# Production Process Optimization of Metal Mines Considering Economic Benefit and Resource Efficiency Using an NSGA-II Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Production Process of Metal Mines

#### 2.1. Exploration Process

#### 2.2. Mining Process

#### 2.3. Beneficiation Process

_{3}, f

_{4}, and f

_{5}depend largely on many factors, such as the rock lithology, mining method, beneficiation method and plant design, in the production process.

## 3. Multi-Objective Optimization Model Considering Economic Profit and Resource Efficiency

#### 3.1. Decision Variables and Constraints

#### 3.1.1. Decision Variables

#### 3.1.2. Constraints

#### 3.2. Objective Function

#### 3.2.1. Economic Benefit Objective

#### 3.2.2. Resource Efficiency Objective

#### 3.3. Multi-Objective Optimization Model

#### 3.4. Development of the NSGA-II Model to Solve the Established Model

^{3}) to O(MN

^{2}). The crowding distance can ensure good distribution with small computational complexity. The fast, non-dominated sorting and crowding distance can make the parent population and child population compete together to produce new parent populations, which both achieves convergence and prevents local optimality. This study used the NSGA-II to optimize the production process of metal mines.

- (a)
- Collect the data related to the production process of a specific metal mine, i.e., the value of each indicator, and the price of concentrate ores.
- (b)
- Determine the relationship between the indications, such as $\phi (x)$, $g(x)$, $f(x)$, ${Q}_{1}={f}_{1}({p}_{1},{p}_{2})$, ${p}_{3}={f}_{2}({p}_{1},{p}_{2})$, ${c}_{2}={f}_{3}({c}_{1})$, ${c}_{3}={f}_{4}({p}_{4})$, ${p}_{5}={f}_{5}({p}_{4},{c}_{3})$.
- (c)
- Determine the decision variables according to the dependency analysis, and the upper and lower boundary values of the decision variables according to the production process of the mine.
- (d)
- The NSGA-II parameters, such as the population size, maximum number of iterations r, crossover probability, mutation probability, crossover index and mutation index, are initialized. Then, n possible individuals are randomly generated as the initial parent population.
- (e)
- The parent population generates a child population with n possible individuals by selection, mutation and crossover.
- (f)
- The parent and child populations are mixed to form a new population with 2n possible individuals.
- (g)
- The profit and resource utilization rate of each individual is calculated in the new population with the input data in (a) and the relationship in (b).
- (h)
- Based on the values of the objective functions, the mixed population is classified based on the non-dominated level, and the crowded distance is calculated.
- (i)
- Based on the non-dominated sorting and the crowding distance calculation results of step (h), the top n possible individuals are retained as a new parent population.
- (j)
- Check the termination condition. If satisfied, the optimization process is terminated and output the optimal decision variables, profit and resource utilization rate; otherwise, goes to step (e).

## 4. Multi-Objective Optimization of Process of the Huogeqi Copper Mine

#### 4.1. Brief Introduction of the Huogeqi Copper Mine

^{3}. The total cost of the ore production is estimated of 34.76 $/t. This is the addition of the mining cost (15.8 $/t) and the beneficiation cost (18.96 $/t) [47].

#### 4.2. Production Indicators of the Huogeqi Copper Mine

#### 4.2.1. Relationship between Ore Weight and Grade

#### 4.2.2. Probability Density of Ore Grade Distribution

#### 4.2.3. Relationship between Dilution Ratio and Loss Ratio

^{−50}. As the significance level of 1.0075 × 10

^{−50}is far less than 0.05, the significance test shows that the dilution ratio has a strong linear relationship with the loss ratio of Cu. The dilution ratio of Cu can thus be obtained by

#### 4.2.4. Relationship between Concentration Ratio and Raw Ore Grade

#### 4.2.5. Concentrate Grade, Concentration Ratio and Raw Ore Grade

#### 4.2.6. Copper Concentrate Transaction Price

#### 4.3. Production Process of the Huogeqi Copper Mine Using the NSGA-II

#### 4.3.1. Parameters of the Huogeqi Copper Mine and NSGA-II Model

#### 4.3.2. Optimization Results Using NSGA-II

## 5. Discussion

#### 5.1. Comparison of Different Optimization Algorithms

#### 5.2. Effect of Decision Variables on the Objective Function

_{0.05}(7,99) = 0.3053, F

_{0.05}(7,99) = 0.3053 and F

_{0.05}(10,99) = 0.3862, respectively, at the 95% confidence interval. The variance analysis F-values for the geological cut-off grade, minimum industrial grade and loss ratio of Cu are 76.38, 51.2 and 2.22, respectively, which are higher than their corresponding tabulated F-values, i.e., 0.3053, 0.3053 and 0.3862. As the P-values of all decision parameters are less than 0.05, the null hypothesis does not stand. Therefore, all decision variables have significant effects on the function of profit. Moreover, the variance analysis results indicate that the profit is mainly affected by the geological cut-off grade of Cu, which has a contribution of 58.84%, and the minimum industrial grade of Cu, which has a contribution of 39.45%; in contrast, the contribution of the loss ratio of Cu (1.71%) is very low.

