Comprehensive Analysis of Milling Performance and Multi-Objective Parameter Optimization for YG6C Milling Tool

Abstract

Numerous conflicting objectives exist in the engineering field, and resolving these conflicts to reduce costs constitutes a problem that demands top-priority consideration. A model for tool wear and a multi-quadratic regression model for milling forces were developed to accurately predict the trends of wear on the rake face of the milling tool and the variations in milling forces. The influence of milling parameters (spindle speed, n; feed rate, v_f; axial milling depth, a_p) on both the wear of the rake face and milling force was analyzed by means of orthogonal experiments. The findings indicated that the impact of these parameters on the wear ranked in the following order: n > v_f > a_p. In contrast, for milling force, F, the ranking was a_p > v_f > n. Utilizing MATLAB’s genetic algorithm, an optimization procedure was conducted with multiple objectives including the wear of the rake face, milling force, and material removal rate; subsequently, a Pareto optimal solution set was generated for milling parameters based on practical processing requirements.

1. Introduction

The processing function of traditional machine tools, such as lathes and milling machines, is relatively straightforward, especially when dealing with small workpieces. In contrast, modern processing equipment, including CNC engraving and milling machines, laser cutting machines, and electrical discharge machining machines, demonstrate significantly higher precision and efficiency in the processing of small workpieces [1,2]. The CNC engraving and milling machine, being a new type of high-performance machine tool with excellent cost-effectiveness, effectively addresses the limitations of traditional machine tools, such as low precision, inefficiency, and inadequate flexibility, and gradually becomes an important branch in modern machine tool processing. It has found wide applications in furniture manufacturing, decoration, small precision mold manufacturing, and hardware production [3,4].

As competition in the global market escalates, companies face a variety of challenges related to improving production efficiency, cutting costs, and enhancing product quality [5]. In this scenario, milling is recognized as a vital process in mechanical manufacturing, making it crucial to optimize its parameters. The careful selection of process parameters, such as spindle speed, feed rate, and milling depth, has a direct and significant impact on key milling performance metrics; these factors not only influence the rate at which the material is removed but also significantly affect tool wear and the surface finish of the final product [6,7]. Therefore, optimizing these parameters systematically can lead to improved production efficiency while reducing resource waste and maximizing economic returns. Additionally, multi-objective optimization techniques facilitate the simultaneous assessment of conflicting goals [8], such as milling productivity, tool lifespan, and milling forces, allowing researchers to perform a thorough analysis of various solutions’ effects and providing a solid foundation for informed decision making [9]. At the same time, in an era characterized by intelligent manufacturing practices, it is essential for businesses to continually adapt their technologies to align with rapidly changing market needs. Thus, accelerating research into both milling parameter optimization and multi-objective methodologies is critical not only for strengthening core competitiveness but also for guiding the entire industry toward enhanced efficiency and sustainable growth. Consequently, the importance and urgency associated with both milling parameter optimization and multi-objective strategies in modern manufacturing are becoming increasingly clear [10,11].

During the milling process, the relationship between tool wear, milling force, and milling efficiency is particularly evident. These three elements are interconnected and together form a complex dynamic system [12]. More specifically, tool wear not only reduces performance but can also negatively impact the surface quality of the workpiece. The level of milling force is directly associated with both the stability and accuracy of machining operations on the machine tool. When excessive forces are present during milling, they can cause vibrations in the machine tool that diminish product surface finish and accelerate tool degradation, ultimately reducing its operational lifespan. Additionally, to improve milling efficiency, it is often necessary to increase feed rate; however, this approach may result in heightened tool wear and greater milling forces [13]. Therefore, while pursuing high-efficiency production presents a notable conflict among these three factors—on one hand, there exists an intention to enhance efficiency for cost savings; on the other hand, it remains crucial to manage both tool wear and milling force to maintain processing quality and ensure equipment safety [14].

