# Design of Flotation Circuits Using Tabu-Search Algorithms: Multispecies, Equipment Design, and Profitability Parameters

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Superstructure

#### 2.2. Mathematical Model

^{3}) of flotation stage $i$, ${F}_{i}$ is the feed stream to stage $i$, ${E}_{g}$ is the gas factor, ${\rho}_{p}$ is the pulp density, ${I}_{F,i}$ is the fixed capital cost to stage $i$, ${I}_{F}$ is the fixed capital cost of circuit, and $FL$ is the Lang factor [45]. Equation (10) is valid for volume between 5 m

^{3}and 200 m

^{3}. The working capital costs are estimated with the following equation:

#### 2.3. Optimization Technique: Tabu-Search Algorithm

## 3. Applications

#### 3.1. Maximization of Revenues

^{3}, 22.63 m

^{3}, 9.98 m

^{3}, 167.59 m

^{3}, and 155.91 m

^{3}, respectively. The final concentrate of the circuit was 14.962 ton/h of chalcopyrite fast, 7.916 ton/h of chalcopyrite slow, 4.938 ton/h of chalcocite fast, 2.633 ton/h of chalcocite slow, 0.008 ton/h of pyrite, 0.017 ton/h of silica, and 0.001 ton/hr of gangue, and its copper grade was 25.70%.

#### 3.2. Maximization of the Net Present Worth

^{3}, 19.84 m

^{3}, 9.60 m

^{3}, 95.72 m

^{3}, and 91.41 m

^{3}, respectively. The final concentrate of circuit contained 14.846 ton/h of chalcopyrite fast, 7.757 ton/h of chalcopyrite slow, 4.739 ton/h of chalcocite fast, 2.164 ton/h of chalcocite slow, 0.009 ton/h of pyrite, 0.019 ton/h of silica, and 0.001 ton/h of gangue, with a copper grade of 26.07%.

#### 3.3. Benchmarking between the Tabu-Search Algorithm and the Baron Solver

#### 3.4. Comparison with Another Approach

## 4. Discussion and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$C1$ | Cleaner stage |

$C2$ | Re-cleaner stage |

${C}_{i}$ | Concentrate stream of the flotation stage $i$ |

${C}_{ik}$ | Mass flow of species $k$ in concentrate ${C}_{i}$ |

${C}_{ijk}$ | Mass flow of species $k$ in the concentrate stream from stage $j$ to stage $i$ |

${C}_{op,i}$ | Operating cost of flotation stage $i$ |

$D$ | Annual depreciation |

${E}_{g}$ | Gas factor |

${F}_{C}$ | Annual cash flows |

$FL$ | Lang factor |

$F{L}_{w}$ | Lang factor for working capital |

$FM$ | Frequency matrix |

$g$ | Grade |

$H$ | Number of hours per year of plant operation |

${I}_{cap}$ | Capital cost |

${I}_{F}$ | Fixed capital cost |

${I}_{w}$ | Working capital cost |

${F}_{ik}$ | Mass flow of species $k$ in feed streams of stage $i$ |

${k}_{max,i,k}$ | Maximum rate constant of the species $k$ in flotation stage $i$ |

${M}_{1k}$ | Mass flow of species $k$ fed to the flotation circuit |

${N}_{i}$ | Number of flotation cell in stage $i$ |

$N\left(x\right)$ | Neighborhood of $x$ |

$n$ | Life time of the project |

${n}_{r}$ | Number of rows of Tabu list |

$P$ | Final concentrate |

${P}_{B}$ | Profits before taxes |

${P}_{k}$ | Kilowatt-hours cost |

$p$ | Fraction of metal paid |

$R$ | Rougher stage |

${R}_{ik}$ | Recovery of stage $i$ for species $k$ |

${R}_{max,i,k}$ | Maximum recovery at infinite time of stage $i$ for species $k$ |

$Rfc$ | Refinery charge |

${r}_{t}$ | Tax rate |

$S1$ | Scavenger stage |

$S2$ | Re-scavenger stage |

${T}_{i}$ | Tail stream of the flotation stage $i$ |

${T}_{ik}$ | Mass flow of the species $k$ in tail ${T}_{i}$ |

${T}_{ijk}$ | Mass flow of species $k$ in the tail stream from stage $j$ to stage $i$ |