_{0.05}(7,99) = 0.3053, F

_{0.05}(7,99) = 0.3053 and F

_{0.05}(10,99) = 0.3862. As the p-values of all decision parameters are less than 0.05, the null hypothesis is rejected. Therefore, for the resource utilization rate, all decision variables are considered significant. Moreover, the variance analysis results indicate that the geological cut-off grade of Cu is the most important decision variable, with a contribution of 54.19%; in contrast, the contributions of the minimum industrial grade and loss ratio of Cu are 27.18% and 18.63%, respectively.

#### 5.3. Sensitivity Analysis of Pareto-Optimal Solutions to Unit Copper Concentrate Price

## 6. Conclusions

- (1)
- The established NSGA-II method is an effective method to approach the multi-objective optimization of the production process of the Huogeqi Copper Mines. It outperforms the MOGA and SPEA2 with lower diversity in solution optimization of the whole production process of metal mines. The Pareto-optimal solutions produced by the NSGA-II method reflect the compromising relationship between the economic benefits and the resource efficiency. The optimization results suggest that the Huogeqi Copper Mine in its current state can be further optimized to obtain a better economic benefit and resource efficiency for sustainable development.
- (2)
- The contributions of decision variables on objective functions show that profit is mainly affected by the geological cut-off grade of Cu (with a contribution of 58.84%) and the minimum industrial grade of Cu (with a contribution of 39.45%), but barely affected by the loss ratio of Cu (with a contribution of 1.71%). With regard to the resource utilization rate, the geological cut-off grade of Cu is the most important decision variable (with a contribution of 54.19%).
- (3)
- The sensitivities of the Pareto-optimal solutions to the unit copper concentrate price show that the Pareto-optimal solutions shift upward towards higher profits with increasing unit copper concentrate prices. The variations of the Pareto-optimal solutions are more sensitive to the unit copper concentrate price at higher profits than those at lower profits.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Flowchart of production process optimization of metal mines using the Non-Dominated Sorting Genetic Algorithm (NSGA-II).

**Figure 5.**Geological ore body distribution in the 1450–1570 stage of Huogeqi Copper Mine under the Xian-80 coordinate system.

**Figure 13.**The Pareto-optimal solutions obtained by NSGA-II, Multi-Objective Genetic Algorithms (MOGA), Improved Strength Pareto Evolutionary Algorithm SPEA2.

Hidden Nodes | Concentrate Grade of Cu | |||
---|---|---|---|---|

Train MARE (%) | Test MARE (%) | Train AMRE (%) | Test AMRE (%) | |

1 | 0.8417 | 0.7491 | 7.3575 | 4.7698 |

2 | 0.3057 | 0.2979 | 1.4701 | 1.0916 |

3 | 0.3049 | 0.2963 | 1.4543 | 1.0597 |

4 | 0.3124 | 0.3019 | 1.6102 | 1.0677 |

5 | 0.3215 | 0.3025 | 1.8151 | 1.4596 |

Grade of Cu (%) | Compensation Price ($·t^{−1}) | Price Coefficient |
---|---|---|

≥23 | 47.4 | 0.86 |

22.00~22.99 | 31.6 | 0.85 |

21.00~21.99 | 15.8 | 0.84 |

20.00~20.99 | 0 | 0.83 |

19.00~19.99 | −15.8 | 0.81 |

18.00~18.99 | −31.6 | 0.795 |

17.00~17.99 | −47.4 | 0.78 |

16.00~16.99 | −63.2 | 0.77 |

Parameter of Huogeqi Copper Mine | Value | NSGA-II Parameter | Value |
---|---|---|---|

Initial value of the geological cut-off grade of Cu ${p}_{a}$ (%) | 0.30 | Number of decision variables | 3 |

Initial value of the minimum industrial grade of Cu ${p}_{b}$ (%) | 0.50 | Number of objective functions | 2 |

Recoverable reserve of the 1450–1570 stage of Cu ${Q}_{0}$ (t) corresponding to ${p}_{a}$ and ${p}_{b}$ | 9 × 10^{6} | Population size | 100 |