The methods of multi-objective optimization have emerged as a significant research focus in the fields of engineering and technology in recent years. Since the 1970s, various effective techniques, including weighted methods, genetic algorithms, particle swarm optimization, and fuzzy logic control, have been developed to address the multiple conflicting objectives inherent in complex systems effectively [15,16,17]. Chen et al. [18] tackled the multi-objective optimization issue resulting from the combined influence of milling parameters and tool geometric parameters through integrating least squares regression modeling and finite element simulation; he successfully accomplished effective control over various milling objectives with the VAEA optimization algorithm. Zhao et al. [19] highlighted a conflict between reducing milling force and lowering tip temperature during cutting tool processing. When cutting speeds are comparable, enhancing cutting efficiency can be accomplished by adjusting both depth of cut and feed rate. Vu [20] made use of the particle swarm optimization algorithm to optimize the surface roughness, cutting temperature, and cutting energy consumption in the course of the hard milling process of AISI H13 steel. The results suggest that the cutting energy consumption can be lowered by 14% following optimization. Mohammed Toufik Amira [21] utilized response surface methodology to assess the impacts of milling parameters on machining characteristics. Relying on genetic algorithms, he attained the most favorable combination of milling parameters that reduced production time and unit production cost. Kamalizadeh [22] put forward a TOPSIS model for multi-criteria decision making, which integrates Taguchi methodology, merging surface roughness and tool wear into a single response after determining the optimal processing parameters. This approach demonstrates that employing variable cutting speeds during machining considerably enhances both tool life and surface roughness in contrast to traditional constant-speed methods.

Multi-objective optimization techniques provide significant benefits in engineering applications; however, many recent studies primarily depend on theoretical models and simulation analyses, lacking sufficient backing from large-scale experimental data, which limits the applicability of certain optimization algorithms in practical production environments. Moreover, differences in milling characteristics due to varying materials and process parameters add further complexity to the optimization procedure. Genetic algorithms serve as a powerful optimization method that can effectively tackle complex issues with multiple peaks by simulating biological evolution processes while being less prone to local optima, thus, enabling efficient solutions for challenges faced in real-world engineering scenarios [23,24].

This research seeks to forecast the trend of variation in the wear width on the rake face and milling force by developing a model for tool wear and a multi-quadratic regression model. Orthogonal experiments are utilized to select key factors (including spindle speed, feed rate, and axial milling depth), thus, allowing for a thorough evaluation of each factor’s impact on the results while differentiating between primary and secondary influences, which aids in optimizing experimental design and improving overall efficiency. Additionally, a genetic algorithm is employed for the multi-objective optimization of material removal rate, the wear of the rake face, and milling force to determine optimal strategies that offer a scientific basis for real-world production practices.

2. Materials and Methods

2.1. Materials

The CNC engraving and milling machine, designated as the SXDK6050D model and manufactured by Taizhou Samsung Machine Tool Manufacturing Factory (Taizhou, China), is renowned for its outstanding precision and high-efficiency performance. This machine is widely applied in mechanical processing, mold manufacturing, and various other industrial fields. Its operational stroke dimensions measure 600 mm × 500 mm × 240 mm, sufficiently meeting the machining demands of diverse workpieces. The dynamometer utilized is the Kistler model 9257B, a high-performance sensor specifically designed for dynamic measurement and monitoring of mechanical parameters during milling operations. This sensor provides precise data, facilitating process optimization and enhancing product quality. Furthermore, the charge amplifier employed is the Kistler model 5070A1210, which cooperates with the dynamometer to enhance signal processing capabilities. In addition to signal amplification, the charge amplifier provides essential support for impedance matching and noise suppression during the milling force measurement process, thereby ensuring the accuracy and reliability of subsequent data acquisition.

The milling tool used in this research is a double-edged straight flute pointed tool, identified as model 2ZJ62030, which is produced by Shanghai Micro-engraving CNC Machinery Co., Ltd. (Shanghai, China). This tool is extensively used in various machining applications due to its outstanding mechanical properties and exceptional wear resistance. The overall length of the tool is 100 mm, with a shank diameter of 6 mm, a milling angle of 20°, and a blade diameter of 3 mm, as depicted in Figure 1. Additionally, this tool is made from YG6C cemented carbide, which demonstrates remarkable hardness and strength properties, enabling it to maintain optimal sharpness and stability over an extended period of use. Detailed performance parameters are presented in

Table 1. The properties of the milling tool.

Model	Chemical Compositions		Density ρ (g/cm3)	Hardness HR (HRA)	Bending Strength σb (MPa)	Melting Point Tm (°C)
	W (wt.%)	Co (wt.%)
YG6C	93	7	14.8	88	1450	1350~1400

. The material of the workpiece is high-density fiberboard (HDF), with dimensions of 120 mm × 80 mm × 8 mm. HDF is a renewable material primarily composed of wood and plant fibers bonded together under heating and pressure using synthetic resin, which imparts excellent mechanical properties and compressive strength to the board. The specific parameters are detailed in

Table 2. The properties of the high-density fiberboard.

Workpiece	Ignition Temperature	Density	Modulus of Elasticity	Bending Strength
	Ti (°C)	ρ (kg/m3)	E (MPa)	σb (MPa)
HDF	230~260	848	4000	35~50

. The layout of the equipment has been meticulously designed to ensure coordination and collaboration among various stages, thereby enhancing overall work efficiency. The equipment and layout are, respectively, depicted in Figure 2 and Figure 3.

