Joint stochastic optimisation of open-pit mine production scheduling with ramp design

Figures & data

Figure 1. Isometric view of a bench depicting a checkpoint (sphere), accessible blocks (shaded volume), road (arrow) and ramp.

Figure 2. Isometric view of a bench depicting checkpoints (spheres), sub-roads (grid of arrows) and a super-road (curved arrow).

Figure 3. Ramp and road generation tree. The children of each node compose the set of all possible roads that can be constructed by adding a single extension.

Table 1. Indices.

$\mathit{i}$	Mining block index
$\mathit{s}$	Simulation index
$\mathit{t}$	Period
$\mathit{d}$	Ramp design index
$\mathit{j}$	Check point index
$\mathit{k}$	Super-road index
$\mathit{l}$	Sub-road index
$\mathit{g}$	Super-road segment index

Table 2. Sets.

$\mathit{CPI}\left(\mathit{i}\right)$	Set of checkpoints with windows overlapping block $\mathit{i}$
$\mathit{AD}{J}^R\left(\mathit{k}\right)$	Set of ramps (described by their location $\mathit{i}$, and design $\mathit{d}$) which terminate adjacent to the beginning of super road $\mathit{k}$
$\mathit{PRED}\left(\mathit{i},\mathit{d}\right)$	Set of predecessor blocks for ramp located at $\mathit{i}$, with design $\mathit{d}$
$\mathit{XSEC}\left(\mathit{i},\mathit{d}\right)$	Set of ramps intersecting ramp at location $\mathit{i}$, with design $\mathit{d}$
$\mathit{SEG}\left(\mathit{k}\right)$	Set of segments composing super-road $\mathit{k}$
$\mathit{AD}{J}^{\mathit{s}-}\left(\mathit{g}\right)$	Set of sub-roads traversing super-road segment $\mathit{g}$
$\mathit{AD}{J}^{\mathit{s}+}\left(\mathit{j}\right)$	Set of super-roads adjacent to checkpoint $\mathit{j}$
S	Set of simulations
I	Set of mining blocks
T	Set of years
D	Set of ramp designs
K	Set of super-roads
L	Set of sub-roads
J	Set of checkpoints

Table 3. Constants.

$v_{i,s}^t$	Economic value of block $\mathit{i}$ in period $\mathit{t}$ for scenario $\mathit{s}$
$\mathit{r}{c}^t$	Cost of placing or removing a ramp in period $\mathit{t}$
$c_o^{t-,+}$	Penalty cost for deviating below $\left(-\right)$ or above $\left(+\right)$ ore target
$c_g^{t-,+}$	Penalty cost for deviating below $\left(-\right)$ or above $\left(+\right)$ metal target

Table 4. Variables.

$b_i^t$	Binary	Mine block $\mathit{i}$ period $\mathit{t}$
$r_i^{t,d}$	Binary	Ramp with design $\mathit{d}$ exists at $\mathit{i}$ period $\mathit{t}$
${\hat{r}}_{\mathit{i}}^{\mathit{td}}$	Binary	Ramp with design $\mathit{d}$ placed/removed at $\mathit{i}$ in period $\mathit{t}$
${\mathrm{c}\mathrm{p}}_j^t$	Binary	Checkpoint $\mathit{j}$ is active in period $\mathit{t}$
$s_k^{t+}$	Binary	Super-road $\mathit{k}\in \mathit{K}$ is active in period $\mathit{t}$
$s_l^{t-}$	Binary	Sub-road $\mathit{l}\in \mathit{L}$ is active in period $\mathit{t}$
$d_s^{to-},d_s^{to+}$	Continuous	Deviation below/above target for ore tonnage in scenario $\mathit{s}$ in period $\mathit{t}$
$d_s^{tg-},d_s^{tg+}$	Continuous	Deviation below/above target for grade in scenario $\mathit{s}$ in period $\mathit{t}$

Figure 4. Bench cross-section containing a ramp, checkpoint (hatched square), mined blocks (white), unmined blocks (shaded) and accessible blocks (hatched circles).

Figure 5. Bench cross-section of accessible and inaccessible ramp placement. Vertically hatched ramp is accessible and horizontally hatched ramp is inaccessible.

Figure 6. Bench cross-section of valid (vertically hatched) and invalid (horizontally hatched) ramp placements.

Figure 7. Bench cross-section with intersecting ramp placements.

Figure 8. Bench cross-section with valid (solid) and invalid (dashed) super-roads.

Figure 9. Plan view of bench with a ramp (R), checkpoints (numbered blocks), sub-roads (straight) and super-roads (curved).

Figure 10. Optimisation loop.

Figure 11. Simulated realisations of gold grades (g/t).

Table 5. Optimisation parameters.

Parameter	Value
Metal Price ($/oz)	1200
Recovery (%)	90%
Mining cost ($/t)	5
Processing cost ($/t)	15
Ramp placement/removal cost ($)	300,000

Figure 12. Cross-section at x = 39.

Figure 13. Cross-section at x = 52.

Figure 14. Plan view of the open pit denoting mined blocks (light colors) and ramps (dark blue) in years 2, 4 and 7.

Figure 15. Plan view of the open pit denoting final ramp placement (dark blue).

Figure 16. Ramp placement profile and cumulative construction costs of optimised ramp design.

Figure 17. Forecasted cumulative discounted cash flow of production schedule and ramp design.

Figure 18. Forecasted gold quantity and grade of life-of-mine production schedule.

Figure 19. Forecasted ore and total tonnages of production schedule.