Multi-Objective Optimization of Wire Electrical Discharge Machining Parameters in Machining of Inconel 625 Using Bio-Inspired Optimization Algorithms

Abstract

Wire Electrical Discharge Machining (WEDM) is highly significant for machining Inconel, which is widely used across industries for its excellent strength and heat resistance. Traditional machining of Inconel poses several difficulties; therefore, WEDM becomes a better option. One of the major challenges in WEDM is achieving optimal Material Removal Rate (MRR) and Surface Roughness (SR) simultaneously. In the present investigation, WEDM operation was conducted on Inconel 625 using a molybdenum wire as the electrode material, with three essential process variables: Pulse ON Time (P_ON), Pulse OFF Time (P_OFF), and peak current (I_P). Multi-response optimization was performed using Response Surface Methodology (RSM) with the Face-centred Central Composite (FCC) design, the Non-dominated Sorting Genetic Algorithm II (NSGA-II), and Particle Swarm Optimization (PSO). The optimal values obtained using RSM were 45.71 µs for P_ON, 8.02 µs for P_OFF, and 4.78 A for I_P, providing an SR of 3.997 µm and an MRR of 5.612 mm³/min. PSO focused more on the quality of the processed surface and gave a better SR of 3.098 µm, but the resulting MRR was 4.437 mm³/min using 20 µs for P_ON, 12 µs for P_OFF, and 3 A for I_P. Conversely, the NSGA-II optimized for maximal machining productivity and generated the highest MRR of 5.907 mm³/min with an SR of 3.867 µm using 49.42 µs for P_ON, 8 µs for P_OFF, and 5 A for I_P. Novelty lies in the use of statistics and metaheuristics to compare the performance of optimization methods in nonlinear parameter response analysis. Although RSM demonstrated high predictive accuracy, both the NSGA-II and PSO improved the search for the trade-off between MRR and SR. Overall, the best compromise between MRR and SR was achieved using the solution based on the NSGA-II results. The obtained results will help in developing advanced machining technologies aimed at precision processing of superalloys that contribute to SDG 9—Industry, Innovation and Infrastructure.

1. Introduction

Wire Electrical Discharge Machining (WEDM) is a widely used method for machining electrically conductive materials. Its precision and versatility have made it indispensable in aerospace, medicine, and die-making industries [1]. The material removal is done using the spark from the wire. The wire does not make direct contact with the workpiece. The key parameters that define the process are Pulse ON Time (P_ON), Pulse OFF Time (P_OFF), peak voltage (V), peak current (I_P), dielectric flow rate (D_F) [2], wire feed, and wire tension [3]. These parameters play a vital role in altering the performance outcomes like Material Removal Rate (MRR) and Surface Roughness (SR). Generally, a higher P_ON and higher I_P increase MRR considerably, but increase the SR value, which is not desirable [4].

MRR and SR are interrelated in the working principles of the WEDM process. In this process, an electrical spark is generated between the wire and the workpiece in the presence of a dielectric fluid. The spark erodes the material from the workpiece, which is simultaneously washed away by the flowing dielectric fluid. When P_ON and voltage (V) increase, the intensity of the spark increases significantly. This increases the amount of material removed and increases the MRR value. Stronger sparks create deeper craters on the surface, resulting in an increased SR value. While it is theoretically possible to balance MRR and SR, achieving both high MRR and best SR is challenging. Therefore, precise adjustment of process parameters is essential to obtain high-quality machining results. The D_F plays a crucial role in WEDM [5]. It helps both the workpiece and the wire to cool down from excess heat and to wash off the material debris. When they are not removed properly, they can stick to the workpiece again and reduce surface quality. A high D_F can effectively wash off the debris, reducing the risk of secondary discharge and enhancing the stability of the sparking process. A lower D_F can lead to the accumulation of debris in the gap between the wire and the workpiece, which leads to short-circuiting, unstable machining, and poor surface finish [6].

Han et al. analysed the impact of pulse duration on surface morphology using thermal and finite element analysis. They found that longer P_ON combined with lower I_P removed material primarily by melting. When a short P_ON is combined with a higher I_P, material is removed primarily through gasification [7]. Optimizing machining parameters is essential in any machining process, as it can reduce production costs and machining time while enhancing surface quality and overall process efficiency. Several techniques have been employed to identify the optimal parameters for the WEDM process. The most common methods include Taguchi’s Design of Experiments (DoE) [8,9], Response Surface Methodology (RSM) [4], Grey Relational Analysis (GRA) [10,11,12,13], Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [14], and Taguchi–Data Envelopment Analysis-based Ranking (DEAR) [15] for multi-response optimization. Machine learning (ML) techniques such as reinforcement learning and deep learning are also being explored in modern machining to predict the outcomes [16].

RSM was used to optimize WEDM process parameters for High-Strength Low-Alloy (HSLA) steel. The five input parameters included P_ON, P_OFF, V, I, and wire tension. A Central Composite Rotatable Design (CCRD) was applied to conduct 32 structured experiments by varying the input parameters. With the help of statistical analysis, including Analysis of Variance (ANOVA) and regression, RSM identified the most significant factors and their interactions [17]. The accuracy and precision of the result can be verified with the help of the coefficient of determination (R²). The R² obtained for MRR was found to be 0.9924, indicating that 99.24% of the variation in MRR can be attributed to the process parameters calculated. The value of R² for the SR model was found to be 0.9766. The predicted R² and adjusted R² were also consistent, having a difference of less than 0.2, with the predicted R² being 0.9509 and the adjusted R² being 0.9853 [18].

Current research has pointed out the efficacy of modern multi-objective optimization techniques in improving WEDM machining efficiency on Inconel superalloys. An ANN-based model in conjunction with the Non-dominated Sorting Genetic Algorithm II (NSGA-II) was proposed for optimizing MRR and SR in machining Inconel 718, resulting in an accuracy rate of 1–2% [19]. The multi-objective teaching–learning-based optimization (TLBO) technique was used for Inconel 686 material, optimizing the process parameters to obtain MRR and SR values of 0.0421 g/min and 3.102 µm, respectively [20]. The best surface finish value of 0.419 µm for Inconel 625 was obtained by using the PSO technique over GA under cryogenic conditions [21]. Ming et al. employed an extreme learning machine-enhanced Cuckoo Search algorithm (CSA) for EDM parameters for Inconel 718, maintaining prediction deviations below 20% [22].

Kumar et al. used the NSGA-II to determine the optimal WEDM parameters during machining of Inconel 718. Input parameters included wire tension, wire speed, I_P, and P_ON. The response parameters were MRR and SR. A Taguchi-based DoE included an L₂₇ orthogonal array, which was used to optimize the WEDM parameters for MRR and SR. The optimal parameters were identified through the Pareto front [23]. The NSGA-II was used to optimize the WEDM parameters for machining Ni_55.8Ti. The input parameters were P_ON, P_OFF, spark gap voltage, and wire feed. MRR and SR were the responses. After NSGA-II optimization, an MRR of 0.021 g/min and an SR of 6.20 μm were attained [24]. The NSGA-II was used to optimize the parameters of a WEDM process on a tough ceramic composite made of ZrB₂ and B₄C. These materials are extremely hard and difficult to machine. The NSGA-II helped the researchers to identify a set of optimal input parameters called the Pareto front, which was then refined with the help of TOPSIS to attain a balanced solution. This helped to overcome the machining challenges and improved the total machining performance [25].

Rajan et al. used Particle Swarm Optimization (PSO) to optimize process parameters of Inconel 718 under the abrasive powder-mixed WEDM (APM-WEDM) process. WEDM process parameters, including P_ON, P_OFF, V, and I_P, were considered. The Pareto optimal solutions were generated and showed a minimal deviation of 2.996% in MRR and 3.553% in SR [26]. The application of ML and multi-objective optimization in recent times has focused on using a hybrid approach to overcome the conflicting requirements for performance in non-conventional machining processes. In WEDM, Zhou et al. developed an ML framework, which compared various regression models, such as support vector regression, random forest, gradient boosting, and artificial neural networks (ANNs). The researchers selected the best-performing ANN to integrate with the NSGA-II to simultaneously optimize SR, MRR, and drum shape error in multi-pass WEDM [27]. Also, Jeet et al. used RSM along with an artificial neural network to optimize MRR and SR in WEDM of the Nimonic-C263 superalloy in relation to spark energy, spark frequency, and I_P [28].

