Prediction of Multi-Hole Copper Electrodes’ Influence on Form Tolerance and Machinability Using Grey Relational Analysis and Adaptive Neuro-Fuzzy Inference System in Electrode Discharge Machining Process

Abstract

Electric discharge machining processes are prominent in the fastest-growing industries because of their accuracy, achievable complex workpiece shapes, and cost-effectiveness. Furthermore, the machining of high-quality difficult-to-machine alloys is becoming critical in the aerospace, manufacturing, and defence industries. While the optimisation of EDM parameters is essential for improving machining outcomes, it is also important to consider the trade-offs between different performances metrics, such as material removal rate and part accuracy. Part accuracy in terms of dimensional and geometric deviations from nominal values was rarely considered in the literature, if not by the authors. Balancing these factors remains a challenge in the field of EDM. Therefore, this work aims to carry out a multi-objective optimisation of both MRR and part accuracy. A Ni-based alloy (Inconel-625) was used that is widely used in creep-resistant turbine blades and vanes and turbine disks in gas turbine engines for aerospace and defence industries. Four performance indices were optimised simultaneously: two related to the performance of the EDM process and two connected with the form deviations of the manufactured surfaces. Multi-hole copper electrodes having different diameters and three process parameters were varied during the experimental tests. Grey relational analysis and the Adaptive Neuro-Fuzzy Inference System method were used for optimisation. Grey relational analysis found that the following values of the process parameter—0.16 mm of multi-hole electrode diameter, 12 Amperes of Peak current, 200 µs of pulse on time and 0.2 kg/m² as dielectric pressure—produce the optimal performance, i.e., a material removal rate of 0.099 mm³/min, an electrode wear rate of 0.0002 g/min, a circularity deviation of 0.0043 mm and a cylindricity deviation of 0.027 mm. From the experimental examination using multi-hole electrodes, it is concluded that the material removal rate increases and the electrode wear rate decreases because of the availability of higher spark discharge areas between the electrode and work material interface. The Adaptive Neuro-Fuzzy Inference System models showed minimum mean percentage error and, therefore, better performance in comparison with regression models.

1. Introduction

Electric discharge machining (EDM) is a process that uses a series of rapid, recurring electrical discharges—or sparks—to erode and remove material from a workpiece. A key advantage of this spark erosion process is its ability to machine any electrically conductive material, no matter how hard it is. This was also proved for non-conductive materials [1]. The rate of metal removal and resulting surface finish can be controlled by proper variation in the energy and duration of spark discharge. De-ionised water is generally used as a dielectric medium to avoid carbon layer formation [2]. During pulse on time, when the electric power is supplied between the two electrodes, material erosion takes place as a consequence of the formation of several sparks. The energy transfer leads to material erosion at the electrodes due to plasma energy distribution between the cathode and anode that is proportional to the energy transferred during the removal stage. At a temperature of 20,000 K, the main mechanism of heat transfer is radiation, which is governed by the Stefan–Boltzmann law [3]. The thermophysical properties of water vapour change drastically at high temperatures (above 2000 K) because of its dissociation to smaller molecules and ionisation into charged particles, which play a key role in heat transfer to the anode and cathode. EDM is very versatile and preferred for manufacturing precise and difficult-to-machine materials [4]. The optimisation of process parameters in electric discharge machining (EDM) is crucial for enhancing machining efficiency and product quality. Various studies have explored different parameters such as peak current, pulse on time, and duty cycle to maximise material removal rate (MRR) while minimising tool wear and surface roughness. The application of advanced optimisation techniques, including Taguchi methods, genetic algorithms, and multi-objective optimisation, has proven effective in achieving these goals [5].

