A Generic Multi-Objective Optimization of Machining Processes Using an End-to-End Evolutionary Algorithm

Abstract

Machining processes have been widely employed in the modern manufacturing industry to transform raw materials into final products, and they are of great importance in improving the environmental impact and production efficiency of this industry. The selection of appropriate machining process parameters can effectively improve the environmental impact and production efficiency of a process. However, most existing studies on the optimization of these parameters have targeted optimization techniques or modeling methods, and have seldom taken into consideration the adaptability of the machining process. Thus, they suffer from poor generalization and flexibility in actual deployment. Based on this, a generic optimization framework based on the end-to-end evolutionary algorithm was proposed in this study, which can be adapted to various machining optimization problems, to guide the operators in selecting the best parameters in an automated way. Firstly, a modeling framework was introduced to guide the operators to develop optimization objectives. Subsequently, a flexible optimization algorithm was employed to generate Pareto front solutions. Finally, the CRITIC-TOPSIS method was employed to provide a final solution from the different Pareto solutions generated. Experiments were conducted on a milling machine to demonstrate the effectiveness and advantages of the proposed method. The results showed that the proposed method is flexible and applicable for the optimization of the different machining steps and objectives.

1. Introduction

The manufacturing industry plays an indispensable role in modern society and has been widely involved in the manufacture of products that meet modern society’s requirements [1]. The machining process is regarded as the most important step in the manufacturing industry and is used to convert raw materials into the desired products [2,3]. Currently, the manufacturing industry is aiming for lower lead times, lower energy consumption, and higher quality owing to the increasing demand for environmental sustainability, competitiveness, and high-end products [4,5]. In general, the quality, energy consumption, and time of machining are directly affected by the process parameters. Hence, it is important to optimize these parameters [6]. However, these goals often conflict with each other, and the machining process is complicated and involves a wide range of chemical and physical reactions. Therefore, extensive research on the multi-objective optimization of machining processes has been conducted and applied in recent years [7,8,9].

As mentioned in the previous paragraph, many researchers have focused on determining suitable process parameters that can balance different objectives [10,11]. In general, existing studies on multi-objective optimization can be broadly classified into two categories: experiment-based and model-based optimization methods [12,13,14]. In the case of experiment-based optimization methods, mathematical relationships are first developed using various experimental techniques. Subsequently, a parameter selection method is employed to determine the most suitable parameters [15,16,17]. Kumar et al. proposed using the Taguchi method and gray relational analysis (GRA) to optimize the surface roughness and delamination of CNC (computer numerical control) drilling parameters for machining HCHCR steel [18]. Xiao et al. proposed the integration of the Taguchi method, the response surface method (RSM), and the modified multi-objective particle swarm optimization (MOPSO) algorithm to minimize energy and time [19]. In their research, the Taguchi method was used to design the experiment, and RSM was employed to develop regression models for the responses based on the experimental data. Akhtar et al. optimized the removal rate of the material (maximum), the roughness of the machined surface (minimum), and cutting force (smallest amount) using the L27 array of the Taguchi method [20]. By using these experimental-based optimization techniques, which have a high generalizability and applicability, the parameters for different machining processes can be readily obtained.

In model-based methods, the relationships between the process parameters and output targets are mostly determined using various physical mathematical formulas or multiple different deep learning algorithms [21,22,23]. Shin et al. proposed a component-based energy modeling methodology to optimize the machining process. In their study, component models that can predict energy on the tool-path level for specific machining configurations were developed [24]. Wang et al. developed a dual-objective optimization model for the selection of milling parameters and minimized the power consumption and processing time [25]. Wu et al. also proposed the use of a deep learning-based data-driven genetic algorithm for the multi-objective optimization of machining process parameters and to determine final solutions. In that study, deep learning was employed to develop data-driven surrogate functions, which were then employed as the optimized objective functions [26]. Cao et al. also proposed a multi-objective decision-making approach for the process parameters of high-speed dry hobbing based on K-means clustering, the multi-objective Runge Kutta optimizer (MORUN), and the analytic hierarchy process (AHP). Through the use of modeling-based optimization techniques, the machining process parameters could be selected prior to the machining process, and the cost and time of machining could be minimized [27].

Accordingly, both model- and experiment-based studies on the optimization of the CNC machining process have achieved huge success. However, most of the existing studies only focused on a specialized optimization process for use within a specific optimization framework and did not consider the dynamic characteristics of machining processes brought about by the manufacturing operations. Accordingly, previously developed optimization models often suffer from poor generalization, which hinders their application in the actual manufacturing industry. In general, problems exist in the following three key aspects.

(1)

During the objective modeling phase of optimization, conventional studies are often conducted under the assumption that modeling methods aimed at different objectives are the same. However, the characteristics of the different objectives vary greatly, resulting in variations in their representation. The lack of guidance in the construction of an objective model system for optimization leads to the poor adaptability of optimization systems to different types of machining processes. Accordingly, to help improve the generalization and flexibility of multi-objective optimization methods, a novel modeling framework that can be readily adapted to different machining optimization objectives is required.

