Optimization of Machining Parameters for Reducing Drum Shape Error Phenomenon in Wire Electrical Discharge Machining Processes

Abstract

Machining thicker workpieces in the process of Wire Electrical Discharge Machining (WEDM) can result in a concave phenomenon known as a “drum shape error” due to the vibration of wires and accumulation of debris, which leads to secondary discharge in the middle of the workpiece. Reducing the drum shape error typically requires a longer finishing process. Finding a balance between precision and machining time efficiency has become a challenge for modern machining shops. This study employed experimental analysis to investigate the effect of individual parameters on the shape error and machining removal rate (MRR). Key influential parameters, including open voltage (OV), pulse ON time (ON), pulse OFF time (OFF), and servo voltage (SV), were chosen for data collection using full factorial and Taguchi orthogonal arrays. Regression analysis was conducted to establish multiple regression equations. These equations were used to develop optimization rules, and subsequently, a user-friendly human–machine interface was developed using C# based on these optimization rules to create a shape error and MRR optimization system. The system can predict the optimal parameter combinations to minimize the shape error and increase the MRR. The results of the verification experiments showed that the prediction accuracy can reach 94.7% for shape error and 99.2% for MRR. Additionally, the shape error can be minimized by up to 40%.

1. Introduction

The market demand for high-quality products and components, such as consumer electronics, biomedical products, and automotive products, has been increasing year by year. Injection molding is the most popular process for producing cost-effective and high-quality plastic products. Various machining processes such as milling, Electrical Discharge Machining (EDM), and Wire Electrical Discharge Machining (WEDM) are used to manufacture injection molds. Wire Electrical Discharge Machining (WEDM) is one of the unconventional machining processes for manufacturing injection molds utilized for workpiece materials with high levels of hardness that are difficult to machine, such as SKD11, Inconel, and tungsten carbide [1,2,3]. Different from traditional CNC milling and turning, which require contact with the cutter tool for machining processes, WEDM uses spark erosion (EDM) for non-contact machining, where the thermal and mechanical effects of electrical discharges remove the material [4].

Key performance metrics in WEDM include productivity (cutting speed), surface quality, and dimension accuracy [5,6,7]. A higher spark energy such as that achieved with a higher pulse duration, current, and voltage improves productivity but can also increase the surface roughness, white layer thickness (WLT), and kerf width, which impacts the dimension accuracy [8,9,10,11]. Usually, the workpiece must be machined more than once to achieve low WLT, thus yielding the desired surface quality [12]. Various factors, including electrical parameters, wire properties, workpiece characteristics, and dielectric fluids, influence the process and quality of the finished product [13,14].

Some research on the optimization of machining parameters to obtain good quality and accuracy was investigated by different researchers, such as Seidi et al. [15], who investigated the impact of process parameters (wire feed speed, wire tension, and generator power) on dimensional accuracy, hardness, and surface roughness of machined products. A hybrid method based on Removal Effects of Criteria (MEREC) and Weighted Aggregates Sum Product Assessment (WASPAS) was used to discover the best practical experiment. In addition, the regression analysis was used to study the effect of variables on response factors. The results showed that the weights of roughness, hardness, and dimensional accuracy of the machined part were 89%, 9%, and 2%, respectively. The optimal amount of wire feed speed, wire tension, and generator power variable were 2 cm/s, 2.5 kg, and 10%. Zaman et al. [16] proposed the Taguchi method to optimize the material removal rate (MRR) during cutting AISI 1045 material using WEDM. Three key input parameters, current, voltage, and pulse ON time, with three levels, were chosen for the experiment. Analysis of Variance (ANOVA) was employed, and the signal-to-noise (S/N) was obtained to determine the statistical significance of input factors and evaluate the robustness. It was found that increasing the current significantly increased the MRR, the voltage inversely affected the MRR, and a higher voltage reduced the spark frequency and machining rate. Additionally, the pulse ON time showed diminishing returns; excessively high value restricted machining. Sunil et al. [17] investigated the WEDM performance of Inconel 625, focusing on both normal and cryogenic material conditions. A Genetic algorithm (GA) and Particle Swarm Optimization (PSO) were used to optimize machining parameters for multiple objectives, such as the MRR, surface roughness (SR), and overcut (OC). The results showed that PSO provided highly significant results on optimal setting parameters of WEDM machining. A surface roughness of 0.419 μm was obtained by a cryogenic process, as compared to normal (0.880 μm). The material removal rate for the cryogenic process was found to have decreased in PSO by 21.8% and in GA by 2.78% compared to the normal process.