Some authors demonstrated that oxygen-assisted EDM is more efficient at lower discharge energies [6]. The authors observed that the material removal rate is improved when oxygen gas is supplied in the gap between the electrode and the work material. Song et al. observed the influence of copper and brass strip electrodes in the EDM process to overcome the electrode wear problem [7]. The authors found that the material removal rate (MRR) is influenced by the peak current, and the copper strip electrode involves a higher MRR. Lin et al. presented the effect of solid and bunched electrodes in the EDM process [8]. The authors found that the use of bunched electrodes improves the spark discharge condition, reduces the tool wear rate and increases the material removal rate. Murugesan and Balamurugan applied the GRA method to optimise the process parameter values in the EDM process on Al-SiC metal matrix composites [9]. The authors performed experiments using multi-hole electrodes and revealed that the machining time decreases with the use of these electrodes. Li et al. concluded that the use of hexagonal-shaped bunched electrodes improves the material removal rate and reduces tool wear [10]. The authors utilised the multi-hole ‘inner flushing technique’ during experimental work. Dhanabalan et al. statistically analysed and found the values of peak current, pulse on time and pulse off time that optimise material removal rate (MRR), electrode wear rate (EWR) and form tolerances using hexagonal and square profile tool electrodes [11]. Kumar and Dhanabalan reported that the form tolerance and material removal rate improve with the use of multi-hole electrodes [12]. The authors conducted experiments on Inconel 718 nickel-based alloy and developed an analytical approach using the fuzzy logic artificial intelligence method. Fantoni et al. proposed a U-shaped electrode that allows reducing the machining time and improving the machined surface quality [13]. In their research, Klocke et al. investigated the effects of suspended Al and Si powders within the dielectric medium [14].

Most of the researchers have utilised various traditional and artificial intelligence optimisation methods to improve the machinability and performance of the machining processes. Ong et al. used artificial neural network to optimise the EDM process [15]. Saffaran et al. proposed the backpropagation artificial intelligence models to reduce the optimisation error in the EDM process [16]. Ubaid et al. concluded that the copper electrode improves machinability and MRR [17]. The authors optimised the EDM process parameters using a hybrid ‘Taguchi-Fuzzy’ artificial intelligence approach. Metaheuristic algorithms like Glowworm Swarm Optimisation (GSO), Grey Wolf Optimiser (GWO), and Whale Optimisation Algorithm (WOA) have been applied to optimise EDM processes. Among these, the GWO algorithm demonstrated superior performance, achieving a minimum surface roughness of 1.7593 µm, indicating its effectiveness in solving EDM optimisation problems [18]. Neural network models, particularly the Backpropagation Neural Model (BPM), have been integrated with metaheuristic techniques like the Butterfly Optimisation Algorithm (BOA) to enhance EDM parameter optimisation. This approach has shown improved prediction rates for EDM parameters, with a maximum prediction rate of 8.93% and a mean prediction rate of 2.83% [19]. A genetic algorithm was used to optimise parameters, resulting in a high MRR of 97.35 mg/min and a low electrode wear rate of 1.81 mg/min [20,21]. Grey relational analysis was used to find the values of the process parameters to optimise the MRR and the surface roughness for Inconel 925 [22] and Inconel 800 [23]. The ANFIS method was used to choose the values of the process parameters to minimise surface roughness in micro-EDM of Inconel 625 [24]. Desirability function approach was used to define the values of the process parameters in wire EDM of Inconel 625 to maximise MRR and minimise the kerf width and the surface roughness [25]. In previous works, the authors studied the influence of circular-shaped brass electrodes on the EDM process performances of Inconel 718 [26,27]. Circularity and cylindricity form tolerances were investigated because they are frequently applied to such parts, which are particularly important in design because they require precision and a higher surface finish [28].

This work focuses on nickel base alloys that are widely used in aerospace and defence industries on account of their high mechanical properties, corrosion resistance properties and so on [29]. Inconel 625 is a high-strength, temperature-resistant and difficult-to-machine nickel-based alloy. The alloy is a hardened nickel–chromium type. It is primarily composed of iron, niobium, and molybdenum, with smaller amounts of aluminium and titanium also present. Inconel 625 alloys have good creep-rupture strength at higher temperatures up to 1300 °F. This alloy’s high strength and toughness make it difficult to machine into intricate shapes using traditional methods, which struggle to achieve reasonable speeds and dimensional accuracy. As a result, electrical discharge machining (EDM) is a versatile and effective solution. EDM can machine this difficult-to-machine alloy with a high surface finish and excellent dimensional accuracy. While the optimisation of EDM parameters is essential for improving machining outcomes, it is also important to consider the trade-offs between different performances metrics, such as process performances and machined part accuracy. Part accuracy in terms of dimensional and geometric deviations from nominal values was rarely considered in the literature, if not by the authors. Balancing these factors remains a challenge in the field of EDM.

Therefore, this work aims to carry out a multi-objective optimisation of both process performances and machined part accuracy. A Ni-based alloy (Inconel-625) was used that is widely used in creep-resistant turbine blades and vanes, turbine disks in gas turbine engines for aerospace and defence industries. Four performance indices were optimised simultaneously: two related to the performance of the EDM process and two connected with the form deviations of the manufactured surfaces. Multi-hole copper electrodes having different diameters and three process parameters were varied during the experimental tests. Grey relational analysis and the Adaptive Neuro-Fuzzy Inference System (ANFIS) method were used for optimisation.

