Question 3 of 11

Question 1.3 (7 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 1.3 is a hand-annotated cash-flow (i,j) block matrix for a 2-D pit cross-section; wall slope 45°, blocks 15×15×15 m. The matrix is printed rotated 90° on the original paper; the bench rows are reproduced below, centre-aligned per the 45° slope, each cell a per-block net cash flow (thousands of dollars, arbitrary units).

Find. (1.3.1) the MC decision rule; (1.3.2/1.3.3) the resulting pit outline and its total cash-flow sum.

1.3.1 — the moving/floating cone rule base. For every block with a positive cash flow, construct the inverted cone of blocks that MUST be removed above it to expose it at a 45° wall slope (in 2-D, the cone is a triangle widening by one block per bench going up). Sum the cash flows of the candidate block plus its overlying cone. If that sum is positive, the cone is “mineable” — mark every block in it as removed and merge it into the current pit outline (overlapping cones already mined are not re-charged). If the sum is negative, the candidate block is left in place (not mined on its own). Re-scan the section, because merging one cone can expose new candidate positive blocks or change the marginal cone cost of a neighbouring one, and repeat until no further positive-sum cone remains. The method is a LOCAL, greedy heuristic — the question explicitly warns not to use a simple adjacent-block cash-flow sum because there is no reliable general algorithm for the true 3-D cone test; a rigorous 2-D equivalent (used to check the greedy result) is to find, independently, the maximum-value stepped outline subject to the same ±1-column-per-bench slope constraint — mathematically the same graph-closure idea Lerchs & Grossmann formalise in Question 3.

Approach. Apply the rigorous stepped-outline equivalent of the cone test (a 1-D envelope optimisation across the columns of Figure 1.3, enforcing the 45° slope as a ±1 column shift per bench) to the reconstructed matrix, and read off the outline and its cash-flow total.

1. Form the column-cumulative cash flow. For each column j, compute the cumulative sum P(bench, j) mining down from the surface bench (1) to a candidate depth — this is the net value of taking a full vertical slice to that depth before any slope constraint is applied.
2. Enforce the 45° slope as a forward envelope. Sweep the columns and cap each column’s achievable mining depth to at most one bench deeper than its neighbour’s already-committed depth, exactly the moving-cone merge rule applied simultaneously to every column instead of block-by-block.
3. Read the optimum. The best achievable column depth profile gives a pit outline that mines only the shallow ore lens straddling columns 10–15 of benches 1–6 (the “2, 2 / 4 / 3 / 3, 4 / 4” cluster bolded in the Given table) plus the minimum overlying cone needed to expose it, leaving the rest of the section un-mined (its own cone cost exceeds its value).

Summing the cash flow of every block inside that outline: $$\boxed{\text{MC optimal pit cash flow} = +2}$$ — a small but positive net value, i.e. this cross-section is only marginally profitable to mine at all under the moving-cone test; most of the section (the -1 to -6 waste blocks with no adjacent ore lens) is correctly left un-mined.