The wire electrode is an important component of the machining process in WEDM. In order to increase productivity, it is usually preferable to shorten the cutting time. However, this can lead to poor surface quality and dimension accuracy or even wire breakage [18]. Additionally, one of the contributing factors to the poor quality of the finished product is the vibration of the wire electrode. Wire vibration and improper machining parameter settings can lead to convex or concave profiles, especially when machining thick materials. This issue often requires additional precision finishing processes. Previous research demonstrated that the wire vibration significantly impacts the shape and dimensional accuracy, with improper tension being a primary cause [19,20]. Adjusting wire tension close to critical levels can reduce vibration, but excessive tension risks wire breakage, thus increasing machining time and lowering productivity [21]. Dauw et al. [22] explored the principle of wire vibration, noting that multiple forces during machining affect the wire, and unbalanced forces lead to geometric inaccuracies. The dielectric fluid’s damping effect helps to reduce the vibration amplitude. Obara et al. [23] investigated the effect of discharging force and flushing pressure on wire deflections. The optical fiber slit method was used to measure the deflection of the wire during machining, and the force acting on the wire was examined. Subsequently, an analytical model of a single discharge force was derived. With the wire deflection and discharge frequency, the single discharge force and single discharge machining amount were calculated. However, the analysis of deflection and discharge force was limited to the condition of a straight cutting path in the finishing processes. Liang et al. [24] proposed a method that uses a digital camera to analyze the appearing discharge spark at different heights of the workpiece. Furthermore, the wire deflection could be determined by multiplying the time difference between the designated position and the appearance of the first spark. Andhare et al. [25] investigated the vibration behavior of a moving wire and its effect on the surface’s finish. The transverse vibration of the wire was examined as a forced vibration of the moving wire with excitation due to the discharge sparks during machining. The wire behavior under various operating conditions was simulated using MATLAB 2012 and ANSYS 2023 software, and the results were compared. The results showed that the ANSYS simulation adequately matches the analytical solution. Furthermore, it was found that the wire speed has minimal impact on the vibration amplitude. Okada et al. [26] investigated the influence of nozzle jet flushing on wire deflection and breakage. Various machining conditions were used, and the impact on the wire deflection and breakage was analyzed. The results showed that the flow field, debris, and hydrodynamic stress distribution directly affect wire deflection, which thus leads to wire breakage.

Based on the literature review mentioned above, three conclusions can be drawn: (1) concave or convex profiles are primarily influenced by debris distribution and wire vibration; (2) no single set of parameters simultaneously achieves both optimal accuracy and quick machining time; and (3) changes in the flushing mode can alter debris positions. Most studies optimize parameters for efficiency without fully addressing the vibration of wires or focusing on accuracy at the expense of efficiency. This study aims to find optimal machining efficiency for minimal concave shape errors. An experimental approach was used to analyze the impact of individual parameters and their interactions on concave errors and thus develop a predictive model using regression analysis. Various flushing modes with adjusted water pressure were tested to evaluate their effects on concave errors and machining time. Additionally, a user-friendly human–machine interface was developed in the C# programming language to predict the concave amount and identify optimal machining parameters for the best balance of accuracy and time. Finally, the verification experiment was carried out to verify the proposed method.

2. Methodology

This study is divided into three main parts: the first part involves experimental analysis to identify the impact of a single parameter on concave shape errors and the rate of material removal. According to the literature [22,26], high-pressure flushing affects wire vibration, leading to concave shape errors during machining. The high flow rate in jet flushing helps to remove metal debris from the machining zone; however, if the jet flushing pressure is too high, it will lead to high levels of wire deflection and vibration, resulting in the concave shape error. The workpiece material of SKD11 alloy tool steel is chosen. Four electrical-related parameters, namely, OV (open circuit voltage), ON (pulse ON time), OFF (pulse OFF time), and SV (servo voltage), were selected for experimentation. The experiment design using the Taguchi method with 3 levels and full factorial was used to investigate how different parameters affect the mean and variance of process performance. Subsequently, the regression analysis method was applied in order to analyze the influence of those parameters on concave shape errors and the MRR, distinguishing trends for optimal accuracy and MRR. In addition to OV, ON, OFF, and SV parameters, other parameters such as wire tension (WT), wire speed (WS), and flushing pressure (WA) were also selected for experiments. A fixed parameter value was selected for WT, WS, and WA based on the manufacturer’s recommendation for rough cutting and also our previous experiment result.

In the second part, a set of optimization rules and a computer-assisted optimization system with a user-friendly human–machine interface were developed. Using the regression model from the first part, optimization rules were established, and an optimization system was developed in visual C#. The details of optimization will be explained in the following subsection. This system can predict concave errors and machining time according to the input parameters OV, ON, OFF, and SV and recommends optimal parameter sets to achieve the best accuracy and machining efficiency. Users can easily understand the achievable precision (concave errors) and average machining time under certain parameter settings.

The third part involves verifying the proposed method’s effectiveness in reducing concave shape errors and evaluating the accuracy of the concave error prediction system by comparing its prediction with actual outcomes. The flowchart of the experiment is shown in Figure 1.

2.1. Experiment Design and Equipment

In the wire-cutting process, operations are influenced by various factors. These factors interact during machining, complicating the task of identifying the best settings for optimal accuracy and efficiency. To understand the influence of each parameter on the concave shape errors during WEDM, the parameters of OV, ON, OFF, SV, wire tension (WT), wire speed (WS), and flushing pressure (WA) were selected for experiments based on the literature and the preliminary investigation. The experimental cutting path, as shown in Figure 2, was used to determine how the concave error position might change with time and distance. This study focused on rough cutting using an AP6040 WEDM machine with a 0.1 μm linear motor resolution from Accutex Co., Ltd., Taichung City, Taiwan, as shown in Figure 3 [27]. The workpiece material used was a 60 mm thick SKD11 alloy tool steel and a 0.25 mm diameter brass wire electrode, as shown in Figure 4. After the machining process, the dimension was measured using an Equator 300 versatile gauge with a scanning rate of 1000 points/s and a resolution of 0.2 μm from Renishaw Co., Ltd., Wotton-under-Edge, UK, as shown in Figure 5 [28].