2. Materials and Methods

The present work adopted an experimental approach. The first step was to design the experiments through Design of Experiments (DOE) tools using various selected input parameters and their ranges. Design of experiments application needs systematic planning of experiments, the layout of experiments and result analysis. The Taguchi method has standardised DOE steps, and this approach drastically reduces the number of experimental trials required to gather necessary information.

In this present experimentation, the most dominating input process parameters were the diameter of the multi-hole electrodes (it ranges from 0.08 mm to 0.24 mm), dielectric pressure, pulse on time, and pulse off time with their three different values (see

Table 1. EDM process parameter values.

EDM Process Parameter	Symbol	Values
Hole diameter of electrode (d) [mm]	A	0.08–0.16–0.20
Peak current (Ip) [A]	B	4.0–8.0–12.0
Pulse on time (Ton) [µs]	C	200–400–600
Dielectric pressure (Dp) [kg/m2]	D	0.10–0.20–0.30

). Three replications were carried out for each of the nine process conditions.

The experiments were performed on a die-sinking electrical discharge machine by Sparkonix Pvt Ltd., Pune, India, as shown in Figure 1. The work material and square profile electrodes were connected with direct current (DC) power source and the “side flushing technique” was utilised to conduct all experimental work. Kerosene was used as a dielectric medium.

The multi-hole copper electrodes that were used in this work are represented in Figure 2, while their diameters are reported in Table 1. The different diameters of the multi-hole electrodes were engraved by laser drilling to increase accuracy.

A circular plate of 4 mm thickness of nickel-based Inconel 625 was used. Dielectric fluid was flushed through the multi-hole electrode, and the centre flushing method was utilised. The chemical composition of Inconel-625 alloy was measured with Bruker SI Turbo alloying Analyzer; it is 58% minimum Ni, 20–23% Cr, 1% Co, 8–10% Mo, 3.2–4.2% Nb, 0.4% Ti, 0.4% Al, 0.1% C, 0.5% Mn, 0.5% Si, and 5% Fe. The hardness of Inconel-625 was 84 HB measured with the Rockwell hardness tester HT-7.

The experiments were performed according to the design of experiments orthogonal array L₉ (3 levels of 4 factors), as shown in

Table 2. Design of experiments.

No.	Hole Diameter (d) [mm]	Peak Current (Ip) [A]	Pulse on Time (Ton) [µs]	Dielectric Pressure (Dp) [kg/m2]
1	0.08	4.0	200	0.10
2	0.08	8.0	400	0.20
3	0.08	12.0	600	0.30
4	0.16	4.0	400	0.30
5	0.16	8.0	600	0.10
6	0.16	12.0	200	0.20
7	0.24	4.0	600	0.20
8	0.24	8.0	200	0.30
9	0.24	12.0	400	0.10

. The results were calculated for four performance parameters: electrode wear rate (EWR), material removal rate (MRR), circularity and cylindricity deviation of the machined holes.

EWR and MRR were calculated using the following relationships:

$$EWR={\displaystyle \frac{\mathrm{weight} \mathrm{of} \mathrm{electrode} \mathrm{before} \mathrm{machining}- \mathrm{weight} \mathrm{of} \mathrm{eletrode} \mathrm{after} \mathrm{machining} }{\mathrm{time}}} {\displaystyle \frac{\mathrm{g}}{\mathrm{m}\mathrm{i}\mathrm{n}}}$$

(1)

$$MRR={\displaystyle \frac{\mathrm{Weight} \mathrm{of} \mathrm{material} \mathrm{removed} \mathrm{by} \mathrm{the} \mathrm{workpiece}}{\mathrm{time}}}{\displaystyle \frac{{\mathrm{m}\mathrm{m}}^3}{\mathrm{m}\mathrm{i}\mathrm{n}}}$$

(2)

An electronic digital weight balance was used to measure the weight of the multi-hole electrode and Inconel-625 alloy. The weight of the electrode and work material was measured thrice before and after experimental trials; the average value was considered in Equations (1) and (2). The TESA Micro Hite 3D coordinate measurement machine was utilised to measure the circularity and cylindricity deviations of the machined holes. A set of 10 points was acquired along each section of the hole, and three sections were measured along the hole height for a total of 30 points for each hole. The least squares method was used to estimate the circularity and cylindricity deviations from the sets of points.

