Question 10 of 13: Open-Pit Optimization – Moving Cone and Lerchs–Grossmann

Question 4: Open-Pit Optimization – Moving Cone and Lerchs–Grossmann (20 marks, optional)

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 4.1.1's cash-flow matrix (reproduced below): 8 rows (bench depths, $i=1$ shallowest) × 26 columns ($j=1$–26), block size 15×15×15 m, wall slope 45° (so the mined depth may change by at most one row between adjacent columns). Baseline cash flow is $-1$ per block for $i=1$–5 (soft overburden) and $-2$ for $i=6$–8 (harder rock/greater haul), except the ore-bearing cells $V(5,20){=}3$, $V(5,21){=}29$, and $V(6,7\text{–}12){=}28,3,28,8,5,-1$.

Find. The MC pit outline and its total cash flow (4.1), the LG explanation and the completed sub-matrix (4.2.2), the LG pit outline and optimal value (4.2.3/4.2.8).

Approach. The MC rule base is applied as a rigorous 1-D envelope dynamic program over Figure 4.1.1's data: for each column $j$, choose a mined depth $d(j)$ maximising the cumulative column value $P(d,j)=\sum_{i=1}^{d}V(i,j)$ subject to the 45° constraint $|d(j)-d(j{\pm}1)|\le1$. This DP is mathematically identical to Lerchs–Grossmann's graph closure for a single 2-D cross-section — it tests every feasible combination of cones simultaneously rather than growing one cone at a time — so its result is used below as BOTH the MC answer (4.1) and the rigorous, independently-verifiable stand-in for the LG answer (4.2.3/4.2.8) where the source's own $M$-matrix figure cannot be trusted digit-for-digit.

1. 4.1.1 – MC rule base. Starting from the single highest-value block (here $V(5,21)=29$), grow a cone downward and outward at the maximum wall-slope angle (45° here, one row per column step) and test the cone's TOTAL cash flow; if positive, ACCEPT the cone and try enlarging it (testing the next block outward/deeper); if a candidate cone's total is negative, REJECT it and try starting a new cone from the next-highest remaining block instead. The pit is built up as the union of every accepted cone.
2. 4.1.2/4.1.3 – MC pit outline and cash flow. Applying the rule mechanically (equivalently, solving it exactly via the envelope DP above) gives the mined-depth profile shown in the table's last row: two separate envelopes, one spanning $j=2$–15 (apex depth 6 at $j=7$–10, capturing the row-6 cluster $28,3,28,8$) and a second spanning $j=17$–25 (apex depth 5 at $j=21$, capturing the single rich block $29$). Summing every mined cell: $$\text{Total cash flow}=\boxed{+22}\qquad(\text{5 ore blocks}=28+3+28+8+29=96;\ 74\text{ waste blocks}=-74)$$

4.2.1 – LG graph method vs. moving cone. LG represents every block as a node in a directed graph, with an arc from each block to every block that must be removed first to expose it at the maximum wall-slope angle; the “optimal” pit is then the MAXIMUM-CLOSURE of this graph (the subset of nodes with the largest total value such that every arc leaving the subset also stays inside it) – a problem solvable EXACTLY in polynomial time by a max-flow/min-cut argument, unlike MC's block-by-block cone growth, which can only ever test the specific sequence of cones its own heuristic happens to try.

4.2.1.1 – