# Parameter Optimization for Computer Numerical Controlled Machining Using Fuzzy and Game Theory

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## Abstract

**:**

## 1. Introduction

## 2. Research Background

#### 2.1. Tool Wear

- 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).

- (1)
- According to the formula, a larger depth of cut and higher cutting speed lead to less tool wear.
- (2)
- From expert experience, higher cutting speed and feed rate lead to less tool wear.
- (3)
- According to the formula, lower cutting speed and feed rate lead to less tool wear.

#### 2.2. Cutting Noise

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

- (1)
- Smaller depth of cut, less noise.
- (2)
- Slower cutting speed, less noise.
- (3)
- Improving the pressure of the tool, less noise.

#### 2.3. Fuzzy Theory

#### 2.4. Game Theory

#### 2.4.1. Elements of a Game

- Player: The actor who makes decisions with the greatest goal of pursuing his own interest.
- Nature: If not a contestant, the action taken is determined by a well-known probability.
- Action set: A collection of all possible actions taken by a contestant.
- Payoff Function: The remuneration that a contestant receives when the results of a game are shown, which is generally affected by all participants.

#### 2.4.2. Information Structure

- The identities of the players.
- The moves could be taken by all players.
- The utility function of all players.

#### 2.4.3. Bargaining Games

- ${d}_{1},{d}_{2}$: the payoff that both players can get when there is no agreement of the bargain.
- ${\mathrm{S}}_{1},{S}_{2}$: the payoff that both players can get when there is a agreement of the bargain.

## 3. Research Design

#### 3.1. Fuzzy Rules Establishment

#### 3.1.1. Tool Wear

#### 3.1.2. Cutting Noise

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

#### 3.3. Combination of Rules and Fuzzy Operation

#### 3.4. Optimal Strategies of Games

#### 3.4.1. Establishment of the Game Model

- (1)
- Cutting speed
- (2)
- Depth of cut
- (3)
- Feed rate
- (4)
- Tool nose runoff

#### 3.4.2. Target of Bargaining Games

#### 3.4.3. Mixed Strategies Game

## 4. Experimental Verification

#### 4.1. Experimental Condition

- Cutting depth: 0.5 mm, 1 mm, and 1.5 mm.
- Cutting speed: The highest CNC lathe rotational speed of the tool was 3000 rpm, the diameter of the medium-carbon steel S45C used in the turning experiment was ∅45 mm, and its highest cutting speed was 339.292 m per minute. The cutting speed was set as 250 m per minute, 200 m per minute, and 150 m per minute, according to the recommendations given by the disposable blade.
- Feed rate: The feed rate of the precision turning experiment was 0.02 mm per revolution, 0.06 mm per revolution, and 0.1 mm per revolution.

#### 4.2. Result of Single Target Production Quality Verification

^{−2}, as shown in Table 15. The median of the cutting noise was 82.83 dB, as shown in Table 16. According to the median and the comparative analysis of the two production qualities, the data obtained in this research was better than the median, which showed that the innovative strategies of both production qualities were optimized, as shown in Table 17.

#### 4.3. Multi-Quality Optimal Strategy

#### 4.3.1. Establish Initial Payoff Matrix Z2

- A: Tool wear
- B: Cutting noise

- A-1: Cutting speed is “low”, cutting depth is “high”, and feed rate is “high”. (Rule9)
- A-2: Cutting speed is “medium”, cutting depth is “medium”, and feed rate is “low”. (Rule13)
- A-3: Cutting speed is “medium”, cutting depth is “high”, and feed rate is “low”. (Rule16)
- A-4: Cutting speed is “medium”, cutting depth is “high”, and feed rate is “high”. (Rule18)
- B-1: Cutting speed is “low”, cutting depth is “high”, and feed rate is “high”. (Rule9)
- B-2: Cutting speed is “medium”, cutting depth is “medium”, and feed rate is “low”. (Rule13)
- B-3: Cutting speed is “medium”, cutting depth is “high”, and feed rate is “low”. (Rule16)
- B-4: Cutting speed is “medium”, cutting depth is “high”, and feed rate is “high”. (Rule18)

#### 4.3.2. Mixed Strategy as the Problem Solver

#### 4.3.3. Analysis of the Results of Multi-Quality Optimization

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Perfect Information | Imperfect Information | |
---|---|---|

Static | Nash Equilibrium | Bayesian Nash Equilibrium |

Dynamic | Sub-dame Perfect Nash Equilibrium | Perfect Bayesian Nash Equilibrium |

Parameter | Cutting Speed | Cutting Depth | Feed Rate | Tool Wear Rate | |
---|---|---|---|---|---|

Rule | |||||

1 | low | low | low | high | |

2 | low | low | moderate | maximum | |

3 | low | low | high | high | |

4 | low | moderate | low | moderate | |

5 | low | moderate | moderate | high | |

6 | low | moderate | high | high | |

7 | low | high | low | minimum | |

8 | low | high | moderate | minimum | |

9 | low | high | high | low | |

10 | moderate | low | low | maximum | |

11 | moderate | low | moderate | maximum | |

12 | moderate | low | high | maximum | |

13 | moderate | moderate | low | moderate | |

14 | moderate | moderate | moderate | moderate | |

15 | moderate | moderate | high | high | |

16 | moderate | high | low | low | |

17 | moderate | high | moderate | minimum | |

18 | moderate | high | high | low | |

19 | high | low | low | high | |

20 | high | low | moderate | maximum | |

21 | high | low | high | maximum | |

22 | high | moderate | low | moderate | |

23 | high | moderate | moderate | low | |

24 | high | moderate | high | low | |

25 | high | high | low | low | |

26 | high | high | moderate | minimum | |

27 | high | high | high | low |

Parameter | Cutting Speed | Cutting Depth | Feed Rate | Cutting Noise | |
---|---|---|---|---|---|

Rule | |||||

1 | low | low | low | minimum | |

2 | low | low | moderate | minimum | |

3 | low | low | high | low | |

4 | low | moderate | low | minimum | |

5 | low | moderate | moderate | low | |

6 | low | moderate | high | moderate | |

7 | low | high | low | low | |

8 | low | high | moderate | low | |

9 | low | high | high | low | |

10 | moderate | low | low | moderate | |

11 | moderate | low | moderate | moderate | |

12 | moderate | low | high | moderate | |

13 | moderate | moderate | low | low | |

14 | moderate | moderate | moderate | high | |

15 | moderate | moderate | high | high | |

16 | moderate | high | low | moderate | |

17 | moderate | high | moderate | moderate | |

18 | moderate | high | high | moderate | |

19 | high | low | low | maximum | |

20 | high | low | moderate | maximum | |

21 | high | low | high | maximum | |

22 | high | moderate | low | maximum | |

23 | high | moderate | moderate | maximum | |

24 | high | moderate | high | maximum | |

25 | high | high | low | maximum | |

26 | high | high | moderate | maximum | |

27 | high | high | high | maximum |

Fuzzy Term | 0 | 1.25 | 2.5 | 3.75 | 5 |

Low | 1 | 0.5 | 0 | 0 | 0 |

Medium | 0 | 0.5 | 1 | 0.5 | 0 |

High | 0 | 0 | 0 | 0.5 | 1 |

Fuzzy Term | 0 | 1.25 | 2.5 | 3.75 | 5 |

Low | 1 | 0.5 | 0 | 0 | 0 |

Medium | 0 | 0.5 | 1 | 0.5 | 0 |

High | 0 | 0 | 0 | 0.5 | 1 |

Fuzzy Term | 0 | 1.25 | 2.5 | 3.75 | 5 |

Low | 1 | 0.5 | 0 | 0 | 0 |

Medium | 0 | 0.5 | 1 | 0.5 | 0 |

High | 0 | 0 | 0 | 0.5 | 1 |