(2)

During the multi-objective optimization stage of the machining process, conventional studies often use fixed optimized dimensions, and it is common practice to establish objective functions and constraints in a stationary state, which cannot be modified during the optimization process. In general, the machining steps and requirements are often changed during daily manufacturing operations, resulting in the state of the objective functions and constraints being dynamic. As a consequence, conventional single-step optimization methods are ineffective for different machining optimization tasks. While it is possible to develop a specialized optimization framework for each machining task, such an approach can be both costly and time-consuming.

(3)

Conventional studies have mostly conducted the multi-objective decision-making process in a subjective and specialized way to determine Pareto front solutions. This is effective when handling low-dimensional Pareto front searches. Unfortunately, in a multi-step machining process, the number of optimized dimensions is high. The conventional way of manually assigning weights and finding a final solution is prone to error, and it is hard for the operators to find the best approximate process parameters. On the basis of this, a novel method that can determine the final parameters for the operators in an objective and automated way is required.

The purpose of this research was to address a crucial gap in the field of optimization techniques by introducing a novel approach that can facilitate the optimization of different machining processes. To achieve this goal, a generic optimization framework based on the end-to-end evolutionary algorithm is presented in this paper, which covers the most common main stage during the optimization of the machining process. This framework allows for dynamic changes to the objective functions and constraints, which is a significant improvement compared with traditional optimization methods that are rigid and inflexible. At first, an objective modeling method was introduced, which covered a physical model, a data-driven model, and a physics-informed machine learning model. On the basis of this, a unified surrogate model was introduced to transfer the developed models into objective functions. In the second step of the method, the problem and constraints were outlined in detail. In this part, the researchers accounted for the potential impact of the different optimization processes. By defining these conditions, an optimization technique was introduced to generate Pareto front solutions. Following this, the intercriteria correlation technique for order preference by similarity to the ideal solution (CRITIC-TOPSIS) was employed to find the final parameters from the high-dimensional Pareto front solutions generated in the previous step.

The contributions of this study are twofold: First, an end-to-end multi optimization framework is introduced, which can help manufacturers to find the best parameters for optimization of the whole machining process with less involvement from experts. Second, a flexible and generic optimization method that can handle most machining process optimization problems is introduced in this study, which can greatly help operators optimize different machining processes.

Overall, this study makes an essential contribution to the field of optimization techniques for machining processes, and it is expected to have a significant impact on the manufacturing industry. The main contributions of this study can be summarized as follows.

(1)

A new approach is suggested for optimizing different CNC processes. This method is better equipped to handle the dynamic characteristics of the process and is therefore more appropriate for optimizing parameters in industrial settings compared with traditional methods.

(2)

A flexible optimization framework is proposed. Compared with conventional optimization studies, the proposed method can not only handle the variations in single-step machining optimization processes, but also those in the multi-step optimization process that was considered in this study.

(3)

A more objective method of determining parameters is proposed. Compared with the conventional method, the proposed method can be used to find the best approximate parameters not only from low-dimensional, but also high-dimensional Pareto front solutions in an automated way.

The rest of the article is organized as follows. In Section 2, the optimization problem is given. In particular, both the local and global optimization problems are given in this part. Next, the proposed optimization method is described in Section 3. Subsequently, the case study is given in Section 4. Then, the results and discussion are given in Section 5. Finally, conclusions are drawn in Section 6.

2. Optimization Problem Analysis

As mentioned above, manufacturing industries are pursuing high quality, high production rates, and a low energy consumption. Massive multi-step CNC machining processes are employed in the manufacturing industry [28]. As reported by many researchers, the factors mostly frequently changed in a factory are the cutting step and the machining requirements. In terms of the optimized parameters, a CNC machining process should be economically feasible and environmentally friendly. Simultaneously, the machining process should yield high-quality products [29,30]. In terms of the optimization process, it should be able to adapt to different cutting steps and machining requirements.

2.1. Analysis of the Optimization of the Parameters of a Machining Process

The number of machining steps is designated $i$ in this study, and the optimized targets are defined as $t$. As a consequence, the optimization problem could be defined as multi-objective optimization problem with $t$ objectives and the i-th machining step.

$$\{\begin{array}{}\mathrm{(1)}& \mathit{min}\left(t_1,t_2\right)\cup \mathit{max}\left(t_3,..,t_z\right),\left[w_1,w_2,w_3\right],\mathrm{1,2},3..,i-2thmachiningstep\mathrm{(2)}& \mathit{min}\left(t_1,t_2\right)\cup \mathit{max}\left(t_3,..,t_z\right),\left[w_4,w_5,w_6\right],i,-1th,i-tthmachiningstep\mathrm{(3)}& \mathit{min}\left(t_1,t_2\right)\cup \mathit{max}\left(t_3,..,t_z\right),\left[w_{3i-2},w_{3i-1},w_{3i}\right],i-thmachiningstep\end{array}$$

where $w_i$ denotes the weights assigned to different objectives in different machining steps. In order to better accommodate the multi-step machining process, an additional machining process optimization step was defined as follows:

$$\min \left(t_1\right)\cup \max \left(t_z\right),\left[w_{i1},w_{i2}\right],machiningprocess$$

