11-07-2017 | Issue 3/2017 Open Access

# A stochastic optimization formulation for the transition from open pit to underground mining

- Journal:
- Optimization and Engineering > Issue 3/2017

- Authors:
- James A. L. MacNeil, Roussos G. Dimitrakopoulos

## 1 Introduction

## 2 Method

### 2.1 The general set up: candidate transition depths

### 2.2 Stochastic integer programming: mine scheduling optimization

### 2.3 Developing risk-management based life-of-mine plans: open pit optimization formulation

- i is the block identifier;
- t is a scheduling time period;
- \(b_{i}^{t} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{Block}}\;i\;{\text{is}}\;{\text{mined}}\;{\text{through}}\;{\text{OP}}\;{\text{in}}\;{\text{period}}\;t;} \hfill \\ 0 \hfill & {\text{Otherwise}} \hfill \\ \end{array} } \right.\)
- \(g_{i}^{s}\) grade of block i in orebody model s;
- \(Rec\) is the mining and processing recovery of the operation;
- T
_{i}is the weight of block i; - \(NR_{i} = T_{i} \times g_{i}^{s} \times Rec \times \left( {{\text{Price}} - {\text{Selling}}\;{\text{Cost}}} \right)\) is the net revenue generated by selling all the metal contained in block i in simulated orebody s;
- MC
_{i}is the cost of mining block i; - PC
_{i}is the processing cost of block i; - \(E\left\{ {V_{i} } \right\} = \left\{ {\begin{array}{*{20}l} {NR_{i} - MC_{i} - PC_{i} } \hfill & {{\text{if}}\;NR_{i} > PC_{i} } \hfill \\ { - MC_{i} } \hfill & {{\text{if}}\;NR_{i} \le PC_{i} } \hfill \\ \end{array} } \right.\) is the economic value of a block i;
- r is the discount rate;
- \(E\left\{ {\left( {NPV_{i}^{t} } \right)} \right\} = \frac{{E\left\{ {V_{i}^{0} } \right\}}}{{\left( {1 + r} \right)^{t} }}\) is the expected NPV if the block i is mined in period t;
- N is the number of selective mining units available for scheduling;
- z is an identifier for the transition depth being considered;
- P
_{z}is the number of production periods scheduled for candidate transition depth z.

- s is a simulated orebody model;
- S is the number of simulated orebody models;
- w and o are target parameters, or type of production targets; w is for the waste target; o if for the ore production target;
- u is the maximum target (upper bound);
- l is the minimum target (lower bound);
- \(d_{su}^{to} ,d_{su}^{tw}\) are the excessive amounts for the target parameters produced;
- \(d_{sl}^{to} , d_{sl}^{tw}\) are the deficient amounts for the target parameters produced;
- \(c_{u}^{to} ,c_{l}^{to} ,c_{u}^{tw} ,c_{l}^{tw}\) are unit costs for \(d_{su}^{to} ,d_{sl}^{to} ,d_{su}^{tw} ,d_{sl}^{tw}\) respectively in the optimization’s objective function.

- W
_{tar}is the targeted amount of waste material to be mined in a given period; - O
_{tar}is the targeted amount of ore material to be mined in a given period; - O
_{si}is the ore tonnage of block i in the orebody model s; - Q
_{UG,tar}is the yearly metal production target during underground mining; - MCap
_{min}is the minimum amount of material required to be mined in a given period; - MCap
_{max}is the maximum amount of material that can possibly be mined in a given period; - l
_{i}is the set of predecessor for block i.

### 2.4 Developing risk-managing life-of-mine plans: underground optimization formulation

_{j}) is defined by considering the relevant geotechnical issues which constrain the sequencing optimization. These precedence relationships are created using the Enhanced Production Scheduler (EPS) software from Datamine (Datamine Software 2013). For the application presented in this paper, the precedence relationships implemented were passed along by industry-based collaborators who operate the mine. Once the optimization for both the OP and UG components is completed for each candidate transition depth, the optimal transition depth can then be identified as the depth z that leads to a maximum value of the expression below.

## 3 Application at a gold deposit

Transition Depth 1 | Transition Depth 2 | Transition Depth 3 | Transition Depth 4 | |
---|---|---|---|---|

Number of OP blocks | 64,255 | 72,585 | 80,915 | 89,245 |

Number of UG stopes | 418 | 356 | 340 | 311 |

Production years through OP | 7 | 8 | 9 | 10 |

Production years through UG | 7 | 6 | 5 | 4 |

Metal price | $900/oz |

Crown pillar height | 60 ft |

Economic discount rate | 10% |

Processing cost/ton | $31.5 |

OP mining cost/ton | $1.5 |

UG mining cost/ton | $135 |

OP mining rate | 18,500,000 t/year |

UG mining rate | 350,000 t/year |

OP mining recovery | 0.95 |

UG mining recovery | 0.92 |