2.1. Grey Relational Analysis (GRA) Method

Multi-criteria decision method (MCDM) involves several objective functions within a pre-determined set of constraints. These methods provide the best alternative optimal solution among all sets of alternatives. In this experimental dataset, the main objectives are to minimise the tool wear, maximise the material removal rate and minimise the form deviations to improve the machinability with close geometrical deviations. The grey relational analysis method is one of the oldest and most reliable MCDM techniques, which is generally used for multi-objective optimisation in various practical applications such as hospitality, marketing, manufacturing, administration, assembly, and selection of materials and processes. The various steps involved in finding the optimal solution using the grey relational analysis method are as follows:

(a)

Data pre-processing or normalisation of results;

(b)

Grey relational coefficient calculations;

(c)

Grey relational grades calculation;

(d)

Calculation for the selection of optimum levels;

(e)

Confirmation and verification.

The mathematical relations used to obtain the optimal solutions through the GRA method are as follows:

(a)

The normalisation or pre-processing of data x_ij is obtained through the standard mathematical relationship given in Equation (3). The values are inside the range 0–1.

$${\mathrm{x}}_{\mathrm{i}\mathrm{j}}^0={\displaystyle \frac{{\mathrm{y}}_{\mathrm{i}\mathrm{j}}-{{\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{j}}\mathrm{y}}_{\mathrm{i}\mathrm{j}}}{{{\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{j}}\mathrm{y}}_{\mathrm{i}\mathrm{j}}-{\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{j}}{\mathrm{y}}_{\mathrm{i}\mathrm{j}}}},$$

(3)

where y_ij represents the ith result of the jth experiment.

(b)

The grey relational coefficient shows the gap between the data and the normalised data. The mathematical relationship for grey relational coefficient calculation is given in Equation (4).

$${\mathsf{\delta }}_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{{\min }_{\mathrm{i}}{\min }_{\mathrm{j}}\left|{\mathrm{x}}_{\mathrm{i}}^0-{\mathrm{x}}_{\mathrm{i}\mathrm{j}}\right|+\mathsf{\xi }{\max }_{\mathrm{i}}{\max }_{\mathrm{j}}\left|{\mathrm{x}}_{\mathrm{i}}^0-{\mathrm{x}}_{\mathrm{i}\mathrm{j}}\right|}{\left|{\mathrm{x}}_{\mathrm{i}}^0-{\mathrm{x}}_{\mathrm{i}\mathrm{j}}\right|+\mathsf{\xi }{\max }_{\mathrm{i}}{\max }_{\mathrm{j}}\left|{\mathrm{x}}_{\mathrm{i}}^0-{\mathrm{x}}_{\mathrm{i}\mathrm{j}}\right|}},0<\xi <1,$$

(4)

where x⁰_i represents the normalised result value.

(c)

Grey relational grade is obtained by calculating the average values of grey relational coefficients. The standard mathematical relationship used for the calculation of grades is given in Equation (5).

$${\alpha }_j={\displaystyle \frac{1}{m}}{\sum }_{i=1}^m{\delta }_{ij}$$

(5)

where α_j represents the value of grades and m represents the number of executions.

(d)

Calculation of optimum levels.

(e)

Confirmation and verification to measure the improvement in the process after obtaining the optimal solutions.

2.2. Adaptive Neuro-Fuzzy Inference System (ANFIS) Hybrid Method

Computational intelligence has drawn the attention of scientists, researchers and engineers in different areas because of its nature-inspired computational methodologies and approaches to addressing complex real-world problems [30]. Artificial Neuro-Fuzzy Inference System (ANFIS) is a metaheuristic intelligence system and a key agent of computational intelligence. It is based on the Takagi and Sugeno approach and was proposed by [31]. An Adaptive Neuro-Fuzzy Inference System (ANFIS) combines neural networks and fuzzy logic. It has five layers: input, fuzzification, AND function, fuzzy inference, and defuzzification [32]. The goal of ANFIS is to create or test the relationship between a system’s inputs and outputs.

The various steps involved in developing a model using ANFIS [33,34,35] are as follows:

(a)

Input: Four input values were utilised from the selected constraints; they are the hole diameter of the electrode, peak current, pulse on time and dielectric pressure.

