# Reducing the Dimensions of the Stochastic Programming Problems of Metallurgical Design Procedures

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Modern Design of Experiments (MDoE)

#### 2.2. Supervised Machine Learning (SML)

#### 2.3. Global Sensitivity Analysis (GSA)

#### 2.4. Generic Framework to Reduce the Uncertainty Space

## 3. Applications

## 4. Conclusions

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Predominant structures in the first and second scenario, (

**a**) Structure 1, (

**b**) Structure 2.

**Figure 2.**Benchmarking of SML tools to predict structure of designed process: (

**a**) accuracy of training for uncertainty space of fifteen input variables; (

**b**) accuracy of training for uncertainty space of twenty input variables. Benchmarking of SML tools to predict NPV, CAPEX, OPEX, REV, PROF, NCF, copper recovery, and copper grade: (

**c**) average ${R}^{2}$ of training for uncertainty space of fifteen input variables; (

**d**) average ${R}^{2}$ of training for testing dataset for uncertainty space of twenty input variables.

**Figure 3.**Benchmarking of SML tools to predict structure of designed process: (

**a**) accuracy of testing for uncertainty space of fifteen input variables; (

**b**) accuracy of testing for uncertainty space of twenty input variables. Benchmarking of SML tools to predict NPV, CAPEX, OPEX, REV, PROF, NCF, copper recovery, and copper grade: (

**c**) average ${R}^{2}$ of testing for uncertainty space of fifteen input variables; (

**d**) average ${R}^{2}$ of testing for testing dataset for uncertainty space of twenty input variables.

**Figure 5.**NPV histogram: (

**a**) free input variables, (

**b**) free influential and fixed noninfluential input variables, (

**c**) free non-influential and fixed influential input variables.

**Figure 6.**Estimation with surrogate models of NPV and structure for designed flotation plants in terms of the function of copper price and chalcopyrite flux fed: (

**a**) first scenario, (

**b**) second scenario.

**Table 1.**Input variables under uncertainty. $N$ represents a normal distribution, $U$ represents an uniform distribution.

Input Variable | Standard Condition | Uncertainty |
---|---|---|

Copper price (1) | 4 MUSD/t | $N$[4,0.3] |

Kilowatt-hours (2) | 0.0002 MUSD | $N$[0.0002,0.00002] |

Cost of mine-crushing-grinding per ton of ore fed to plant (3) | 0.003 MUSD/t | $N$[0.003,0.0004] |

Chalcopyrite fast mass flux fed (4) | 3 t/h, | $N$[3,0.3] |

Chalcopyrite slow mass flux fed (5) | 2 t/h | $N$[2,0.3] |

Chalcocite fast mass flux fed (6) | 1 t/h | $N$[1,0.1] |

Chalcocite slow mass flux fed (7) | 1 t/h | $N$[0.4,0.04] |

Pyrite fast mass flux fed (8) | 5 t/h | $N$[5,0.2] |

Pyrite slow mass flux fed (9) | 3.5 t/h | $N$[3.5,0.3] |

Quartz mass flux fed (10) | 150 t/h | $N$[150,3] |

Gangue mass flux fed (11) | 300 t/h | $N$[300,3] |

Chalcopyrite fast copper grade fed (12) | 34% | $N$[0.34.0.01] |

Chalcopyrite slow copper grade fed (13) | 25% | $N$[0.25,0.01] |

Chalcocite fast copper grade fed (14) | 18% | $N$[0.18,0.01] |

Chalcocite slow copper grade fed (15) | 10% | $N$[0.1,0.01] |

Number of cells in the rougher stage (16) | 5 | $U$[3,10] |

Number of cells in the cleaner stage (17) | 5 | $U$[3,10] |

Number of cells in the recleaner stage (18) | 5 | $U$[3,10] |

Number of cells in the scavenger stage (19) | 5 | $U$[3,10] |

Number of cells in the rescavenger stage (20) | 5 | $U$[3,10] |

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Lucay, F.A. Reducing the Dimensions of the Stochastic Programming Problems of Metallurgical Design Procedures. *Minerals* **2021**, *11*, 1302.
https://doi.org/10.3390/min11121302

**AMA Style**

Lucay FA. Reducing the Dimensions of the Stochastic Programming Problems of Metallurgical Design Procedures. *Minerals*. 2021; 11(12):1302.
https://doi.org/10.3390/min11121302

**Chicago/Turabian Style**

Lucay, Freddy A. 2021. "Reducing the Dimensions of the Stochastic Programming Problems of Metallurgical Design Procedures" *Minerals* 11, no. 12: 1302.
https://doi.org/10.3390/min11121302