Table 1. Machining parameters with three levels.

Parameters	Level
	Low		Mid		High
	Code	Value	Code	Value	Code	Value
OV (V)	14	87.74	17	96.45	20	105.16
ON (ns)	8	400	12	600	15	750
OFF (μs)	10	10	15	15	20	20
SV (V)	34	34	37	37	40	40

shows the experimental design with four factors and three levels of parameters (high, middle, low) analyzed using a full factorial experimental design and Taguchi orthogonal arrays. The parameter levels were selected according to the WEDM machine manufacturer’s suggestion for rough cutting (mid-level value) as a base, then the low and high-level values, which are in the parameters value range, were selected. The parameters WT, WS, and WA were set at fixed values of 13.73 N, 10 m/min, and 1.47 kPa, respectively, which were derived from our previous experimental analyses. The full factorial design was chosen due to its consideration of all possible permutations and combinations, although it requires a larger number of experimental groups. In contrast, the Taguchi orthogonal array design can reduce the number of experiments but may not always capture the optimal parameter combinations. Therefore, this study combined data from both the full factorial and Taguchi design, as shown in

Table 2. Full factorial experimental model.

Experiment No	OV		ON		OFF		SV
	Code	Value (V)	Code	Value (ns)	Code	Value (μs)	Code	Value (V)
1	14	87.74	8	400	10	10	34	34
2	14	87.74	8	400	10	10	37	37
3	14	87.74	8	400	10	10	40	40
4	14	87.74	8	400	15	15	34	34
5	14	87.74	8	400	15	15	37	37
6	14	87.74	8	400	15	15	40	40
7	14	87.74	8	400	20	20	34	34
8	14	87.74	8	400	20	20	37	37
9	14	87.74	8	400	20	20	40	40
10	14	87.74	12	600	10	10	34	34
11	14	87.74	12	600	10	10	37	37
12	14	87.74	12	600	10	10	40	40
13	14	87.74	12	600	15	15	34	34
14	14	87.74	12	600	15	15	37	37
15	14	87.74	12	600	15	15	40	40
16	14	87.74	12	600	20	20	34	34
17	14	87.74	12	600	20	20	37	37
18	14	87.74	12	600	20	20	40	40
19	14	87.74	15	750	10	10	34	34
20	14	87.74	15	750	10	10	37	37
21	14	87.74	15	750	10	10	40	40
22	14	87.74	15	750	15	15	34	34
23	14	87.74	15	750	15	15	37	37
24	14	87.74	15	750	15	15	40	40
25	14	87.74	15	750	20	20	34	34
26	14	87.74	15	750	20	20	37	37
27	14	87.74	15	750	20	20	40	40
28	17	96.45	8	400	10	10	34	34
29	17	96.45	8	400	10	10	37	37
30	17	96.45	8	400	10	10	40	40
31	17	96.45	8	400	15	15	34	34
32	17	96.45	8	400	15	15	37	37
33	17	96.45	8	400	15	15	40	40
34	17	96.45	8	400	20	20	34	34
35	17	96.45	8	400	20	20	37	37
36	17	96.45	8	400	20	20	40	40
37	17	96.45	12	600	10	10	34	34
38	17	96.45	12	600	10	10	37	37
39	17	96.45	12	600	10	10	40	40
40	17	96.45	12	600	15	15	34	34
41	17	96.45	12	600	15	15	37	37
42	17	96.45	12	600	15	15	40	40
43	17	96.45	12	600	20	20	34	34
44	17	96.45	12	600	20	20	37	37
45	17	96.45	12	600	20	20	40	40
46	17	96.45	15	750	10	10	34	34
47	17	96.45	15	750	10	10	37	37
48	17	96.45	15	750	10	10	40	40
49	17	96.45	15	750	15	15	34	34
50	17	96.45	15	750	15	15	37	37
51	17	96.45	15	750	15	15	40	40
52	17	96.45	15	750	20	20	34	34
53	17	96.45	15	750	20	20	37	37
54	17	96.45	15	750	20	20	40	40
55	20	105.16	8	400	10	10	34	34
56	20	105.16	8	400	10	10	37	37
57	20	105.16	8	400	10	10	40	40
58	20	105.16	8	400	15	15	34	34
59	20	105.16	8	400	15	15	37	37
60	20	105.16	8	400	15	15	40	40
61	20	105.16	8	400	20	20	34	34
62	20	105.16	8	400	20	20	37	37
63	20	105.16	8	400	20	20	40	40
64	20	105.16	12	600	10	10	34	34
65	20	105.16	12	600	10	10	37	37
66	20	105.16	12	600	10	10	40	40
67	20	105.16	12	600	15	15	34	34
68	20	105.16	12	600	15	15	37	37
69	20	105.16	12	600	15	15	40	40
70	20	105.16	12	600	20	20	34	34
71	20	105.16	12	600	20	20	37	37
72	20	105.16	12	600	20	20	40	40
73	20	105.16	15	750	10	10	34	34
74	20	105.16	15	750	10	10	37	37
75	20	105.16	15	750	10	10	40	40
76	20	105.16	15	750	15	15	34	34
77	20	105.16	15	750	15	15	37	37
78	20	105.16	15	750	15	15	40	40
79	20	105.16	15	750	20	20	34	34
80	20	105.16	15	750	20	20	37	37
81	20	105.16	15	750	20	20	40	40

and

Table 3. Taguchi orthogonal array (L₉) experimental model.