$TL$ | Tabu list |

$Trc$ | Treatment charge |

${V}_{i}$ | Cell volumen in stage $i$ |

$W$ | Final tail |

${W}_{NP}$ | Net present worth |

${x}_{best}$ | Best neighbor of $N\left(x\right)$ |

Greek symbols | |

${\alpha}_{ij}$ | Decision variables |

${\beta}_{ij}$ | Decision variables |

$\gamma $ | Penalty parameter |

${\rho}_{p}$ | Pulp density |

$\mu $ | Grade deduction |

${\tau}_{i}$ | Cell residence time in stage $i$ |

## Appendix A

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**Figure 1.**Equipment superstructure. R: rougher stage, C1: cleaner stage, C2: re-cleaner stage, S1: scavenger stage, S2: re-scavenger stage, M: stream mixer, D: stream splitter, ${\alpha}_{ij}\in \left\{0,1\right\}$ decision variables indicating the destination of the concentrate stream from stage $i$, ${\beta}_{ij}\in \left\{0,1\right\}$ decision variables indicating the destination of the tail stream from stage $i$.

**Figure 2.**Block diagram of Tabu-search algorithm. ITE: iteration of algorithm, TL: Tabu list, FM: frequency matrix, D: iteration related to diversification, I: iteration related to intensification, ${\left({\alpha}_{ij},\text{}{\beta}_{ij}\right)}_{i,j}$: circuit structure, ${\left({N}_{t},\text{}{\tau}_{t}\right)}_{t}$: number of equipment and operating conditions of circuit, $N\left(x\right)$: neighborhood of the solution $x$, ${x}_{best}$: best neighbor of $N\left(x\right)$, $f$: objective function, ${D}_{max}:$ number of iteration related to the implementation of diversification, ${I}_{max}:$ number of iteration related to the implementation of intensification.

**Figure 5.**Structures of circuits obtained by maximization of net present worth (

**a**,

**c**) or revenues (

**c**,

**d**,

**i**) in the case 3.3. Structures (

**a**,

**b**,

**e**–

**h**) are obtained by maximization revenues in the case 3.4.

Species | Copper Grade wt % | Feed (t/h) |
---|---|---|

Chalcopyrite fast (Cpf) | 0.35 | 15 |

Chalcopyrite slow (Cpy) | 0.25 | 8 |

Chalcocite fast (Cf) | 0.1 | 5 |

Chalcocite slow (Cs) | 0.07 | 3 |

Pyrite (P) | 0.0 | 4 |

Silica (S) | 0.0 | 200 |

Gangue (G) | 0.0 | 300 |

${\mathit{k}}_{\mathit{m}\mathit{a}\mathit{x},\mathit{i},\mathit{k}}$ | ${\mathit{R}}_{\mathit{m}\mathit{a}\mathit{x},\mathit{i},\mathit{k}}$ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Stage\Species | Cpf | Cpy | Cf | Cs | P | S | G | Cpg | Cpy | Cf | Cs | P | S | F |

R | 1.85 | 1.50 | 1.00 | 0.70 | 0.80 | 0.60 | 0.30 | 0.90 | 0.85 | 0.85 | 0.75 | 0.80 | 0.60 | 0.20 |

C1 | 1.30 | 1.00 | 0.80 | 0.40 | 0.70 | 0.30 | 0.20 | 0.75 | 0.70 | 0.70 | 0.60 | 0.60 | 0.50 | 0.15 |

C2 | 1.30 | 1.00 | 0.80 | 0.40 | 0.70 | 0.30 | 0.20 | 0.70 | 0.65 | 0.65 | 0.50 | 0.60 | 0.50 | 0.15 |

S1 | 1.85 | 1.50 | 1.00 | 0.70 | 0.80 | 0.60 | 0.30 | 0.90 | 0.85 | 0.85 | 0.75 | 0.80 | 0.60 | 0.20 |

S2 | 1.85 | 1.50 | 1.00 | 0.70 | 0.80 | 0.60 | 0.30 | 0.90 | 0.85 | 0.85 | 0.75 | 0.80 | 0.60 | 0.20 |

**Table 3.**Benchmarking between the Tabu-search algorithm and the Baron Solver (revenues, Bof = bornite fast).

Case | 1 | 2 | 3 | 4 | 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Algorithm | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu search | Baron |

Species | Cpf, Cps, S, G | Cpf, Cps, P, S, G | Cpf, Cps, Cf, P, S, G | Cpf, Cps, Cf, Cs, P, S, G | Cpf, Cps, Cf, Cs, Bof, P, S, G | |||||