No. | Minimal | Small | Moderate | Large | Greatest |
---|---|---|---|---|---|

0 | 1 | 0 | 0 | 0 | 0 |

1 | 0.84 | 0 | 0 | 0 | 0 |

2 | 0.68 | 0 | 0 | 0 | 0 |

3 | 0.52 | 0 | 0 | 0 | 0 |

4 | 0.36 | 0 | 0 | 0 | 0 |

5 | 0.2 | 0.04 | 0 | 0 | 0 |

6 | 0.04 | 0.2 | 0 | 0 | 0 |

7 | 0 | 0.36 | 0 | 0 | 0 |

8 | 0 | 0.52 | 0 | 0 | 0 |

9 | 0 | 0.84 | 0 | 0 | 0 |

10 | 0 | 1 | 0 | 0 | 0 |

11 | 0 | 0.68 | 0 | 0 | 0 |

12 | 0 | 0.52 | 0 | 0 | 0 |

13 | 0 | 0.36 | 0 | 0 | 0 |

14 | 0 | 0.2 | 0.04 | 0 | 0 |

15 | 0 | 0.04 | 0.2 | 0 | 0 |

16 | 0 | 0 | 0.36 | 0 | 0 |

17 | 0 | 0 | 0.52 | 0 | 0 |

18 | 0 | 0 | 0.68 | 0 | 0 |

19 | 0 | 0 | 0.84 | 0 | 0 |

20 | 0 | 0 | 1 | 0 | 0 |

21 | 0 | 0 | 0.84 | 0 | 0 |

22 | 0 | 0 | 0.68 | 0 | 0 |

23 | 0 | 0 | 0.52 | 0 | 0 |

24 | 0 | 0 | 0.36 | 0 | 0 |

25 | 0 | 0 | 0.2 | 0 | 0 |

26 | 0 | 0 | 0.04 | 0.04 | 0 |

27 | 0 | 0 | 0 | 0.2 | 0 |

28 | 0 | 0 | 0 | 0.36 | 0 |

29 | 0 | 0 | 0 | 0.52 | 0 |

30 | 0 | 0 | 0 | 0.68 | 0 |

31 | 0 | 0 | 0 | 0.84 | 0 |

32 | 0 | 0 | 0 | 1 | 0 |

33 | 0 | 0 | 0 | 0.84 | 0 |

34 | 0 | 0 | 0 | 0.68 | 0.04 |

35 | 0 | 0 | 0 | 0.52 | 0.2 |

36 | 0 | 0 | 0 | 0.36 | 0.36 |

37 | 0 | 0 | 0 | 0.2 | 0.52 |

38 | 0 | 0 | 0 | 0.04 | 0.68 |

39 | 0 | 0 | 0 | 0 | 0.84 |

40 | 0 | 0 | 0 | 0 | 1 |

Cutting Speed (m/min) | Depth of Cut (mm) | Feed Rate (mm/rev) | Tool Nose Runoff (mm) | Tool Wear (μm ^{−2}) |
---|---|---|---|---|

2 | 2 | 2 | 2 | 4.38 |

3 | 2 | 2 | 1 | 4.13 |

3 | 2 | 2 | 2 | 3.87 |

1 | 2 | 2 | 1 | 4.21 |

2 | 3 | 2 | 2 | 2.97 |

1 | 2 | 2 | 3 | 4.13 |

2 | 2 | 1 | 1 | 4.38 |

2 | 2 | 2 | 1 | 4.04 |

2 | 1 | 3 | 3 | 4.13 |

1 | 2 | 1 | 3 | 4.55 |

1 | 3 | 3 | 2 | 3.38 |

Cutting Speed (m/min) | Depth of Cut (mm) | Feed Rate (mm/rev) | Tool Nose Runoff (mm) | Cutting Noise (dB) |
---|---|---|---|---|

2 | 2 | 2 | 2 | 82.83 |

1 | 2 | 2 | 2 | 81.73 |

3 | 2 | 2 | 2 | 85.97 |

2 | 1 | 2 | 2 | 82.61 |

2 | 3 | 2 | 2 | 82.91 |

2 | 2 | 1 | 2 | 82.55 |

2 | 2 | 3 | 2 | 82.93 |

2 | 2 | 2 | 1 | 82.79 |

2 | 2 | 2 | 3 | 82.81 |

1 | 1 | 1 | 1 | 81.5 |

1 | 3 | 3 | 2 | 81.94 |

B | |||||
---|---|---|---|---|---|

B-1 | B-2 | B-3 | B-4 | ||

A | A-1 | ||||

A-2 | |||||

A-3 | |||||

A-4 |

B | |||||
---|---|---|---|---|---|

B-1 | B-2 | B-3 | B-4 | ||

A | A-1 | (P_{a1},P_{b1}) | (P_{a2},P_{b2}) | (P_{a3},P_{b3}) | (P_{a4},P_{b4}) |

A-2 | (P_{a5},P_{b5}) | (P_{a6},P_{b6}) | (P_{a7},P_{b7}) | (P_{a8},P_{b8}) | |

A-3 | (P_{a9},P_{b9}) | (P_{a10},P_{b10}) | (P_{a11},P_{b11}) | (P_{a12},P_{b12}) | |

A-4 | (P_{a13},P_{b13}) | (P_{a14},P_{b14}) | (P_{a15},P_{b15}) | (P_{a16},P_{b16}) |

_{a}: the payoff value of quality A under different situation; P

_{b}: the payoff value of quality B under different situation.