The constraints are defined as follows.

$$a_{p_{min}}\le a_p\le a_{p_{max}},$$

(4)

$${a_e}_{min}\le a_e\le {a_e}_{max},$$

(5)

$$f_{min}\le f\le f_{max},$$

(6)

$$r_{min}\le r\le r_{max}.$$

(7)

In the $i$-th step of the machining process, the constraints refer to the first, second, …, and $i-1th$ machining steps, which are mostly focused on the process parameters. On the basis of the limitations of the machining tools, cutting tools, and safety, the constraints of the parameters can be set as described above.

For the $i-th$ machining step, additional constraints should be considered as follows:

$$t_{z+1}\le t_{cons}$$

(8)

On the basis of the abovementioned definitions, the number of objectives was found to be $3\times i+2$. These objectives included $3\times i$ local optimization objectives and two global objectives. After defining the objectives and constraints of optimization, in light of the analysis above, both global and local optimization problems for the different machining processes were considered during the optimization. In other words, by defining the optimized conditions and constraints, the proposed method can be adapted to different types of machining processes.

2.2. Optimization Problem in Machining Process

As mentioned above, the optimized method should adapt to different cutting steps and different machining requirements. The optimized results of the machining process should be economically feasible and environmentally friendly. Simultaneously, the machining process should yield a high-quality product. On the basis of this, this study proposed two case studies to demonstrate the effectiveness and generalizability of the proposed method [31]. In the first case, the optimization problem was defined as a multi-objective optimization problem with three conflicting objectives: maximum production rate, minimum energy consumption, and maximum cutting force. The number of machining steps was defined as four. In the second case, the optimization of a single machining step was defined as bi-objective optimization with two conflicting objectives: production rate optimization and energy consumption optimization. The maximum cutting force and surface roughness were used as the constraints. Accordingly, the optimized objectives and constraints could be defined. In the first case, the optimization problem was defined as follows:

$$\{\begin{array}{}\mathrm{(9)}& \mathit{min}\left(E_c,C_f\right)\cup \mathit{max}\left(MRR\right),\left[w_1,w_2,w_3\right],\mathrm{1,2}thmachiningstep\mathrm{(10)}& \mathit{min}\left(E_c,C_f\right)\cup \mathit{max}\left(MRR\right),\left[w_4,w_5,w_6\right],3machiningstep\mathrm{(11)}& \mathit{min}\left(E_c,C_f\right)\cup \mathit{max}\left(MRR\right),\left[w_{3i-2},w_{3i-1},w_{3i}\right],4-thmachiningstep\end{array}$$

subject to

$$\{\begin{array}{}\mathrm{(12)}& & a_{p_{min}}\le a_p\le a_{p_{max}},\mathrm{(13)}& & {a_e}_{min}\le a_e\le {a_e}_{max},\mathrm{(14)}& & f_{min}\le f\le f_{max},\mathrm{(15)}& & r_{min}\le r\le r_{max}.\end{array}$$

$$R_i\le R_{max}.$$

(16)

where $R_i$ denotes the surface roughness in the $i-th$ machining step, and MRR denotes the material removal rate. To provide a more detailed description of multi-step optimization, the differences between single-step and multi-step optimization are presented in this section. We assumed that each machining step consists of three typical stages: initial, middle, and final. In the initial stage, such as in the rough milling process, the machining process focuses more on the material removal rate and energy consumption. For optimization in the middle stage, the machining process focuses more on the cutting force and energy consumption. In the final cutting stage, the machining process emphasizes the machined quality.

In terms of the second case, the optimization was defined as follows:

$$\mathit{min}\left(E_c\right)\cup \mathit{max}\left(MRR\right)$$

subject to

$$\{\begin{array}{}\mathrm{(17)}& & a_{p_{min}}\le a_p\le a_{p_{max}},\mathrm{(18)}& & {a_e}_{min}\le a_e\le {a_e}_{max},\mathrm{(19)}& & f_{min}\le f\le f_{max},\mathrm{(20)}& & r_{min}\le r\le r_{max}.\end{array}$$

$$R\le R_{max}$$

(21)

$$C_f<{C_f}_{max}$$

(22)

3. Proposed Optimization Method

3.1. Overall Framework

The overall framework of the end-to-end evolutionary algorithm method is shown in Figure 1. The figure shows that the proposed method comprises three parts: modeling, multi-objective decision making (MODM), and multi-attribute decision making (MADM). In the modeling stage, the physical model, the data-driven model using machine learning, the physics-informed machine learning model, and empirical functions were employed as the basis to represent the target variation. The surrogate model was followed with the these models to transfer the predictive functions into optimized objectives. For MODM, the non-dominated sorting genetic algorithm II (NSGA-II) was used. By using NSGA-II, the sets of parameters and Pareto fronts for each machining step could be obtained. Then, the criteria importance, obtained through the intercriteria correlation technique for order preference according to the similarity to the ideal solution (CRITIC-TOPSIS), was used in this study to provide a final solution for each target. In particular, the effects of local and global parameters were considered. Thus, the identified parameters could be fitted for the entire machining process. In this way, an end-to-end evolutionary algorithm was developed.