Applying the above-mentioned approach to the Inconel 718, Kumar & Dupadu took one step further compared to classical Pareto approach by establishing a link between the optimization of the WEDM process and the behaviour of discharge signals during the process, giving more physical meaning to the optimization process [29]. Altogether, these papers demonstrate a clear tendency to combine evolutionary algorithms and multi-criteria decision making (MCDM) approaches to make process optimization more robust and interpretable. To attain the best optimal parameters, researchers have used population-based, non-traditional optimization algorithms. These include the Genetic Algorithm (GA) [21], PSO [21], the NSGA-II [30], the Firefly algorithm (FA) [2], TLBO [31], Ant Colony Optimization (ACO), Artificial Bee Colony (ABC) methods, the Chicken Swarm Algorithm, Biogeography-Based Optimization (BBO), and the Sheep Flock Algorithm (SFO) [32]. The different works using the WEDM process parameter optimization are presented in Table 1.

Table 1. Summary of the literature related to WEDM and optimization.

References	Material	Wire Electrode	Process Parameters	Responses Measured	Optimization Method	Key Findings
[33]	Inconel 718	Molybdenum wire	IP, PON, POFF	Machining time (MT), MRR, SR	Taguchi DoE, overall evaluation criteria (OEC)	IP was the most influential factor for all responses. Optimal settings (2 A, 40 µs TON, and 9 µs TOFF) for balanced MRR and SR.
[34]	Ti6Al4V	Molybdenum wire	IP, PON, POFF	Cutting speed (CS), MRR, SR	RSM, fuzzy AHP–fuzzy TOPSIS (MCDM)	The optimal process parameters after integrating fuzzy AHP coupled with the fuzzy TOPSIS approach were found to be PON = 40 µs, POFF = 15 µs, and IP = 2 A.
[35]	Inconel 718	Novel Cu-Ni-B4C nanocomposite electrode	IP, PON, POFF	MRR, electrode wear rate (EWR), SR	RSM, CNN, RNN, LSTM	With optimal settings (PON: 50 μs, POFF: 90 μs, and IP: 8 A), it increased MRR to 0.0118 g min−1, reducing EWR to 0.001 g min−1 and SR to 3.108 μm. CNN has the better prediction accuracy compared to others.
[36].	Inconel Grades (Generic Model)	Not specified	IP, PON, POFF, servo voltage (SV), chemical composition	MRR, SR	Machine learning (ML): random forest (RF), SVM, KNN, linear regression	Novel approach using chemical composition as input. RF was found to be the most accurate ML algorithm for predicting WEDM responses on untested Inconel grades.
[37].	Inconel 690	Molybdenum (Mo) wire	PON, wire speed (WS), servo sensitivity, gap voltage (GV)	CS, SR	Taguchi, ANN-MOJAYA (Hybrid AI)	TON was the most significant parameter for both CS and SR. Hybrid ANN-MOJAYA successfully optimized the conflicting objectives.
[38]	Inconel 825	Brass wire (0.25 mm)	PON, POFF, GV, wire feed	CS, gap current	RSM-BBD	PON, POFF, and SV have major influencing factors on responses, whereas wire feed was insignificant.
[8]	Inconel 625	Brass wire	IP, PON, POFF, SV	SR, MRR	Taguchi DoE	IP and PON were the most significant factors for MRR (72.75%) and SR (11.68%), respectively.
[39]	Inconel 718	Zinc diffusion-coated brass wire	IP, PON, POFF, SV, wire tension (WT)	MRR, kerf width (KW), SR	Taguchi–DEAR	PON was the most significant factor for quality. The use of DEAR provided a comprehensive ranking of all experimental runs based on overall efficiency.
[40]	Inconel 625	Zinc-coated brass wire	PON, POFF, SV	MRR, KW, SR	RSM	PON was the most significant factor (64.14% contribution) for the composite desirability. The optimal setting (PON = 18 µm, POFF = 53 µm, and SV = 23 V) improved all three responses simultaneously.
[41]	Inconel 600	Untreated zinc and cryogenically treated zinc wire	IP, PON, POFF	MRR, SR	Taguchi DoE	Cryogenically treated zinc wire showed superior MRR and SR compared to untreated wire, highlighting the importance of wire pretreatment.
[42].	Inconel 825	Zinc-coated wire	IP, PON, POFF, WT, GV	SR, MRR, and dimensional shift (DS)	Taguchi DoE, GRA	PON, POFF, and GV were found to significantly influence MRR, SR, and DS, respectively.
[43].	Ti–6Al–4 V Alloy	Zinc-coated copper wire (0.25 mm diameter)	IP, PON, POFF, SV	MRR, SR	GRA	IP was the most significant parameter for MRR and SR. GRA successfully optimized the multi-response problem.
[44]	Inconel 718	Brass wire	IP, infeed rate (Z)	Surface topography parameters (Ra, Rz, white layer thickness (WLT)), hardness	Experimental and microstructural analysis	The study focused on finishing cuts. IP was the main parameter shaping the surface topography and WLT in the finishing WEDM process.
[45].	Inconel 718	Zinc-coated brass and Brass	PON, flushing pressure (Fp), WT, discharge current (Id)	MRR, SR, corner deviation	GRA–teaching–learning-based optimization (TLBO) algorithm, GRA-GA, GRA-JAYA	TLBO, a modern metaheuristic algorithm, was effectively used to find optimal parameters (PON = 26 µs, Fp = 8 kg/mm2, WT = 12 N, and Id = 16 A), showing better performance than traditional optimization methods.
[46]	Inconel 718	Molybdenum wire	IP, PON, POFF	MRR, SR, MT	Multi-objective genetic algorithm (MOGA), ANN	MOGA was used to generate a Pareto optimal front, providing a set of trade-off solutions (IP 2 A, PON 50 μs, and POFF 9 μs) for MRR, MT, and SR. IP was found to be the most significant parameter. The prediction accuracy of the ANN was better than multiple linear regression.
[21].	Inconel 625 and Cryogenic Inconel 625	Zinc-coated Brass wire	IP, PON, POFF, wire feed, WT	MRR, SR, overcut (OC)	Taguchi DoE, PSO, GA	Cryogenic treatment significantly reduces SR and MRR and increases the OC. PSO showed faster convergence and higher precision than the GA and Taguchi methods.
[19].	Inconel 718	Molybdenum	IP, PON, POFF, wire feed	MRR, SR	ANN + NSGA-II	Wire feed and PON were the significant parameters for MRR and SR. Using the ANN + NSGA-II hybrid model, the Pareto front optimal setting was obtained, yielding an SR of 1.55 µm and an MRR of 66.06 mm2/min.
[47]	EN8 Steel	Brass	PON, POFF, SV, WT	CS, SR, gap current	Taguchi DoE, nature-inspired metaheuristic algorithms, multi-objective ant lion optimization (MOALO), dragonfly algorithm (MODA), grasshopper optimization algorithm (MOGOA), grey wolf optimizer (MOGWO), non-dominated sorting moth flame optimization (NSMFO), non-dominated sorting whale optimization algorithm (NSWOA)	PON was found to be the most significant parameter for MRR and SR. A comparative study showed that nature-inspired metaheuristic algorithms yielded good Pareto fronts, providing manufacturers with flexible optimal trade-off solutions. Among the six algorithms considered, MOGOA emerges as the best metaheuristic with respect to the range of the Pareto fronts generated.
[13]	Tool Steels	AC Brass LP 1000	IP, PON, POFF, discharge voltage (U)	SR, MRR	Full factorial DoE (L16), GRA	Results showed that Low IP, PON, and U and high TOFF produced favourable SR and MRR. An optimum SR of 1.50 μm and an MRR of 12.50 mm3/min were obtained at IP = 2 A, PON = 32 μs, U = 90 V, and POFF = 20 μs.
[48]	9Cr18MoV Stainless Steel	Molybdenum	IP, PON, POFF, wire feed speed (WS)	Cutting efficiency (CE). SR	L16 Taguchi DoE, GRA, AHP	Optimal multi-objective parameter combination was PON = 28 μs, POFF = 3 μs, IP = 9 A, and WS = Level 3.
[49]	Al6061/ Mg/MoS2 Metal Matrix Composite	Brass wire coated with zinc	IP, PON, VG	MRR, SR, EWR	RSM-BBD, NSGA-II	The optimal parameters were PON = 105 μs, VG = 35 V, and IP = 31 A, producing an MRR of 7.51 mm3/min and an SR of 1.97 μm. Compared with non-optimized conditions, the optimized process achieved a 15% increase in MRR and a 20% decrease in SR, exhibiting the effectiveness of the BBD–NSGA-II approach.
[50]	AISI D2 and DC53 Tool Steels	Molybdenum	IP, VG, spark gap, type of material	Linear cutting speed (CS), MRR	Taguchi DoE, ANN, NSGA-II	Optimization with the NSGA-II greatly enhanced CS and MRR with increases in CS and MRR of 63.17% and 75.36% for curved profiles and 74.10% and 73.38% for flat profiles