2.2. Methods

Overall framework: (1) The spindle speed, n, feed rate, v_f, and axial milling depth, a_p, are selected as the optimization parameters, following a 3-factor 3-level design, as detailed in

Table 3. The experimental milling factors and levels.

Level	Experimental Milling Factors
	n (r/min)	vf (mm/min)	ap (mm)
1	12,000	1000	3
2	15,000	1500	4
3	18,000	2000	5

. Through orthogonal experiments, the wear width of the rake face and the milling force under different parameter combinations are determined. (2) By fitting the tool wear model and milling force model, predictive functions for both phenomena are derived. (3) Finally, a genetic algorithm is employed to optimize the multi-objective function, yielding the Pareto optimal solution.

The design for measuring the wear width of the tool: for each set of experimental parameters, the wear width of the tool is measured after accumulating 30 min of machining. Each set of experiments is repeated three times, and the average of the three measurement results is taken as the final result for that set of experiments.

Steps for measuring the wear width of the tool: (1) Clean the tool surface with ethanol to ensure it is free from contaminants, thereby enhancing measurement accuracy. (2) Place the tool on stage of the ZL-SMZ16X stereomicroscope (manufactured by Zi Ling Optical Instrument Co., Ltd., Guangzhou, China) and adjust the focus until the tool’s rake face is clearly visible in the field of view. (3) Identify the starting and ending points of the wear area and use the accompanying software and digital ruler to accurately measure and calculate the wear width.

2.2.1. The Model for Tool Wear

The wear width of the rake face, referred to as b, is a crucial indicator that significantly impacts machining quality. Research demonstrates that b exhibits a complex relationship with spindle speed, n, feed rate, v_f, and axial milling depth, a_p, which can be mathematically expressed as a power function [18].

$$b={\mathrm{C}}_{\mathrm{F}}\cdot n^{{\alpha }_1}\cdot v_{\mathrm{f}}^{{\alpha }_2}\cdot a_{\mathrm{p}}^{{\alpha }_3}$$

(1)

where C_F denotes the processing coefficient, which is influenced by a variety of factors, including the chosen processing material, operational conditions, and equipment performance. Furthermore, α₁, α₂, and α₃ are significant power indices associated with each processing variable. To enhance subsequent data analysis, we can apply logarithmic transformation to both sides of the equation, thereby converting the complex power function into a more manageable first-order linear function.

$$\lg b=\lg {\mathrm{C}}_{\mathrm{F}}+{\alpha }_1\lg n+{\alpha }_2\lg v_{\mathrm{f}}+{\alpha }_3\lg a_{\mathrm{p}}$$

(2)

By defining y = lgb, α₀ = lgC_F, x₁ = lgn, x₂ = lgv_f, x₃ = lga_p, we can derive the following results:

$$\mathrm{y}={\alpha }_0+{\alpha }_1{\mathrm{x}}_1+{\alpha }_2{\mathrm{x}}_2+{\alpha }_3{\mathrm{x}}_3$$

(3)

Through substituting the outcomes gained from the orthogonal experiment into Equation (3), the following system of linear equations can be obtained:

$$\left\{\begin{array}{l}\mathrm{y}={\alpha }_0+{\alpha }_1{\mathrm{x}}_{11}+{\alpha }_2{\mathrm{x}}_{12}+{\alpha }_3{\mathrm{x}}_{13}\\ \mathrm{y}={\alpha }_0+{\alpha }_1{\mathrm{x}}_{21}+{\alpha }_2{\mathrm{x}}_{22}+{\alpha }_3{\mathrm{x}}_{23}\\ \cdots \\ \mathrm{y}={\alpha }_0+{\alpha }_1{\mathrm{x}}_{91}+{\alpha }_2{\mathrm{x}}_{92}+{\alpha }_3{\mathrm{x}}_{93}\end{array}\right.$$

(4)

where x_i1, x_i2, and x_i3, respectively, represent x₁, x₂, and x₃ in the ith group of experiments. By undertaking a fitting analysis, the specific numerical values of the coefficients within the equation set can be determined. The wear condition of the tool’s rake face is assessed through the signal-to-noise ratio S/N (η) in the Taguchi method. Throughout the entire process of tool processing, there is typically a requirement that the wear degree of the tool should be as minimal as possible; that is, the interference caused by the noise to the signal source should be reduced to the greatest extent possible. This situation pertains to the problem of minimizing the characteristic, and the corresponding calculation formula is as follows [25]:

$$\mathrm{S}/\mathrm{N}\left(\mathrm{\eta }\right)=-10\log \left({\displaystyle \frac{1}{\mathrm{k}}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{k}}{b}_{\mathrm{i}}^2}\right)$$

(5)

where b_i indicates the wear width of the rake face for the ith group during the kth experiment, where k refers to the number of measurements.

