**1. Introduction**

Under strict international environmental regulations, although there are various cutting conditions related to environmental protection quality, tool wear and cutting noise are always considered preferentially as because of their green environmental protection quality in the practice of machining of cutting. There are often sophisticated nonlinear relationships in the problem of parameter optimization in multi-quality precision CNC production. The industry often selects appropriate machining parameters that rely on the program of the numerically controlled machine tool or the technicians' experience, but the results are not necessarily optimal and are not guaranteed to be optimal under multi-quality (more than two target qualities). Most of the cutting parameter optimization literature obviously does not meet the needs of the industry as it either considers only a single quality (only one target quality) or has overly costly research.

According to the research on cutting parameters, using Analytic Hierarchy Process (AHP) to combine the innovative thinking model of Teoriya Resheniya Izobretatelskikh Zadatch (TRIZ) and the concept of green production reduces the impact on the environment [1]. The optimal turning parameters obtained by using fuzzy semantic quantification can indeed be used as a method of analyzing parameters for practical cutting operations under environmental and cost considerations [2]. Considering the problem of cutting noise, Lan, Chuang, and Chen analyzed the combination of the optimal factor level with the Taguchi method [3]. However, the research only explored the noise target and thus was research of a single quality. Zhang et al. analyzed the influence of cutting parameters on noise with the variance. The cutting cost model was proposed after the analysis results showed that the cutting depth was the main factor affecting the cutting noise. However, the results of the study also applied only to a single quality material [4]. Hossein and Kops's research showed that the cutting temperature increases under larger cutting depth and higher cutting speed, which in turn shortens the tool life. However, the research took time to carry out the cutting work and belonged solely to the research of a single quality and not a multi-quality research [5]. The research of Schultheiss et al. which pointed out that reducing the tool wear can shorten the time of the production cycle and reduce energy consumption, was also a single quality research and not a multi-quality research [6]. Weng obtained the optimal cutting parameters by using fuzzy quantification. The parameters can reach 10% of the level prior the whole experiment under the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) arrangement. This proved that the experiment is unnecessary in reaching this level and can result in cost and labor savings. While this research obtained optimized combinations of multi-quality parameters, it did not take into account the conflict of quality objectives [7]. Li et al. established a multi-quality optimization model scheme by applying game theory to machining cutting parameters. The research showed that game theory was suitable for multi-quality optimization design, but failed to obtain the best combination [8]. Zhou et al. reduced and optimized the carbon footprint of the cutting process through game theory, but multi-quality was not taken into account [9]. Tian et al. considered the tool wear conditions and optimized the cutting parameters through game theory. The research discussed the conditions of tool wear, which was also a single-quality optimization research [10]. The above-mentioned local or abroad researches on cutting parameters are either discussions of only a single quality or optimization plans with specific conditions. Not only is there no further explanation and analysis of the conflict between the production qualities, but there is a need to be achieved through the actual operation of the cutting equipment, which is a waste of material resources, time, and labor and has an influence on the surrounding environment. Different control parameters are required when the processing conditions (materials, equipment, and tools) are different, which troubles the CNC industry. Therefore, developing a set of general production optimal mechanisms with green innovation by analyzing the inference method of green product design without equipment operation will be positive for the competitiveness and development of the precision CNC turning industry.

Based on the shortcomings of the above-mentioned researches, this research integrate fuzzy theory and game theory. Through the method of semantic quantification, a set all-purpose prediction models provides the fuzzy value of each goal for the selection of cutting parameters without actual cutting by the machine. The research also resolves the conflict problem between production qualities and control parameters by using game theory. The best quality strategy was obtained through statistic to help improve the understanding of engineering science by technicians as a consideration in the design or manufacture of future products. Through the result of this research, a set of optimal, all-purpose economic prospective parameter analysis methods could be provided to the technicians to enhance the overall competitiveness of the automated CNC cutting industry.

#### **2. Research Background**

#### *2.1. Tool Wear*

From Taylor's tool life formula, the wear of high-speed steel tools refers to the use time in the upper limit of the low wear rate area, which was used to record the characteristics of High Speed Steel (HSS) tools and to obtain Formula (1) after rearrangement [11]. The relationship between feed rate and cutting speed must be properly matched during cutting. For instance, friction phenomenon instead of cutting might occur with overly slow speed. However, unexpected high-speed might break the cutting edge or roughen the transient surface.

$$T V^{1/n} f^{1/m} d^{1/l} = \mathcal{C}'\tag{1}$$

*T*: tool function.

*V*: cutting speed.

*f*: feed rate.

*D*: diameter of milling cutter.