Current studies on WEDM of Inconel materials and Titanium alloys have mainly revolved around material-dependent studies, employing classical Taguchi-based or RSM-based multi-response optimization approaches, such as GRA, TOPSIS, or DEAR, which provide the optimal result as a single point, thus ignoring the material-dependent trade-offs between productivity and surface integrity. Though current studies have encouraged the usage of evolutionary approaches, the scope remains largely restricted to individual materials or wire electrodes. Further, the integration of statistically verified response models with Pareto optimization remains unexplored, while the treatment of recent advancements in various methods for optimization remains unreported. The RSM, PSO, and NSGA-II optimization algorithms are well-known methods and have been used separately or together in previous investigations on the WEDM process. The focus of this research is to systematically compare and combine these methods for multi-response optimization of process parameters, considering SR and MRR. Optimization methods are compared based on the same experimental data, process parameter constraints, and performance criteria, and finally, experimental verification of the chosen optimum is done. In summary, the main contribution of this study can be seen in the systematic comparison and combination of the known optimization methods rather than introducing a new one.

This work presents an integrated comparative multi-response optimization framework for WEDM, which combines RSM with the NSGA-II and PSO. Both the NSGA-II and PSO are thoroughly analysed using the same framework to understand convergence performance, population diversity, and robustness, which are not often discussed in the previous WEDM literature. This paper aims to answer appropriate research questions that are geared towards improving process optimization in WEDM, as shown in Table 2.

Table 2. Research questions.

Research Question	Methodological Role
RQ1: How can statistically reliable models be developed for WEDM responses?	RSM is used to establish regression-based predictive models and quantify parameter significance and interactions.
RQ2: How can trade-offs between MRR and SR be effectively captured?	The NSGA-II is employed to generate Pareto optimal solutions without aggregating objectives.
RQ3: Can global optima be efficiently identified with faster convergence?	PSO is applied for comparative multi-response optimization and convergence analysis.
RQ4: Which optimization strategy offers superior performance and decision flexibility?	Comparative evaluation of the NSGA-II vs. PSO based on convergence and solution robustness.

2. Materials and Methods

2.1. Research Framework

The research framework for the execution of multi-response optimization of WEDM is presented in Figure 1. Inconel 625 was considered as a workpiece material. The process parameters, such as Pulse ON Time (P_ON), Pulse OFF Time (P_OFF), and peak current (I_P), were considered to evaluate the Surface Roughness (SR) and Material Removal Rate (MRR). RSM with a Face-centred Central Composite (FCC) design was employed to design the experiments. The experiments were performed on a CNC Concord Wire-cut—DK7731 WEDM, with a molybdenum wire of 0.18 mm diameter as the electrode. The objective function was obtained from RSM, and optimal parameters were obtained using the NSGA-II and the PSO algorithm. The results were verified with the RSM desirability function through a verification test.

Figure 1. WEDM research framework.

2.2. Material Selection

Inconel 625, a nickel-based superalloy known for its excellent temperature strength, corrosion resistance, and outstanding fatigue performance, was selected as the workpiece for this study. Due to this, the enhanced properties of Inconel 625 are most suitable for aerospace, marine, and nuclear applications. Though Inconel 625 renders the alloy difficult to cut using conventional methods, this justifies the adoption of the WEDM process. The workpiece was prepared as a rectangular block, as shown in Figure 2.

Figure 2. Inconel 625 workpiece.

2.3. Experimental Setup

The experimental analysis was carried out on Inconel 625 using a CNC Concord Wire-cut—DK7731 WEDM (Figure 3). The Inconel 625 specimens were prepared and clamped on the WEDM table to ensure stable cutting throughout the experiment. A molybdenum wire electrode with a diameter of 0.18 mm was used due to its enhanced electrical conductivity, improved flushing characteristics, and improved spark stability during the machining of nickel-based superalloys. A mixture of JR3A wire-cut coolant gel and deionized water in a ratio of 1:40 was used as a dielectric medium to ensure efficient removal and discharge conditions during the experiment. The WEDM control parameters, such as P_ON, P_OFF, and I_P, were considered major process parameters. These parameters were chosen because they directly influence discharge and energy during the machining process. Other parameters like wire tension (10 N), wire feed speed (20 m/s), servo/gap voltage (250 V), and dielectric flow rate (2 LPM) were kept constant during the experiment. The purpose of keeping these parameters constant was to evaluate the influence of P_ON, P_OFF, and I_P on Surface Roughness and MRR. Each control parameter was varied at the selected levels within the stable cutting range of the machine to avoid breakage of wire and arcing. RSM-based FCCD was employed to design the experiments. The selected control parameter range and its levels are presented in Table 3. One trial of WEDM machining process was carried out by each experimental condition as per the experiment design. The same experimental setting and procedure of measuring was kept constant during the experiments to provide consistency of data for modelling and optimization of the response.

Figure 3. Experimental setup.

Table 3. WEDM control parameters and their levels.

S. No.	Parameter	Symbol	Units	Level 1	Level 2	Level 3
1	Pulse ON	PON	μs	20	40	60
2	Pulse OFF	POFF	μs	8	10	12
3	Peak Current	Ip	A	3	4	5

The SR and MRR were considered as the responses. SR measurements were then taken from three distinct representative points on the same specimen. The SR value was measured using a Mitutoyo Surftest SJ-410 Surface Roughness tester (make: Mitutoyo, model: SJ-410). Average Surface Roughness (Ra), which is the arithmetic average roughness of the machined surface, was considered as the surface quality variable. All measurements were done at a rate of 0.5 mm/s with a cut-off length (λc) of 0.8 mm. The stylus travelled perpendicularly to the main direction of the WEDM cutting process. For every experiment, three measurements were taken from different locations on the same machined surface, and the mean value was taken as the SR output. The measurement of SR is shown in Figure 4. MRR was computed based on the diameter of the wire used for machining. It is important to observe that the kerf width obtained by WEDM operation can become wider than the wire itself due to the presence of the discharge gap around the wire. The MRR value obtained is the estimated MRR based on the diameter of the wire. This method has been consistently used throughout all experimental settings for the comparative analysis of WEDM process parameters. The MRR value was computed using the formula given in Equation (1):

$$\mathrm{M}\mathrm{R}\mathrm{R}=\mathrm{F}\times D_w\times \mathrm{H}$$

(1)

where F in mm/min is the machine feed rate (workpiece length/machining time);

D_w in mm is the wire diameter (0.18 mm);

H in mm is the thickness of the workpiece (20 mm).

Figure 4. Measurement of SR.

Figure 4. Measurement of SR.

