Question 9 of 13: Open-Pit Optimization – Moving Cone and Lerchs–Grossmann

Question 3: Open-Pit Optimization – Moving Cone and Lerchs–Grossmann (15 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. Figure 3.1.1: an 8-bench (i = 1 shallowest .. 8 deepest) × 26-column (j = 1..26) cash-flow block model, 15×15×15 m blocks, 45° wall slope. Benches i = 1–4 are uniform waste at –1 per block; i = 7–8 are uniform waste at –2 per block. The two non-uniform benches are:

i.e. one richer mineralized zone straddles j = 7–11 on bench 6 (peak blocks +24 at j = 7 and j = 9), and one isolated rich block sits on bench 5 at j = 21 (+29, with a +3 neighbour at j = 20).

Find. The moving-cone pit outline and its cash-flow total (3.1); the completed P sub-matrix, the LG pit outline and optimal value (3.2).

Approach. Build P(i,j) as the straight cumulative sum down each column (the mechanical step the exam figures walk through), then solve the 45° slope-constrained pit-outline optimization exactly with the classical envelope dynamic program that is the modern equivalent of Lerchs & Grossmann’s 1965 graph-theoretic construction – this both answers 3.2 directly and lets 3.1’s moving-cone result be checked against a known-exact benchmark.

1. 3.1.1 – Moving-cone rule base. For a candidate ore block, construct the inverted cone of every block that must be removed to expose it under the given wall slope – for a 45° slope and unit (one-block) benches, mining block (i, j) requires blocks (i−1, j−1), (i−1, j) and (i−1, j+1) to already be removed, so the cone widens by exactly one column on each side per shallower bench. Sum the cash flow of every block in the cone: if the sum is $\ge 0$, the cone is economic and is added to the pit; the method then moves to the next candidate block (working from the richest block outward, per the question’s instruction not to pre-sum adjacent blocks) and repeats, merging and re-testing overlapping cones, until no further profitable cone can be added.
2. 3.1.1 (applied) – single-block cone tests. Testing the cone rooted at the single richest block first, per the rule base: $$\text{cone}(6,7)=\sum = -11 \qquad \text{cone}(6,9)=\sum=-11 \qquad \text{cone}(5,21)=\sum=+5$$ A full symmetric cone rooted on either +24 block alone is not economic (the waste above it, widening a full column per bench for 5–6 benches, costs more than the ore is worth) – this is the classic weakness of the naive single-block moving cone that Lerchs–Grossmann was developed to overcome. The isolated +29 block’s cone, by contrast, is profitable standing alone (sum +5).
3. 3.1.1/3.1.2 – merging into the flat-bottomed cone. Because blocks (6,7), (6,8), (6,9), (6,10) and (6,11) are contiguous on the same bench, the moving-cone method next tests the union of their cones as one wider, flat-bottomed shape rather than five separate symmetric cones (which would over-count the shared overburden down the middle). Trimming the flat-bottomed cone bench by bench from its widest test width until the marginal column no longer adds value converges on: $$\text{depth profile, } j=2\text{–}14: [1,2,3,4,5,6,6,6,5,4,3,2,1] \quad\Rightarrow\quad \boxed{\text{cone value} = +6}$$ Re-testing the isolated block’s cone the same way (trimming its outer columns) converges on its own natural taper, j = 17–25, depth profile [1,2,3,4,5,4,3,2,1], value $+5$ (unchanged from the single-cone test, since it was already optimal in isolation).
4. 3.1.2 – pit outline (Fig. 3.2, below). The two cones do not overlap (columns 15–16 remain unmined, depth 0), so the moving-cone pit is simply their union: mine columns 2–14 down to the stepped profile above (flat bottom at depth 6 across j = 7–9), and columns 17–25 down to the stepped profile peaking at depth 5 at j = 21.
5. 3.1.3 – total cash flow of the moving-cone pit. $$CF_{total} = CF_{lobe\,1} + CF_{lobe\,2} = 6 + 5 = \boxed{+11}$$
6. 3.2.1 – LG graph method vs. moving cone. Lerchs & Grossmann (1965) recast the pit-outline problem as finding the maximum-weight closure of a directed precedence graph (each block a node, weighted by its own cash flow; an arc from each block to every block that must be removed first under the slope constraint) – solved exactly by graph algorithms rather than by the moving cone’s local, block-by-block heuristic. Three cases illustrate why the graph formulation is necessary:
   (i) Two-bottom large pit – two separate rich zones (as in this very matrix, benches 5 and 6) each generate their own locally optimal sub-pit; a moving cone tested only on the single richest block in the whole model could miss the second, separate positive lobe entirely, whereas LG’s graph search evaluates every block’s closure and finds both.
   (ii) Deep flat-lying seam – a thin, deep, laterally extensive seam (unlike this section’s compact ore) needs a very wide, gently tapered pit to reach economically; individual moving cones rooted block-by-block along the seam systematically under-value it (each cone alone carries the full taper cost) where LG’s closure recognizes the shared, jointly-amortized stripping cost across the whole seam length.
   (iii) “Half-donut” pit (a horizontal cut leaving a bottom bench and a central untouched “island”/mountain) – arises where a central column of blocks is uneconomic to remove (negative closure) even though it sits between two profitable zones; only a true closure algorithm correctly leaves the island in place rather than assuming the pit must be simply connected, which a naive cone-merging heuristic can get wrong.
7. 3.2.2 – P(i,j) sub-matrix (cumulative column sums), j = 6–12. $P(i,j)=\sum_{k=1}^{i}V(k,j)$, the running sum down each column from the surface (used by the exam’s Figure 3.2.1.3.b/c as the input to the LG graph transform):

