preferred (best/ideal) with real investigational data. The average of the grey coefficient is used to estimate the grey grade. This grade is generally varies between 0 and 1. When the value is close to 1, it signifies that the solution approaches the ideal condition. In the final acquired data set the parametric combinations which have the maximum grey relation grade, that combination will be termed as the optimized solution.

The normalized equation for the condition where the maximum value is required, like MRR that can be expressed as:

$$X\_{\vec{\eta}} = \frac{Y\_{\vec{\eta}} - \text{Min}\left[Y\_{\vec{\eta}}, i = 1, 2, \dots \dots n\right]}{\text{Max}\left[Y\_{\vec{\eta}}, i = 1, 2, \dots \dots n\right] - \text{Min}\left[Y\_{\vec{\eta}}, i = 1, 2, \dots n\right]} \tag{5}$$

If lower value for the better performance such as TWR then it is expressed as,

$$X\_{\vec{\eta}} = \frac{\text{Max}\left[Y\_{\vec{\eta}}, i = 1, 2, \dots \dots n\right] - Y\_{\vec{\eta}}}{\text{Max}\left[Y\_{\vec{\eta}}, i = 1, 2, \dots \dots n\right] - \text{Min}\left[Y\_{\vec{\eta}}, i = 1, 2, \dots n\right]} \tag{6}$$

To find out the single solution for these two contradictory processes the GRA is performed. When the grey grade is 1, that solution gives the optimized single parametric combination.

#### **6. Fuzzy set theory**

To find out the ambiguous solution in decision-making problem, the Fuzzy set theory can be used as a powerful tool. Instead of using numerical values, assign of weights for linguistic assessment is more useful [15]. During the consideration of the decision makers' fuzzy rating, fuzzy decision matrix can be achieved from a decision matrix and finally it can be converted into weighted normalized fuzzy decision matrix. A fuzzy set can be described by a membership function *μ*^ *<sup>d</sup>*ð Þ *x* while converting X. A degree of membership of x in ^ *d* can be plots individual element x in X to a real number in the period of 0 to 1. In this case triangular fuzzy number (TFN), can be defined as a triplet (d1, d2, … , dn) and the membership function is defined [16].

The translation method of fuzzy number into the non-fuzzy number, that is, a crisp value is identified as defuzzification. In this current research work 'centroid of area' technique for defining Best Non-Fuzzy Performance (BNP) value is applied.

*Nature Inspired Metaheuristic Approach for Best Tool Work Combination for EDM Process DOI: http://dx.doi.org/10.5772/intechopen.96725*

## **7. Multi-criteria decision making (MCDM) analysis**

In this chapter the two contradictory responses i.e. MRR and TWR have got a dissimilar level of rank. In this case the maximized MRR is the primary objective rather than the lower TWR.

**Table 3** is tabulated by using the weightage from the fuzzy set theory.

The closed value to the 1 gives the ideal solution between the comparative sequences. In this case, 16 simulated data from the genetic algorithm have been used for further evaluation. In this case, the criteria for decision making have been set as Maximum MRR and minimum TWR.

#### **7.1 Optimization of the parameters**

The specified weights for MRR and TWR are 84.4% and 15.6% respectively as calculated by using Entropy method [17]. **Table 4** represents the grey relation coefficient and grades corresponding to parametric settings and responses for the material in **Table 2**.


#### **Table 3.**

MRR and TWR can be found out. For MRR the range is between 1045.3295 cm<sup>3</sup>

gm/min respectively. Therefore, in order to arrive at an optimal or near optimal parametric combination which will consecutively satisfy the contradictory nature of

**5. Multi objective solution using Grey Relation Analysis (GRA)**

the model for the unknown of partially known or unspecified data.

The different parametric combinations with respective responses as obtained

GRA can be employed to simultaneously find out the optimized solution for several contradictory responses [13]. This theory has been proposed by Deng in 1982's [14]. In modern research work, this theory is a very essential tool to design

GRA form the link between preferred (best/ideal) with real investigational data. The average of the grey coefficient is used to estimate the grey grade. This grade is generally varies between 0 and 1. When the value is close to 1, it signifies that the solution approaches the ideal condition. In the final acquired data set the parametric combinations which have the maximum grey relation grade, that combination will

*Max Yij*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … … *<sup>n</sup>* � *Min Yij*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … *<sup>n</sup>* (5)

*Max Yij*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … … *<sup>n</sup>* � *Min Yij*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … *<sup>n</sup>* (6)

The normalized equation for the condition where the maximum value is

If lower value for the better performance such as TWR then it is expressed as,

To find out the single solution for these two contradictory processes the GRA is

To find out the ambiguous solution in decision-making problem, the Fuzzy set theory can be used as a powerful tool. Instead of using numerical values, assign of weights for linguistic assessment is more useful [15]. During the consideration of the decision makers' fuzzy rating, fuzzy decision matrix can be achieved from a decision matrix and finally it can be converted into weighted normalized fuzzy decision matrix. A fuzzy set can be described by a membership function *μ*^

X to a real number in the period of 0 to 1. In this case triangular fuzzy number (TFN), can be defined as a triplet (d1, d2, … , dn) and the membership function is

The translation method of fuzzy number into the non-fuzzy number, that is, a crisp value is identified as defuzzification. In this current research work 'centroid of area' technique for defining Best Non-Fuzzy Performance (BNP) value is applied.