2.2.2. The Regression Model for Milling Force

Through meticulous data analysis, a multi-quadratic regression model was set up using rigorous mathematical theory and scientific calculation methods. It can comprehensively consider the interactions among various complex factors. This model offers a powerful tool and reliable basis for accurately analyzing and predicting the trend of milling force changes, thereby providing crucial technical support for achieving efficient and precise milling processing. The multi-quadratic regression model is capable of presenting the following in a clear and accurate polynomial form [26]:

$$\mathrm{y}={\mathrm{\beta }}_0+{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{k}}{\mathrm{\beta }}_{\mathrm{j}}}{\mathrm{x}}_{\mathrm{j}}+{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{k}}{\mathrm{\beta }}_{\mathrm{j}\mathrm{j}}}{\mathrm{x}}_{\mathrm{j}}^2+{\displaystyle \sum _{\mathrm{j}>\mathrm{i}}{\mathrm{\beta }}_{\mathrm{j}\mathrm{i}}}{\mathrm{x}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{i}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left(\mathrm{i}=1,2,\cdots ,\mathrm{k}-1\right)$$

(6)

where x_i and x_j represent the independent variables, while y represents the dependent variable. The coefficient β is yet to be determined. In this study, the independent variables include spindle speed, n, feed rate, v_f, and axial milling depth, a_p. Therefore, the milling force F can be expressed as:

$$F={\mathrm{\beta }}_0+{\mathrm{\beta }}_{1}n+{\mathrm{\beta }}_2{v}_{\mathrm{f}}+{\mathrm{\beta }}_3{a}_{\mathrm{p}}+{\mathrm{\beta }}_{12}n{v}_{\mathrm{f}}+{\mathrm{\beta }}_{13}n{a}_{\mathrm{p}}+{\mathrm{\beta }}_{23}{v}_{\mathrm{f}}{a}_{\mathrm{p}}+{\mathrm{\beta }}_{11}{n}^2+{\mathrm{\beta }}_{22}{v}_{\mathrm{f}}^2+{\mathrm{\beta }}_{33}{a}_{\mathrm{p}}^2$$

(7)

2.2.3. The Principles of Genetic Algorithms

The fundamental concept of the genetic algorithm approach is illustrated in Figure 4. Firstly, a random initial population composed of multiple individuals is generated. Through non-dominated sorting, the likelihood of selecting individuals with higher fitness is significantly greater than that of selecting individuals with lower fitness. Subsequently, the first-generation offspring population is produced through crossover and mutation. Starting from the second generation, the parent population and the offspring population are combined, and non-dominated sorting is employed to calculate the crowding degree of the individuals, with suitable individuals being selected to generate the new parent population. Ultimately, through the continuous repetition of the aforementioned selection, crossover, and mutation processes, the population gradually evolves, and the fitness is constantly enhanced, eventually converging to an optimal or near-optimal solution.

3. Results

3.1. The Results and Analysis for Tool Wear

The experiment utilized an L⁹ (3⁴) orthogonal design to examine the effects of various milling parameters on the wear width, b, of the rake face. To ensure data accuracy and reliability, the wear width, b, was measured independently three times under each experimental condition. This methodology not only aids in minimizing random errors but also provides a more accurate representation of actual wear conditions. Ultimately, we calculated the average of these three measurements, with the results from the orthogonal experiments presented in

Table 4. Orthogonal Experimental Results of tool wear.

No	Parameters			Results
	n (r/min)	vf (mm/min)	ap (mm)	b (mm)	S/N (η)
1	12,000	1000	3	0.0445	13.5067
2	12,000	1500	4	0.0611	12.1325
3	12,000	2000	5	0.0702	11.5428
4	15,000	1000	4	0.0582	12.3582
5	15,000	1500	5	0.0781	11.0679
6	15,000	2000	3	0.0842	10.7417
7	18,000	1000	5	0.0782	11.0735
8	18,000	1500	3	0.0872	10.5899
9	18,000	2000	4	0.1023	9.8970

.

The signal-to-noise ratio S/N (η) for the tool wear condition presented in Table 4 was analyzed using range analysis and variance analysis, with the results detailed in

Table 5. Range Analysis of S/N (η).