Revenue, USD/year | 132,318,860 | 132,323,459 | 132,852,410 | 132,854,057 | 130,981,546 | 130,981,549 | 130,960,079 | 130,958,748 | 129,185,242 | 130,335,529 |

Net present worth, USD | 612,961,580 | 612,702,132 | 618,316,980 | 617,392,845 | 598,070,573 | 598,081,544 | 595,475,455 | 599,377,620 | 671,113,149 | 692,374,613 |

Profit before taxes, USD/year | 94,235,451 | 94,208,659 | 94,972,404 | 94,849,038 | 92,078,958 | 92,080,433 | 91,716,855 | 92,213,521 | 101,068,398 | 103,931,519 |

Total capital investment, USD | 52,137,292 | 52,317,036 | 51,261,610 | 51,471,333 | 52,917,663 | 52,915,338 | 53,265,087 | 52,005,813 | 24,608,590 | 20,166,465 |

Total annual cost, USD/year | 35,259,307 | 35,280,960 | 35,103,337 | 35,216,989 | 36,036,215 | 36,034,869 | 36,358,032 | 35,928,246 | 26,783,878 | 25,311,660 |

${V}_{R}$, ${\mathrm{m}}^{3}$ | 182.822 | 182.54 | 181.825 | 185.596 | 195.500 | 195.46 | 197.704 | 197.60 | 119.350 | 111.037 |

${V}_{C1}$, ${\mathrm{m}}^{3}$ | 21.222 | 20.31 | 22.805 | 22.872 | 22.442 | 22.44 | 22.712 | 22.63 | 25.841 | 20.083 |

${V}_{C2}$, ${\mathrm{m}}^{3}$ | 7.761 | 7.15 | 8.973 | 8.800 | 9.964 | 9.96 | 9.984 | 9.98 | 14.138 | 10.287 |

${V}_{S1}$, ${\mathrm{m}}^{3}$ | 163.211 | 163.17 | 163.697 | 163.700 | 165.782 | 165.78 | 167.565 | 167.59 | 159.679 | 101.602 |

${V}_{S2}$, ${\mathrm{m}}^{3}$ | 153.893 | 153.88 | 153.971 | 153.972 | 154.464 | 154.46 | 155.908 | 155.91 | 151.817 | 155.590 |

${\tau}_{R}$, min | 5.000 | 5.000 | 4.900 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 3.300 | 3.000 |

${\tau}_{C1}$, min | 3.130 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 4.300 | 3.000 |

${\tau}_{C2}$, min | 3.320 | 3.000 | 3.070 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 4.400 | 3.000 |

${\tau}_{S1}$, min | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 4.400 | 3.000 |

${\tau}_{S2}$, min | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 | 4.800 | 4.920 |

${N}_{R}$ | 15.000 | 15.000 | 14.000 | 14.000 | 15.000 | 15.000 | 15.000 | 14.000 | 3.000 | 3.000 |

${N}_{C1}$ | 4.000 | 5.000 | 5.000 | 5.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 |

${N}_{C2}$ | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 |

${N}_{S1}$ | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 9.000 | 3.000 |

${N}_{S2}$ | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 | 10.000 | 13.000 |

Grade Cu | 0.3123 | 0.312 | 0.274 | 0.274 | 0.257 | 0.257 | 0.257 | 0.257 | 0.250 | 0.250 |

Circuit structure | Figure 4 | Figure 4 | Figure 4 | Figure 4 | Figure 3 | Figure 3 | Figure 3 | Figure 3 | Figure 5d | Figure 5c |

Time, s | 46.610 | 233.000 | 176.580 | 423.030 | 287.670 | 9026.190 | 601.630 | 14,640.00 | 433.930 | 10,257.020 |

Iterations of algorithm | 1000 | - | 1000 | - | 1500 | - | 2000 | - | 3000 | - |

Neighborhood size | 40 | - | 130 | - | 150 | - | 170 | - | 60 | - |

Iterations of diversification | 50 | - | 20 | - | 30 | - | 30 | - | 20 | - |

Iteration of intensification | 10 | - | 40 | - | 40 | - | 50 | - | 50 | - |

No. rows of Tabu list | 100 | - | 50 | - | 50 | - | 50 | - | 50 | - |

**Table 4.**Benchmarking between the Tabu-search algorithm and the Baron solver (net present worth; Bof = bornite fast, Bos = bornite slow).