*n*, *m*: constant of tool material properties (acquired by experiment or experience).

*l*: cutting length.

*C* : cutting speed of tool life in 1 minute (supplied by tool manufacturer).


### *2.2. Cutting Noise*

All the noise values produced by the measurement experiment, including the noise values produced by motor idling and cutting experiments, were substituted in the formula, since their differences are lower than three Decibels, as shown in Formula (2) [12].

$$LPC = 10\log\left[10^{\frac{l\_{PB}}{10}} - 10^{\frac{l\_{PA}}{10}}\right] \tag{2}$$

*LPB*: the measured value of motor running with no cutting. *LPA*: the measured value of motor running with cutting.


#### *2.3. Fuzzy Theory*

In 1965, Professor Zadeh of the University of California, Berkeley proposed fuzzy theory, which is a kind of fuzzy concept quantification based on fuzzy sets. It is mainly focused on making a correct judgment without going through complicated calculation processes of the fuzzy message of the human brain or incomplete information [13]. The language 'IF ... THEN ... ' is used in fuzzy theory to represent the fuzzy relationship. A language represents a qualitative conditional sentence and an uncertain rule, which is quantified by fuzzy mathematical tools. Fuzzy logic control was used to convert the input language to a fuzzy set. The fuzzy logic control architecture included the fuzzification interface, interface engine, defuzzification interface, and the fuzzy rule-based system, as shown in Figure 1 [14].

**Figure 1.** Architecture of fuzzy logic control.

#### *2.4. Game Theory*

Game theory was proposed in 1928 [15,16] before being promoted by the economist John Nash. Von Neumann and Oskar Morgenstern co-authored *The Theory of Games and Economic Behavior* in 1944, which analyzed game theory and economic behavior in detail and explained zero-sum games, the league, and the cooperative game, which further laid the theoretical foundation of game theory. Shapley contributed significantly to the development of core theory in game theory through the development of a prisoner's dilemma game [17]. Nash proved the Nash equilibrium existence theorem in 1953, which set a milestone for the current non-cooperative game theory.

Game theory is a state of confrontation for two or more contestants in a rational situation, with the pursuit of their own interests as the greatest goal. The conflict and cooperative relationship between rational contestants, using mathematical model simulation, has been widely used in various types of study.

#### 2.4.1. Elements of a Game

The setting of contestants is rational in a game; however, the result could be quite the contrary or could have a Pareto principle, which is just in line with the current economic development trend since the smart contestant sits first in the best strategy of others to greater their own payoff function. Therefore, the result of the game is not necessarily rational or efficient but is closer to the economic situation. The main elements of the game are:


#### 2.4.2. Information Structure

The information structure was divided into four types by Rasmusen, which are perfect information, complete information, certain information, and symmetric information [18]. Within a game with perfect information, each information set is a single node, which means that the players are clear about at which decision point the decision is made. If not, the game is called an imperfect information game. A game in which the players sit aware of the following three situations is called a complete information game. If not, the game is called an incomplete information game.


Within a game with certain information, players will not act naturally after acting. If not, the game is called an uncertain information game. In a game with symmetric information, the information a player gets at the move node or at the end is the same with other players. If not, the game is called an asymmetric information game. Based on the player's simultaneous move (static game) or sequential move (dynamic game), and prior information (strategy and playoff) a player has or does not have, the game is divided into four types, as shown in Table 1.



#### 2.4.3. Bargaining Games

The largest difference between bargaining game theory and decision theory is that the problems faced by a group of decision makers in a given situation can solve many economic problems. Therefore, game theory, which is widely used by economic, political, and financial experts, not only has the rigor of a mathematical model, but also simplifies the complex interaction phenomena in a real environment, and provides the strategic behavior analysis method for decision makers. In 1950, Nash assumed that a group of axioms would only get a solution to a set of bargaining models based on a non-cooperative game, which was divided into four parts [19,20].

#### 1. Pareto efficiency

The outcome of the contestants' bargaining is beneficial to both parties; in other words, there is no other bargaining outcome that can increase the interests of all participants at the same time.

2. Independence of the irrelevant alternatives

Add things that do not matter in the game, and the outcome of the bargain is not affected.

3. Symmetry

If there is symmetry in the contestants' negotiation questions, the two contestants will receive an equal result.

4. Invariance under strategically equivalent representations

The utility function after the monotonic transformation still indicates that the participants have the same preference, and the monotonic transformed utility function does not affect the bargaining result.