3.2. Objective Function Modeling Using the Proposed Method

As mentioned above, the optimization problem was conducted on the basis of the cutting force, surface roughness, energy consumption, and material removal rate. Thus, in order to carry out the optimization process, a predictive function for these objectives was developed. To demonstrate the generalization of the proposed method, the modeling method toward each objective is presented in this article.

3.2.1. Data-Driven Model of Energy Consumption

Considering that the energy consumption can be readily obtained using a power meter, it is complex to develop a highly accurate physical prediction model. Thus, a data-driven prediction model for the energy consumption of the machining process was employed to represent the energy consumption behavior using machine learning. In the data-driven energy consumption model, the data, namely, the input data, the parameters of the machining process, and the corresponding output data on energy consumption, had to be collected first. Then, machine learning algorithms were used to determine the relationship between the energy consumption and process parameters. Multiple machine learning algorithms exist. Considering the non-linear complexity of predicting energy consumption, the convolutional neural network (CNN) was used in this study to represent the energy consumption. The details of this machine learning strategy have been reported in many studies, so only a simplified introduction is given in this study [32,33].

The detailed structure of the CNN is presented in Figure 2. It can be seen from the figure that the CNN is composed of mostly convolutional and pooling layers. Convolutional and pooling operations were employed in this study to extract the features of energy consumption, and then the extracted features were transferred to the fully connected layers to realize the prediction process.

3.2.2. Physical Modeling of the Cutting Force

A flowchart of the cutting force modeling procedure is introduced in this part, based on earlier studies [34,35,36]. In this study, flat-end milling with N cutters was selected. Thus, the modeling procedure was developed on the basis of this. As shown in the flowchart in Figure 3, the main procedure related to the cutting force was to calculate the micro-element cutting force, which could be obtained by using the following physical formula.

$$\{\begin{array}{}\mathrm{(23)}& {dF}_{t,j}\left({\mathrm{\varnothing }}_j\left(z\right)\right)=K_{tc}{h}_j\left({\mathrm{\varnothing }}_j\left(z\right)\right)·dz+K_{te}·dz\mathrm{(24)}& {dF}_{r,j}\left({\mathrm{\varnothing }}_j\left(z\right)\right)=K_{rc}{h}_j\left({\mathrm{\varnothing }}_j\left(z\right)\right)·dz+K_{re}·dz\mathrm{(25)}& {dF}_{a,j}\left({\mathrm{\varnothing }}_j\left(z\right)\right)=K_{ac}{h}_j\left({\mathrm{\varnothing }}_j\left(z\right)\right)·dz+K_{ae}·dz\end{array}$$

where $K_{tc}$, $K_{rc}$, and $K_{ac}$ denote the shear force coefficient in the tangential, radial, and axial directions, respectively; $K_{te}$, $K_{re}$, and $K_{ae}$ denote the plowing force in the tangential, radial, and axial directions, respectively; $h_j\left({\mathrm{\varnothing }}_j\left(z\right)\right)$ is the instantaneous cutting thickness; and $dz$ denotes the axial cutting depth of infinitesimal elements.

3.2.3. Physics-Informed Machine Learning for Surface Roughness

A physics-guided deep learning modeling method was employed in this study to develop objective functions for surface roughness and energy consumption. The physics-informed machine learning model was developed for data-insufficient and complex modeling targets. The process of physics-guided deep learning modeling is composed of two stages: physical modeling and physics-guided training.

Using this modeling method, physical knowledge of the variation in surface roughness and energy consumption can be integrated with dynamic data-driven predictions even with limited data. The details of the proposed method can be found in previously published work [37], and a simplified flowchart of this modeling process is provided in Figure 4. Considering that this neural network has been reported in numerous studies, it is not described in this paper [38,39,40]. By using this physics-guided deep learning prediction model, machining process optimization can be realized quickly and effectively.

3.2.4. Empirical Functions for the Material Removal Rate

In general, empirical functions are mostly developed for targets that have few requirements regarding the machining process. It would be a waste of resources and time to develop a highly accurate prediction model using physical modeling or data-driven modeling. Considering that milling is the most widely used operation in modern manufacturing, end milling was employed as a case study. Accordingly, the material removal rate (MRR) can be represented using a physical formula, as follows:

$$MRR={\displaystyle \frac{a_p\times a_e\times f\times N\times r}{60\times 1000}}={\displaystyle \frac{a_p\times a_e\times f\times N\times {\nu }_c}{60\times \pi \times d_0}},$$

(26)

where $f$, $a_p$, $a_e$, and ${\nu }_c$ denote the process parameters, i.e., the feed rate, depth of cutting, width of cutting, and cutting speed; $N$ is the number of cutters; and $d_0$ is the tool’s diameter.