*Xij* <sup>¼</sup> *Yij* � *Min Yij*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … … *<sup>n</sup>*

*Xij* <sup>¼</sup> *Max Yij*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … … *<sup>n</sup>* � *Yij*

performed. When the grey grade is 1, that solution gives the optimized single

the responses Fuzzy Gray Relational Analysis is conducted.

*Computational Optimization Techniques and Applications*

through GA are shown in **Table 2** below.

be termed as the optimized solution.

parametric combination.

**6. Fuzzy set theory**

defined [16].

**146**

required, like MRR that can be expressed as:

converting X. A degree of membership of x in ^

/min and for TWR, it varies between 149.402 gm/min to 1216.031

to 1437.789 cm3

/min

*<sup>d</sup>*ð Þ *x* while

*d* can be plots individual element x in

*Weight criteria for deference responses.*


#### **Table 4.**

*Grey relation co-efficient along with grades and ranks.*


#### **Table 5.**

*Results of machining performance using the initial and optimal machining parameters.*

Grey relation coefficient, relation grade, and the ranks have been displayed in **Table 4**. From this table, it is obvious that the experimental run number 4 has achieved maximum gray relation grade. As has been discuses earlier, this experimental run satisfies the condition for the optimized multi-response parameter. So, experiment 4, which have parametric combination Tool Density 15.7 g/cm<sup>3</sup> , Workpiece density of 14.17 g/cm<sup>3</sup> , Current 86.24 amp, POT 7836.89 μSec, and Machining time of 199.16 min is the best parametric combination for having high MRR and low TWR.

The confirmation experiment performed with the above optimal combination results in grey relational grade MRR and TWR is obtained as 1385.75 cm<sup>3</sup> /min and 256.84 gm/min respectively. It is observed that MRR and TWR improve significantly by using optimal machining variables combinations. **Table 5** shows the validation results while machining at optimizing condition.

#### **8. Conclusion**

The experimental study indicates that while machining different workpieces like HCHCr, HDS, and OHNS using different EDM tools like Cu, Al, and Br, the responses are dependent on tool material, workpiece material, pulse on time, and machining time. While analyzing the response data individually applying, ANN considering four control parameters in order to achieve maximum MRR and minimum TWR the training, validation, and testing data indicates that the values of R, are almost 1.

A multi-objective response is developed which is optimized using GA. Since the objectives of the responses are contradictory in nature, therefore, using GA the optimum values of responses are obtained within a range. In the case of MRR varies from 1045.3295 cm3 /min to 1437.789 cm<sup>3</sup> /min and TWR varies between 149.402 gm/min to 1216.031 gm/min respectively.

**Author details**

West Bengal, India

**149**

Goutam Kumar Bose and Pritam Pain\*

provided the original work is properly cited.

Department of Mechanical Engineering, Haldia Institute of Technology, Haldia,

*Nature Inspired Metaheuristic Approach for Best Tool Work Combination for EDM Process*

*DOI: http://dx.doi.org/10.5772/intechopen.96725*

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: pritam.me.dscsdec@gmail.com

The GRA establishes the ranks of output for different variables combinations and establishes optimal combinations for a complex process like EDM process. For evaluating the optimum parametric combination during machining her Tool Density of 15.7 g/cm3, Workpiece density of 14.17 g/cm<sup>3</sup> , Current of 86.24 amp, POT of 7836.89 μSec and Machining time of 199.16 min is the best among all the other combinations for having high MRR and low TWR. If any tool material and workpiece have the exact tool density as obtained from the GA that will give the best tool and workpiece combination for machining. As our research is limited to the three types of tool material and three types of the workpiece material, hence it can be concluded that Cu is the best tool for machining OHNS workpiece material by EDM when 20 amp current and POT is 800 μSec.

Therefore, this experimental analysis for estimating the optimum EDM parametric combination during machining with a different tool and work materials can act as valuable and an effective guideline for machining of die and mould.

*Nature Inspired Metaheuristic Approach for Best Tool Work Combination for EDM Process DOI: http://dx.doi.org/10.5772/intechopen.96725*