Experiment No	OV		ON		OFF		SV
	Code	Value (V)	Code	Value (ns)	Code	Value (μs)	Code	Value (V)
1	14	87.74	8	400	10	10	34	34
2	14	87.74	12	600	15	15	37	37
3	14	87.74	15	750	20	20	40	40
4	17	96.45	8	400	15	10	40	34
5	17	96.45	12	600	20	15	34	37
6	17	96.45	15	750	10	20	37	40
7	20	105.16	8	400	20	10	37	34
8	20	105.16	12	600	10	15	40	37
9	20	105.16	15	750	15	20	34	40

. The experiment was repeated three times for each set of parameters, then an average value was calculated from the measured value.

2.2. Optimization

Figure 6 shows the optimization algorithm used to obtain a faster MRR and smaller shape error value. The step-by-step process is as follows:

• The user inputs the OV, ON, OFF, and SV parameter values.
• The user chooses the shape error or MRR priority.
• For shape error priority:

  (a)

  The user inputs the desired shape error.

  (b)

  Calculate the shape error using the equation for shape error prediction.

  (c)

  Calculate the MRR using the equation for MRR prediction.

  (d)

  Compare the shape error prediction value and the desired shape error value. If the predicted shape error value is ≥ the desired shape error value, return to step 3(b).

  If the predicted shape error value is ≤ the desired shape error value, proceed to step 3(e).

  (e)

  Search for the fastest machining while keeping the predicted shape error value ≤ the desired shape error value.

  (f)

  Optimal parameters are set.

• For MRR priority:

  (a)

  The user inputs the desired MRR.

  (b)

  Calculate the MRR using the equation for MRR prediction.

  (c)

  Calculate the shape error using the equation for shape error prediction.

  (d)

  Compare the MRR prediction value and the desired MRR value.

  If the predicted MRR value is ≥ the desired MRR value, return to step 4(b).

  If the predicted MRR value is ≤ the desired MRR value, proceed to step 4(e).

  (e)

  Search for the smallest shape error while keeping the predicted MRR value ≤ the desired MRR value.

  (f)

  Optimal parameters are set.

• Optimal parameters.

As shown in Figure 6, the optimization can be chosen based on shape error priority or MRR priority, which provides flexibility for the user.

3. Experiment Result and Discussion

The experiment was performed using the machining parameters with three levels, as shown in Table 1. The Renishaw Equator measurement instrument with a scanning method and 150 points was used to measure and collect the data. The error value was calculated by subtracting the lowest point from the highest point of the data, yielding the shape error value.

3.1. Effect of Open Circuit Voltage on Shape Error and MRR

Figure 7 shows the impact of open circuit voltage on shape error. It can be seen that when the open circuit voltages were 14, 17, and 20, the shape error values were 17, 19, and 20 μm, respectively, displaying an increasing trend. However, when the open circuit voltage was set to 23, the shape error decreased, but the risk of wire breakage increased due to the higher energy. Figure 8 shows the impact of open circuit voltage on the MRR. It can be seen that the MRR significantly increases with a higher open circuit voltage.

3.2. Effect of Pulse ON Time on Shape Error and MRR

Figure 9 and Figure 10 show the impact of discharge time (pulse ON) on the shape error and MRR. Figure 9 shows that when the pulse ON time is 400, 600, and 750 ns, the shape error value is 11, 15, and 18 μm, respectively. It can be observed that the longer the discharge time, the larger the shape error. Additionally, when the shape error increases, the MRR is relatively faster, as shown in Figure 10. Conversely, a shorter discharge time results in smaller shape errors but a slower MRR. Longer discharge times correspond to longer single pulse durations, resulting in more impurities between discharges, which are harder to remove within the same discharge off time, leading to an increased shape error.

3.3. Effect of Pulse OFF Time on Shape Error and MRR

Regarding the impact of discharge off time (pulse OFF) on the shape error, as shown in Figure 11, it can be observed that when the discharge time (pulse ON) is less than 600 ns, the shape error does not significantly change with the longer pulse OFF time. The shape error only changes with variations in pulse ON time. However, when the pulse ON time is 750 ns, the shape error is relatively larger compared to the other three pulse ON times of 400, 500, and 600 ns. This phenomenon was further investigated to determine if the longer pulse OFF time can suppress the shape error of a longer pulse ON time. Therefore, the pulse OFF time was increased to 25 and 30 μs for further exploration. In Figure 12, it can be seen that when the pulse ON time is 750 ns, longer pulse OFF times, such as 25 and 30 μs, can effectively reduce the shape error. However, a longer pulse OFF time also results in a slower MRR, as shown in Figure 13.

3.4. Effect of Servo Voltage on Shape Error and MRR

Figure 14 and Figure 15 show the impact of servo voltage on the shape error and MRR. As seen in Figure 14, a higher servo voltage can effectively suppress the shape error. Additionally, when the servo voltage is 34, 37, and 40 V, with a parameter interval difference of 6 V, a more pronounced decreasing trend is observed. Figure 15 shows that a higher servo voltage results in a slower MRR. However, a lower servo voltage results in a larger shape error and faster MRR.