3.2.5. Surrogate Model for the Objectives

The surrogate model is a computational method for complex engineering problems. It is mainly used in computational tasks that have a large amount of actual calculations and are not easy to solve. At present, the commonly used method of constructing a surrogate model is mainly to combine a set of decision variables with real output values and to replace the original input and output response method by constructing a high-precision approximate mapping relationship that can quickly respond to the current input and output. Therefore, the surrogate model can be considered as a function that can quickly approximate a mapping relationship, and this function can be used to replace the original calculation process.

The surrogate model is mostly employed for conducting computationally efficient calculations for the objective functions. Considering that there are multiple different modeling methods, the surrogate model was employed to unify the predictive functions. In this study, a similarity fitting method was used to develop the surrogate model, and approximate functions using the predictive model were used to develop the optimized functions.

3.3. MODM Using NSGA-II

NSGA-II was employed in this study to realize multi-objective decision making. NSGA-II is a multi-objective evolutionary algorithm that was first proposed by Deb in 2002 and is based on a fast, non-dominated sorting principle [41]. In recent years, many researchers have applied NSGA-II for decision making in various areas, and it shows great potential. The steps of the NSGA-II algorithm are presented in the flowchart in Figure 5. The pseudo code for the NSGA-II is given in Algorithm 1.

Algorithm 1 Document clustering using NSGA-II

Begin t 0 CN RANDOM (K) C {CN1,CN2,…CNj} Initialize population P(t) While (not termination condition) do  $P^{\prime }\left(t\right)$ compute objective functions P(t)  $F^{\prime }\left(l\right)$ $\mathrm{fast}\mathrm{nondominated}\mathrm{sort}{P}^{\prime }\left(t\right)$  $A^{\prime }\left(l\right)$ $\mathrm{fitness}\mathrm{assignment}{F}^{\prime }\left(t\right)$  $\mathrm{Offspring}\mathrm{Q}\left(\mathrm{t}\right)$ $\mathrm{evolution}\mathrm{operate}{A}^{\prime }\left(l\right)$  R(t) P(t)+ Q(t)  $R^{\prime }\left(t\right)$ fast nondominated sort R(t)  $P(t+1)$ fitness assignment  t t+1  end end

3.4. Weight Determination Using CRITIC

As mentioned above, a Pareto front solution was generated using NSGA-II. To obtain the final solution for the machining process, it was necessary to assign a weight to each objective. Generally, weight determination methods can be classified into two categories: subjective and objective. There were eleven weights to be determined. Because it is difficult for operators to provide subjective preferences in terms of the importance of weights $\mathrm{M}\mathrm{R}\mathrm{R},{\mathrm{E}}_{\mathrm{c}},{\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{C}}_{\mathrm{f}}$, an objective weight determination method was employed in this study. The CRITIC method was proposed by Diakoulaki et al. in 1995, and has recently been widely employed for weight determination [42,43]. Within the CRITIC method, standard deviation is used to calculate the discreteness of alternative criteria, and a correlation coefficient is used to represent conflicts between criteria.

The calculation procedure of the CRITIC method is as follows:

$$w_j={\displaystyle \frac{{\sigma }_j{\sum }_{i=1}^n\left(1-r_{ij}\right)}{{\sum }_{j=1}^m\left({\sigma }_j{\sum }_{i=1}^n\left(1-r_{ij}\right)\right)}}$$

(27)

$${\sigma }_j=\sqrt{{\sum }_{i=1}^m{\displaystyle \frac{(a_{ij}-\overline{a_j}{)}^2}{m-1}}}$$

(28)

$$r_{ij}={\displaystyle \frac{{\sum }_{x=1}^m\left(a_{ix}-\overline{a_i}\right)\left(a_{jx}-\overline{a_j}\right)}{\sqrt{{\sum }_{x=1}^m(a_{ix}-\overline{a_i}{)}^2}\sqrt{{\sum }_{x=1}^m(a_{jx}-\overline{a_j}{)}^2}}}$$

(29)

where $w_j$ denotes the weight for the $j-th$ objectives, ${\sigma }_j$ is the standard deviation of the $j-th$ objectives, and $r_{ij}$ is the correlation coefficient between objective $i$ and $j$. In particular, the $MRR,E_c$, and $C_f$ in each step are regarded as different objectives. This way, the local optimization characteristics of the target could be obtained.

3.5. MADM Using TOPSIS

After determining the weights for each objective, TOPSIS was employed to find the final solutions for the operators. TOPSIS is a ranking method that was first proposed by Hwang and Yoon in 1981, and it has been improved by many researchers in recent years [44]. The principle of TOPSIS is to sort all of the possible solutions by measuring the distance from the evaluated object to the ideal solution and the distance to the negative ideal solution. By sorting the distance to the ideal solution and from the negative ideal solution, the best object can be found.