Item	Ki	n (r/min)	vf (mm/min)	ap (mm)
S/N (η)	K1	12.394	12.313	11.613
	K2	11.389	11.263	11.463
	K3	10.520	10.727	11.228
	Range	1.874	1.586	0.385

Primary and secondary factors: n > v_f > a_p.

and

Table 6. Analysis of variance for S/N (η).

Item	Factors	Sum of Squared Deviations	F	Fcritical value	Significance
S/N (η)	n	5.276	83.746	19	*
	vf	3.903	61.952	19	*
	ap	0.226	3.587	19	-

* indicates a significant correlation.

. The range analysis indicates that the influence of milling parameters on the signal-to-noise ratio S/N (η) of tool wear, follows this order: spindle speed n > feed rate v_f > axial milling depth a_p. According to Equation (5), the smaller the tool wear, the higher its signal-to-noise ratio. As depicted in Figure 5, to minimize the wear width of the rake face while ensuring compliance with milling force and tool life requirements, the optimal milling parameters are identified as n = 12,000 r/min, v_f = 1000 mm/min, and a_p = 3 mm.

During the process of data analysis, the F-test approach with statistical significance was employed scientifically and rationally, with the significance level fixed at α = 0.05, in an attempt to form objective and reasonable judgments through calculating and comparing the relevant data and, thus, guaranteeing that the conclusions reached are highly reliable. The analysis of variance results reveal that F (n) > F (v_f) > F_{critical value} > F (a_p), suggesting that there exists a significant relationship among the spindle speed n, the feed rate v_f, and the signal-to-noise ratio S/N (η) of tool wear. The degree of significant correlation is that the spindle speed is greater than the feed rate. This outcome is in accordance with the conclusion of the range analysis.

The data of Table 4 were fitted through Equation (1), and the resulting outputs are displayed in

Table 7. The fitting results of tool wear.

CF	α1	α2	α3	R2
6.68 × 10−8	1.043	0.505	0.152	0.989

. It can be discerned from the table that the coefficient of determination, R², is 0.989. The determination, R², is a highly significant indicator in statistics and data analysis, gauging the predictive ability of the independent variable for the dependent variable. The value of R² spans from 0 to 1, and the nearer it is to 1, the more precise the model’s prediction. Therefore, it can be inferred that the model has satisfactory fitting effects.

By incorporating the fitted parameters into Equation (1), the wear width, b, of the rake face can be articulated as follows:

$$b=6.68\times 10^{-8}{n}^{1.043}{v}_{\mathrm{f}}^{0.505}{a}_{\mathrm{p}}^{0.152}$$

(8)

According to Equation (8), the exponents α₁, α₂, and α₃ corresponding to spindle speed, feed rate, and axial milling depth are all positive values that adhere to the relationship α₁ > α₂ > α₃. The result is consistent with the previous analysis of variance and was anticipated, as it aligns with established expectations regarding the influence of these parameters on tool wear. This finding suggests that spindle speed exerts the most pronounced influence on the wear width of rake face, followed by feed rate; in contrast, axial milling depth has a comparatively minor effect. As illustrated in Figure 6, the correlation between variations in these factors and the extent of wear on the rake face is clearly delineated. The primary factors influencing the wear of the milling tool are spindle speed and feed rate. This is because as the spindle speed increases, the cutting speed also rises, leading to an accelerated relative motion between the tool and the workpiece, thereby exacerbating tool wear. Additionally, an increase in feed rate results in higher contact stress between the tool and the workpiece, causing a rapid rise in milling forces, which further accelerates tool wear. While augmenting axial milling depth may also enhance material removal rates per unit time, its effect on leading edge wear is significantly less pronounced compared to that of feed rate.

3.2. The Results and Analysis for Milling Force

The milling performance during the machining process is notably influenced by a diverse array of milling parameters.

Table 8. The results of orthogonal experiment for milling force.

No	Parameters			Results
	n (r/min)	vf (mm/min)	ap (mm)	F (N)
1	12,000	1000	3	40.02
2	12,000	1000	4	43.92
3	12,000	1000	5	45.71
4	12,000	1500	3	42.41
5	12,000	1500	4	50.35
6	12,000	1500	5	53.51
7	12,000	2000	3	46.84
8	12,000	2000	4	55.12
9	12,000	2000	5	58.57
10	15,000	1000	3	39.84
11	15,000	1000	4	45.44
12	15,000	1000	5	49.65
13	15,000	1500	3	49.31
14	15,000	1500	4	48.36
15	15,000	1500	5	52.63
16	15,000	2000	3	47.27
17	15,000	2000	4	51.78
18	15,000	2000	5	56.53
19	18,000	1000	3	38.65
20	18,000	1000	4	44.71
21	18,000	1000	5	55.03
22	18,000	1500	3	40.15
23	18,000	1500	4	52.87
24	18,000	1500	5	60.94
25	18,000	2000	3	48.26
26	18,000	2000	4	53.12
27	18,000	2000	5	60.34

precisely presents the alterations in the milling force, F, under the different spindle speed, n, feed rate, v_f, and axial milling depth, a_p.

