Question 11 of 11: Block Economics and Ultimate Pit Optimization (Moving Cone vs. Lerchs–Grossmann)

Question 6: Block Economics and Ultimate Pit Optimization (Moving Cone vs. Lerchs–Grossmann) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Revenue/tonne = 60.0×%Cu−8.0; mining cost $2.0/t mined (all material); mill+other cost $6.0/t MILLED (ore only). Block-profit matrix (Figure 6.1): 8 benches × 26 columns, 15 m cubes, wall slope 45°.

Find. 6.1: revenue and cash flow per tonne for three grades. 6.2/6.3: the pit outline, total cash flow and ore/waste block counts under the Moving-Cone and Lerchs–Grossmann methods.

Approach. Compute the three per-tonne block values directly from the given revenue/cost model (6.1). Then, on the VERIFIED Figure 6.1 profit grid, run the Moving-Cone algorithm exactly as specified (single-block target, no parcel-combining) to get an achievable pit (6.2), and independently solve the SAME 45°-slope-constrained optimization exactly (an envelope dynamic program that is mathematically equivalent to the Lerchs–Grossmann graph closure for a single 2-D cross-section) to get the true optimal pit for comparison (6.3).

1. 6.1.1 — 0.4% Cu block. $\text{Rev} = 60.0(0.4)-8.0 = \$16.00/\text{t}$. This is an ORE block (milled): $$CF = \text{Rev} - 2.0 - 6.0 = 16.00-2.00-6.00 = \boxed{\$8.00/\text{t}}$$
2. 6.1.2 — 0.15% Cu block. $\text{Rev} = 60.0(0.15)-8.0 = \$1.00/\text{t}$. If milled: $$CF = 1.00-2.00-6.00 = \boxed{-\$7.00/\text{t}}$$ — negative even though revenue is positive, because the $6.00/t milling cost is not covered at this grade; this block is sub-economic and would in practice be sent to waste (or a low-grade stockpile) rather than milled.
3. 6.1.3 — 0% Cu waste block. $\text{Rev} = 60.0(0)-8.0 = -\$8.00/\text{t}$ (the model’s $8/t baseline deduction, e.g. smelter/treatment terms, applies to any tonne notionally valued this way). A true waste block is NOT sent to the mill, so only the mining cost is actually incurred: $$CF = -8.00 - 2.00 = \boxed{-\$10.00/\text{t}}$$

6.2.1 — The five steps of the Moving/Floating Cone method. (1) Scan the entire block model for the single most profitable block not yet evaluated. (2) Construct the MINIMUM cone of blocks above it — following the wall-slope constraint (here 45°, one bench of lateral offset per bench of depth) — needed to expose and support mining down to that block. (3) Sum the net value of the whole cone (the target block plus every supporting block in it). (4) If the cone’s total value is ≥ 0, ACCEPT it — add every block in the cone to the pit; if negative, REJECT that target block (leave it and its cone unmined) and move on. (5) Repeat from step 1 with the next most valuable remaining block, continuing until no remaining candidate block can be added with a non-negative cone value — at which point the process has converged and mining stops.

6.2.2/6.2.3 — Applying MC to Figure 6.1 (single-block target, no parcel-combining). Running the algorithm exactly as specified — taking ONE target block at a time in descending order of its own value, accepting a cone only if its own net sum is non-negative, never combining several blocks into a joint parcel — on the verified 8×26 profit grid converges after 3 rounds (the two rich pockets, value 24/3/24/4/5 at bench 6 and 3/29 at bench 5, each pull in their own supporting waste cone):

6.3.1–6.3.3 — Lerchs–Grossmann (optimal) solution. Figure 6.2 is the exam’s own worked LG output table for this section; a candidate reads the total pit cash flow directly from it. Independently, the SAME optimization (maximize total block value subject to the 45°-slope envelope constraint) was solved exactly here by dynamic programming over the verified Figure 6.1 grid.

6.3.2 — Pit outline. The optimal (DP/LG-equivalent) outline follows column depths (benches mined, 0 = none, 8 = full column) of 0,1,2,3,4,5,6,6,6,5,4,3,2,1,0,0,1,2,3,4,5,4,3,2,1,0 across the 26 columns — two 45° cones, a larger one centred on the columns–7–11 pocket (value 24,3,24,4,5) reaching 6 benches deep, and a smaller one on the columns 20–21 pocket (value 3,29) reaching 5 benches deep, exactly matching the two positive clusters visible in Figure 6.1 (shown as the red outline on the figure above).

6.3.4 — Differences between the MC and LG solutions. On this exact block set, MC (greedy, value +9, 74 blocks) undercuts the true optimum (LG, value +11, 73 blocks) by 2 units of cash flow despite moving MORE material — the two column-depth profiles differ by exactly one bench at column 8 (MC mines 7 benches deep there, LG only 6): MC’s single-block-at-a-time greedy rule locked in an extra bench of waste while chasing the column–7 pocket's own local value, a commitment the true optimum does not make once the WHOLE two-cone shape is considered jointly. This is the textbook failure mode of Moving Cone: because it commits to and never undoes a cone once accepted, it can be trapped into a locally-attractive but globally sub-optimal shape, and its answer can depend on the order candidate blocks are processed. Lerchs–Grossmann's graph-theoretic closure (equivalent to a max-flow/min-cut formulation) considers ALL feasible pit shapes simultaneously and is mathematically GUARANTEED to find the true maximum-value pit for a given block model and slope constraint — in general LG’s optimal value is always ≥ MC’s, exactly as observed here (11 ≥ 9).