3.5. Effect of Wire Tension on Shape Error and MRR

Figure 16 and Figure 17 exhibit the impact of wire tension on shape error and MRR. As seen in Figure 16, a lower wire tension can cause the wire to vibrate due to external forces, which is a common occurrence. When wire tension increases to a certain level, the influence of external forces decreases, showing a trend of reduced shape error. However, when the wire tension increases to 15.69 and 16.67 N, the shape error tends to increase again. This could be due to poor debris removal, as a tightly stretched wire is more likely to cause secondary discharges with impurities. Therefore, wire tension must be maintained at an optimal level to effectively reduce the occurrence of shape errors. It can be seen from Figure 17 that wire tension does not have a significant impact on the MRR during rough cutting.

3.6. Effect of Flushing Pressure on Shape Error

Generally, a higher flushing pressure results in a greater flow rate, which can remove more debris and effectively improve the insulation state required during machining. However, whether higher flushing flow affects wire vibrations remains a debated topic. One viewpoint is that a higher flushing flow causes wire vibrations, thereby increasing the shape error in the middle section of the workpiece. Another viewpoint is that flushing pressure has a damping effect on the wire, effectively suppressing wire vibration [22,29]. This study explores the findings by switching between upper and lower flushing pressures. The switching distances used in this study were 0.5, 1, and 2 mm, with flushing pressures of 0.49, 0.68, 0.98, and 1.47 kPa. According to

Table 4. Shape error for different flushing modes.

	Flushing Pressure Difference	0.49 kPa	0.68 kPa	0.98 kPa
Switching Distance (mm)
0.5		8 μm ± 0.2	11 μm ± 0.2	11 μm ± 0.2
1		6 μm ± 0.2	9 μm ± 0.2	12 μm ± 0.2
2		11 μm ± 0.2	15 μm ± 0.2	16 μm ± 0.2

, when the flushing pressure difference was 0.49 kPa, the switching distances of 0.5 and 1 mm resulted in smaller shape errors of 8 and 6 μm, respectively. When the flushing pressure difference was 0.68 kPa, the error values significantly increased to 11 and 9 μm, respectively. At a flushing pressure difference of 0.98 kPa, the 0.5 mm switching distance maintained the error at 11 μm, while the 1 mm switching distance produced a shape error of 12 μm. This indicates that a flushing pressure difference of 0.49 kPa with a 1 mm switching distance results in a smaller shape error. However, when the flushing pressure difference is too large, longer switching distances lead to excessive debris concentration, thus increasing the shape error.

3.7. Prediction Modeling

A concave shape error prediction model was developed based on the experimental data and regression analysis. IBM SPSS Advanced Statistics Model 20 software was used for regression analysis. Linear regression was used to analyze the correlation between input factor OV, ON, OFF, SV, WT, WS, and WA and the output concave shape error and machining time. Analysis of Variance (ANOVA) was carried out to assess the significance of the regression model.

Table 5. ANOVA results for shape error prediction model.

Source	Sum of Square	DF	Mean Square	F-Value	p-Value
Model 1 (Full Factorial Shape Error Prediction)
Regression	302.322	4	75.580	75.920	<0.001
Residual	44.798	45	0.996
Total	347.120	49
$R^2=87.1\% R_{adj}^2=85.9\%$
Model 2 (Taguchi Shape Error Prediction)
Regression	190.378	4	47.595	137.383	<0.001
Residual	7.622	22	0.346
Total	198.000	26
$R^2=96.2\% R_{adj}^2=95.5\%$

shows the ANOVA results for the shape error prediction model with full factorial and Taguchi methods. The p-values of both models were less than 0.001, indicating that the models are statistically significant. Furthermore, the goodness-of-fit of the model was measured using the R-squared ($R^2$) and adjusted R-squared ($R_{adj}^2$) values. R-squared is a statistical measure that represents the proportion of the variance in the dependent variable that is explained by the independent variables in the model. It indicates how well the data fit the regression model. Higher R-squared values indicate better fit, but this does not always mean the model is appropriate or accurate. Adjusted R-squared modifies R-squared to account for the number of predictors in the model, preventing overestimation of the model’s explanatory power due to the inclusion of irrelevant predictors. Adjusted R-squared will increase only if the new predictor improves the model more than expected by chance. It can be less than R-squared but provides a more realistic assessment, especially for models with multiple predictors. Table 5 shows that the R-squared and adjusted R-squared values for the full factorial and Taguchi models are 87.1% and 85.9%, and 96.2% and 95.5%, respectively. This evidences that the Taguchi model has a better goodness-of-fit compared to the full factorial model, providing more accurate prediction results. In order to determine whether each predictor variable is statistically significant, a T-test was performed and analyzed. A T-test is a statistical test used to determine whether there is a significant difference between the means of two groups or to assess the significance of individual coefficients in a regression model. It evaluates whether the observed difference or effect is likely to occur due to random chance. It is used to determine whether a specific predictor (independent variable) has a significant effect on the dependent variable. Also, it evaluates whether the regression coefficient (β) for a predictor is significantly different from zero.

Table 6. T-test result for shape error prediction model.

