Question 5 of 13: 5 (14 marks, compulsory)

Question 1.5 (14 marks, compulsory)

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.

1.5.1 – Moving/floating cone rule base (2-D section). Working on a single vertical cross-section, block-model cash flows (revenue for ore blocks, cost for waste blocks), the method proceeds: (1) starting from the highest-value block in the section, provisionally place an inverted cone (in 2-D, a symmetric wedge) whose sides honour the maximum stable wall slope, apex at that block and opening upward to surface; (2) sum the cash flow of every block inside the cone; (3) if the cone total is positive, accept the block and all blocks above it within the cone as part of the pit – the waste is “paid for” by the ore at the apex; (4) if the cone total is negative, reject that apex block (leave it, and its cone, unmined) and move to the next-highest-value block not yet tested; (5) repeat for every ore block in the section, in descending order of value, each time re-testing against the current pit outline (a cone can be re-tested and re-accepted once neighbouring cones have already added waste-stripping capacity nearby); (6) the process stops when no remaining untested block yields a positive cone – the union of all accepted cones is the 2-D “floating cone” pit outline. This is exactly the block-by-block logic mechanised by the depth-profile dynamic program used in Question 5.2.

1.5.2 – Deficiencies the Lerchs–Grossmann (1964) algorithm fixed. The moving/floating cone method (a) is not guaranteed to find the true global optimum – because each cone is tested independently and only against the CURRENT partial outline, the final accepted pit depends on the order in which blocks are tested, and a genuinely optimal combination of two adjacent, individually-negative cones (whose UNION is positive once shared waste is counted once, not twice) can be missed entirely; (b) double-counts shared waste blocks when two overlapping cones are each tested and accepted separately, overstating the true stripping cost of the combined shape; and (c) has no rigorous mathematical proof of optimality – it is a heuristic, not an algorithm with a guaranteed bound. Lerchs & Grossmann reformulated the block model as a directed graph (each block a node, slope-constraint precedence arcs to the blocks that must be removed first) and applied max-flow/min-cut (graph closure) theory, which guarantees the mathematically optimal ultimate pit for a given block-value model in one pass, with no double-counting and no dependence on the order blocks are tested.

1.5.3 – Why Lerchs–Grossmann is still not “optimal” for a 10–20 year mine. Lerchs–Grossmann optimizes a SINGLE, static block-value model – it answers “what is the best ultimate pit shape if every block is mined and sold today, at today’s prices, costs and discount rate.” A real 10–20 year mine life makes that framing incomplete in at least three ways it cannot resolve: (i) it takes no account of the TIME VALUE of money or scheduling – it does not say WHEN each block is mined, only whether it is inside the ultimate pit, so the true NPV-optimal outline (which, as in Question 1.2, prefers a smaller pit that front-loads value) can differ materially from the tonnage/undiscounted-value-optimal LG outline; (ii) it assumes a single fixed price/cost/slope scenario, whereas metal price, exchange rate and even the achievable slope angle (geotechnical data improves as the pit deepens) all change over a 10–20 year horizon – the real answer is a family of nested pit envelopes (parametrized on price) scheduled through time, not one outline; and (iii) it ignores operational constraints (equipment fleet size, mill capacity, blending requirements) that govern the achievable MINING SEQUENCE, which is exactly what pushback/phase design and NPV scheduling (built ON TOP of the LG outline) exist to add.