Overall Dynamic Optimization of Metal Mine Technical Indicators Considering Spatial Distribution of Ore Grade Using AADE

In optimizing the production of a metal mine, either the overall dynamic relations between technical indicators or the spatial distribution of the ore grade are usually considered, but few studies have considered both factors together. These two factors in combination have a greater effect on the optimization of mine production in terms of economic benefit and resource utilization than they do individually. We proposed an overall dynamic optimization model of technical indicators of metal mine production that considers the spatial distribution of the ore grade to better optimize the technical indicators and improve sustainable development of mineral resources. We incorporated an adaptive mutation strategy and adaptive control parameters into a differential evolution algorithm (AADE) in order to overcome the drawbacks of the differential evolution algorithm in solving this optimization model. The adaptive mutation strategy and adaptive control parameters were used to increase the rate of convergence and improve the search for a global maximum. To assess the performance of AADE, we used a real case and four test functions (the Sphere, Griewank, Rastrigin and Rosenbrock functions) in tests that compared AADE with a standard genetic algorithm, a standard differential evolution algorithm and the recently developed adaptive differential evolution algorithm. The results indicate that the optimization model we created is better aligned with mine production processes than current optimization models. In optimizing the technical indicators of metal mine production to maximize economic benefits, AADE performed significantly better than the other three algorithms tested in terms of convergence rate and global search ability.

the security of such resources has great strategic value. The 23 rapid development of China's economy has resulted in tens 24 of thousands of mines having been built, and the quantity of 25 solid minerals mined in China is the highest in the world. 26 The associate editor coordinating the review of this manuscript and approving it for publication was Ehab Elsayed Elattar .
However, in a time of product shortages in China, the pur- 27 suit of high quantities of mineral resources and increased 28 development speed have resulted in a wide range of produc-29 tion methods, low levels of technology use, and insufficient 30 investment in and attention to the sustainable development of 31 mineral resources. Consequently, mineral resources are not 32 fully utilized, and waste is a serious problem. It is therefore 33 essential to develop efficient mining techniques for mineral 34 resources. 35 The term ''mineral resources'' refers to geological bodies 36  Such an approach must consider two aspects, the overall 94 dynamic relations between the technical indicators and the 95 spatial distribution of the ore grade. 96 The production of metal mines is a multi-factor, multi-97 level, multi-constrained, complex dynamic process, so the 98 optimization of technical indicators is a complex nonlinear 99 optimization problem. It has been well documented that stan-100 dard optimization methods cannot easily solve complex non-101 linear optimization problems [8], [9], [10]. Many intelligent 102 evolutionary methods have been proposed to solve complex 103 nonlinear optimization problems, such as the genetic algo-104 rithm (GA) [11], the particle swarm algorithm (PS) [12], [13] 105 and the differential evolution algorithm (DE) [14], [15]. DE is 106 an easy-to-use algorithm that has few control parameters, is 107 low in computational complexity, and shows good conver-108 gence. It is therefore used to solve many complex nonlinear 109 optimization problems [16], [17], [18]. However, the standard 110 DE algorithm has two drawbacks when used to optimize the 111 overall dynamic behavior of technical indicators of metal 112 mine production, considering the ore grade distribution. 113 First, the mutation strategy, determination of which is the 114 most important step in this algorithm, has a great effect 115 on the performance of DE. The mutation strategies used in 116 standard DE algorithms are DE/rand/ * and DE/best/ * [19]. 117 Many studies have shown that DE/rand/ * has a global search 118 capability but converges slowly. DE/best/ * , in contrast, con-119 verges rapidly but has a tendency to fix on a local optimum 120 [20], [21].

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Second, DE has two important control parameters that 122 affect its accuracy and its convergence rate: the scale factor 123 F and the crossover rate CR [22], [23]. In the standard DE 124 algorithm, the values of these two control parameters are 125 preset and remain unchanged during evolution [24]. However, 126 researchers have found that the optimal control parameter 127 values are generally different for various problems or even 128 for various evolutionary stages of the same problem [25]. It is 129 therefore difficult to determine the optimal control parameter 130 values to solve the overall dynamic optimization problem 131 for metal mine technical indicators when considering the ore 132 grade distribution.