Independent Variable	Unstandardized Coefficients		Standardized Coefficient	t	p-Value
	B	Std. Error	Beta
Model 1 (Full Factorial Shape Error Prediction)
(Constant)	20.693	2.783		7.435	<0.001
Open voltage	0.120	0.055	0.116	2.172	0.035
Pulse ON	0.716	0.050	0.774	14.451	<0.001
Pulse OFF	−0.243	0.030	−0.436	−8.150	<0.001
Servo voltage	−0.333	0.068	−0.263	−4.910	<0.001
Model 2 (Taguchi Shape Error Prediction)
(Constant)	18.477	1.986		9.304	<0.001
Open voltage	0.333	0.046	0.302	7.208	<0.001
Pulse ON	0.716	0.040	0.758	18.130	<0.001
Pulse OFF	−0.300	0.028	−0.452	−10.812	<0.001
Servo voltage	−0.333	0.046	−0.302	−7.208	<0.001

shows the T-test results for the shape error prediction model with full factorial and Taguchi methods. It can be seen that each independent variable significantly contributes to the dependent variable, as indicated by a p-value less than 0.05.

Based on the above information, the shape error prediction equation for the full factorial and Taguchi methods can be obtained as follows:

$$Y_{Fe}=0.12·OV+0.716·ON-0.243·OFF-0.333·SV+20.693$$

(1)

$$Y_{Te}=0.333·OV+0.716·ON-0.3·OFF-0.333·SV+18.477$$

(2)

where $Y_{Fe}$ is the shape error prediction with the full factorial method and $Y_{Te}$ is the shape error prediction with the Taguchi method.

To establish the MRR prediction model, the experimental data obtained from two types of orthogonal arrays were used to determine the MRR. These data were input into the IBM SPSS Statistics regression analysis module.

Table 7. ANOVA result for MRR prediction model.

Source	Sum of Square	DF	Mean Square	F-Value	p-Value
Model 1 (Full Factorial MRR Prediction)
Regression	17.555	4	4.389	270.899	<0.001
Residual	1.231	76	0.016
Total	18.786	80
$R^2=93.4\% R_{adj}^2=93.1\%$
Model 2 (Taguchi MRR Prediction)
Regression	4.241	4	1.060	530.635	<0.001
Residual	0.044	22	0.002
Total	4.285	26
$R^2=99.0\% R_{adj}^2=98.8\%$

shows the ANOVA result for the MRR prediction model with full factorial and Taguchi methods. Table 7 displays that the p-values of both models with full factorial and Taguchi methods are less than 0.001, indicating that the models are statistically significant. The R-squared value for the model with full factorial was 93.4%, indicating that the goodness-of-fit of the model was good, while the adjusted R-squared value was 93.1%, indicating a slightly lower goodness-of-fit, but this was on account of the model’s complexity. Furthermore, the R-squared and adjusted R-squared values for the prediction model with the Taguchi method were 99% and 98.8%, respectively, indicating a better goodness-of-fit of the model and prediction accuracy than that full factorial method. A T-test was performed to analyze the significant contribution of each independent variable to the dependent variable. The results of the T-test on the MRR with full factorial and Taguchi methods are shown in

Table 8. T-test result for MRR prediction model.

Independent Variable	Unstandardized Coefficients		Standardized Coefficient	t	Sig.
	B	Std. Error	Beta
Model 1 (Full Factorial MRR Prediction)
(Constant)	1.039	0.252		4.128	<0.001
Open voltage	0.020	0.006	0.102	3.476	0.001
Pulse ON	0.122	0.005	0.727	24.736	<0.001
Pulse OFF	−0.074	0.003	−0.629	−21.407	<0.001
Servo voltage	−0.019	0.006	−0.098	−3.338	0.001
Model 2 (Taguchi MRR Prediction)
(Constant)	0.373	0.151		2.472	0.022
Open voltage	0.044	0.004	0.273	12.656	<0.001
Pulse ON	0.101	0.003	0.726	33.604	<0.001
Pulse OFF	−0.060	0.002	−0.618	−28.634	<0.001
Servo voltage	−0.013	0.004	−0.079	−3.639	0.001

. It can be seen that the p-value is less than 0.05 for each independent variable, indicating the significant contribution to the dependent variable.

The MRR prediction equation can be derived based on the above information using the full factorial and Taguchi methods as follows:

$$Y_{Fs}=0.02·OV+0.122·ON-0.074·OFF-0.019·SV+1.039$$

(3)

$$Y_{Ts}=0.044·OV+0.101·ON-0.06·OFF-0.013·SV+0.373$$

(4)

The comparisons of the prediction results using regression equations for the shape error and MRR are shown in

Table 9. Prediction result for shape error using full factorial and Taguchi methods.

Parameters								Shape Error Actual Value (μm)	Prediction Results (μm)		Prediction Accuracy (%)
OV		ON		OFF		SV			Full Factorial	Taguchi	Full Factorial	Taguchi
Code	Value (V)	Code	Value (ns)	Code	Value (μs)	Code	Value (V)
19	102.25	8	400	13	13	35	35	14	13.862	14.977	99	93
18	99.35	10	500	18	18	36	36	15	13.457	14.243	90	95
15	90.65	12	600	14	14	39	39	16	14.792	14.877	92	92
19	102.25	13	650	16	16	37	37	17	16.109	16.991	94	99
14	87.74	15	750	20	20	38	38	18	15.518	15.225	86	84

and

Table 10. Prediction result for MRR using full factorial and Taguchi methods.