Case | 1 | 2 | 3 | 4 | 5 | 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Algorithm | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu | Baron |

Species | Cpf, Cps, S, G | Cpf, Cps, P, S, G | Cpf, Cps, Cf, P, S, G | Cpf, Cps, Cf, Cs, P, S, G | Cpf, Cps, Cf, Cs, Bof, P, S, G | Cpf, Cps, Cf, Cs, Bof, Bos, P, S, G | ||||||

Net present worth, USD | 735,935,440 | 735,938,754 | 737,697,720 | 737,697,912 | 726,139,400 | 726,139,411 | 724,828,320 | 724,828,404 | 723,842,180 | 723,842,865 | 719,710,450 | 719,712,493 |

Revenue, USD/year | 130,918,800 | 130,920,451 | 131,435,920 | 131,433,456 | 129,853,889 | 129,853,889 | 129,830,340 | 129,830,357 | 129,839,150 | 129,840,240 | 129,548,420 | 129,551,233 |

Profit before taxes, USD/year | 109,609,160 | 109,606,145 | 109,882,230 | 109,881,842 | 108,168,025 | 108,168,032 | 107,977,400 | 107,977,414 | 107,833,120 | 107,833,404 | 107,267,450 | 107,268,177 |

Total capital inv., USD | 8,162,377 | 8,108,261 | 8,345,130 | 8,338,775 | 8,329,184 | 8,329,184 | 8,386,189 | 8,386,191 | 8,415,244 | 8,418,023 | 9,135,362 | 9,141,936 |

Total annual cost, USD/year | 20,867,504 | 20,875,109 | 21,101,667 | 21,099,930 | 21,234,694 | 21,234,694 | 21,398,690 | 21,398,690 | 21,550,197 | 21,550,859 | 21,786,145 | 21,787,868 |

${V}_{R}$, ${\mathrm{m}}^{3}$ | 101.244 | 101.240 | 103.058 | 103.070 | 105.819 | 105.820 | 106.847 | 106.850 | 111.845 | 111.807 | 111.390 | 111.325 |

${V}_{C1}$, ${\mathrm{m}}^{3}$ | 15.985 | 17.190 | 18.905 | 18.700 | 19.595 | 19.600 | 19.847 | 19.840 | 25.510 | 25.520 | 21.292 | 21.620 |

${V}_{C2}$, ${\mathrm{m}}^{3}$ | 6.619 | 8.390 | 8.112 | 8.110 | 9.578 | 9.580 | 9.600 | 9.600 | 10.266 | 10.412 | 10.252 | 10.260 |

${V}_{S1}$, ${\mathrm{m}}^{3}$ | 93.280 | 93.280 | 93.795 | 93.800 | 94.817 | 94.820 | 95.722 | 95.720 | 97.236 | 97.225 | 99.287 | 99.265 |

${V}_{S2}$, ${\mathrm{m}}^{3}$ | 90.073 | 90.070 | 90.219 | 90.220 | 90.615 | 90.620 | 91.412 | 91.410 | 91.901 | 91.898 | 92.295 | 92.322 |

${\tau}_{R}$, min | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 |

${\tau}_{C1}$, min | 3.000 | 3.200 | 3.080 | 3.040 | 3.000 | 3.000 | 3.000 | 3.000 | 3.960 | 3.973 | 3.200 | 3.256 |

${\tau}_{C2}$, min | 3.000 | 3.730 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.051 | 3.000 | 3.000 |

${\tau}_{S1}$, min | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 |

${\tau}_{S2}$, min | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.001 |

${N}_{R}$ | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 |

${N}_{C1}$ | 4.000 | 4.000 | 4.000 | 4.000 | 4.000 | 4.000 | 4.000 | 4.000 | 3.000 | 3.000 | 3.000 | 3.000 |

${N}_{C2}$ | 4.000 | 3.000 | 4.000 | 4.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 |

${N}_{S1}$ | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 4.000 | 4.000 |

${N}_{S2}$ | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 | 3.000 |

Grade Cu | 0.3125 | 0.3130 | 0.276 | 0.276 | 0.2608 | 0.261 | 0.2607 | 0.261 | 0.250 | 0.250 | 0.250 | 0.250 |

Circuit structure | Figure 5a | Figure 5a | Figure 5a | Figure 5a | Figure 4 | Figure 4 | Figure 4 | Figure 4 | Figure 3 | Figure 3 | Figure 5c | Figure 5c |

Time, s | 106.260 | 94.200 | 424.720 | 294.920 | 464.990 | 533.400 | 527.610 | 355.200 | 710.040 | 1082.640 | 875.980 | 4299.860 |