(b)

Fuzzification: The standard mathematical relationship used for the membership function of a bell-shaped function is given in Equation (6).

$$\mu A_i(x)={\displaystyle \frac{1}{1+\left[\right(\frac{x-c_i}{a_i}{)}^2]\times b_i}},$$

(6)

where a_i, b_i and c_i represent the constraint set. A bell-shaped function was used to provide a smooth and gradual transition between fuzzy sets. Therefore, this functions allow for more accurate modelling of complex systems. By applying the Fuzzy operator, the multiplication of incoming signals is obtained through the standard mathematical relationship given in Equation (7).

$$i=\mu A_i\left(x\right)\times \mu B_i\left(y\right)\times \mu C_i\left(z\right),$$

(7)

where the value of i = 1, 2, 3.

(c)

Perform AND operator: The normalisation of each node is calculated by the mathematical relationship in Equation (8).

$${\overline{w}}_i={\displaystyle \frac{w_i}{{\sum }_i{w}_i}},i=\mathrm{1,2},3,$$

(8)

(d)

Fuzzy inference and defuzzification: In defuzzification, the fuzzy quantity is converted by the rules and membership functions combination to predict the accurate output results. The standard mathematical relationship used to calculate the defuzzification is given in Equation (9).

$${\overline{w}}_i\times f_i={\overline{w}}_i.(p_i.x+q_i.y+r_i.z+s_i),$$

(9)

where p_i, q_i, and r_i represent the consequent constraint set at each node.

(e)

Output: The output of ANFIS structure is calculated by the sum of all incoming signals. The mathematical relationship used to calculate the output is given in Equation (10).

$$Output={\sum }_i{\overline{w}}_i.f_i={\displaystyle \frac{{\sum }_i{\overline{w}}_i.f_i}{{\sum }_i{\overline{w}}_i}},$$

(10)

3. Results and Discussion

3.1. GRA Method

The average values (µ) and the standard deviation values (σ) of the selected performance variables, electrode wear rate (EWR), material removal rate (MRR), circularity and cylindricity deviations, obtained by the experimental tests, are presented in

Table 3. Measured average values of performance parameters.

No.	MRR (mm3/min)		EWR (g/min)		Circularity (mm)		Cylindricity (mm)
	µ	σ	µ	σ	µ	σ	µ	σ
1	0.009	0.0002	0.002	0.0002	0.0183	0.002	0.0441	0.002
2	0.045	0.001	0.001	0.0002	0.1042	0.03	0.0125	0.001
3	0.086	0.003	0.002	0.0001	0.0216	0.001	0.0613	0.002
4	0.013	0.003	0.0005	0.00001	0.085	0.002	0.0533	0.003
5	0.048	0.002	0.0017	0.0003	0.0625	0.002	0.0003	0.00001
6	0.099	0.003	0.0002	0.00002	0.0043	0.0003	0.0272	0.001
7	0.022	0.002	0.0027	0.0002	0.1294	0.02	0.0482	0.003
8	0.059	0.003	0.0074	0.0003	0.0011	0.0003	0.0528	0.002
9	0.074	0.003	0.0078	0.0002	0.032	0.001	0.0345	0.003

. The standard deviations connected with these values were an order of magnitude smaller than the average shown in Table 3.

Table 4. Calculated normalised values, grey relational coefficients, and grade values using GRA.

No.	Normalisation				Grey Relational Coefficients				Grades
	MRR	EWR	Circularity	Cylindricity	MRR	EWR	Circularity	Cylindricity
1	0	0.763	0.866	0.282	0.333	0.679	0.789	0.410	0.553
2	0.400	0.895	0.196	0.800	0.455	0.826	0.384	0.714	0.595
3.	0.856	0.763	0.840	0	0.776	0.679	0.758	0.333	0.636
4	0.044	0.961	0.346	0.131	0.344	0.927	0.433	0.365	0.517
5	0.433	0.803	0.521	1	0.469	0.717	0.511	1.000	0.674
6	1	1	0.975	0.559	1.000	1.000	0.952	0.531	0.871
7	0.144	0.671	0	0.215	0.369	0.603	0.333	0.389	0.424
8	0.556	0.053	1	0.139	0.529	0.345	1.000	0.367	0.561
9	0.722	0	0.759	0.439	0.643	0.333	0.675	0.471	0.531

shows the values of the calculated normalised data, grey relational coefficients, and grade values.

Table 5. GRA grades response values.