A range and variance analysis was conducted on the milling force, F, presented in Table 8, with the results detailed in

Table 9. Range Analysis of milling force.

Item	Ki	n (r/min)	vf (mm/min)	ap (mm)
F (N)	K1	48.496	44.776	43.642
	K2	48.981	50.060	49.514
	K3	50.453	53.094	54.773
	Range	1.957	8.318	11.131

Primary and secondary factors: a_p > v_f > n.

and

Table 10. Analysis of variance for milling force.

Item	Factors	Sum of Squared Deviations	F	Fcritical value	Significance
F (N)	n	18.708	11.463	19	-
	vf	319.011	195.472	19	*
	ap	558.122	341.987	19	*

* indicates a significant correlation.

. The range analysis indicates that the influence of milling parameters on milling force F follows a descending order: axial milling depth a_p > feed rate v_f > spindle speed n. As depicted in Figure 7, to minimize the milling force exerted on the tool while ensuring milling efficiency and tool life, the optimal milling parameters are identified as n = 12,000 r/min, v_f = 1000 mm/min, and a_p = 3 mm. By setting the significance level at α = 0.05, the analysis of variance results reveal that F (a_p) > F (v_f) > F_{critical value} > F (n), which implies that the significance order of the influence of milling parameters on milling force F is a_p > v_f > n.

The data presented in Table 8 were fitted using Equation (7) and analyzed with 1stopt software (version 15.0), yielding the fitting results of the multi-quadratic regression model as shown in

Table 11. The fitting results of milling force.

β0	β1	β2	β3	β4	β5
33.681	−2.980 × 10−3	2.738 × 10−2	−0.873	−4.217 × 10−7	5.736 × 10−4
β6	β7	β8	β9	R2
1.917 × 10−4	5.481 × 10−8	−4.500 × 10−6	−0.307	0.951

. The coefficient of determination, R², amounts to 0.904. This value implies that variations in the milling parameters have a significant and predictable impact on the milling force, F. Additionally, the fitting model demonstrates excellent performance and can effectively predict alterations in the milling force of the tool, providing valuable references and theoretical support for the actual milling processing procedures.

When the fitting results are substituted into Equation (7), the following can be derived:

$$\begin{array}{l}F=33.681-2.980\times {10}^{-3}n+2.738\times {10}^{-2}{v}_{\mathrm{f}}-0.873a_{\mathrm{p}}-4.217\times {10}^{-7}n{v}_{\mathrm{f}}+5.736\times {10}^{-4}n{a}_{\mathrm{p}}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+1.917\times {10}^{-4}{v}_{\mathrm{f}}{a}_{\mathrm{p}}+5.481\times {10}^{-8}{n}^2-4.5\times {10}^{-6}{v}_{\mathrm{f}}^2-0.307a_{\mathrm{p}}^2\end{array}$$

(9)

Figure 8a–c illustrate the relationship between milling force, F, and the spindle speed, n, feed rate, v_f, and axial milling depth, a_p. As observed in Figure 8a,b, the milling force exhibits an increase with rising feed rate and axial milling depth. This phenomenon can be attributed to the fact that an increase in feed rate results in a greater machining travel distance and volume per unit time. Consequently, this not only heightens friction during the milling process but also leads to a larger quantity of material being removed, necessitating a higher force to overcome these resistances. Furthermore, as the axial milling depth increases, the tool penetrates deeper into the workpiece, resulting in elevated shearing stresses throughout the milling operation. Therefore, when both parameters are simultaneously increased, there is a corresponding rise in the force required to be applied to the tool. Figure 8c indicates that there exists a negative correlation between milling force, F, and spindle speed, n. When maintaining constant axial milling depth and feed rate conditions, an increase in spindle speed leads to more rotations per minute; however, it concurrently reduces the amount of material processed per rotation. The relatively lower feed of the cutter implies that contact time with the workpiece surface is diminished overall, thereby decreasing contact stress levels. Additionally, high-speed rotation may induce heat accumulation which could further influence material properties while reducing the necessary force exerted on the milling tool.

