Question 3 of 11: The Floating-Cone Principle and Lerchs–Grossmann

Question 1.3: The Floating-Cone Principle and Lerchs–Grossmann (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.

1.3.1 — the governing principle. A block is mined if and only if extracting it — together with every block that must be removed to physically reach it under the pit's slope constraint — yields a non-negative net economic value: the sum of the block's own value plus the value (usually negative) of the overlying/adjacent waste required to expose it. This is a marginal test applied to the whole removal path, not to the block in isolation: a rich ore block can be uneconomic to mine if the overburden needed to expose it is too thick or too costly relative to that ore's value, and conversely a marginally negative waste block is mined anyway if it sits on the removal path to a sufficiently valuable block below or beside it.

1.3.2 — the floating-cone method. Each block is first assigned an economic value: for an ore block this is (grade × recovery × metal price) × block tonnage, less mining cost, processing cost and the ore's own selling/refining costs — i.e. a genuine profit per block, which can still be negative for very low-grade ore that is nonetheless above the mill's economic cut-off; for a waste block it is simply the negative of its mining (and, where applicable, rehandle) cost, since waste generates no revenue. The algorithm then proceeds stage by stage: (1) select the highest-value unmined ore block remaining in the model as the cone apex; (2) construct the “cone” of material that must be removed above/around it to satisfy the pit-slope angle — in 2-D cross-section this cone widens by one column each row moved toward surface, in 3-D it is a true inverted cone (or frustum, on a bench-by-bench basis); (3) sum the net value of every block in that cone, crediting any other positive (ore) block that happens to fall inside the same cone footprint, per Lizotte's rule quoted in the question; (4) if the cone's net value is positive, extract the whole cone and mark every block in it as mined; if negative, leave that apex block in place for now; (5) repeat from step 1 over the remaining unmined ore blocks until no positive-value cone remains. The procedure is “floating” because the cone is re-evaluated fresh at each stage, sliding to a new apex once the previous cone (if accepted) has been removed.

1.3.3 — the exact method: Lerchs–Grossmann. The Lerchs–Grossmann (LG) graph-closure algorithm, developed by Helmut Lerchs and Ingo Grossmann and published in the CIM Bulletin in 1965 (with wider dissemination and computer implementation through the late 1960s, which is presumably the “1967/68” the question refers to), reformulates the block model as a directed graph in which every block is a node and every “block $b$ requires block $a$ to be removed first” slope-precedence relation is a directed arc $a\to b$. The pit-limit problem becomes: find the maximum-weight closure of this graph — a subset of nodes containing every predecessor of every node it contains — that maximizes the sum of node (block) values. This is solved exactly, in polynomial time, using a graph-theoretic max-flow/min-cut-equivalent procedure (a specialised network algorithm building nested spanning trees of “strong” and “weak” arcs), which guarantees the globally optimal pit outline for the given block economics and slope constraints — not merely a good one.

1.3.4 — two configurations where floating cone is sub-optimal. (1) Cones that must merge but are evaluated independently. Two ore blocks close enough together that their individually-required slope cones overlap can each be individually marginal or even individually negative in isolation, while the combined, wider excavation that serves both (sharing the waste stripping between them) is strongly positive — floating cone, which only ever tests one apex-centred cone at a time, can reject both and leave value in the ground that LG's global closure would correctly capture. (2) Order-dependence. Because floating cone commits each accepted cone permanently before moving to the next apex, evaluating a small, high-standalone-value cone before a larger cone that would have absorbed it “for free” (per Lizotte's own inclusion rule) strands the larger cone at a lower, sometimes negative, marginal value once its would-be free ore has already been claimed — the final pit outline, and its total value, then depends on an arbitrary tie-break in apex-selection order rather than on the deposit's true economics. Check: Question 1.5 works a concrete instance of exactly this second failure mode — evaluating the small block at row 2, column 6 before the two larger cones strands both larger cones at negative marginal value and halts the heuristic at a total pit value of only +1.0, versus the +5.5 reached (matching the Lerchs–Grossmann-equivalent optimum) by processing the largest blocks first.