2.4. Bio-Inspired Algorithms

2.4.1. Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is an Multi-Objective Evolutionary Algorithm utilized for solving optimization problems with at least two conflicting objectives [30]. In this study, the NSGA-II was adopted to optimize the multi-objective functions (i.e., maximizing MRR and minimizing SR) encountered in the WEDM process. Strategies employed by the NSGA-II include fast non-dominated sorting, crowding distance, and elitism, which enable the preservation of diversity between solutions and convergence towards the Pareto optimal front. The algorithm starts with generating an initial population of size n, where n is the number of individuals/solutions that can be retained in each generation. Comparisons between solutions are then made to establish the optimal solution set. Non-dominated sorting guarantees that solutions are sorted in terms of dominance relationships, provided that the criterion below, as indicated in Equation (2), is satisfied [51]:

$$\begin{array}{l}\mathrm{Objective} 1\left[\mathrm{i}\right] > \mathrm{Objective} 1\left[\mathrm{j}\right] \mathrm{and}\\ \mathrm{Objective} 2\left[\mathrm{i}\right] \ge \mathrm{Objective} 2\left[\mathrm{j}\right] \mathrm{and}\\ Or\\ \mathrm{Objective} 1\left[\mathrm{i}\right] \ge \mathrm{Objective} 1\left[\mathrm{j}\right] \mathrm{and}\\ \mathrm{Objective} 2\left[\mathrm{i}\right] > \mathrm{Objective} 2\left[\mathrm{j}\right] \mathrm{and}\end{array}$$

(2)

where i ≠ j, i, and j denote different solution numbers.

For sorting ‘n’ non-dominated solutions, those solutions are selected based on satisfying the above conditions, and then they are ranked 1; otherwise, these solutions are considered as dominated solutions. Likewise, the next non-domination sorting is performed on dominated solutions and ranks 2, and so on. Such iterations will go on until each solution gets its respective ranking. Also, based on the rankings, fitness value is allotted to the solutions. Crowding distance (CD) is calculated for each of the non-dominated solutions, and a high CD value results in better diversification of the population. The comparative analysis between tournament selection and the CD approach was used for selecting the parent solutions. Offspring solutions obtained from mutation and crossover operators, along with the initial population and offspring solutions, were used to achieve the non-dominated solutions. If the population size exceeded the maximum size, then a new population was created. Otherwise, the process continues similarly until a Pareto front of non-dominated solutions is achieved [19]. The flowchart for the NSGA-II algorithm is shown in Figure 5 [12]. The NAGA-II parameters are shown in Table 4.

Figure 5. Flowchart for the NSGA-II algorithm.

Table 4. NSGA-II parameters and their range.

Parameter	Description	Typical Range
Population Size (N)	Number of candidate solutions in each generation. Larger N improves diversity.	50–150
Number of Generations	Iterations of evolution. Higher values allow better convergence.	50–150
Crossover Probability (Pc)	Probability of combining two parents to create offspring. Promotes exploration.	0.8–0.9
Mutation Probability (Pm)	Probability of random changes in offspring. Prevents premature convergence.	0.01–0.2
Selection Method	Binary tournament based on rank and crowding distance.	Fixed
Crowding Distance	Measure of solution density in objective space; ensures diversity.	Computed dynamically
Elitism	Best solutions are preserved across generations.	Always enabled

2.4.2. Particle Swarm Optimization (PSO)

The Particle Swarm Optimization (PSO) algorithm is specifically designed for optimizing hard numerical problems and is inspired by human social behaviour [52]. For PSO, there are a number of simple objects known as particles that are distributed throughout the search space, where each of the particles is assigned an evaluation of the objective function at its position. Then, each particle is responsible for determining how it will move throughout the search space. At the end of this cycle, the swarm, which functions much like a group of birds searching for food, is expected to be closer to the fitness function’s optimum point [53].

The dynamics of each individual particle in the search space are modified by dynamically changing the velocities of each particle depending upon its own flying experience and the flying experience of other particles within the search space. The position vector and the velocity vector of the ith particle in the D-dimensional search space are denoted by: X_i = (x_i1, x_i2, …, x_iD) and V_i = (v_i1, v_i2, …, v_iD). According to the fitness function selected by the user, the position that gives the best fitness value p_best to each particle is selected, which can be denoted by P_i = (p_i1, p_i2, …, p_iD) and the global best value g_best so far obtained from any particle within the search space in the PSO algorithm. The Equations (3) and (4) are used to calculate the new velocities and positions of each particle:

$$v_{id}=w\times v_{id}+c_1\times r_1()\times (P_{id}-x_{id})+c_2\times r_2()\times (P_{gd}-x_{id})$$

(3)

$$x_{id}=x_{id}+v_{id}$$

(4)

P_best and g_best are indicated by P_id and P_gd. In this case, w is the weight of the inertia factor that determines the ability to explore globally and locally. The constants c₁ and c₂ are the cognitive and social factors, respectively, while r₁ and r₂ are random numbers within the range of 0 to 1. The inertia factor weight is set at a high level initially, but it decreases as the process of searching continues [54]. The PSO algorithm parameters and their ranges are shown in Table 5. Figure 6 shows the flowchart of the PSO algorithm.

Table 5. PSO parameters and their range.

Parameter	Description	Typical Rang
Swarm Size (N)	Number of particles in the swarm.	30–100
Number of Iterations	How many times do particles update their positions?	100–150
Inertia Weight (w)	Controls the balance between exploration (global search) and exploitation (local refinement).	0.4–0.7
Cognitive Coefficient (c1)	Weight for particle’s own best-known position (self-learning).	1.5–2.0
Social Coefficient (c2)	Weight for the swarm’s best-known position.	1.5–2.0
Velocity Limits (Vmax)	The maximum step size a particle can move. Prevents divergence.	Problem-dependent
Topology	Defines how particles share information (global best vs. local best neighbourhoods).	Global or local

Figure 6. Flowchart for the PSO algorithm.

3. Results and Discussion

The experimental data obtained from the WEDM trials form the basis for subsequent modelling and optimization. Table 6 presents the measured responses (SR and MRR) under varying process parameters such as P_ON, P_OFF, and I_P. The above findings can then serve as an important basis for constructing the required predictive models using RSM and for formulating the search spaces in metaheuristic techniques such as PSO and the NSGA-II.

Table 6. Experimental results.

Run Order	PON (µs)	POFF (µs)	IP (A)	SR (µm)	MRR (mm3/min)
1	40	10	3	4.412	4.783
2	40	12	4	4.321	4.741
3	60	12	3	4.651	4.512
4	40	10	4	4.551	4.521
5	60	12	5	4.601	3.971
6	40	10	4	4.535	4.489
7	60	10	4	4.741	4.162
8	60	8	5	3.732	5.560
9	20	12	3	3.102	4.371
10	40	10	4	4.530	4.231
11	40	10	5	4.395	5.101
12	40	10	4	4.530	4.462
13	40	10	4	4.530	4.486
14	40	8	4	4.222	5.036
15	20	8	3	3.712	3.710
16	20	8	5	3.702	4.806
17	20	10	4	3.978	3.743
18	60	8	3	4.457	4.081
19	40	10	4	4.521	4.465
20	20	12	5	3.738	3.648

3.1. ANOVA

Analysis of Variance (ANOVA) is a method for determining the importance of factors of WEDM and the interactions between them by partitioning the total variability among different sources [55]. ANOVA is also used to assess the adequacy of the regression model obtained through RSM for WEDM parameter optimization and to determine the impact of each factor, such as P_ON, P_OFF, and I_P, on SR and MRR. The ANOVA for SR and MRR is discussed in Table 7.

Table 7. ANOVA for SR and MRR.