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To overcome the preceding drawbacks of the DE, we intro-134 duced an adaptive mutation strategy and adaptive control 135 parameters into the standard DE algorithm and designed an 136 adaptive mutation operator and adaptive control parameter for 137 DE (AADE) to be incorporated into the algorithm in order to 138 solve the overall dynamic optimization problem. Mine production consists of a geological process, a mining 145 process, and a beneficiation process, each of which has main 146 technical indicators, as shown in Figure 1; the technical 147  indicators are defined or described in detail in Table 1. 148 We model the three processes in the following sections.
where p a and p b are respectively the original boundary grade 158 and original industrial grade; Q 0 is the geological reserve 159 calculated using mining software with boundary grade and 160 industrial grade p a and p b ; ϕ(x) is the probability function 161 that the ore with a grade between the boundary grade and 162 industrial grade is mined; z is a constant value, depending on 163 the geological properties of the ore; g(x) is a function of ore 164 weight and grade; and c(x) is the probability density function 165 of the ore grade distribution.

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The ore characteristics and mining methods of each mining 168 area are basically the same for an individual mine. In this case, 169 there was some correlation between the dilution rate and the 170 loss rate in the ore production process [1], and the relationship 171 model is: 172 The dilution rate is the ratio of the difference between the 174 ore average grade and the mining grade to the ore average 175 grade: Rearranging equation (5), the mining grade can be calcu-178 lated by equation (6): Depending on the amount of metal conserved during min-181 ing, it can be calculated that: Combining equations (6) and (7), the equation to calculate 184 the mining amount is: The concentration ratio is the ratio of the mining amount to 188 the concentrate amount: For an individual mine, the characteristics of the min-191 ing ore, the processing technology, the equipment, and the 192 agent can be assumed to be constant. In this case, there 193 are relationships between ore concentration ratio and mining 194 grade, between concentrate grade and mining grade, between 195 concentrate grade and ore concentration ratio, and between 196 concentrate grade and concentrate selling price [26], [27], 197 [28], [29], [30]. These relationships are modeled by equations 198 (10-12):  on the economic benefit is therefore: where θ is the total net present value; θ i is the net present

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(1) Calculate the mining time t i of optimized zone i: where Q 2,i is the total mining quantity of optimized zone i, 242 and Qz is the annual production capacity.

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(2) The mining start time T 1,i of optimized zone i is the 244 sum of the mining times of each previously optimized zone, 245 and is calculated by equation (15):

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(3) The mining end time T 2,i of optimized zone i is: (4) The total profit G i of optimized zone i is: where Q 3,i is the concentrate amount in optimized zone i; q i 252 is the selling price of the concentrate in optimized zone i; and 253 h is the total production cost per unit of ore.

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(5) The average annual profit g i of optimized zone i is: (6) Total net present value of optimized zone i is:

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(1) The boundary grade is not higher than the industrial grade: 262 where p 1,i is the boundary grade of optimized zone i, and p 2,i 264 is the industrial grade of optimized zone i.

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(2) The concentrate grade is not less than the minimum 266 smelting grade: where p 5,i is the concentrate grade of optimized zone i, and 269 p y is the lowest smelting grade. Combining the above objective function and constraints, the 272 technical indicator optimization model for metal mine pro-273 duction to maximize economic benefit is: proportion factor F or crossover rate CR. 294 We therefore introduced the adaptive mutation operator   The production requirements of the Yinshan copper mine set 315 the boundary grade and the industrial grade of each optimized 316 unit in the range 0.05% to 0.45%. The parameter values for 317 the Yinshan copper mine were as follows. The lower bound 318 and upper bound of the boundary grade and industrial grade 319 were respectively 0.05% and 0.45%; the lower bound of the 320 smelting grade was 16%; the annual production capacity of 321 the mine was 1.5 Mt; the z value in equation (2)  A scatter plot of the orebody weight and copper grade was 329 plotted from the 204 items of orebody weight and grade 330 data collected from the Yinshan copper mine, as shown in 331 Figure 3. There was no correlation between orebody weight 332 and copper grade. The results show that there was no rela-333 tionship between orebody weight and copper grade and that 334 orebody weight was not affected by copper grade. We there-335 fore used the average value of the orebody weight for 336 the ore weight function of the Yinshan copper mine. The 337 VOLUME 10, 2022 mathematical function is: were plotted, as shown in Figure 5. It can be seen from 357 Figure 5 that there was no correlation between copper dilution 358 rate and loss rate. The correlation coefficient between copper 359 dilution rate and loss rate was −0.0771, and the F-test was 360 significant at a level of 0.5449, which was >0.05. There was 361 therefore no relationship between copper dilution rate and 362 loss rate. In the subsequent optimization, the values used for 363 both dilution rate (2%) and loss rate (9%) were the planned 364 values.