The general procedure of the TOPSIS is shown in Figure 6, and it can be seen from the figure that weights are required in the decision-making process. As mentioned earlier, these weights were determined using the CRITIC method. On the basis of this, the weighted matrix was as follows:

$$V_{ij}=\left[\begin{array}{c}{v}_{11}{v}_{12}\cdots v_{1n}\\ v_{21}{v}_{22}\cdots v_{2n}\\ ⋮⋮⋮⋮\\ v_{m1}{v}_{m2}\cdots v_{mn}\end{array}\right]=\left[\begin{array}{c}{a}_{11}·w_1{a}_{12}·w_2\cdots a_{1n}·w_n\\ a_{21}·w_1{a}_{22}·w_2\cdots a_{2n}·w_n\\ ⋮⋮⋮⋮\\ a_{m1}·w_1{a}_{m2}·w_2\cdots a_{mn}·w_n\end{array}\right]$$

(30)

The basic steps of the TOPSIS are as follows.

Step 1: Determine the ideal solution $z_j^{+}$ and the negative ideal solutions.

$$z_j^{+}=\{\begin{array}{c}{\max }_{1⩽i⩽m}\left({\nu }_{ij}\right),if{f}_j\in F_1\\ {\min }_{1⩽i⩽m}\left({\nu }_{ij}\right),if{f}_j\in F_2\end{array}$$

(31)

where $F_1$ is the set of benefit objectives, $F_2$ is the set of cost objectives, and $1⩽j⩽n$.

Step 2: Determine the distance of each possible solution, i.e., the Pareto front solutions, from the ideal solution and the negative ideal solutions.

$$S_i^{+}={\sum }_{j=1}^n({\nu }_{ij}-z_j^{+}{)}^{2}i=\mathrm{1,2},\cdots ,m$$

(32)

$$S_i^{-}={\sum }_{j=1}^n({\nu }_{ij}-z_j^{-}{)}^{2}i=\mathrm{1,2},\cdots ,m$$

(33)

Step 3: Determine the relative closeness to the ideal solution.

$$C_j^{+}={\displaystyle \frac{s_j^{+}}{s_j^{+}+s_j^{-}}}$$

(34)

On the basis of this, all possible solutions can be sorted according to the $C_j^{+}$ value; the larger the $C_j^{+}$ value, the better the solution. To make the proposed method more transparent, the pseudo-code of TOPSIS is presented in Algorithm 2.

Algorithm 2 Steps of the TOPSIS method in pseudo-code format.
1.	Construction of the normalized decision matrix
	$r_{ij}={\displaystyle \frac{x_{ij}}{\sqrt{{\sum }_{i=1}^m{x}_{ij}^2}}}\forall j$ where $r_{ij}$ denotes the elements of normalized decision matrix
2.	Construction of the weighted normalized decision matrix
	$v_{ij}=r_{ij}\ast {\omega }_j\forall i,j$
	$\mathrm{where}{\omega }_j$$\mathrm{denotes}\mathrm{the}\mathrm{assigned}\mathrm{weight}\mathrm{attributed}\mathrm{to}j$
3.	$\mathrm{Determination}\mathrm{of}\mathrm{the}\mathrm{ideal}({\mathcal{A}}^{+}$$\left)\mathrm{and}\mathrm{negative}\mathrm{ideal}\right({\mathcal{A}}^{-}$) solutions
	${\mathcal{A}}^{+}=\left\{\left(\begin{array}{c}max{v}_{ij}\\ j\end{array}|i\in I\right),\left(\begin{array}{c}min{v}_{ij}\\ j\end{array}|i\in I^{\prime }\right);\forall j\right\}=\left\{v_1^{+},v_2^{+},\dots \right\}$ ${\mathcal{A}}^{-}=\left\{\left(\begin{array}{c}min{v}_{ij}\\ j\end{array}|i\in I\right),\left(\begin{array}{c}max{v}_{ij}\\ j\end{array}|i\in I^{\prime }\right);\forall j\right\}=\left\{v_1^{-},v_2^{-},\dots \right\}$
	where $I$ and $I^{\prime }$ are associated with the benefit and cost attributes, respectively.
4.	Calculation of the separation measure
	$S_i^{+}=\sqrt{\sum _{i=1}^n{(v_{ij}-v^{+})}^2}\forall j$ $S_i^{-}=\sqrt{\sum _{i=1}^n{(v_{ij}-v^{-})}^2}\forall j$
5.	Calculation of the relative closeness to the ideal solution
	$C_j^{+}={\displaystyle \frac{S_j^{+}}{S_j^{+}+S_j^{-}}}$
	Ranking of alternatives according to the value of $C_j^{+}$

4. Case Study

In this study, two experiments were conducted on an actual milling machine to verify the effectiveness and robustness of the proposed method. In the first experiment, four machining steps were conducted, with three conflicting objectives. The second experiment involved single-step experiments, with two conflicting objectives. The programming process was based on Python, and pytorch was employed in this study to model the process. The experimental research was conducted on four Nvidia GTX 4090 graphics cards on a Windows server. The details of the experiments are described below.