Source	Sum of Squares		df	Mean Square		F-Value		p-Value		% Contribution
	SR	MRR		SR	MRR	SR	MRR	SR	MRR	SR	MRR
Model	3.56	4.32	9	0.3957	0.4798	363.45	23.24	<0.0001	<0.0001	--	--
A-PON	1.56	0.4032	1	1.56	0.4032	1433.38	19.52	<0.0001	0.0013	57.39	8.20
B-POFF	0.0346	0.3802	1	0.0346	0.3802	31.82	18.41	0.0002	0.0016	1.27	7.73
C-IP	0.0028	0.2656	1	0.0028	0.2656	2.53	12.86	0.1427	0.005	0.10	5.40
AB	0.335	0.0546	1	0.335	0.0546	307.67	2.64	<0.0001	0.1351	12.32	1.11
AC	0.2452	0.0398	1	0.2452	0.0398	225.23	1.93	<0.0001	0.1951	9.02	0.81
BC	0.2181	1.84	1	0.2181	1.84	200.35	89.19	<0.0001	<0.0001	8.02	37.43
A2	0.081	1.33	1	0.081	1.33	74.37	64.4	<0.0001	<0.0001	2.98	27.05
B2	0.1857	0.1591	1	0.1857	0.1591	170.61	7.7	<0.0001	0.0196	6.83	3.24
C2	0.0449	0.237	1	0.0449	0.237	41.25	11.48	<0.0001	0.0069	1.65	4.82
Residual	0.0109	0.2065	10	0.0011	0.0207					0.40	4.20
Lack of Fit	0.0015	0.1507	5	0.0003	0.0301	0.1615	2.7	0.9665	0.1497
Pure Error	0.0094	0.0558	5	0.0019	0.0112
Cor Total	3.57	4.52	19
Fit statistics	Std. Dev.		Mean		R2		Predicted R2	Adjusted R2	Adeq. Precision		C.V.%
SR	0.0121		4.25		0.9996		0.9973	0.9992	193.6630		0.2845
MRR	0.1437		4.44		0.9544		0.8191	0.9133	18.9601		3.23

3.1.1. ANOVA for Surface Roughness

Table 7 shows the output from the ANOVA analysis performed on the response variable SR. The significance of the quadratic RSM model on the SR can be seen clearly since the computed F-value and the corresponding p-value are 363.45 and less than 0.0001, respectively, proving that the model effectively models the functional behaviour of the WEDM process parameters versus SR. This means that the probability of getting such a high value by chance is less than 0.01%. The lack of fit analysis revealed the insignificance of this phenomenon (F = 0.1615, p = 0.9665).

Amongst the individual parameters of the processes, P_ON was found to be the most dominating parameter influencing SR, with a contribution percentage of 57.39%. This is because the higher the value of P_ON, the higher the thermal energy released during each discharge, which leads to the formation of deep craters and rough surfaces. Similar results were found in the literature [56,57]. Linear significance of the parameter P_OFF was found to be statistically significant (p = 0.0002) with a contribution percentage of 1.27%, whereas I_P had no significance at all with p = 0.1427. However, the interaction parameters of I_P such as AC and BC (P_ON × I_P, P_OFF × I_P) contributed 9.02% and 8.02% to the response variable, respectively.

The coefficient of AB (P_ON × P_OFF) had a relative contribution to SR of 12.32% (p < 0.0001), further emphasizing the significance of balancing pulse time as an important factor affecting SR. It was also evident that all three quadratic coefficients of A² (2.98%), B² (6.83%), and C² (1.65%) contributed significantly to the SR (p < 0.0001), thus establishing the nonlinear behaviour of the SR system and proving the appropriateness of using a second-order RSM model.

Model fitting statistics provide additional evidence on the appropriateness of the constructed model. The high value of the coefficient of determination R² = 0.9996 suggests that 99.96% of the total variance in SR is accounted for by the model. An adjusted R² of 0.9992 and predicted R² of 0.9973 indicate a high correlation between predicted values and observed data with little difference (0.05), showing the model is free from overfitting and has good predictive ability. The adequate precision value of 193.6630, exceeding the minimum adequate value of 4, shows an adequate signal-to-noise ratio, making the model ideal for exploring the design space in the PSO and NSGA-II optimization process. Additionally, the small Coefficient of Variation (C.V. = 0.2845%) ensures experimental reproducibility.

3.1.2. ANOVA for Material Removal Rate

The ANOVA analysis for MRR is shown in Table 7. The second-order polynomial model for MRR is extremely significant (F-value = 23.24, p-value < 0.0001). The lack of fit is insignificant (F = 2.70, p-value = 0.1497) and indicates that there is no error due to the lack of fit in the experiment. Thus, the model is adequate and can be used for predictions inside the design space. In contrast to SR, MRR follows a completely different hierarchy of parameters. The interaction of BC (P_OFF × I_P) is the primary source of variance, accounting for 37.43% (p < 0.0001). The squared A-term (P_ON²) accounts for 27.05% (p < 0.0001) of the variance, indicating that MRR is mainly controlled by the interactive and nonlinear behaviour of discharge energy parameters rather than individual linear parameters. The contributions of P_ON, P_OFF, and I_P to MRR variance are 8.20% (p = 0.0013), 7.73% (p = 0.0016), and 5.40% (p = 0.0050), respectively. The linear terms are statistically significant. Interestingly, the linear contribution of I_P to MRR is significant despite having a negligible effect on SR.

However, the terms AB (P_ON × P_OFF) and AC (P_ON × I_P) proved to be insignificant for MRR (with values of p = 0.1351 and p = 0.1951, respectively) in stark contrast to their very high significance for SR. It is interesting to note that there is a clear difference in the significance levels of the interaction terms in both cases, thus suggesting that there is a completely different pattern of sensitivity in the two parameters and hence justifying the multi-response optimization process. The terms B² and C² were significant (p = 0.0196 and p = 0.0069).

According to the obtained model fit statistics, the model shows good predictive capability. The value of R² being 0.9544 indicates that 95.44% of the total MRR variation can be accounted for by the developed model. Although the adjusted R² and predicted R² values of 0.9133 and 0.8191 are less than those in the SR model, they are still satisfactory enough for any engineering application. An adequate precision value of 18.9601 (above 4) implies an adequate signal-to-noise ratio for reliable navigation of the design space. The C.V. value of 3.23% also ensures that experiments can be reproduced accurately. The residual contribution for MRR (4.20%) is greater than that of SR (0.40%), indicating the potential involvement of other factors in the material removal process.

3.1.3. Comparative Parametric Influence on SR and MRR

From the ANOVA outputs, it is evident that there is a conflicting relationship between the parametric effects on SR and MRR, as shown in Table 7. In the case of SR, P_ON stands out as the most influential factor (57.39%) while making up 8.20% of MRR. On the other hand, the factors BC interaction and A² have strong influences on MRR (64.48%), but their contribution to SR is low (11.00%). I_P shows different tendencies since its linear effect on SR is very low (0.10%, p = 0.1427), whereas its effect on MRR is high (5.40%, p = 0.0050).

The balancing between the two response variables, because of their contrasting parametric effects, creates the basic rationale for adopting the approach of multi-response optimization. While PSO can be used to identify the global optimum for each of the two responses separately, the NSGA-II is adopted to create a Pareto optimal front that allows the determination of a balanced solution between SR and MRR. Both objectives have equal weightage. The regression equations obtained from the ANOVA can be used as an objective function for PSO and the NSGA-II.

SR = −1.94838 + 0.038747 ∗ P_ON + 0.814464 ∗ P_OFF + 0.56298 ∗ I_P + 0.005116 ∗ P_ON ∗ P_OFF − 0.008754 ∗ P_ON ∗ I_P + 0.082562 ∗ P_OFF ∗ I_P − 0.000439 ∗ P_ON ∗ P_ON − 0.065995 ∗ P_OFF ∗ P_OFF − 0.131882 ∗ I_P ∗ I_P

MRR = 2.51729 + 0.155662 ∗ P_ON − 0.25796 ∗ P_OFF + 0.072363 ∗ I_P − 0.002065 ∗ P_ON ∗ P_OFF + 0.003527 ∗ P_ON ∗ I_P − 0.239902 ∗ P_OFF ∗ I_P − 0.001739 ∗ P_ON ∗ P_ON − 0.060134 ∗ P_OFF ∗ P_OFF − 0.293567 ∗ I_P ∗ I_P

Objective Function = 0.5 ∗ SR + (1/(0.5 ∗ MRR))

3.2. Effect of Process Parameters on SR

The 3D response surface diagrams developed using RSM show a graphical representation of the effect of the interaction of the WEDM process parameters, such as P_ON, P_OFF, and I_P, on the output response SR (Figure 7). These diagrams help to understand the effects of individual and combined parameter effects on SR, and thus, optimal values of machining parameters that minimize Surface Roughness can be determined.

Figure 7. Effect of process parameters on SR (a) P_ON vs P_OFF, (b) P_ON vs I_P, (c) P_OFF vs I_P.