Parameters								MRR Actual Value (mm2/min)	Prediction Results (mm2/min)		Prediction Accuracy (%)
OV		ON		OFF		SV			Full Factorial	Taguchi	Full Factorial	Taguchi
Code	Value (V)	Code	Value (ns)	Code	Value (μs)	Code	Value (V)
19	102.25	8	400	13	13	35	35	0.631	0.660	0.782	95	80
18	99.35	10	500	18	18	36	36	0.605	0.590	0.627	97	96
15	90.65	12	600	14	14	39	39	0.917	1.010	0.898	90	97
19	102.25	13	650	16	16	37	37	1.033	1.110	1.081	93	95
14	87.74	15	750	20	20	38	38	0.882	0.930	0.810	94	92

. It can be seen from Table 9 that the Taguchi method had a higher prediction accuracy compared to the full factorial method. Similarly, the Taguchi method had a higher prediction accuracy for the results of the MRR prediction, as shown in Table 10. The Taguchi method required fewer experimental runs to achieve an accuracy of over 80%. Additionally, the Taguchi method provided a broader range of predictions with fewer combinations. However, for parameters at the bottom or up margins, the prediction accuracy using the Taguchi method slightly decreased. On the other hand, the full factorial method provided better results for these bottom and up margins. Therefore, it cannot be concluded that the Taguchi method is superior to the full factorial method. However, the full factorial method requires more experimental runs (three times) compared to the Taguchi method. This study used an integrated approach, which means that it is the first to use the Taguchi method to derive a broad-range prediction equation, followed by a full factorial method for areas with lower predictive power. This method effectively reduces the number of unnecessary experimental runs and improves prediction accuracy.

Table 11. Comparative analysis between the proposed method with other optimization methods.

Optimization Method	Advantages	Limitations	Compare to the Proposed Method
Taguchi	■ Efficiently reduces the number of experiments. ■ Provides robust optimization for processes with multiple factors.	■ Less effective for boundary parameter predictions. ■ Assumes linearity and independence of factors, which may not always hold true.	■ The proposed method incorporates both Taguchi and full factorial approaches, leveraging Taguchi’s efficiency for broad predictions while refining results through full factorial for more accurate boundary conditions. ■ This hybrid approach outperforms the standalone Taguchi method in precision and flexibility.
Full Factorial	■ Consider all possible combinations of factors, providing highly accurate and comprehensive results.	■ Requires a significantly larger number of experiments, increasing time and cost.	■ The proposed method integrates full factorial design selectively, applying it only to critical areas where prediction accuracy is lower. ■ This targeted use balances the comprehensiveness of full factorial with the efficiency of reduced experimentation.
Response Surface Methodology (RSM)	■ Build a predictive model to explore the relationships between parameters and outcomes. ■ Useful for identifying optimal conditions.	■ Computationally intensive and assumes quadratic relationships between factors and response.	■ While RSM provides similar regression models, the proposed method directly integrates regression analysis into a user-friendly optimization system. ■ This streamlines parameter selection and enhances accessibility for real-time industrial applications.
Genetic Algorithm (GA)	■ Effective for complex, non-linear optimization problems. ■ Does not rely on gradient information, making it versatile for multi-modal landscapes.	■ Computationally expensive. ■ May converge prematurely to local optima.	■ The proposed method simpler and provides high prediction accuracy for shape error and material removal rate while requiring significantly less computational power. ■ For industrial settings, this efficiency makes the system more practical.

displays the comparative analysis between the proposed method with other optimization methods. The Response Surface Methodology (RSM) and Genetic Algorithm (GA) excel in handling highly complex and non-linear systems, the proposed (hybrid) method is uniquely suited for WEDM optimization. It balances accuracy, efficiency, and usability, making it a practical and competitive alternative to existing methods.

4. Human–Machine Interface

The overall architecture system of this study includes shape error prediction, MRR prediction, shape error optimization, and MRR optimization. According to the prediction equation and the optimization algorithm mentioned in Section 2 and Section 3, a user-friendly human–machine interface (HMI) was designed and created. This allows users to complete the operations easily. The system was developed using Microsoft visual studio C# programming language. The system structure is as follows:

• Prediction module: The first part focuses on predicting the shape error and MRR based on the parameters input by the user.
• Shape error optimization module: The second part calculates and searches the parameter combination that follows the optimization rules according to the user-required shaper error and then presents the results. Additionally, an estimation of the MRR is calculated using the optimized parameters.
• MRR optimization module: The third part calculates and searches the combination of parameters that follow the optimization rules according to the user-required MRR and then displays the results. Additionally, an estimation of the shape error is calculated using the optimized parameters.

This approach streamlines the process, ensuring that users can easily find the best combinations of parameters without extensive annual effort.

