# Toward the Operability of Flotation Systems under Uncertainty

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## Abstract

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## 1. Introduction

- Is it possible to determine the structures presenting favorable conditions for operating the flotation systems?
- Does the selection of flotation equipment and metal price influence the operability of the flotation systems?

**Table 1.**Flotation system design methodologies, NLP = nonlinear programming, LP = linear programming, and MINLP = mixed integer nonlinear programming. Uncertainty described by distribution functions.

Reference | Model Type | Cell/Bank/Approximate Model | Grinding | Operational and Metal Price Uncertainty | Selection of Flotation Equipment |
---|---|---|---|---|---|

Mehrotra and Kapur, 1974 [19] | NLP | Bank | No | No–No | No |

Reuter et al., 1988 [20] | LP | Bank | No | No–No | No |

Reuter and Van Deventer, 1990 [21] | LP | Bank | Yes | No–No | No |

Schena et al., 1996 [22] | MINLP | Bank | Yes | No–No | No |

Schena et al., 1997 [9] | MINLP | Bank | No | No–No | No |

Guria et al., 2005 [23] | NLP | Cell | No | No–No | No |

Guria et al., 2005 [24] | NLP | Cell | No | No–No | No |

Cisternas et al., 2006 [10] | MINLP | Bank | Yes | No–No | Yes |

Méndez et al., 2009 [25] | MINLP | Bank | Yes | No–No | Yes |

Ghobadi et al., 2011 [26] | MINLP | Bank | No | No–No | No |

Maldonado et al., 2011 [27] | NLP | Bank | No | No–No | No |

Hu et al., 2013 [1] | MINLP | Cell | No | No–No | No |

Montenegro et al., 2013 [13] | MILP | Approximate | No | Yes–No | No |

Cisternas et al., 2014 [16] | MINLP | Bank | No | No–No | No |

Jamett et al., 2015 [14] | MINLP | Bank | No | Yes–No | No |

Cisternas et al., 2015 [4] | MILP | Approximate | No | Yes–No | No |

Acosta-Flores et al., 2018 [7] | MINLP | Bank–Cell | No | No–No | No |

Lucay et al., 2019 [28] | MINLP | Bank | No | No–No | No |

Liang et al., 2020 [15] | MINLP | Cell | No | Yes–No | No |

Acosta-Flores et al., 2020 [8] | MILP | Approximate | Yes | Yes–No | No |

## 2. Strategy

#### 2.1. Uncertainty Analysis (UA)

#### 2.2. Superstructure

#### 2.3. Modeling of Design Alternatives

#### 2.4. Optimization Algorithms

## 3. Applications

#### 3.1. Uncertainty in Grinding and Flotation Stages, and the Selection of Equipment in the Recleaner Stage

#### 3.2. Uncertainty in Regrinding and Flotation Stages and Selection of Equipment in Cleaner and Recleaner Stages

## 4. Conclusions

- Using mathematical programming and uncertainty analysis, we determined structures presenting favorable conditions for facing operational and economic uncertainty and consequently conditions favoring flexibility/resilience to determine an optimal operation region;
- The selection of flotation equipment and metal price influenced the percentages of structures in the optimal set. A higher percentage of optimal solutions of one structure implies a greater capacity to face operational and metal price changes. A high copper price reduced the number of primal optimal structures and promoted the appearance of new structures.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Graphical representation of results obtained by selecting equipment in the recleaner stage, with a copper price equal to 3500 USD/ton.

**Figure 4.**Graphical representation of results obtained by using a copper price equal to 3500 USD/ton, fixed bank (

**a**) and fixed column (

**b**).

**Figure 5.**Graphical representation of the results obtained by selecting equipment in the recleaner stage, (

**a**) copper price uncertainty described by U(3000,4000) USD/t, (

**b**) copper price uncertainty U(5000,7000) USD/t.

**Figure 6.**Graphical representation of the percentage of optimal structures (

**a**) without uncertainty in the copper price, and (

**b**) with a copper price uncertainty described by U[000,000,3,4] USD/t.

**Figure 7.**Structure 2 using columns in the cleaner and recleaner stages, with copper price uncertainty described by U(3000,4000) USD/t.

**Figure 8.**Graphical representation of the percentage of optimal structures considering the selection of equipment in cleaner and recleaner stages, with copper price uncertainty described by U(5000,7000).

Stages | $\mathit{R}$ | $\mathit{G}{\mathit{r}}_{1}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | $\mathit{G}{\mathit{r}}_{2}$ | $\mathit{C}\mathit{S}$ | $\mathit{W}$ | $\mathit{P}$ |
---|---|---|---|---|---|---|---|---|---|---|

$R$ | o | o | x | |||||||

$G{r}_{1}$ | o | o | ||||||||

${C}_{1}$ | x | o | x | x | x | o | ||||

${C}_{2}$ | x | x | x | x | o | |||||

${S}_{1}$ | o | o | o | o | x | o | o | |||

${S}_{2}$ | o | o | o | o | o | x | ||||

$G{r}_{2}$ | o | o | o | o | o | |||||

$CS$ | o | x | o | o |

Stages | CPY.f1 | CPY.f2 | CPY.f3 | MIX.f1 | MIX.f2 | MIX.f3 | SC.f1 | SC.f2 | SC.f3 |
---|---|---|---|---|---|---|---|---|---|