Process Parameters	Level 1	Level 2	Level 3
A	0.595	0.687	0.505
B	0.498	0.610	0.679
C	0.661	0.547	0.578
D	0.586	0.630	0.571
Average grey relational value = 0.59566

shows the optimum levels. The standard mathematical relationship used to calculate the estimated grey relational grade is shown in Equation (11).

$$\widehat{\mathsf{\alpha }}={\mathsf{\alpha }}_{\mathrm{m}}+{\sum }_{\mathrm{i}=1}^{\mathrm{q}}{\overline{(\mathsf{\alpha }}}_{\mathrm{i}}-{\mathsf{\alpha }}_{\mathrm{m}}),$$

(11)

where q represents the number of constraints and α_m represents the mean value of grades.

The optimal values were compared with the experimental ones, as shown in

Table 6. Comparison between optimal values and experimental data.

	Optimal Value	Experimentation
Level	A2B3C1D2	A2B3C1D2
MRR [mm3/min]	0.048	0.099
EWR [g/min]	0.0017	0.0002
Circularity [mm]	0.0625	0.0043
Cylindricity [mm]	0.0003	0.0272
Grade	0.67417	0.87096
Improvement in grey relational grade: 0.1968

, to validate the training of the GRA method. The differences in the optimal values compared to experimental data are 0.051 mm³/min for MRR, 0.0015 g/min for EWR, 0.0582 mm for circularity and 0.027 mm for cylindricity. These values are so small as to be considered negligible. These differences between GRA predictions and experimental results exist because GRA is a powerful analytical tool for decision-making with limited data. It excels at identifying the most influential parameters and trends, but its capacities are based on simplifications of a complex system, and the results will always be affected by the inherent uncertainties of both the method and the experimental data.

3.2. ANFIS Method

The ANFIS model was developed using nine training data points and randomly selected six testing data points from the available dataset. Matlab R2023b software was utilised for the selection of optimal parameters by training and testing of data. The selected input parameters for the ANFIS model to obtain the optimal solutions are represented in

Table 7. Input constraints for the ANFIS model.

Output Response	EWR	MRR	Circularity	Cylindricity
Training Method	Hybrid	Hybrid	Hybrid	Hybrid
MF’s	trimf	trimf	trimf	trimf
No. of Mf’s	3 3 3 3	3 3 3 3	3 3 3 3	3 3 3 3
No. of epochs	300	300	300	300
Output function	Constant	Constant	Constant	Constant

.

The ANFIS model structure with five layers obtained for the prediction of optimal performance parameters is represented in Figure 3. It has five layers: input, fuzzification, AND function, fuzzy inference, and defuzzification.

3.2.1. EWR Analysis

The four input functions of multi-hole electrode diameter, peak current, pulse on time and dielectric pressure were considered to obtain the optimal solution for electrode wear rate. Triangular membership functions (‘trimf’ MF’s) have the lowest testing error and lowest mean absolute percentage error with inclined and declined features. Therefore, the triangular membership function was selected rather than other membership functions. A total of 300 epochs were executed for data training. Testing data was selected randomly from the experimental dataset. The values obtained for training and testing error were 3.855 × 10⁻⁸ and 2.6904 × 10^−6, respectively. The comparison between the experimental and predicted values is represented in Figure 4a.

The 3D effect of multi-hole electrode diameter and pulse on time on electrode wear rate (generated through the ANFIS model) is presented in Figure 4b, which shows the reduction in EWR as electrode-hole diameter increases up to 0.16 mm. This is due to the reduction in current density at the active surface and improved heat dissipation from the larger electrode body, leading to less intense thermal erosion per unit area. The trend of EWR with pulse on time increases up to 400 µs and then decreases. Too short pulse on time values involve less energy per pulse, leading to higher EWR if the energy is too concentrated and causes localised severe wear without efficient material removal from the workpiece. Longer pulse on time values can sometimes lead to a decrease in EWR due to the formation of protective layers and/or reduced effective current density, but might also lead to less stable machining.

It was found that the diameter of the multi-hole electrode and peak current are the most significant parameters for electrode wear rate because of higher spark discharge area availability near the tool and work material interface, as shown in Figure 4b Surface view which clearly indicates that the electrode wear rate is higher at 8 amps of peak current and lower diameter values of the internally engraved multi-hole electrode. As the diameter of multi-hole electrodes increases and the value of peak current decreases or increases from 8 amperes, the electrode wear rate decreases because of the availability of a higher spark discharge area. Beyond the diameter of 0.16 mm up to 0.24 mm, the electrode wear rate remains constant.

