Comprehensive Utilization of Mineral Resources: Optimal Blending of Polymetallic Ore Using an Improved NSGAIII Algorithm
Abstract
:1. Introduction
 (1)
 A multiobjective ore blending mathematical model for the polymetallic mineral is established. The grade of several metals and the grindability of the ore are considered objectives. The oxidation rate and harmful substances that affect the recovery rate of the mineral beneficiation are considered the constraints. These factors are not considered by the existing traditional ore blending model, but they are a confirmed impact on beneficiation [11,14].
 (2)
 A novel algorithm is proposed. To solve the multiobjective ore blending mathematical model, a novel algorithm is proposed based on the NSGAIII algorithm which is improved by using a mating pool mechanism, a general normalization process, and the referencepoint generation method, named as constrained NSGAIII algorithm based on a matching mechanism (CMNSGAIII).
 (3)
 The model and CMNSGAIII are applied to obtain a daily ore blending plan for a large openpit mine in China. We analyze the overall situation of the calculated target values, the target demand of each unloading point, and the finally obtained planning. The results show that the CMNSGAIII scheme can obtain a satisfactory ore blending plan.
2. Literature Review of Ore Blending
3. Mathematical Model of Polymetallic Ore Blending Schedule Problem
3.1. Problem Assumption and Description
 In a mining face (or a loading point), the various components in the geology are uniform distribution. We can get the average value of each component through laboratory tests and calculations. The type of rock at a mining face (or a loading point) is unique.
 The implementation of the ore blending is strictly in accordance with the ore blending schedule.
 The relationship between the beneficiation recovery rate and the homogeneous ore has been established by laboratory tests. As long as the ore is blended to the required quality, the best beneficiation recovery rate can be obtained.
3.2. Mathematical Formulation of Polymetallic Ore Blending Schedule Problem
4. The Proposed CMNSGAIII Algorithm
4.1. The Framework of the Proposed CMNSGAIII Algorithm
Algorithm 1. Framework of CMNSGAIII 
Input: N (population size) Output: approximated Paretooptimal front
Repeat ${S}_{t}={S}_{t}$∪${F}_{i}$ and i = i + 1 until ${S}_{t}$ ≥ N, Last front to be included: ${F}_{l}={F}_{i}$ if ${S}_{t}$ = N then, ${P}_{t+1}={S}_{t}$, break else ${P}_{t+1}={{{\displaystyle \cup}}^{\text{}}}_{\mathrm{j}=1}^{\mathrm{l}1}{F}_{j}$
Normalize objectives: ${{S}^{\prime}}_{t}$ = Normalization (${S}_{t}$) Association: ($\mathsf{\pi}\left(\mathrm{s}\right),\mathrm{d}\left(\mathrm{s}\right)$) = Associate (W, ${S}_{t}$) %$\text{}\mathsf{\pi}\left(\mathrm{s}\right)$: closest reference point, d: distance between s and π(s) Compute niche count: ${\rho}_{j}={\displaystyle \sum}_{\mathrm{s}\in {S}_{t}/{F}_{l}}\left(\left(\mathsf{\pi}\left(\mathrm{s}\right)=\mathrm{j}\right)?\text{}1\text{}:\text{}0\right)$, j ∈ W Choose K members one at a time from ${F}_{l}$ to construct ${P}_{t+1}$: Niching (K, ${\rho}_{j}$, π, d, W, ${F}_{l}$, ${P}_{t+1}$) Transformation reference point set W: W = ReferencePointTransformation (W) end if 
4.2. ASF MatingPool Scheme and Normalization
4.2.1. ASF MatingPool Scheme
Algorithm 2. ASF MatingPool (${P}_{t}$) 
%$ASF\_rank\left(i\right)$: the maximum ASF value of solution i, $\mathsf{\epsilon}={10}^{4}$