3.3. Results and Analysis of Multi-Objective Optimization

In material processing, raising the feed rate can significantly improve milling efficiency; however, it also leads to an increase in the milling force applied to the tool, which in turn accelerates wear. As a result, there is a certain level of conflict among material removal rate, milling force, and tool wear. To effectively tackle this challenge, it is essential to find a balance between these three elements using multi-objective optimization methods.

The prediction model of the wear width of the rake face and the milling force are as follows:

$$\left\{\begin{array}{l}b=6.68\times {10}^{-8}{n}^{1.043}{v}_{\mathrm{f}}^{0.505}{a}_{\mathrm{p}}^{0.152}\\ F=33.681-2.980\times {10}^{-3}n+2.738\times {10}^{-2}{v}_{\mathrm{f}}-0.873a_{\mathrm{p}}-4.217\times {10}^{-7}n{v}_{\mathrm{f}}+5.736\times {10}^{-4}n{a}_{\mathrm{p}}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+1.917\times {10}^{-4}{v}_{\mathrm{f}}{a}_{\mathrm{p}}+5.481\times {10}^{-8}{n}^2-4.5\times {10}^{-6}{v}_{\mathrm{f}}^2-0.307a_{\mathrm{p}}^2\end{array}\right.$$

(10)

Material removal rate, Q, can be represented as [27]:

$$Q={\displaystyle \frac{1}{1000}}{v}_{\mathrm{f}}\cdot a_{\mathrm{p}}\cdot a_e$$

(11)

The following outlines the objective function intended for optimization:

$$\left\{\begin{array}{l}\mathrm{miny}\left(1\right)=6.68\times {10}^{-8}{\mathrm{x}}_1^{1.043}\cdot {\mathrm{x}}_2^{0.505}\cdot {\mathrm{x}}_3^{0.152}\\ \mathrm{miny}\left(2\right)=33.681-2.980\times {10}^{-3}{\mathrm{x}}_1+2.738\times {10}^{-2}{\mathrm{x}}_2-0.873{\mathrm{x}}_3-4.217\times {10}^{-7}{\mathrm{x}}_1{\mathrm{x}}_2+5.736\times {10}^{-4}{\mathrm{x}}_1{\mathrm{x}}_3\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+1.917\times {10}^{-4}{\mathrm{x}}_2{\mathrm{x}}_3+5.481\times {10}^{-8}{\mathrm{x}}_1^2-4.5\times {10}^{-6}{\mathrm{x}}_2^2-0.307{\mathrm{x}}_3^2\\ \mathrm{miny}(3)=-{\displaystyle \frac{1}{1000}}\cdot {\mathrm{x}}_2\cdot {\mathrm{x}}_3\end{array}\right.$$

(12)

The constraints are as follows:

$$\left\{\begin{array}{l}12000\le {\mathrm{x}}_1\le 18000\\ 1000\le {\mathrm{x}}_2\le 2000\\ 3\le {\mathrm{x}}_3\le 5\end{array}\right.$$

(13)

where y(1), y(2), and y(3) denote the wear width, b, of the rake face, the milling force, F, and the material removal rate, −Q, respectively. The reason why y(3) equals the material removal rate, Q, multiplied by −1 is that, during milling, our goal is to maximize the material removal rate. However, the genetic algorithm tool, gamultiobj, is designed for solving multi-objective minimization problems. By multiplying the material removal rate by −1, we can convert the maximization problem into a minimization problem, which can be effectively optimized using the genetic algorithm.

Similarly, x₁, x₂, and x₃ represent spindle speed, n, feed rate, v_f, and axial milling depth, a_p. The radial milling depth, a_e, is set to 1 mm. This is because an excessively large a_e can compromise the surface quality of the workpiece, while an overly small a_e may reduce machining efficiency. Consequently, following the factory’s recommended value, a_e is ultimately set to 1 mm.

The multi-objective genetic algorithm NSGA-II was adopted in MATLAB to optimize the objective functions. The Pareto fractions were set at 0.3, the population size was 200, the number of generations was 300, the number of stall generations was 200, and the function tolerance was established at 1 × 10⁻¹⁰. When the algorithm was executed, the variations in the Pareto front related to the wear width, b, of the rake face, the milling force, F, and the material removal rate, Q, were acquired as shown in Figure 9. The figure reveals that there exists a striped scattering distribution among the milling force, the wear width, and the material removal rate. As the milling force grows, both the tool wear and the material removal rate increase correspondingly.

Table 12. The optimum solution for a portion of the Pareto front.