The plot for the response surface (Figure 7a) shows that SR varies considerably with an increase in P_ON, while the influence of P_OFF is somewhat moderate but opposing. For P_ON ≈ 20 µs, all SR values lie in the range 3.0 to 3.5 µm, regardless of the values of P_OFF. With increased P_ON values up to 60 µs, SR becomes very high, about 4.8 to 5.0 µm, especially for lower P_OFF values. The contour plot, at the bottom, clearly shows an elliptic shape with a minimum SR area at low P_ON and high P_OFF values, showing a considerable interaction between the two variables.

The drastic slope exhibited by the surface with respect to the P_ON axis clearly indicates its significant influence (57.39%) on SR. Higher durations for P_ON result in prolonged periods for the electric spark, hence transferring greater amounts of thermal energy into the work surface per spark event. Because of greater energy input, deeper and bigger craters are formed upon resolidification, which adversely affects the quality of the surface. However, increasing the value of P_OFF helps to some extent by providing sufficient time for the dielectric fluid to flush out the debris.

The 3D surface plot in Figure 7b illustrates that SR increases with increasing P_ON across all levels of I_P. At low P_ON (≈20 µs), SR remains relatively low (≈3.0–3.5 µm) regardless of the I_P level. As P_ON increases to 60 µs, SR escalates to nearly 4.5–5.0 µm even at moderate I_P settings. The influence of I_P on SR is comparatively moderate in its linear trend; however, a noticeable interaction is observed at higher P_ON combined with higher I_P, and the surface deterioration is more pronounced [8]. The contour ellipses at the base indicate that the minimum SR region is confined to the zone of low P_ON and low-to-mid I_P values, confirming the significance of the AC interaction (9.02% contribution from ANOVA).

The combined effect of P_ON and I_P on SR reflects the cumulative discharge energy delivered to the machined surface. While the parameter P_ON controls the period of spark occurrence, I_P affects the strength of the discharge current. With the increase in both P_ON and I_P, there is an increase in heat energy released during the operation cycles, thus leading to the formation of larger and deeper craters and higher values of recast layer and Surface Roughness. In contrast to P_ON, the lower gradient value in the I_P axis implies that the linear effect of I_P on SR is insignificant (0.10%). Therefore, maintaining low P_ON with moderate I_P is advisable to minimize SR.

The response surface for the P_OFF–I_P (Figure 7c) interaction on SR exhibits a relatively flatter and more gradual curvature compared to plots (a) and (b), indicating a less steep but still meaningful combined influence. SR values range from approximately 3.7 µm to 4.7 µm across the design space. SR tends to increase at lower P_OFF values combined with higher I_P levels, while the minimum SR region, as confirmed by the near-circular to elliptical contour pattern at the base, is located at intermediate-to-high P_OFF values and low-to-moderate I_P levels. The broader contour spread indicates a wider feasible region for achieving acceptable SR values when P_ON is controlled at its centre level.

The BC interaction (P_OFF × I_P) contributes 8.02% to SR variation, as confirmed by ANOVA. At lower P_OFF settings, insufficient inter-pulse delay restricts the dielectric fluid from effectively flushing away the molten debris and resolidified particles from the machining zone. Where this happens to be in conjunction with a higher I_P value, the accumulation of the deposited material, along with the increased discharge energy, will act together to deteriorate the Surface Roughness. An elevated P_OFF will ensure sufficient cooling and cleaning during the intervals between pulses and thereby help in minimizing the thickness of the recast layer. It is also noted that the consistent nature of the curve indicates the importance of the quadratic effect of both B² and C² factors found through ANOVA.

Together, the three-dimensional response surface plots infer the following:

• The P_ON is the most significant factor affecting the SR in the WEDM process, having the maximum gradient among the three response surface graphs, considering the high percentage contribution of 57.39% from the ANOVA analysis. Lowering the P_ON is considered the best way to achieve an improved surface finish.
• Two-way interaction effects between all three factors (AB, AC, and BC) are evident from the plots, which clearly show that none of the factors can be optimized individually because the response surfaces are twisted and not flat, implying interaction effects.
• The P_OFF moderates the response of the SR; higher values of the P_OFF will result in lower SR since more efficient dielectric flushing and better inter-pulse recovery will occur. The SR is independent of changes in the P_ON and I_P.
• The IP affects the SR through interactions with other factors, especially with the P_ON and P_OFF. There is no linear relationship between I_P and SR, as observed from the response surface in Figure 7b,c.
• The curvatures present in all three graphs confirm that the proposed quadratic model is adequate for optimizing the SR using the RSM method combined with PSO and the NSGA-II.

3.3. Effect of Process Parameters on MRR

The three-dimensional response surface plots generated through RSM provide a visual representation of the relationship between WEDM process parameters and the resulting MRR, as presented in Figure 8. The plot of the response surface (Figure 8a) shows the existence of a nonlinear interaction between the input variables P_ON and P_OFF and the output response, MRR. When P_ON is lower (approximately 20 µs), the MRR value tends to be low at about 3.5–4.0 mm³/min regardless of the value of P_OFF. As the P_ON value increases toward the intermediate level (approximately 40 µs), the value of the MRR increases significantly to reach about 4.5–5.0 mm³/min. Above this intermediate value of P^ON, a plateau followed by a small decrease in the MRR value can be noticed, especially when P_OFF values are low. The presence of a clear saddle-shaped response surface indicates the domination of the quadratic effect of A² (with 27.05%) identified from ANOVA. The contour pattern shows parallel bands for P_OFF at the bottom, which indicates the moderate independent effect of P_OFF on MRR at constant P_ON values.

Figure 8. Effect of process parameters on MRR (a) P_ON vs P_OFF, (b) P_ON vs I_P, (c) P_OFF vs I_P.

The nonlinear relationship between MRR and P_ON is due to the two effects caused by P_ON duration in WEDM. For medium values of P_ON, the energy released during the discharge would be enough to melt the workpiece material and achieve optimum material removal. Nevertheless, when P_ON is too high, there will be an increase in heat generation, leading to excessive build-up of the recast layer, thus lowering MRR. Higher values of P_OFF for very high P_ON ensure more time between the pulses for regaining dielectric strength and removal of debris, hence decreasing the rate of reduction in MRR. The relationship is similar to the presence of a large A² value for the quadratic effect of P_ON on MRR (27.05%) and a low A term for the linear effect (8.20%) as seen from the ANOVA results.

The three-dimensional surface graph presented in Figure 8b shows that MRR reacts nonlinearly to the variation in both P_ON and I_P. At lower values of P_ON (about 20 µs) and I_P (about 3 A), the lowest value of MRR can be achieved, about 3.5 mm³/min. As P_ON or I_P increases from lower to intermediate values, MRR steadily rises. In an intermediate range of P_ON (about 35–45 µs) together with high I_P values (about 4.5–5 A), the highest values of MRR (about 5.0–5.5 mm³/min) are obtained. However, MRR decreases as P_ON further increases beyond an intermediate level, even when I_P is kept at a relatively high level. The surface slope of MRR with respect to the P_ON parameter appears to be greater than that for I_P, demonstrating the larger nonlinearity effect of P_ON on MRR compared to I_P.

P_ON vs. I_P interaction is associated with the joint effect of discharge duration and discharge intensity on the material erosion rate in the WEDM process. An increase in I_P implies higher discharge current density and, hence, higher instantaneous thermal energy input to the workpiece, leading to higher energy density for material melting and vaporization per pulse. By combining these factors with an optimal P_ON, the maximum amount of material can be effectively removed [8]. But with a P_ON too high, even at increased I_P values, energy saturation effects begin to play out, and any further increase in discharge duration results not in further effective material erosion but, rather, in the erosion of the wire electrode, short circuits, and deposition of eroded material back into the workpiece. This phenomenon accounts for decreasing MRR along the high P_ON-high I_P region of the response surface. Although the AC interaction term is not statistically significant (p = 0.1951), it is clearly visible on the surface in the form of a slight twisting of the latter, whereas significant effects of the two factors themselves (p = 0.0013 for P_ON; p = 0.0050 for I_P) are readily observable.

