Application of Mining Width-Constrained Open Pit Mine Production Scheduling Problem to the Medium-Term Planning of Radomiro Tomic Mine: A Case Study

Abstract

This article presents a novel approach to address the mining width-constrained open pit mine production scheduling problem in the context of medium-term planning. A mathematical formulation is proposed to incorporate mining width constraints into the production scheduling process, aiming to maximize the NPV of the schedule while ensuring enough room for the operation of mining equipment. To tackle the computational challenges posed by large-scale instances of the problem, we propose a method based on variable fixing and horizontal precedence generation. In this study, we apply the developed model to real-world scenarios from Radomiro Tomic short-term mine planning problems such as optimizing the timing of major truck maintenance and the impact of external factors, like the delay in the production of the Chuquicamata underground project. Remarkable improvements are observed with the mining width-constrained model. Specifically, the mining width satisfiability is enhanced from 2 to 60% compared to the traditional open pit mine production scheduling model, underscoring the significance of incorporating these constraints. The proposed method showed good results reaching optimality gaps within 5%.

Appendix. Formulation of \({\mathcal{P}}^{*}\)

The following describes the formulation of the subproblem ${\mathcal{P}}^{*}$; we use the same notation and sets described in Sect. 2.

Variables.

The set of variables are associated with the extraction of the blocks.

$$z_{b,d,t}=\left\{\begin{array}{cc}1,\,&\mathrm{if}\;\mathrm{block}\;b\;\mathrm{is}\;\mathrm{sent}\;\mathrm{to}\;\mathrm{destination}\;\mathrm d\;\mathrm{by}\;\mathrm{period}\;\mathrm t\\0,&otherwise\end{array}\right.$$

Objective Function

$${\text{max}}\sum_{b\in \mathcal{B}}\sum_{d\in \mathcal{D}}\sum_{t\in \mathcal{T}}{p}_{b,d,t}^{*}\cdot {(z}_{b,d,t}-{z}_{b,d,t-1})$$

The objective function aims to maximize the discounted cashflow or net present value.

Constraints

$$\begin{array}{cc}\sum\limits_{b\in \mathcal{B}}{a}_{b,d}\cdot {(z}_{b,d,t}-{z}_{b,d,t-1})\le {C}_{d,t}& t\in \mathcal{T}, d\in \mathcal{D}\end{array}$$

(8)

$$\begin{array}{cc}{z}_{a,D,t}\le {z}_{b,D,t}& \left(a,b\right)\in \mathcal{A}\cup \mathcal{H}, t\in \mathcal{T}\end{array}$$

(9)

$$\begin{array}{cc}{z}_{b,D,t}\le {z}_{b,0,t+1}& b\in \mathcal{B}, t\in \{1,..,T-1\}\end{array}$$

(10)

$$\begin{array}{cc}{z}_{b,d,t}\le {z}_{b,d+1,t}& b\in \mathcal{B}, t\in \mathcal{T}, d\in \{1,..,D-1\}\end{array}$$

(11)

$$\begin{array}{cc}{z}_{b,d,0}=0 & b\in \mathcal{B},d\in \mathcal{D}\end{array}$$

(12)

$$\begin{array}{cc}{z}_{i,D,t}\le {z}_{j,0,t+1}& \left(i,j\right)\in \mathcal{S}\end{array}$$

(13)

$$\begin{array}{cc}\sum\limits_{d\in \mathcal{D}}\sum\limits_{t\in \mathcal{T}}{(z}_{b,d,t}-{z}_{b,d,t-1})\le 1& b\in \mathcal{B}\end{array}$$

(14)

Constraint (8) represents the resource constraint per destination per period; in our case, the resource associated with each block consists of its tonnage. This constraint can directly be extended to multiple resource constraints per destination. Constraint (9) ensures that, to extract a block b by period t, all its vertical and horizontal predecessors must also be extracted by period t. Constraint (10) forces that if a block is mined by a period t, then it will continue as mined for all subsequent periods. Constraint (11) is analogue to Constraint (10) but regarding the destinations. Constraint (12) ensures that no blocks can be extracted at period 0. Constraint (13) ensures that all the blocks that could violate the sinking rate constraint from block i are extracted at least a period earlier than block j itself. Constraint (14) ensures that the blocks are selected on at most one period one destination.