No	n/(r/min)	vf/(mm/min)	ap/(mm)	b/(mm)	F/(N)	Q/(cm3/min)
1	13,750.264	1999.996	5.000	0.082	57.563	10.000
2	13,461.229	1999.777	4.976	0.080	57.305	9.950
3	12,000.000	1000.000	3.000	0.046	39.484	3.000
4	12,436.256	1006.978	3.099	0.049	39.883	3.120
5	13,191.062	1770.687	3.718	0.071	50.352	6.583
6	13,316.196	1709.255	4.132	0.071	51.856	7.063

demonstrates several optimal solutions along the Pareto front. To guarantee processing efficiency, it is suggested to select either Group 1 or Group 2 milling parameters from the table; on the contrary, for minimizing tool wear and extending its service life, the parameters of Groups 3 and 4 are recommended. Moreover, to reach a balance among the milling force, the reduction in tool wear, and processing efficiency, considerations should be given to the milling parameters of Groups 5 and 6.

Select parameters of Group 1, Group 3, and Group 5 from Table 12 for milling validation, and the validation results are presented in

Table 13. The comparison between measured values and predicted values.

No	b/(mm)			F/(N)			Q/ (cm3/min)
	Predicted Values	Measured Values	Error %	Predicted Values	Measured Values	Error %
1	0.082	0.078	5.13	57.563	55.49	3.74	10.000
3	0.046	0.045	2.22	39.484	41.46	1.31	3.000
5	0.071	0.068	4.41	50.352	46.83	7.52	6.583

Note: Items No. 1, 3, and 5 in Table 13 correspond, respectively, to items No. 1, 3 and 5 in Table 12.

. It can be observed from the data in the table that, compared with the predicted values, the error in the wear width of the cutting tool’s front face is controlled within 5% during actual machining, and the error range of the milling force is approximately 7.5%. The predicted model has achieved the expected goal.

4. Discussion

In this paper, the principal research parameters are the spindle speed, n, the feed rate, v_f, and the axial milling depth, a_p. These parameters are crucial process variables in CNC milling and directly exert an influence on tool wear and machining quality. Specifically, the spindle speed, n, determines the frequency of interaction between the tool and the workpiece, while the feed rate, v_f, is associated with the material removal rate and surface roughness. Additionally, the axial milling depth, a_p, has an impact on the distribution of cutting forces and the loading condition of the tool. Thus, appropriately setting these parameters holds significant value for tool machining.

Through orthogonal experiments, representative level combinations were chosen, and the patterns of the wear of the rake face and milling force alterations under diverse parameter combinations were disclosed through in-depth data analysis. The primary and secondary factors influencing the wear of the rake face and milling force were identified by employing range analysis. The findings indicate that the influence of milling parameters on the wear width follows this order: spindle speed n > feed speed v_f > axial milling depth a_p. In terms of milling force, the influence of the parameters is ranked as follows: axial milling depth a_p > feed speed v_f > spindle speed n. A model for tool wear and a multi-quadratic regression model for milling force were put forward to predict the variations in tool wear and milling force. The fitting results demonstrate that the coefficient of determination R² for the tool wear model and the milling force model are 0.989 and 0.904, respectively, indicating that these models exhibit high predictive accuracy and reliability.

In response to the conflicting problems regarding milling force, tool wear, and milling efficiency in mechanical processing, a multi-objective optimization strategy based on genetic algorithms was put forward, effectively resolving the contradiction. Considering the variations in emphasis during actual processing, specific milling parameter combinations were presented. The results indicate that, depending on the processing priorities, the following milling parameter combinations are recommended: To prioritize milling efficiency, use n = 13,750.264 r/min, v_f = 1999.996 mm/min, and a_p = 5 mm; to minimize tool wear, use n = 12,000 r/min, v_f = 1000 mm/min, and a_p = 3 mm; to balance efficiency and wear, use n = 13,191.062 r/min, v_f = 1770.6876 mm/min, and a_p = 3.718 mm.

In the future, we need to persistently pay attention to the potential influence of emerging technologies, such as big data and artificial intelligence, on multi-objective optimization and contemplate how to address the new issues brought about by technological transformation. Additionally, we should explore effective strategies and methods for multi-objective optimization under circumstances with uncertain factors such as parameter fluctuations and environmental changes.

5. Conclusions

A model for tool wear and a multi-quadratic regression model for milling force were developed, enabling effective predictions of variations in both tool wear and milling force. Employing the genetic algorithm within MATLAB, multi-objective optimization was conducted on the wear width of rake face, milling force, and material removal rate. Based on real machining conditions, a set of Pareto front optimal solutions was generated for milling parameters under varying machining requirements, thereby offering valuable insights for practical machining applications.