However, the contour plot for the effect of interaction between P_OFF and I_P on MRR (Figure 8c) presents the most pronounced curvature compared to other contour plots. This interaction produces an apparent valley-type contour plot, which shows that MRR attains its minimum value (approximately 3.5–3.8 mm³/min) at the highest levels of P_OFF (≈12 µs) and lowest levels of I_P (≈3 A). On the other hand, there is a sharp increment in MRR values when P_OFF is decreased, and I_P is increased simultaneously until it reaches its maximum values at around 5.5–5.8 mm³/min. It can be observed from Figure 8c that there are densely packed contour lines at the lower levels of P_OFF on the I_P axis. This represents that there is a high sensitivity of MRR to I_P under the condition of lower inter-pulse delays.

The high correlation between BC can be attributed to the dual effect of P_OFF time and I_P on material removal efficiency. In the case of high I_P, high discharge energy will occur for each pulse, leading to high material melting and evaporation. At the same time, when P_OFF is kept low, the highest pulse frequency will be achieved since a shorter pulse-to-pulse interval allows the generation of sparks at a higher frequency. The combination of high energy and high frequency leads to high MRR. On the contrary, P_OFF with low I_P gives the least MRR since high P_OFF causes low energy and long pulse intervals, hence a lower number of sparks generated within a particular period. The statistical significance of B² (p = 0.0196) and C² (p = 0.0069), as indicated by the nature of the curved saddle-shaped surface, shows the quadratic effect of the independent variables. To achieve optimum MRR, P_OFF and I_P should be low and high, respectively.

The response surface plots of MRR indicate the following:

• The MRR values show strong nonlinearity about all WEDM variables (with marked quadratic nature especially for the P_ON variable—A², 27.05%) and thus, is different from the SR response variable which is dominated by linearity.
• The interaction between BC and P_OFF × I_P is the most important parameter for MRR (37.43%), indicating that pulse frequency and discharge current need to be optimized together instead of individually for maximizing MRR.
• Optimal intermediate P_ON needs to be achieved; otherwise, the MRR value will decrease because of energy saturation, recast formation, and degradation of the wire electrode.
• The effect of I_P on MRR (5.40%, p = 0.0050) is more pronounced than its effect on SR (0.10%, p = 0.1427), thereby highlighting the conflicting nature of sensitivity parameters for these two responses and providing justification for applying the NSGA-II technique to multi-objective optimization.
• The optimal settings of the control variables for minimal SR and maximal MRR—low P_ON and high P_OFF for minimal SR, and intermediate P_ON, low P_OFF, and high I_P for maximal MRR—further support the fundamental conflict in WEDM between workpiece quality and productivity and provide effective resolution through Pareto optimal solutions.

3.4. Process Parameter Optimization

There are many performance measures in the WEDM process, like MRR and SR. There tends to be some contradiction in terms of the measures. Therefore, the use of RSM models along with the PSO algorithm and the NSGA-II was employed for multi-response optimization. First, RSM was utilized for both modelling and optimization of the multiple responses. The accuracy of RSM is high in terms of creating predictive equations, while PSO and the NSGA-II were able to search through the design space and find the best possible parameters that would optimize the trade-off between maximizing MRR and minimizing SR.

3.4.1. Multi-Response Optimization Using RSM Desirability Function

Multi-response optimization was done by utilizing the desirability function technique, which aimed at optimizing both MRR and SR simultaneously. In this method, several responses are combined in terms of a dimensionless scale varying from 0, which means completely undesirable, to 1, which is the ideal response. The desirability function for each response is determined considering the goal of each response. The desirability function for all responses is found out by taking the geometric mean of the individual desirability function. Optimization was used to determine the optimized parameter values and presented in Figure 9. Optimal machining parameters are presented in Table 8. With the help of optimized parameter values, the following response values can be obtained. The MRR value was found to be 5.664 mm³/min and the SR value was 3.99 μm.

Figure 9. RSM: Optimal parameters from the desirability function.

Table 8. Optimal machining parameters.

Parameter	Optimal Value
	RSM	PSO	NSGA-II
Pulse-on time (PON), μs	45.71	20	49.42
Pulse-off time (POFF), μs	8.02	12	8.00
Peak current (IP), A	4.78	3	5.00
SR, μm	3.99	3.098	3.867
MRR, mm3/min	5.66	4.437	5.907
Optimization Strategy	Balanced productivity	Quality-focused	Maximum productivity

3.4.2. Multi-Response Optimization Using PSO

Particle Swarm Optimization (PSO) was used to achieve optimization of the WEDM process parameters using a multi-response approach. As opposed to the desirability approach where several responses were combined to give one objective function, PSO can provide the optimal solutions. The PSO algorithm was parameterized as follows:

•

Swarm size: 50 particles

•

Maximum iterations: 100

•

Inertia weight (w): 0.7

•

Cognitive coefficient (c₁): 1.5

•

Social coefficient (c₂): 1.5

The settings were chosen in such a way as to maintain the right balance between exploration and exploitation ability. Figure 10 shows the convergence graph for PSO. In Figure 10, the convergence features of the PSO algorithm are presented, with the best SR being the best result obtained by any individual agent, and the best MRR being the average result of the swarm.

Figure 10. Convergence plot—PSO.

The PSO convergence features show very high optimization efficiency at all stages of its operation. Specifically, fast convergence in the beginning (after just 20 iterations) proves that the algorithm has efficiently found the promising areas in the search space with further exploration, leading to the discovery of high-quality solutions. Next, it is noticeable that the best and average fitness curves converge after about 40 iterations; thus, no oscillatory behaviour occurs, which can indicate problems with parameter stability and sensitivity. Moreover, consistently low values of the gap between the two curves for further iterations ensure the efficient convergence of the whole swarm towards the optimum rather than splitting into several local optima. These findings testify to the efficiency of the selected population-based search approach. Thus, the absence of an oscillatory curve and the stable fitness growth prove that the chosen PSO parameters have successfully provided a good balance between global and local search efficiency.

3.4.3. Multi-Objective Optimization Using NSGA-II

The NSGA-II, a multi-objective elitist evolutionary optimization approach, was applied to investigate the trade-off relationship between SR and MRR. The application of the NSGA-II involves the use of non-domination ranking, crowd distance computation, and elitism. This algorithm was configured with optimized parameters:

•

Population size: 100 individuals

•

Number of generations: 100

•

Crossover probability (Cx): 0.9

•

Mutation probability (Mut): 0.1

This selection was done to obtain sufficient genetic diversity for effective searching of the solution region in the search space, while preventing premature convergence.

Figure 11a shows the convergence of the NSGA-II algorithm in the number of 100 generations, considering both the average SR and average MRR of the population. The algorithm showed a high speed of improvement in the early stages of iterations when both indicators were significantly improved by 20–25 iterations, which is the result of efficient searching of good areas in the multi-objective problem space. After iteration 20, both values became quite steady with low fluctuations, forming a stable plateau for their convergence without oscillations. The average value of the SR of the population converged in the range of 3.8–4.0 µm, whereas the average MRR reached a value of 4.8–5.0 mm³/min. The stable convergence curves with low variance in later stages show that the population converged steadily without significant deviation.

Figure 11. NSGA-II: (a) convergence plot (b) Pareto front.

The Pareto front illustrated in Figure 11b is derived from the NSGA-II and consists of various non-dominated solutions along the complete trade-off range between surface quality and productivity. A wide-ranging Pareto front is observed, as seen by SR values ranging from 3.1 to 3.9 μm and MRR values ranging from 4.4 to 6.0 mm³/min, providing roughly 40–50 equally distributed solutions to consider during decision making. The convex and nonlinear nature of the trade-off plot denotes an increasingly difficult trade-off between both objective functions as either end value of the scale is approached, resulting in two separate operation ranges: one where surface quality prevails and the other where production rate prevails. The optimal compromise point was selected from the Pareto front through the knee point approach, as shown in Table 5, which is the optimum trade-off between conflicting objectives where there cannot be any improvement in one without compromising on the other.