4.1. Experimental Setup

The experimental setup is illustrated in Figure 7. As shown in the figure, the cutting force was measured using a force sensor, and the energy consumption was measured using HONKI Power meters. As mentioned above, the main purpose of this study was to demonstrate the effectiveness of the proposed method for different machining steps and objective functions. On the basis of this, other factors such as the tool path and machining steps were unified in this study.

The overall tool path is shown in Figure 8, and the tool’s geometry is also depicted in the figure. Considering that the purpose of this study was to demonstrate the effectiveness of the proposed method in different machining processes, other machining conditions such as the tool path and tool conditions were kept the same.

The process of identifying the crucial parameters for NSGA-II is also illustrated in this section. In general, the population size, probability of crossover, probability of mutation, and distribution indices for crossover and mutation must be set during the optimization process. This was identified in this study by referring to previous work [7,38,45].

4.2. Statistical Analysis of the Prediction Model

The data used for the training process are also discussed in this part. The energy consumption and the surface roughness data were collected from real-world industrial processes. In this data collection process, the sampling frequency of the power meter was 12.8 kHz, and 1.23 GB of data were collected on energy consumption in this study. In terms of the surface roughness, a total of 124 workpieces were utilized in this study, which was not enough to develop a robust model for the prediction stage. To address this data insufficiency, a physical model was employed. Specifically, during the neural network training process, the hyperparameters in the model were trained using a trial-and-error strategy.

To illustrate the performance of the proposed method more clearly, a statistical analysis of the final results of the prediction model is presented here. The metrics employed for evaluating the data-driven model are as follows:

$$\mathrm{M}\mathrm{P}\mathrm{A}=1-{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{\left|y_{mea}-y_{pre}\right|}{y_{mea}}}$$

(35)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_{mea}-y_{pre}\right|$$

(36)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_{mea}-y_{pre}\right)}^2}$$

(37)

where $y_{mea}$ and $y_{pre}$ denote the measured and predicted values of the corresponding targets. The corresponding results for energy consumption and surface roughness are presented in

Table 1. Statistical analysis of the different predictive models.

Type	MAE	RMSE	MPA
PIM for surface roughness	122.33	231.47	92.33%
CNN for energy consumption	260.60	273.52	94.60%

. It can be seen from Table 1 that both the physics-informed machine learning model (PIM) and the purely machine learning model, namely, the CNN, achieved adequate performance under these conditions. Regarding the PIM, it can be seen that the model achieved a 92.33% prediction accuracy, and the energy consumption model achieved an accuracy of 94.60%. Both of them can be adapted to different working conditions.

4.3. Determination of the Boundary Conditions of the Machining Process

The process parameters were optimized in this study. To realize the optimization process, the settings in NSGA-II needed to be modified according to the actual machining conditions. First, the optimization objectives and constraint conditions for each machining step were identified. The constraints for the first three machining steps were developed for the process parameters, and the surface roughness was added as a constraint in the last step. Considering the limitations of the actual operating conditions of the machining tool, the constraints of the process parameters are as follows:

$$\{\begin{array}{}\mathrm{(38)}& & 0<a_p\le 4,\mathrm{(39)}& & 0<a_e\le 10,\mathrm{(40)}& & 0.16\le f\times N\le 0.5,\mathrm{(41)}& & 30\le {\nu }_c\le 100.\end{array}$$

The constraints of surface roughness were identified on the basis of the actual production requirements. In addition to the abovementioned constraints, the process parameters were restricted by the number of decimal points, which were allowed to exceed two. The constraint on the surface roughness was as follows:

$$R_a\le 0.8\mathsf{\mu }\mathrm{m}$$

(42)

In this study, there were four machining steps. Accordingly, fourteen optimization objectives were included in the first case. In particular, twelve optimization objectives were employed for local optimization and two optimization objectives were used for the entire machining process. In terms of the second case, the process parameter limitations were kept the same as in the first case, while an additional constraint on the cutting force was set as follows:

$$C_f\le 1000N$$

(43)

5. Results and Discussion

The final generated Pareto front solutions are depicted in Figure 9. As noted above, fourteen objectives were involved in this study, which were the same as the objectives defined in the experimental setup. It can be seen from Figure 9 that it is hard for the operators to give a final decision regarding the process parameters. To make these results clearer, the initially generated Pareto parameters were transferred into five individual Pareto solutions, which correspond to the first, second, third, and fourth machining steps and the whole machining process.

The generated individual Pareto solutions for each machining step are shown in Figure 10. Figure 10a–d correspond to the first, second, third, and fourth machining steps, and Figure 10e shows the energy consumption and material removal rate of the whole machining process. It can be seen from Figure 9 that the distributions of the Pareto front solutions varied across the different machining steps. This is because, compared with the traditional optimization process, the determination of the optimized parameters in each machining step should both consider the whole machining process and the variation in each localized machining step. As a result, these differences in the localized machining requirements cause variations in the localized optimized process parameters across the machining steps.