No. rows Tabu list | 50 | - | 50 | - | 50 | - | 50 | - | 50 | - | 50 | - |

No. iterations of algorithm | 2000 | - | 2000 | - | 2000 | - | 2000 | - | 2000 | - | 2000 | - |

Iteration of diversification | 30 | - | 30 | - | 30 | - | 30 | - | 30 | - | 30 | - |

Iteration of intensification | 50 | - | 40 | - | 40 | - | 50 | - | 50 | - | 50 | - |

Neighborhood size | 70 | - | 130 | - | 150 | - | 170 | - | 190 | - | 210 | - |

**Table 5.**Benchmarking between the Tabu algorithm and the Baron solver (revenues; reduced mathematical model; Pf = pyrite fast, Ps = pyrite slow).

Case | 1 | 2 | 3 | 4 | 5 | 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Algorithm | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu | Baron | Tabu | Baron |

Species | Cpf, S, G | Cpf, Cps, S, G | Cpf, Cps, Pf, S, G | Cpf, Cps, Pf, Ps, S, G | Cpf, Cps, Cf, Pf, Ps, S, G | Cpf, Cps, Cf, Cs, Pf, Ps, S, G | ||||||

Revenue, USD/year | 26,455,569 | 26,455,571 | 38,755,316 | did not converge after 5 days | 38,568,418 | did not converge after 5 days | 38,567,734 | did not converge after 5 days | 36,714,037 | did not converge after 5 days | There is no solution | There is no solution |

${\tau}_{R}$, min | 5.000 | 5.000 | 5.000 | - | 5.000 | - | 5000 | - | 3.260 | - | - | - |

${\tau}_{C1}$, min | 3.000 | 3.000 | 3.090 | - | 3.000 | - | 3000 | - | 3.000 | - | - | - |

${\tau}_{C2}$, min | 3.000 | 3.000 | 3.000 | - | 3.000 | - | 3000 | - | 3.000 | - | - | - |

${\tau}_{S1}$, min | 5.000 | 5.000 | 5.000 | - | 5.000 | - | 5000 | - | 5.000 | - | - | - |

${\tau}_{S2}$, min | 5.000 | 5.000 | 5.000 | - | 5.000 | - | 5000 | - | 3.800 | - | - | - |

${N}_{R}$ | 15.000 | 15.000 | 15.000 | - | 15.000 | - | 15,000 | - | 5.000 | - | - | - |

${N}_{C1}$ | 5.000 | 5.000 | 3.000 | - | 4.000 | - | 4000 | - | 3.000 | - | - | - |

${N}_{C2}$ | 3.000 | 3.000 | 3.000 | - | 3.000 | - | 3000 | - | 3.000 | - | - | - |

${N}_{S1}$ | 15.000 | 15.000 | 15.000 | - | 15.000 | - | 15,000 | - | 6.000 | - | - | - |

${N}_{S2}$ | 15.000 | 15.000 | 15.000 | - | 15.000 | - | 15,000 | - | 6.000 | - | - | - |

Grade Cu | 0.345 | 0.345 | 0.304 | - | 0.303 | - | 0.303 | - | 0.250 | - | - | - |

Circuit structure | Figure 3 | Figure 3 | Figure 3 | - | Figure 4 | - | Figure 4 | - | Figure 5i | - | - | - |

Time, s | 54.900 | 1961.830 | 120.040 | - | 165.340 | - | 216.390 | - | 472.720 | - | - | - |

No. rows Tabu list | 100 | - | 50 | - | 50 | - | 100 | - | 50 | - | - | - |

No. iterations of algorithm | 1000 | - | 1000 | - | 1000 | - | 1000 | - | 1000 | - | - | - |

Iteration of diversification | 30 | - | 20 | - | 20 | - | 20 | - | 20 | - | - | - |

Iteration of intensification | 100 | - | 40 | - | 40 | - | 40 | - | 40 | - | - | - |