3.4.4. Comparison of RSM-Based Desirability Optimization and Metaheuristic Optimization Using PSO and NSGA-II

RSM was initially used to generate the relationship between WEDM process parameters (P_ON, P_OFF, and I_P) and machining responses (SR and MRR). The regression equations from the RSM process were used as surrogate objective functions during the optimization process. For an RSM-based approach, a desirability function was used to arrive at an optimal solution. For the PSO algorithm, the regression equations were used in the objective function in a single-objective weighted-sum approach. In the NSGA-II approach, the regression equations were used as conflicting objective functions to generate non-dominated solutions. This means that the comparison involves different approaches to solving similar RSM-based response equations.

The performance comparison of the three approaches shows differences in terms of performance characteristics and approaches to parameter selection. The surface quality is best for the PSO, with a predicted Surface Roughness of 3.098 μm. This algorithm demonstrates the highest performance, showing a difference of 19.9% compared to the NSGA-II and a difference of 22.5% compared to the RSM. The NSGA-II algorithm shows better productivity with the value of MRR equal to 5.907 mm³/min, which is 4.2% better than in the case of RSM and 33.1% better than for PSO. In turn, RSM showed good results in terms of productivity, but the results were rather low in the case of SR: 3.998 μm and 5.668 mm³/min, correspondingly. However, its productivity remains higher by 27.7% than in the case of the PSO algorithm.

This high productivity is because of the differences in the philosophies of the two machining processes, which are dictated by the selection strategies employed for the WEDM parameters. The NSGA-II uses a very aggressive machining process that utilized maximum energy inputs. It makes use of a high P_ON of 49.42 μs, a minimum P_OFF of 8.00 μs, and a maximum I_P of 5 A, giving it the highest duty cycle (P_ON/[P_ON + P_OFF]) of 86.1%. Similarly, the RSM uses another efficient high-energy input process, which utilizes a P_ON of 45.71 μs, P_OFF of 8.02 μs, and I_P of 4.78 A, producing an 85.0% duty cycle. On the other hand, the PSO uses a very conservative machining process, which involves a low P_ON of 20.00 μs, a high P_OFF of 12.00 μs, and a low I_P of 3.00 A, giving it the lowest duty cycle of 62.5%.

• Trade-off Analysis

The three optimization methodologies lie in different areas of the objective function space:

Quality-oriented (low SR, low MRR) and productivity-oriented: (high SR, high MRR).

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PSO-based optimization: Obtains 75.1% of the maximum MRR while improving the surface quality by 80.1% (compared to the NSGA-II).

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RSM-based desirability optimization: Obtains 95.9% of the maximum MRR at an acceptable level of surface quality (worse by 3.4% when compared with the NSGA-II).

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NSGA-II-based multi-objective optimization: Maximum MRR but with an SR value within reasonable limits.

3.5. Confirmation Experiments

To assess the performance of the developed models with respect to prediction, confirmation tests were done by employing optimum settings for parameters that had been found using the RSM-based desirability function approach, PSO, and the NSGA-II, respectively. The predicted and actual values of the SR and MRR are shown in Table 9. An independent WEDM machining test was performed for each of the chosen combinations of parameter settings. For each sample machined, three different SR values were measured at different spots on the same machined surface, and the average of these three values served as the experimental SR value. Good agreement was found between the predicted and experimental values since the deviation was not above 2.5%. For example, the NSGA-II solution produced a predicted MRR of 5.956 mm³/min and a deviation of 0.823%, while the PSO solution resulted in SR = 3.057 µm and a deviation of 1.341%. The RSM desirability solution gave SR = 3.997 µm and MRR = 5.612 mm³/min. These were the values of the compromise solution based on the desired criteria.

Table 9. Confirmation test results for optimized parameter settings.

Algorithm	SR (µm)		%Deviation	MRR (mm3/min)		%Deviation
	Experimental	Predicted		Experimental	Predicted
RSM	3.997	3.990	0.175	5.612	5.660	0.855
PSO	3.057	3.098	1.341	4.421	4.437	0.362
NSGA-II	3.961	3.867	2.373	5.956	5.907	0.823

The solution obtained by the NSGA-II has a smaller SR and a greater MRR than the solution chosen based on the RSM, and the NSGA-II dominates the RSM solution in objective space. Thus, the RSM solution cannot be recognized as a globally better solution than the NSGA-II solution. On the contrary, it is possible to speak about the impact of optimization and selection of the solution. The method for obtaining the compromise solution using RSM and desirability shows one compromise solution; the PSO algorithm produces one solution using weighted objective function, while the NSGA-II produces a number of non-dominated solutions.

3.6. Integrated Decision Framework

In terms of comparative analysis, the hierarchy of performance levels depends on the optimization criteria; thus, when quality is the focus, PSO emerges as more effective algorithm than the NSGA-II and RSM, but productivity can be optimized better with the NSGA-II than RSM, and finally, with PSO. In cases where the goal is balanced performance, the NSGA-II and RSM prove equally efficient, far exceeding the capabilities of PSO. Based on the above conclusions, it can be determined that the NSGA-II is best suited for situations where the main goal is maximum productivity, along with reasonable quality levels, and PSO yields the best results in processes where precision is essential.

However, it is important to state that due to their nature of multi-objective solutions, both PSO and the NSGA-II present an additional benefit for the user since they clearly demonstrate possible compromises between objectives, making it possible for the user to choose an option based on their own preferences rather than taking one aggregated option. Due to the complementarity of the three methods described above, it can be concluded that a combination of RSM to conduct the preliminary screening, followed by more elaborate multi-objective algorithms, may prove to be the most promising method. The decision framework is presented in Table 10.

Table 10. Integrated Decision Framework.

Application Requirement	Primary Objective	Recommended Approach	Expected Outcome
Precision finishing	Ra < 3.2 μm	PSO	Best surface quality
High-volume production	MRR > 5.8 mm3/min	NSGA-II	Maximum productivity
Balanced performance	Ra ~ 4.0 μm, MRR ~ 5.7 mm3/min	RSM	Efficient operation
Research exploration	Complete trade-off understanding	NSGA-II	Comprehensive Pareto
Quick implementation	Minimal setup, fast results	RSM	Instant optimization

3.7. Practical Implications

Validation results provide clear grounds for decision making for engineers and technicians:

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The NSGA-II should be used in cases where machining time and the volume of materials used are crucial parameters; Surface Roughness values of SR ≈ 4 µm should be expected, but MRR will reach its maximal value, MRR ≈ 6 mm³/min.

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PSO should be applied in cases where surface integrity becomes vital, such as in finishing operations and cases where strict demands for Surface Roughness are set; MRR will remain quite high but somewhat lower, while SR ≈ 3 µm could be achieved, even if machining takes somewhat longer.

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RSM should be used if versatility is a crucial feature of the working environment: it is a balanced alternative that ensures reliable results with no need for further calculations.

As the correlation between theoretical and actual data does not exceed 1.5%, all the presented models can be safely applied in practice; however, each of them can only be used under specific conditions.

4. Conclusions

The investigation of optimization of WEDM processes for processing Inconel 625 using RSM, the NSGA-II, and PSO was the main purpose of this research work. The results show that the developed mathematical regression models are statistically significant, and that they are characterized by high values of coefficients of determination (R² = 0.9996 for SR and 0.9544 for MRR). Moreover, the lack of fit was insignificant, which means that the created models can be considered adequate and applicable for predictions. According to the ANOVA analysis, it can be concluded that the most significant parameter influencing SR is P_ON, with a contribution of 57.39%. Other significant parameters include AB and AC, with contributions of 12.32% and 9.02%, respectively. As for MRR, the interaction between P_OFF and I_P (BC) is the most important one, with a contribution of 37.43%.

Among the various optimization algorithms, the NSGA-II gave a balanced optimum solution (P_ON = 48.42 μs, P_OFF = 8 μs, I_P = 5 A) with an excellent compromise between MRR (5.956 mm³/min) and SR (3.961 μm). PSO gave an excellent surface quality, while RSM optimized the productivity of the system. The analysis reveals that the interactions among parameters have a significant impact on the performance of the WEDM process.

In future research, other variables like wire tension, flushing pressure, and feed speed could be considered for better optimization. The combination of machine learning and metaheuristics can help to improve predictive accuracy and even allow real-time control. Other investigations on various superalloys and sustainability issues, like energy usage and tool life, are also possible.