   i	j=6	j=7	j=8	j=9	j=10	j=11	j=12
   0	0	0	0	0	0	0	0
   1	−1	−1	−1	−1	−1	−1	−1
   2	−2	−2	−2	−2	−2	−2	−2
   3	−3	−3	−3	−3	−3	−3	−3
   4	−4	−4	−4	−4	−4	−4	−4
   5	−5	−5	−5	−5	−5	−5	−5
   6	−7	19	−2	19	−1	0	−6
   7	−9	17	−4	17	−3	−2	−8
   8	−11	15	−6	15	−5	−4	−10

   The two columns the exam figure leaves blank for the candidate to fill (j = 8, 9) are bolded above: P(6,8) = −2, P(6,9) = +19.
8. 3.2.3/3.2.4 – LG pit outline and optimal value. Applying the 45°-slope envelope program (the graph-theoretic closure, worked column by column left to right exactly as the exam’s M-matrix construction describes) over the full 26-column section reproduces the moving-cone outline exactly – both lobes, same depth profile as step 4 above – because in this particular matrix the two positive zones are already well separated and individually well-shaped, so no merging or trimming beyond what the moving cone already converged to is available. Reading the running left-to-right M-table for just the j = 6–11 window that the exam figure isolates, its local maximum is $M(6,9) = 21$ (the cumulative value of columns 1–9 mined under the optimal envelope, i.e. including the mandatory taper-in cost from the section’s left edge) – this is the cell the exam’s Figure 3.2.1.3.c “LG solution read directly off the table” refers to for that lobe. The standalone value of that lobe, net of its own taper on both sides (comparable to the moving-cone figure in step 5), is $+6$, and the grand total optimal pit value across the whole given section, combining both lobes, is: $$CF_{LG,\,total} = \boxed{+11}$$ identical to the moving-cone answer (3.1.3) – in this data set the moving cone, correctly iterated to a merged, trimmed flat-bottomed shape rather than tested as single blind cones, already reaches the LG global optimum; the two methods diverge in general (case (i)–(iii) above) but happen to agree here because neither lobe interacts with the other or requires excluding an interior island.
9. 3.2.5 – adding the time value of money. LG’s graph closure maximizes undiscounted total cash flow and is indifferent to the order blocks are mined in, whereas a real pit is mined bench by bench over years, so early cash flow is worth more than late cash flow. The standard fix is to re-run the optimization as nested Lerchs–Grossmann pit envelopes at a series of revenue factors (0.2, 0.4, …, 1.0 of the base price), which generates a family of pits from smallest/highest-grade to largest/full extent; a mine-scheduling algorithm (or, in modern commercial software, an integrated NPV-scheduling optimizer, e.g. Whittle/Milawa 4-D scheduling) then sequences those nested envelopes over time to maximize discounted NPV rather than undiscounted total value, mining the highest-value envelopes first even where that is not the tonnage-maximizing sequence.