Open Pit Limits: 24-MMP-B8 Rock Slope Engineering, Undated paper, Q5

Question 5: Open Pit Limits (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.

Block model parameters
Quantity	Value
Block volume	1,000 m³
Density (ore and waste)	2.5 t/m³ (tonnage/block = 2,500 t)
Cutoff grade	1.5%
Combined mining + milling + overhead cost	$50/t
Net processed mineral value	$2,800/t contained metal
Block model size	5 rows (depth) × 8 columns

Fig. 5.1 — Geological block model (top row = surface bench; darker green = higher grade).

Find. The 2-D pit outline that maximizes total profit, obeying a 45° maximum wall slope (adjacent columns' pit floors may differ in depth by at most one block).

Approach. Convert every grade block to a dollar value, then find the pit outline (one mined depth per column, changing by at most one block between adjacent columns, anchored to zero depth at both edges of the model) that maximizes total profit — the 2-D equivalent of Lerchs-Grossman graph closure, solved here by an envelope dynamic program over cumulative column profit.

Check: block economic value equals tonnage times (grade fraction times $2,800 minus $50), i.e. the $2,800/tonne figure is a metal price applied to the tonnes of contained metal in each block (tonnage × grade fraction), not a flat dollar-per-tonne-of-rock revenue — this reading is confirmed by the resulting block values landing on clean round numbers (−$125,000, −$55,000, $15,000, $85,000, $155,000 for grades 0–4), which would not happen under an alternative unit interpretation. Waste-grade blocks (0% and 1%, below the 1.5% cutoff) are assigned the SAME cost-minus-revenue formula rather than a separate waste-only mining cost, the standard simplified convention for a 2-D Lerchs-Grossman teaching exercise absent a stated separate waste mining rate — a 0%-grade block naturally values at −$125,000 (pure cost, zero revenue) under this formula, so the cutoff grade is used to LABEL the optimal pit's ore/waste tonnage after solving, not to change the optimization itself.

Block economic value. Each block is $1{,}000\ \text{m}^3\times2.5\ \text{t/m}^3=2{,}500$ t. $$v(g) = 2{,}500\times\left[\frac{g}{100}\times2800 - 50\right] = 2{,}500(28g - 50)\ \text{\$}$$ $$\boxed{v(0)=-\$125{,}000,\ v(1)=-\$55{,}000,\ v(2)=\$15{,}000,\ v(3)=\$85{,}000,\ v(4)=\$155{,}000}$$
Cumulative column profit. For each of the 8 columns, sum block values from the surface (row 1) down to depth $d$ (0–5 blocks); e.g. column 5 (grades 4,4,2,3,2 top-to-bottom) gives $P(1)=155\text{k}$, $P(2)=310\text{k}$, $P(3)=325\text{k}$, $P(4)=410\text{k}$, $P(5)=425\text{k}$ (all $\times\$1000$).
Envelope DP for the maximum-profit pit outline. Choosing a mined depth $d_c\in\{0,\ldots,5\}$ per column $c$, subject to $|d_c-d_{c-1}|\le1$ (45° slope) and $d_0,d_7\le1$ at the model's outer edges, maximize $\sum_c P_c(d_c)$ by dynamic programming across columns (analogous to the floating-cone/envelope DP already used for this discipline's other pit-limit questions): $$\boxed{d = [0,\,0,\,1,\,2,\,2,\,1,\,0,\,0]\ \text{(columns 1--8, blocks mined from surface)}}$$
Optimal pit profit. Summing $P_c(d_c)$ over all 8 columns: $$\boxed{\text{Total pit profit} = 15{,}000+85{,}000+155{,}000+155{,}000+155{,}000+85{,}000 = \$650{,}000}$$
Ore/waste tonnage within the optimal pit. The pit mines 6 blocks (columns 3–6, depths 1/2/2/1), all at grade $\ge2\%$ (above the 1.5% cutoff), so every mined block is ore: $$\boxed{\text{Ore} = 6\times2{,}500 = 15{,}000\ \text{t},\quad \text{Waste} = 0\ \text{t},\quad \text{Overall strip ratio} = 0}$$ The optimizer naturally excludes every column whose best-achievable cumulative profit is negative (columns 1, 2, 7, 8 stay at $d=0$) and stops each ore column exactly where the next block's marginal value would turn the cumulative sum down (e.g. column 5 stops at $d=2$ even though extending it further would itself still be profitable in isolation ($P(3)=325\text{k}$, $P(4)=410\text{k}$) — it is the slope constraint linking column 5 to column 6, whose own best cumulative profit is only $P(1)=85\text{k}$ (its $P(2)=-40\text{k}$ turns negative), that holds column 5 back to $d=2$, not any weakness in column 5's own economics).

Fig. 5.2 — Profit block model ($, ×1000) with the optimal pit outline (red) from the envelope DP: green = mined ore, pink = left in place (negative-value waste).

Question 5 — final results
Item	Result
Optimal pit floor depth per column (1–8)	0, 0, 1, 2, 2, 1, 0, 0
Total pit profit	$650,000
Ore mined / waste mined	15,000 t / 0 t
Blocks mined	6 (columns 3–6)

Topic. 2-D Lerchs-Grossman open-pit limit optimization by envelope dynamic programming.

Key relations. Block economic value $v=\text{tonnage}\times[(\text{grade}/100)\times\text{price}-\text{cost}]$; cumulative column profit $P_c(d)=\sum_{i=1}^{d}v_i$; optimal outline maximizes $\sum_cP_c(d_c)$ subject to a maximum-slope constraint $|d_c-d_{c\pm1}|\le1$ — mathematically equivalent to the Lerchs-Grossman graph-closure algorithm restricted to a single 2-D cross-section.

Why this works. The slope constraint means a deep, high-grade column can only be mined if its neighbours are also excavated down to within one block of the same depth (to keep pit walls stable), so a rich block surrounded by waste may not be economic to reach even though its own value is strongly positive — the DP captures this by forcing every column's chosen depth to be consistent with its neighbours, rather than optimizing each column independently.

Common pitfall. Optimizing each column's depth independently (ignoring the slope constraint, which would over-mine isolated rich blocks and produce an unmineable wall); and applying the cutoff grade as a hard stop on the DP itself rather than as a post-hoc ore/waste label — the optimizer must be allowed to mine through a marginal or sub-cutoff block if doing so is needed to reach higher-value material below it.

Source. Hustrulid, Kuchta & Martin, Open Pit Mine Planning and Design (Lerchs-Grossman pit optimization, floating-cone/envelope methods).

Question 5 of 5: Open Pit Limits

Question 5: Open Pit Limits (20 marks)