To further demonstrate the effectiveness of the method proposed in this paper, the final solutions of the parameters for the multi-step machining process are presented in

Table 2. Comparison of the parameters of the multi-step machining process.

Step	${\mathit{v}}_{\mathit{c}}$	$\mathit{f}$ (10−2 )	${\mathit{a}}_{\mathit{e}}$	${\mathit{a}}_{\mathit{p}}$
1	97.997	0.497	9.310	3.899
2	99.809	0.490	9.741	3.903
3	99.698	0.496	9.997	3.861
4	97.758	0.192	9.979	3.104

. In the proposed method, CRITIC-TOPSIS, was used to give a final ranking of all possible process parameters, and then the best approximate process parameters were selected for each machining step. The final results of the response value in Table 2 are illustrated in Figure 11. It can be seen from this figure that the energy consumption had a gradually upward trend, which may have been caused by the extra machining time. The cutting force is constant throughout the whole cutting process. It can also be seen from the figure that the cutting force had an upward trend in the first three steps, and decreased in the fourth cutting step. In general, the final cutting step is the last machining process, and it requires moderate process parameters to maintain the machining quality. Consequently, the cutting force decreases sharply in the final cutting step to maintain the machining quality. Accordingly, the change in the material removal rate was not stable during the whole cutting process; this was probably caused by the variation in the weights assigned to the MRR in different cutting steps, which resulted in differences in the different cutting steps.

The Pareto fronts given for the second case are presented in Figure 12. It can be seen in the figure that the distribution is two-dimensional, and the characteristics of the distribution are completely different from those of the first case. In general, although the selection of the parameters in the second case would be easier than the first case, it is hard for operators to make a quick decision between multiple different solutions. The final solution given by CRITIC-TOPSIS is given in

Table 3. Comparison of the results for the multi-step machining process.

Step	${\mathit{v}}_{\mathit{c}}$	$\mathit{f}$ (10−2)	${\mathit{a}}_{\mathit{e}}$	${\mathit{a}}_{\mathit{p}}$	${\mathit{E}}_{\mathit{c}}$	MRR
1	97.348	0.429	9.958	3.223	11.92	8.900

, and the corresponding response values are also presented in the figure. The final results indicated that the proposed method can also be effective in the optimization of a single step with different objectives.

It can be seen from the above figure that the proposed method has great potential for handling machining process parameter optimization. For this, the proposed method can be effective for generating Pareto front solutions in multi-step optimization tasks, and it can adapt to variations in the number of objectives and changing constraints. These results indicate that the proposed method can be generalized over different machining processes, and it is flexible when faced with changes in the optimization module according to the machining environment.

In conclusion, the results presented above show that the proposed method can be successfully applied to single-step and multi-step machining processes, and it also can be adapted to different objectives. This success regarding processes with different machining steps and objectives indicates that the proposed method is flexible and robust enough to handle different working conditions, making it more applicable and suitable for the modern manufacturing industry. As mentioned above, the machining process requirements usually exhibit dynamic fluctuations over time and over different machining stages.

6. Conclusions

Machining processes have been widely employed in the manufacturing industry, directly affecting its environmental impact and the product quality. It is of great importance to find the optimum parameters to improve the environmental impact, efficiency, and quality of the machining process. However, most existing studies on machining process parameter optimization have focused on optimization techniques or modeling methods, and have seldom considered their adaptability in terms of the machining process. Consequently, they suffer from poor generalization and flexibility in actual use. Thus, the main objective of this study was to provide a novel optimization method with high generalizability and effectiveness for the machining process. To address the crucial flaws existing in the conventional optimization method, a novel approach that can facilitate the optimization of machining process parameters in a flexible and generic way was proposed.

In this study, an end-to-end optimization method, which comprises three steps, was proposed. The proposed three-step method was based on a flexible optimization framework, which allowed for dynamic changes to the objective functions and constraints in each localized machining step. First, an objective model for the different machining steps was developed. Specifically, within the modeling stage, different objective functions for the optimization of the machining process were developed and transferred into a surrogate model. Then, the MODM method using NSGA-II was employed to find a solution for the different machining steps. Considering the complexity of the Pareto front solutions generated in the former step, an MAMD method using CRITIC-TOPSIS was used.

In general, the main contributions of this study are threefold. The first is the new approach, which can handle the dynamic and global nature of the machining process. The proposed method is more appropriate for optimizing the parameters in industrial settings compared with traditional methods. Secondly, a flexible optimization framework is proposed. Compared with conventional optimization studies, the proposed method can optimize not only single-step machining processes, but also the multi-step process that was considered in this study. Third, a more objective method for determining final Pareto front solutions was proposed in this study. By using this proposed method, it will be easier and quicker for operators to find the best approximate solution from a complex range of Pareto front solutions.

Although the proposed method makes a great contribution to the field of optimization, it does not consider real-time information from the machining process. Future work may make some breakthroughs in the integration of real-time monitoring systems, predictive maintenance, or the use of digital twins in the optimization process, to further enhance the applicability and advantages of handling the problem of real-time optimization in modern manufacturing.