3.2.2. MRR Analysis

The four considered input functions were the diameter of the multi-hole electrode, peak current, pulse on time and dielectric pressure. Triangular membership functions (‘trimf’ MFs) were selected, and 300 epochs were executed for data training. The values obtained for training data and testing data error were 6.598 × 10⁻⁷ and 5.2582 × 10⁻⁷, respectively. The comparison between the experimental and predicted MRR values is represented in Figure 5a.

The 3D effect generated through the ANFIS model for multi-hole electrode diameter and dielectric pressure on MRR is shown in Figure 5b, which shows the reduction in MRR when the electrode-hole diameter increases. It can be due to lower effective energy concentration at the point of discharge and challenges in maintaining a clean and stable machining gap. MRR value increases when dielectric fluid pressure increases up to a certain value and then decreases. This happens because, at very low- and high-pressure rates, the dielectric fluid does not have sufficient time for ionisation and spark generation. It was found that the multi-hole electrode diameter and dielectric pressure are the most significant parameters for the material removal rate.

The surface view, shown in Figure 5b, clearly indicates that the value of MRR is lower at the first level of dielectric pressure and the third level of multi-hole electrode diameter. As the value of dielectric pressure increases from the first level to the second level, the MRR value increases up to a certain value from which it remains constant and finally decreases. The highest value of MRR is connected with the first level of multi-hole electrode diameter.

3.2.3. Circularity Deviation Analysis

The ANFIS model for circularity deviation analysis includes four input functions of multi-hole electrode diameter, peak current, pulse on time and dielectric pressure to obtain the optimal solutions. Triangular membership functions (‘trimf’ MFs) were selected, and a data training curve was obtained after 300 epochs. The values obtained for training data and testing data error were 1.553 × 10⁻⁶ and 6.869 × 10⁻⁷, respectively. The comparison between the experimental and predicted values is represented in Figure 6a.

Circularity deviation increases gradually with the increase in pulse-on time up to the second level. As the pulse on time value increases from the second to the third level, circularity deviation decreases gradually.

It was found that the pulse on time and peak current significantly affect the circularity deviation. The surface view, shown in Figure 6b, clearly indicates that the circularity deviation value is highest at level 3 of peak current and level 1 of pulse on time. As the value of pulse on time increases from the second level to the third level and peak current decreases from the third to the second level, circularity deviation decreases gradually. These two parameters dictate the energy delivered per spark and the thermal conditions in the machining gap, which directly impact the shape and quality of the eroded material. High peak currents can lead to less stable discharge conditions, increased stray discharges, and more aggressive material removal, which can cause uneven erosion around the periphery of the hole, thus worsening circularity. The increased heat and material removal can also make it harder to maintain a precise, perfectly circular path. Shorter, more controlled pulses can lead to more precise and uniform material removal, contributing to better circularity. Long pulse on time values involve a high energy per discharge that leads to significant electrode wear. However, this wear can become more uniform across the electrode face, and the large plasma cloud can help stabilise the process, reducing some of the erratic side-wall erosion. While the overall process is less efficient, the circularity deviation may not be at its worst.

3.2.4. Cylindricity Deviation Analysis

The ANFIS model for cylindrical deviation analysis includes four input functions, namely the diameter of the multi-hole electrode, peak current, pulse on time and dielectric pressure, to obtain the optimal solutions. Triangular membership functions (‘trimf’ MFs) were selected, and a data training curve was obtained after 300 epochs. The values obtained for training data and testing data error were 6.303 × 10⁻⁷ and 1.219 × 10⁻⁷, respectively. The comparison between the experimental and predicted values is illustrated in Figure 7a. The 3D effect generated through the ANFIS model of electrode hole diameter on cylindricity deviation is shown in Figure 7b. It clearly represents that the cylindricity deviation declines with the increase in the multi-hole electrode diameter.

It was found that the pulse on time and multi-hole electrode diameter have the greatest effect on cylindricity deviation. The surface view, shown in Figure 7b, clearly indicates that the cylindricity deviation value is highest at level 2 of pulse on time and level 1 of multi-hole electrode diameter. This is probably due to the following considerations.

Low pulse on time values involve an energy per discharge that is low. This results in minimal electrode wear, and the efficient flushing of debris ensures a stable process with no secondary sparking. The result is a highly accurate, cylindrical hole with low deviation.