B | |||
---|---|---|---|

B-1 | B-2 | ||

A | A-1 | (P_{a1},P_{b1}) | (P_{a2},P_{b2}) |

A-2 | (P_{a3},P_{b3}) | (P_{a4},P_{b4}) |

Player | Optimal Strategy | Adoption Probability (%) |
---|---|---|

Tool wear (S) | ||

Cutting noise (Z) |

Control Parameter | Level 1 | Level 2 | Level 3 |
---|---|---|---|

A: Depth of cut (mm) | 0.5 | 1 | 1.5 |

B: Cutting speed (m/min) | 150 | 200 | 250 |

C: Feed rate (mm/rev) | 0.02 | 0.06 | 0.1 |

D: Tool nose runoff (mm) | −0.1 | ±0.03 | 0.1 |

Cutting Speed (m/min) | Depth of Cut (mm) | Feed Rate (mm/rev) | Tool Nose Runoff (mm) | Tool Wear (μm ^{−2}) |
---|---|---|---|---|

200 | 1 | 0.06 | ±0.03 | 4.38 |

Cutting Speed (m/min) | Depth of Cut (mm) | Feed Rate (mm/rev) | Tool Nose Runoff (mm) | Cutting Noise (dB) |
---|---|---|---|---|

200 | 1 | 0.06 | ±0.03 | 82.83 |

Tool Wear | Cutting Speed(m/min) | Depth of Cut(mm) | Feed Rate(mm/rev) | Tool Nose Runoff(mm) | (μm^{−2}) |

Cutting speed | 250 | 1 | 0.06 | ±0.03 | 3.87 |

Depth of cut | 200 | 1.5 | 0.06 | ±0.03 | 2.97 |

Feed rate | 200 | 1 | 0.02 | ±0.03 | 4.55 |

Median | 200 | 1 | 0.06 | ±0.03 | 4.38 |

Cutting Noise | Cutting Speed(m/min) | Depth of Cut(mm) | Feed Rate(mm/rev) | Tool Nose Runoff(mm) | (dB) |

Cutting speed | 150 | 1 | 0.06 | ±0.03 | 81.73 |

Depth of cut | 200 | 0.5 | 0.06 | ±0.03 | 82.61 |

Feed rate | 200 | 1 | 0.02 | ±0.03 | 82.55 |

Median | 200 | 1 | 0.06 | ±0.03 | 82.83 |

B | |||||
---|---|---|---|---|---|

B-1 | B-2 | B-3 | B-4 | ||

A | A-1 | (9.966,9.966) | (9.966,9.966) | (9.966,20) | (9.966,20) |

A-2 | (20,9.966) | (20,9.966) | (20,20) | (20,20) | |

A-3 | (9.966,9.966) | (9.966,9.966) | (9.966,20) | (9.966,20) | |

A-4 | (1.769,9.966) | (1.769,9.966) | (1.769,20) | (1.769,20) |

B | |||
---|---|---|---|

B-1 | B-2 | ||

A | A-1 | (9.966,9.966) | (9.966,9.966) |

A-3 | (9.966,9.966) | (9.966,9.966) |

B | |||
---|---|---|---|

B-1 | B-2 | ||

A | A-1 | (3.38,81.94) | (3.38,82.55) |

A-3 | (4.38,81.94) | (4.38,82.55) |

Player | Optimal Strategy | Adoption Probability (%) |
---|---|---|

Tool wear (S) | Increasing the cutting depth | 100 |

Cutting noise (Z) | Reducing the cutting speed | 100 |

Cutting Speed (m/min) | Depth of Cut (mm) | Feed Rate (mm/rev) | Tool Nose Runoff (mm) | Comparison | ||
---|---|---|---|---|---|---|

Multi-Quality Optimization | 150 | 1.5 | 0.1 | ±0.03 | Tool wear | 3.38 (μm^{−2}) |

Cutting noise | 81.94 (dB) | |||||

Median | 200 | 1 | 0.06 | ±0.03 | Tool wear | 4.38 (μm^{−2}) |

Cutting noise | 82.83 (dB) |

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## Share and Cite

**MDPI and ACS Style**

Chuang, K.-C.; Lan, T.-S.; Zhang, L.-P.; Chen, Y.-M.; Dai, X.-J.
Parameter Optimization for Computer Numerical Controlled Machining Using Fuzzy and Game Theory. *Symmetry* **2019**, *11*, 1450.
https://doi.org/10.3390/sym11121450

**AMA Style**

Chuang K-C, Lan T-S, Zhang L-P, Chen Y-M, Dai X-J.
Parameter Optimization for Computer Numerical Controlled Machining Using Fuzzy and Game Theory. *Symmetry*. 2019; 11(12):1450.
https://doi.org/10.3390/sym11121450

**Chicago/Turabian Style**

Chuang, Kai-Chi, Tian-Syung Lan, Lie-Ping Zhang, Yee-Ming Chen, and Xuan-Jun Dai.
2019. "Parameter Optimization for Computer Numerical Controlled Machining Using Fuzzy and Game Theory" *Symmetry* 11, no. 12: 1450.
https://doi.org/10.3390/sym11121450