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FIGURE 5. Scatter plot of the copper loss rate and dilution rate of the Yinshan copper mine.  functions used in the training function, input layer and output 391 layer were respectively traingdm, tansig and purelin. The 392 learning rate was 0.1, training error accuracy was 0.001, and 393 the maximum number of iterations was 2500. The fitting 394 diagram of copper concentrate grade produced by the BP 395 neural network is shown in Figure 7. Figure 7 shows that for the fitting, the coefficient of deter-397 mination R 2 was 0.939, mean absolute error MAE was 0.258, 398 and root mean square error RMSE was 0.307. R 2 was >0.9, 399 and both MAE and RMSE were <0.5. These results indicate 400 that the BP neural network well fitted the model of the 401 relationships of copper mining grade with concentrate grade 402 and concentrate ratio with concentrate grade in the Yinshan 403 copper mine.

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The market transaction price of copper ore concentrate is 406 based mainly on a 20% concentrate grade (#1 grade), and 407 the price of any other copper ore concentrate grade is a 408 function of this price. The price adjustment factors and the 409 compensation prices are shown in Table 3. The copper ore 410 concentrate price q is calculated by equation (25): where k 1 is the sales price of #1 copper concentrate; λ is the 413 price adjustment coefficient; k 2 is the compensation price.  Table 4.  Table 5.   from each other, indicating that the boundary grades and 462 industrial grades differ between optimized zones. If the spa-463 tial distribution of ore grades is not considered and one overall 464 grade distribution is adopted, the boundary grade and the 465 industrial grade are both the same. Obviously, if the same 466 boundary grade and industrial grade are used in high-grade 467 and low-grade zones, mineral resources will be wasted. Con-468 sidering differences in the spatial distribution of grades pro-469 duces a model that is more in line with actual production and 470 is therefore very important.

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The optimized scheme was compared with the current 472 scheme; the technical index values and net present value of 473 the current scheme are shown in Table 6.   To ensure we were comparing like with like, the parameters 490 of the four algorithms were set to be as consistent as far as 491 possible. The adaptive control parameters ψ, ϕ, δ l and δ u 492 of AADE were respectively set to 0.7, 0.1, 0, and 1. The 493 mutation rate and crossover rate of GA were respectively set 494 to 0.7 and 0.5. The proportion factor F and crossover rate CR 495 of DE were respectively set to 0.7 and 0.5. The distribution 496 function and the adaptive control parameters of ADE were set 497 to be the same as those of AADE. The population size of all 498 four algorithms was set to 50, and the maximum number of 499 iterations was set to 100.

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To avoid any random effects in algorithm operation, the 501 four algorithms were independently run 31 times each, and 502 the total net present value of each run was recorded. The 503 results are shown in Table 7.