R | (0.665,0.735) | (0.855,0.945) | (0.760,0.840) | (0.380,0.420) | (0.665,0.735) | (0.570,0.630) | (0.048,0.053) | (0.095,0.105) | (0.048,0.053) |

C1, cell | (0.475,0.525) | (0.665,0.735) | (0.475,0.525) | (0.190,0.210) | (0.475,0.525) | (0.285,0.315) | (0.048,0.053) | (0.057,0.063) | (0.048,0.053) |

C2, cell | (0.475,0.525) | (0.665,0.735) | (0.475,0.525) | (0.190,0.210) | (0.475,0.525) | (0.285,0.315) | (0.048,0.053) | (0.057,0.063) | (0.048,0.053) |

C1, col | (0.285,0.315) | (0.380,0.420) | (0.285,0.315) | (0.190,0.210) | (0.285,0.315) | (0.190,0.210) | (0.024,0.026) | (0.024,0.026) | (0.024,0.026) |

C2, col | (0.285,0.315) | (0.380,0.420) | (0.285,0.315) | (0.190,0.210) | (0.285,0.315) | (0.190,0.210) | (0.024,0.026) | (0.024,0.026) | (0.024,0.026) |

S1 | (0.665,0.735) | (0.855,0.945) | (0.760,0.840) | (0.380,0.420) | (0.665,0.735) | (,0570,0.630) | (0.048,0.053) | (0.095,0.105) | (0.048,0.053) |

S2 | (0.665,0.735) | (0.855,0.945) | (0.760,0.840) | (0.380,0.420) | (0.665,0.735) | (,0570,0.630) | (0.095,0.105) | (0.190,0.210) | (0.095,0.105) |

CS | (0.665,0.735) | (0.855,0.945) | (0.760,0.840) | (0.380,0.420) | (0.665,0.735) | (,0570,0.630) | (0.095,0.105) | (0.190,0.210) | (0.095,0.105) |

CPY.f1 | CPY.f2 | CPY.f3 | MIX.f1 | MIX.f2 | MIX.f3 | SC.f1 | SC.f2 | SC.f3 | |
---|---|---|---|---|---|---|---|---|---|

CPY.f1 | (0.05,0.15) | (0.35,0.45) | (0.45,0.55) | ||||||

CPY.f2 | (0.15,0.25) | (0.75,0.85) | |||||||

CPY.f3 | 1 | ||||||||

MIX.f1 | (0.05,0.15) | (0.05,0.15) | (0.25,0.30) | (0.25,0.30) | (0.05,0.15) | (0.00,0.10) | (0.00,0.075) | (0.00,0.075) | |

MIX.f2 | (0.05,0.15) | (0.55,0.65) | (0.05,0.15) | (0.15,0.25) | |||||

MIX.f3 | 1 | ||||||||

SC.f1 | 1 | ||||||||

SC.f2 | (0.190,0.210) | (0.095,0.105) |

CPY.f1 | CPY.f2 | CPY.f3 | MIX.f1 | MIX.f2 | MIX.f3 | SC.f1 | SC.f2 | SC.f3 | |
---|---|---|---|---|---|---|---|---|---|

CPY.f1 | (0.048,0.053) | (0.285,0.315) | (0.615,0.682) | ||||||

CPY.f2 | (0.285,0.315) | (0.665,0.735) | |||||||

CPY.f3 | 1 | ||||||||

MIX.f1 | (0.19,0.21) | (0.19,0.21) | (0.095,0.105) | (0.19,0.21) | (0.19,0.21) | (0.0475,0.0525) | (0.0237,0.0262) | (0.0237,0.0262) | |

MIX.f2 | (0.095,0.105) | (0.285,0.315) | (0.095,0.105) | (0.21,0.19) | (0.19,0.21) | (0.095,0.105) | |||

MIX.f3 | (0.19,0.21) | (0.38,0.42) | (0.38,0.42) | ||||||

SC.f1 | (0.095,0.105) | (0.38,0.42) | (0.475,0.525) | ||||||

SC.f2 | (0.665,0.735) | (0.285,0.315) |

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**MDPI and ACS Style**

Lucay, F.A.; Acosta-Flores, R.; Gálvez, E.D.; Cisternas, L.A.
Toward the Operability of Flotation Systems under Uncertainty. *Minerals* **2021**, *11*, 646.
https://doi.org/10.3390/min11060646

**AMA Style**

Lucay FA, Acosta-Flores R, Gálvez ED, Cisternas LA.
Toward the Operability of Flotation Systems under Uncertainty. *Minerals*. 2021; 11(6):646.
https://doi.org/10.3390/min11060646

**Chicago/Turabian Style**

Lucay, Freddy A., Renato Acosta-Flores, Edelmira D. Gálvez, and Luis A. Cisternas.
2021. "Toward the Operability of Flotation Systems under Uncertainty" *Minerals* 11, no. 6: 646.
https://doi.org/10.3390/min11060646