High pulse on time values involve a high energy per discharge that leads to significant electrode wear. However, this wear can become more uniform across the electrode face, and the large plasma cloud can help stabilise the process, reducing some of the erratic side-wall erosion. While the overall process is less efficient, the cylindricity deviation may not be at its worst.

Intermediate pulse on time values indicate that the energy is high enough to cause significant, but non-uniform, electrode wear, leading to pronounced tapering. At the same time, the pulse duration is too long for the dielectric fluid to efficiently flush away debris, which leads to frequent and unpredictable secondary sparking on the side walls of the hole. This combination of uneven electrode wear and erratic erosion of the hole walls results in the maximum cylindricity deviation.

A larger electrode diameter hampers the ability to effectively flush debris, leading to unstable machining and secondary sparking. It also exacerbates the issue of uneven corner wear, causing the electrode to taper. Both of these factors directly contribute to an increase in the cylindricity deviation of the machined hole.

4. Empirical Modelling Using Regression Analysis

Regression analysis is a statistical tool used for the mathematical representation of a relationship between input variables and performance parameters. In regression statistics, ‘R²’ represents the coefficient of determination or covariance. It is the proportion of variability in performance parameters that is explained by the regression curve. The regression model is suitable if the value of ‘R2’ lies near 1 [36].

The regression model calculations for the considered performance parameters are given in Equations (12)–(15).

$$\mathrm{M}\mathrm{R}\mathrm{R}=-0.0314+0.0312\ast \mathrm{A}+0.009\ast \mathrm{B}-0.000009\ast \mathrm{C}+0.0450\ast \mathrm{D},$$

(12)

$$\mathrm{E}\mathrm{W}\mathrm{R}=-0.0015+0.0269\ast \mathrm{A}+0.0002\ast \mathrm{B}-0.000003\mathrm{C}-0.0027\ast \mathrm{D},$$

(13)

$$\mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=0.0415+0.038\ast \mathrm{A}-0.0073\ast \mathrm{B}+0.00016\mathrm{C}-0.008\ast \mathrm{D},$$

(14)

$$\mathrm{C}\mathrm{y}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=0.0141+0.037\ast \mathrm{A}-0.001\ast \mathrm{B}-0.000012\mathrm{C}+0.147\ast \mathrm{D}$$

(15)

Their R² values are very close to the unit.

Figure 8 shows the results predicted by ANFIS and those obtained experimentally; as can be seen, they are almost completely overlapping. The results predicted by the regression Equations (12)–(15) are reported in Figure 8 too; as it can be seen, they are farther from the experimental results than the ANFIS results are, except for the MRR results.

5. Conclusions

This work shows a multi-objective optimisation of both EDM performances and machined part geometric accuracy. A Ni-based alloy (Inconel-625) was used that is widely used in creep-resistant turbine blades and vanes and turbine disks in gas turbine engines for aerospace and defence industries. Grey relational analysis and the ANFIS method were used for optimisation. Grey relational analysis found that the following values of the process parameter—0.16mm of multi-hole electrode diameter, 12 Amperes of Peak current, 200 µs of pulse on time and 0.2 kg/m² as dielectric pressure—produce the maximum process performances and the better machined part accuracy, i.e., a material removal rate of 0.099 mm³/min, an electrode wear rate of 0.0002 g/min, a circularity deviation of 0.0043 mm and a cylindricity deviation of 0.027 mm.

From experimental results, it was found that the use of multi-hole electrodes increases the material removal rate because a higher spark discharge area is available between the electrode and the work material. Moreover, the use of multi-hole electrodes in the EDM process reduces the electrode wear rate, circularity and cylindricity deviations.

ANFIS (Adaptive Neuro-Fuzzy Inference System) models have been developed for the prediction of performance parameters, namely electrode wear rate, material removal rate and form deviations. The four most dominating process variables were considered as the input constraints for ANFIS models. To measure the reliability of the developed models, the models were compared with full factorial regression empirical models. It was found that the ANFIS models have lower mean absolute percentage error, and therefore, these models can be utilised for reliable results. A limit of the proposed approach is connected to using the average values of the datasets instead of all the data for training, thus not considering the variability of the considered process. To remove this limit is a matter of further study.

The obtained results allow aerospace, defence and advanced machining industries to increase the material removal rate and the accuracy of nickel-based Inconel-625 alloy.

Further studies involve the analysis of the influence of different dimensions of datasets on the values of the performance parameters. Other studies will focus on other hard-to-work materials and other shapes always obtained through the EDM process.