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The maximum total net present value, the average total net 505 present value, and the minimum total net present value was 506 calculated for each of each algorithm. The results are shown 507 in Table 8. 508 Table 8 shows that the maximum total net present value 509 (CNY 49 429.32 × 10 4 ), the average total net present value 510 (CNY 49 316.79 × 10 4 ), and the minimum total net present 511 VOLUME 10, 2022    (26):  Table 7 shows that the standard deviation of the AADE 534 solution was 47.31. Using this value and the average total net 535 present value shown in Table 8, the t-statistics for the indi-536 vidual comparisons of AADE with the other three algorithms 537 were calculated using equation (26); the results are shown in 538 Table 9.
539 Table 9 shows that the t-statistic for AADE when compared 540 with each of the other three algorithms was in all cases greater 541 than t 0.025,30 = 2.042. This shows that AADE performed 542 significantly better than GA, DE and ADE in optimizing 543 the technical production in terms of maximizing economic 544 benefit. In the light of these results, we conclude that AADE 545 is more effective and therefore preferable in optimizing the 546 dynamic technical indicators when considering the spatial 547 distribution of the ore grade. This advantage of AADE is 548 mainly due to the adaptive mutation strategy and the use 549 of control parameters, which improves both the convergence 550 rate and the global search capability at the same time.

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We used four commonly used single-objective continuous 553 test functions to test the performance of AADE: the Sphere, 554 Griewank, Rastrigin and Rosenbrock functions. The four test 555 functions can be divided into three categories: the Sphere 556 function is a unimodal convex function; the Griewank and 557 Rastrigin functions are multimodal functions with many local 558 optimum points, and how to step out from a local optimum 559 point to find the global optimum point is a major challenge 560 for any optimization algorithm; and the Rosenbrock function 561 is a non-convex function for which the global minimum is 562 in a flat valley, which makes it very difficult to find the 563 global optimum points. The overall dynamic optimization of 564 the technical indicators basically falls into one of the three 565 categories. We therefore chose the four test functions to assess 566 the performance of AADE.

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Each of the four functions was used as the objective func-568 tion to be minimized; the equations are shown in Table 10. 569 In calculating the minimum value, the variable dimension 570 was 10 dimensions. The three-dimensional graphs of the four 571 functions are shown in Figure 9. 572 AADE was tested using each the four single-objective 573 continuous test functions, as were GA, DE and ADE, and the 574 results were compared. The parameters of the four algorithms 575 were set as follows. The maximum number of iterations was 576 set to 100 (Sphere), 1000 (Griewank), 500 (Rastrigin), and 577 1000 (Rosenbrock). All other parameter values were the same 578 as those described in the preceding section. Each algorithm 579 was independently run 31 times for each test function, and the 580 worst value, mean value and optimum value of each algorithm 581   One-tailed t-tests were performed for the four test func-590 tions to see if the advantage of AADE in solving com-591 plex single-objective continuous optimization problems was 592 significant. The standard deviations of AADE results were 593 0.000 578 (Sphere), 0.053 592 (Griewank), 0.175 768 (Ras-594 trigin) and 6.400 64 × 10 −10 (Rosenbrock). Using these 595 standard deviation values and the average values of the algo-596 rithms, shown in Table 11, the t-statistics for AADE com-597 pared with the other three algorithms were calculated using 598 equation (26); the results are shown in Table 12.      Therefore, the AADE algorithm has wide applicability.

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The results presented above support the following conclu- with the metal mine production process than the cur-616 rently used optimization model.

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2) Optimizing the technical indicators for metal mine 618 production in terms of economic benefit is a complex 619 nonlinear single-objective optimization problem that 620 the AADE algorithm is proposed to solve. AADE uses 621 adaptive control parameters and a mutation strategy 622 based on the DE algorithm to improve the optimal 623 performance of DE.

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3) The total net present value of the optimization scheme 625 is greater than that of the currently used scheme, and 626 the optimized technical production indicators are better 627 aligned with the actual operation of the mine, which 628 should guide mine production and planning. 629 4) AADE performs significantly better than GA, DE and 630 ADE in optimizing the technical indicators of metal 631 mine production in terms of economic benefits. 632 We created an effective optimization model and a method 633 for optimizing metal mine technical indicators that considers 634 the overall dynamic relationships between technical indica-635 tors and the spatial distribution of the ore grade. There are 636 two aspects of the model that can be improved in future as an Associate Professor, in 2021. His research experiences in more than 825 five research projects during the last ten years allow him authoring more 826 than 10 refereed journal articles. His research interests include intelligence 827 optimization and prediction, resource management, and risk assessment of 828 engineering.