4.2.2. Normalization
4.3. ReferencePoint Generation and Transformation
4.4. Constraints Handling
5. Case Study
5.1. The Situation of the Polymetallic OpenPit Mine and Experimental Data
5.2. Parameter Settings and Experimental Environment
5.3. Results
5.4. Discussion
6. Conclusions
 There is a conflict between the various metals in the ore when the associated metal ores are blended, i.e., for the grade of one metal in the blended ore to meet its beneficiation grade, the grade of the other metal in the ore will deviate from its beneficiation grade. So, the blending of associated ores is a multiobjective optimization problem, and it cannot be converted into a singleobjective optimization problem by assigning weights to each objective. If it is transformed into a singleobjective optimization problem, the optimal result will be a Pareto local optimum solution.
 Through the experiment, compared with the NSGAIII algorithm, the result obtained by the CMNSGAIII algorithm is better. It shows that the CMNSGAIII algorithm has better optimization solving performance. Moreover, the proposed model and algorithm can provide different optimal ore blending schemes available for the production of the mine and ensure a relatively minimal deviation from the required target by testing on the case.
 When the quality of the ore is not strictly required, i.e., it is allowed to fluctuate within a certain range, the method proposed in this paper can provide a theoretical basis for determining the fluctuation range of various metals in the ore.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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$C{a}_{j}$  Capacity at the jth unloading point (crushing station or stockpiling station). 
${E}_{s}$  Production capacity of each shovel in each production cycle. 
${E}_{v}$  Transportation capability of each truck in each production cycle. 
${g}_{ki}$  Ore grade of the kth metal at the ith loading point (or mining face). 
${G}_{kj}$  Target ore grade of the kth metal at the jth crushing station. 
${h}_{S}$  Percentage of the sth harmful substance. 
${h}_{S\_i}$  Percentage of the sth harmful substance at the ith loading point (or mining face). 
$K$  The number of metal species. 
$M$  The number of loading points (or mining faces). 
$NC$  The number of crushing stations. 
$NS$  The number of stockpiling stations. 
${N}_{s}$  The number of shovels in the openpit mine. 
${N}_{v}$  The number of trucks in the openpit mine. 
${O}_{i}$  Oxidation rate at the ith loading points (or mining faces). 
${O}_{j}$  Allowable oxidation rate at the jth unloading points (crushing station or stockpiling station). 
${P}_{i}$  The minimum required production of the ith ore loading point (or mining face). 
${P}_{f\_i}$  The maximum quantity of ore at the ith ore loading point (or mining face). 
$R$  The number of types of rock. 
${R}_{n}$  The number of loading points (or mining face) of the rth type of rock. 
${R}_{j}^{NC}$  The minimum production of the jth crushing station. 
${T}_{r}$  The minimum ore transportation volume from the ith loading point (or mining face) to the jth unloading point (crushing station or stockpiling station) 
${\mu}_{rj}$  Target percentage of the rth type of rock at the jth crushing station. 
${x}_{ij}$  Quantity from the ith loading point (or mining face) to the jth unloading point (crushing station or stockpiling station). 
Loading Point  Ore Quantity (ton)  Mo Grade (%)  W Grade (%)  Cu Grade (%)  Oxidation Rate (%)  Rock Type 

No. 1  6000  0.083  0.118  0.006  0.204  skarn 
No. 2  3500  0.098  0.034  0.010  0.058  skarn 
No. 3  4800  0.170  0.134  0.017  0.094  skarn 
No. 4  5000  0.055  0.017  0.006  0.046  halleflinta 
No. 5  6000  0.087  0.157  0.010  0.250  skarn 
No. 6  5500  0.114  0.092  0.010  0.074  gillebackite 
No. 7  6000  0.095  0.081  0.015  0.080  skarn 
No. 8  4000  0.066  0.069  0.017  0.197  skarn 
No. 9  4600  0.095  0.081  0.015  0.080  skarn 
No. 10  5000  0.075  0.021  0.009  0.028  halleflinta 
No. 11  8400  0.118  0.157  0.021  0.220  skarn 
No. 12  6800  0.093  0.177  0.020  0.283  skarn 
No. 13  7000  0.066  0.086  0.017  0.197  skarn 
No. 14  6000  0.069  0.079  0.009  0.157  gillebackite 
Unloading Point  Capacity  Mo Grade (%)  W Grade (%)  Cu Grade (%)  Upper Oxidation Rate (%)  

Lower  Upper  
1#  4150  4400  0.085  0.065  0.011  0.1 
2#  5950  6200  0.095  0.103  0.013  0.147 
3#  6550  7200  0.095  0.103  0.013  0.147 
4#  3400  4000  0.088  0.068  0.011  0.104 
Parameters  NSGAIII  CMNSGAIII 

Initial population size  n = 84  n = 84 
Reference points  H = 78  H = 81 
Maximum iterations  gen = 100,000  gen = 100,000 
Decision variable  D = 56  D = 56 
Crossover probability  ${P}_{c}$ = 1  ${P}_{c}$ = 1 
Mutation probability  ${P}_{m}=1/D$  ${P}_{m}=1/D$ 
Distribution index  η = 20  η = 20 
Unloading Point  Loading Point (ton)  Total  Ore Grade (%)  

1  2  3  4  5  6  7  8  9  10  11  12  13  14  Mo  W  Cu  
1#  0  494  0  309  0  658  0  0  0  1728  0  750  0  311  4249  0.0850  0.0650  0.0110 
2#  947  310  303  0  0  795  315  0  570  483  678  653  389  662  6105  0.0956  0.1025  0.0130 
3#  623  301  785  710  513  436  965  310  0  0  616  472  311  729  6771  0.0956  0.1013  0.0130 
4#  537  1067  0  332  0  0  555  0  308  380  0  303  0  0  3480  0.0879  0.0680  0.0110 
Total  2107  2171  1088  1351  513  1889  1835  310  877  2591  1294  2178  701  1701  20,605       
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Chen, L.; Gu, Q.; Wang, R.; Feng, Z.; Zhang, C. Comprehensive Utilization of Mineral Resources: Optimal Blending of Polymetallic Ore Using an Improved NSGAIII Algorithm. Sustainability 2022, 14, 10766. https://doi.org/10.3390/su141710766
Chen L, Gu Q, Wang R, Feng Z, Zhang C. Comprehensive Utilization of Mineral Resources: Optimal Blending of Polymetallic Ore Using an Improved NSGAIII Algorithm. Sustainability. 2022; 14(17):10766. https://doi.org/10.3390/su141710766
Chicago/Turabian StyleChen, Lu, Qinghua Gu, Rui Wang, Zhidong Feng, and Chao Zhang. 2022. "Comprehensive Utilization of Mineral Resources: Optimal Blending of Polymetallic Ore Using an Improved NSGAIII Algorithm" Sustainability 14, no. 17: 10766. https://doi.org/10.3390/su141710766