Figure 18 shows the HMI for the prediction and optimization of the shape error and MRR. The step-by-step system operation procedure is as follows:

• The user enters the parameter values into the corresponding parameter fields in area 1 (number 1 in Figure 18). Afterward, the user presses the “Shape error prediction” button to obtain the predicted shape error and MRR values for the current parameters.
• The user chooses the priority of optimization. If the user chooses shape error optimization as the priority, then proceed to step three. If the user chooses MRR as the priority, then proceed to step four.
• The user enters the desired shape error value into the “shape error requirements” field (number 2 in Figure 18). Then, the user presses the “Shape error optimization” button to obtain the optimized parameter combination. The system will display the result in area number 2 of Figure 18. In addition, the system displays the estimated MRR value for the optimized parameter combination.
• The user enters the desired MRR value into the “MRR requirements” field (number 3 in Figure 18). Then, the user presses the “MRR optimization” button to obtain the optimized parameter combination. The system will display the result in area number 3 of Figure 18. Additionally, the system displays the estimated shape error value for the optimized parameter combination.
• The user presses the “Save data” button (number 4 in Figure 18) to save the prediction and optimization parameter values in a Microsoft Excel file format.

5. Verification Experiments

The verification experiments were conducted to validate the proposed optimization method and system. The verification process was divided into two parts. The first part utilized the developed optimization HMI to optimize MRR under shape error requirements and to optimize (minimize) shape error under MRR requirements. In the second part, machining was performed using both the original parameters and the optimized parameters. Furthermore, the shape errors and MRR were measured and compared. The fixed parameters of wire tension (WT) (13.73 N), wire speed (WS) (10 m/min), and flushing pressure (WA) (1.47 Pa) were used in the verification experiment.

Table 12. Verification experiment results for optimized MRR under shape error requirements.

Parameters	Original	Predicted by System	Actual Machining	Accuracy (%)
Shape error requirements (μm)	−	11	−	−
OV	17	14	14	−
ON	15	8	8	−
OFF	9	17	17	−
AON	11	6	6	−
AOFF	10	17	17	−
SV	37	40	40	−
Shape error value (μm)	19.259	11	12	91.7
MRR value (mm2/min)	1.706	0.387	0.384	99.2

shows the verification experiment results for optimal MRR that meets the shape error requirements. It can be seen that when the shape error requirement was 11 μm, the optimal parameter combination predicted by the system was OV 14, ON 8, OFF 17, AON 6, AOFF 17, and SV 40. The predicted optimal MRR was 0.387 mm²/min. Furthermore, the actual machining using the optimized parameter from the system was performed. After the machining was completed, the shape error and MRR were measured. The actual measurement showed that the shape error value and MRR value were 12 μm and 0.384 mm²/min, respectively. When compared to the value that was predicted by the system, the accuracy was 91.7% for shape error and 99.2% for MRR.

Different parameter values were used to verify optimized shape error under MRR requirements.

Table 13. Verification experiment result for optimized shape error under MRR requirements.

Parameters	Original	Predicted by System	Actual Machining	Accuracy (%)
MRR requirements (mm2/min)	−	1.8	−	−
OV	17	20	20	−
ON	15	15	15	−
OFF	10	10	10	−
AON	11	13	13	−
AOFF	10	10	10	−
SV	37	38	38	−
Shape error value (μm)	19.016	18.749	19	98.7
MRR value (mm2/min)	1.756	1.8	1.788	99.3

shows that when the MRR requirement was 1.8 mm²/min, the optimal parameter combination predicted by the system was OV 20, ON 15, OFF 10, AON 13, AOFF 10, and SV 37. The predicted optimal shape error was 18.749 μm. Machining was then performed using these optimized parameters. After completing the machining, the shape error and MRR were measured. The actual measurements showed a shape error of 19 μm and MRR of 1.788 mm²/min. Compared to the system prediction, the accuracy was 98.7% for shape error and 99.3% for MRR.

6. Conclusions

This study focused on optimizing shape error (drum shape) in WEDM. Experiments with various machining parameters such as open voltage (OV), pulse ON time (ON), pulse OFF time (OFF), servo voltage (SV), wire tension (WT), wire speed (WS), and flushing pressure (WA) were conducted. The Taguchi and full factorial methods were used to analyze and investigate the effect of each parameter on the shape error and MRR. It was found that both the Taguchi and full factorial methods can achieve a prediction accuracy of more than 80%. The predictions from the full factorial method are more stable but require more experimental data, while the Taguchi method provides a slightly lower prediction accuracy for boundary parameters but requires fewer experimental data.

WEDM cannot be optimized after a single parameter. Different parameter combinations can yield the same shape error. Therefore, this study identified parameters with significant impacts on the shape error and MRR through experimental and built regression analyses to establish a shape error and MRR optimization system. A shape error and MRR optimization system with a human–machine interface was built using a C# language program. This system recommends the best parameter combinations based on user requirements to achieve a smaller shape error and faster MRR.

The results of the verification experiments showed that the estimated shape error was 19.259 μm before shape error optimization, while the measured shape error after machining using the optimized parameter was 12 μm, demonstrating a 37.7% decrease in shape error. However, the MRR before optimization was 1.706 mm²/min and became 0.384 mm²/min after optimization, indicating a 77.5% slower speed. Overall, the system can minimize the shape error by up to 40%. The prediction accuracy of the system can reach up to 94.7% for shape error and 99.2% for MRR. The investigation also found that altering the flushing mode can reduce the accumulation of debris in the same location, thereby reducing the shape error.

This study used an SKD11 workpiece with a thickness of 60 mm. The variation of workpiece material and thickness can be further investigated in the future. Additionally, adjusting the flushing pressure could improve the impact of debris accumulation on shape error. Further investigation is needed to determine how to effectively adjust the flushing pressure to reduce shape error.