Nash's suggestion, as shown in Formula (3), proved that the bargaining solution exists and is unique if these four axioms are satisfied.

$$\max\_{s1s2} (S\_1 - d\_1)(S\_2 - d\_2) \tag{3}$$

*d*1, *d*2: the payoff that both players can get when there is no agreement of the bargain.

S1, *S*2: the payoff that both players can get when there is a agreement of the bargain.

Only if the result of the bargain is better than the one before the bargain can the players be motivated to bargain, so that (*d*1, *d*2) ≤ (*s*1,*s*2).

Bargaining game theory has been used universally in economics, international relationships, calculator science, military strategy, and other disciplines. Some general topics that had used bargaining game theory are about the efficiency and rationality of solving supplier selection problems [21], the demand-response resource allocation between distribution networks [22], the reduction of environmental risks to enterprises in production processes [23], the solutions to the upload transmission power optimization problems in the multilateral bargaining model [24], and more.

### **3. Research Design**

Precision CNC cutting was taken as an example in this research, while tool wear and cutting noise were selected as the green production quality of CNC cutting. The depth of cut, cutting speed, feed rate, and tool nose runoff were taken as control parameters. Fuzzy theory was used to define the semantic rules of the relationship of control parameters and production quality to carry out the fuzzy quantification. The quantified output values were introduced into game theory to resolve the conflict among the two production qualities and four control parameters. The strategy probability statistics of the game result and the strategy option with the highest sum of probability as the best strategy of that production quality were taken.

#### *3.1. Fuzzy Rules Establishment*

In the selection of the fuzzy membership function, different membership functions, based on each rule, were compared by entering the three factors: Cutting speed, cutting depth, and feed rate. The minimum membership function was calculated by the intersection, and the maximum value of the union was selected as the output part of the set to calculate the value of the center of gravity of the largest area in order to obtain the fuzzy value. The triangular membership function was used as the fuzzy pattern and the defuzzification was calculated by the center of gravity. Tool wear and cutting noise were chosen as the production qualities in this research. According to the literature, relevant cutting experience level range, and the suggestion of cutting parameters from tool manuals, was determined as low, medium, or high. The cutting characteristics of the target were obtained by using semantic quantification and were divided into five levels: Greatest, large, moderate, small, and minimal.

#### 3.1.1. Tool Wear

Tool wear is a vital factor affecting cutting quality in precision machining. Changing the cutting tool before the end of the tool's life may result in higher production cost, lower production efficiency, and many disposals of tool inserts, which cause environmental pollution. Therefore, this study established fuzzy rules using cutting speed, cutting depth, and feed rate to minimize the tool wear, as shown in Table 2.


**Table 2.** Tool wear fuzzy rule table.


**Table 2.** *Cont.*

#### 3.1.2. Cutting Noise

The noise during the cutting process is mainly caused by the vibration phenomenon, which not only interferes with the entire cutting process, but also seriously influences the quality of the work piece. The noise might even influence the mood of the technicians during work, which has a certain negative impact on production quality. In order to reduce the vibration frequency, it is necessary to reduce the cutting speed, depth of cutting, and feed rate of the tool, which in turn reduces productivity. Therefore, the fuzzy rules were established with cutting speed, depth of cutting, and feed rate as the factors based on the semantic considerations, as shown in Table 3.


**Table 3.** Cutting noise fuzzy rule table.


**Table 3.** *Cont.*

#### *3.2. Variability of the Input and Output Domains*

The operation had three inputs and one output. The input target was the control factor, and the output target was the default result. The input domain of the variables was in the interval [0,5] and was divided into five equal parts. The output domain of the variables was in the interval [0,40] and was divided into 40 equal parts.

1. Input target (1): The degree of membership of cutting speed as the control factor (Figure 2).

**Figure 2.** Degree of membership of the cutting speed.

Fuzzy terms: The degree of membership presented in Figure 2 is listed in Table 4.

**Table 4.** Input membership values of cutting speed.


2. Input target (2): The degree of membership of cutting depth as the control factor (Figure 3).

**Figure 3.** Degree of membership of the cutting depth.

Fuzzy terms: The degree of membership presented in Figure 3 is listed in Table 5.

**Table 5.** Input membership values of cutting depth.


3. Input target (3): The degree of membership of feed rate as the control factor (Figure 4).

**Figure 4.** Degree of membership of the feed rate.

Fuzzy terms: The degree of membership presented in Figure 4 is listed in Table 6.

**Table 6.** Input membership values of feed rate.


4. Output target: Membership functions of the output variable (Figure 5).

**Figure 5.** Degree of membership of output variables.

Fuzzy terms: The degree of membership presented in Figure 5 is detailed in Table 7.


**Table 7.** Output membership values.