Constant $m$ | 0.66 | Maximum number of iterations ${N}_{\mathrm{max}}$ | 100 |

Unit mining cost ${h}_{1}$ ($/t) | 15.8 | Crossover index ${\eta}_{c}$ (SBX) | 20 |

Unit beneficiation cost ${h}_{2}$ ($/t) | 18.96 | Mutation index ${\eta}_{w}$ (polynomial mutation) | 20 |

Unit #1 copper price ${q}_{1}$ ($/t) | 7114.16 | Crossover probabilities | 0.5 |

Lower bound of geological cut-off grade of Cu ${p}_{1\mathrm{min}}$ (%) | 0.10 | Mutation probabilities | 1/3 |

Upper bound of geological cut-off grade of Cu ${p}_{1\mathrm{max}}$ (%) | 0.90 | ||

Lower bound of minimum industrial grade of Cu ${p}_{2\mathrm{min}}$ (%) | 0.10 | ||

Upper bound of minimum industrial grade of Cu ${p}_{2\mathrm{max}}$ (%) | 0.90 | ||

Lower bound of loss ratio of Cu ${c}_{1\mathrm{min}}$ (%) | 6 | ||

Upper bound of loss ratio of Cu ${c}_{1\mathrm{min}}$ (%) | 12 | ||

Lower bound of melted grade of Cu ${p}_{\mathrm{melter}}$ (%) | 16 |

Parameters | Case A | Case B | Case C | Current Case |
---|---|---|---|---|

Profits ($) | 2.9317 × 10^{8} | 2.5776 × 10^{8} | −5.49 × 10^{7} | 2.503 × 10^{8} |

Resource utilization rate | 0.6689 | 0.7578 | 0.8416 | 0.7383 |

Geological cut-off grade of Cu (%) | 0.582 | 0.366 | 0.117 | 0.3 |

Minimum industrial grade of Cu (%) | 0.647 | 0.410 | 0.135 | 0.5 |

Loss ratio of Cu (%) | 6.018 | 6.006 | 6 | 8 |

Factors | Degrees of Freedom | Sum of Squares | Mean Squares | F | P | Contribution (%) |
---|---|---|---|---|---|---|

Profit | ||||||

Geological cut-off grade of Cu | 7 | 8.04912 × 10^{16} | 1.14987 × 10^{16} | 76.38 | 0 | 58.84 |

Minimum industrial grade of Cu | 7 | 5.39543 × 10^{16} | 7.70776 × 10^{15} | 51.2 | 0 | 39.45 |

Loss ratio of Cu | 10 | 3.33574 × 10^{15} | 3.33574 × 10^{14} | 2.22 | 0.0256 | 1.71 |

Error | 75 | 1.12915 × 10^{16} | 1.50554 × 10^{14} | |||

Total | 99 | 2.55872 × 10^{17} | ||||

Resource utilization rate | ||||||

Geological cut-off grade of Cu | 7 | 1.32481 × 10^{9} | 189,258,746.7 | 2543.23 | 0 | 54.19 |

Minimum industrial grade of Cu | 7 | 6.64489 × 10^{8} | 94,926,983.4 | 1275.61 | 0 | 27.18 |

Loss ratio of Cu | 10 | 6.50715 × 10^{8} | 65,071,498.6 | 874.42 | 0 | 18.63 |

Error | 75 | 5.58125 × 10^{6} | 74,416.7 | |||

Total | 99 | 5.35262 × 10^{9} |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Gu, X.; Liu, Z.; Wang, Q.; Xu, X.; Zheng, M.
Production Process Optimization of Metal Mines Considering Economic Benefit and Resource Efficiency Using an NSGA-II Model. *Processes* **2018**, *6*, 228.
https://doi.org/10.3390/pr6110228

**AMA Style**

Wang X, Gu X, Liu Z, Wang Q, Xu X, Zheng M.
Production Process Optimization of Metal Mines Considering Economic Benefit and Resource Efficiency Using an NSGA-II Model. *Processes*. 2018; 6(11):228.
https://doi.org/10.3390/pr6110228

**Chicago/Turabian Style**

Wang, Xunhong, Xiaowei Gu, Zaobao Liu, Qing Wang, Xiaochuan Xu, and Minggui Zheng.
2018. "Production Process Optimization of Metal Mines Considering Economic Benefit and Resource Efficiency Using an NSGA-II Model" *Processes* 6, no. 11: 228.
https://doi.org/10.3390/pr6110228