Neighborhood size | 90 | - | 110 | - | 130 | - | 150 | - | 170 | - | - | - |

Algorithm | Tabu Search | Baron | |||
---|---|---|---|---|---|

Revenue, USD $1000/year | 49,792 | 49,306 | 49,783 | 49,543 | 49,792 |

${\tau}_{R}$, min | 6.000 | 6.000 | 6.000 | 6.000 | 6.000 |

${\tau}_{C1}$, min | 3.020 | 6.000 | 2.349 | 6.000 | 2.978 |

${\tau}_{C2}$, min | 0.500 | 0.500 | 0.500 | 0.500 | 0.500 |

${\tau}_{S1}$, min | 6.000 | 2.424 | 6.000 | 1.816 | 6.000 |

${\tau}_{S2}$, min | 6.000 | 2.424 | 6.000 | 6.000 | 6.000 |

${N}_{R}$ | 15.000 | 15.000 | 15.000 | 15,000 | 15.000 |

${N}_{C1}$ | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

${N}_{C2}$ | 5.000 | 2.000 | 6.000 | 2.000 | 5.000 |

${N}_{S1}$ | 15.000 | 15.000 | 15.000 | 15.000 | 15.000 |

${N}_{S2}$ | 15.000 | 15.000 | 15.000 | 10.000 | 15.000 |

Grade Cu | 0.222 | 0.220 | 0.222 | 0.222 | 0.222 |

Circuit structure | Figure 4 | Figure 5g (circuit 1) * | Figure 5a (circuit 2) * | Figure 5f (circuit 3) * | Figure 4 (circuit 4) * |

Time, s | 798.06 | 605,789.65 | 628,906.07 | 630,906.7 | 80,193.67 |

Algorithm | Best Design | Secondary Designs | ||
---|---|---|---|---|

Revenue, USD/year | 49,792,2192 | 49,698,998 | 49,670,988 | 49,575,119 |

${\tau}_{R}$, min | 6.000 | 4.190 | 4.780 | 3.200 |

${\tau}_{C1}$, min | 3.020 | 5.160 | 5.680 | 5.700 |

${\tau}_{C2}$, min | 0.500 | 5.390 | 4.260 | 6.000 |

${\tau}_{S1}$, min | 6.000 | 5.850 | 5.640 | 5.950 |

${\tau}_{S2}$, min | 6.000 | 6.000 | 5.200 | 5.210 |

${N}_{R}$ | 15.000 | 15.000 | 15.000 | 9.000 |

${N}_{C1}$ | 8.000 | 4.000 | 3.000 | 2.000 |

${N}_{C2}$ | 5.000 | 2.000 | 2.000 | 3.000 |

${N}_{S1}$ | 15.000 | 15.000 | 15.000 | 7.000 |

${N}_{S2}$ | 15.000 | 15.000 | 15.000 | 11.000 |

Grade Cu | 0.222 | 0.222 | 0.222 | 0.222 |

Circuit structure | Figure 4 | Figure 5b | Figure 5h | Figure 5e |

Algorithm | Tabu Search | Baron Solver |
---|---|---|

Advantages | The convergence is fast. The algorithm always provides a solution when the mathematical model is well defined. The algorithm is flexible, i.e., it allows the use of other methods, such as linear programming algorithms. The algorithm provides good quality solutions. The algorithm provides secondary designs. | The solver provides a global optimal design when converged. The solver does not need to adjust parameters for providing a solution. The obtained designs do not change if the solver is run again. |

Disadvantages | Some algorithm parameters, such as neighborhood size and number of iterations of the algorithm, among others, must be adjusted for finding a good quality solution. Penalty parameters must be used for satisfying the constraints of the mathematical model. The obtained designs could change if the algorithm is run again. The algorithm must incorporate diversification and intensification for finding a good quality solution. | Depending on the mathematical model and the number of species, the convergence is slow or the algorithm does not converge. The variables of the model must be bounded for guaranteeing the finding of global optimal design, i.e., experience in circuit design is required. The solver provides a single design. The obtained designs depend on the version of the solver. |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lucay, F.A.; Gálvez, E.D.; Cisternas, L.A. Design of Flotation Circuits Using Tabu-Search Algorithms: Multispecies, Equipment Design, and Profitability Parameters. *Minerals* **2019**, *9*, 181.
https://doi.org/10.3390/min9030181

**AMA Style**

Lucay FA, Gálvez ED, Cisternas LA. Design of Flotation Circuits Using Tabu-Search Algorithms: Multispecies, Equipment Design, and Profitability Parameters. *Minerals*. 2019; 9(3):181.
https://doi.org/10.3390/min9030181

**Chicago/Turabian Style**

Lucay, Freddy A., Edelmira D. Gálvez, and Luis A. Cisternas. 2019. "Design of Flotation Circuits Using Tabu-Search Algorithms: Multispecies, Equipment Design, and Profitability Parameters" *Minerals* 9, no. 3: 181.
https://doi.org/10.3390/min9030181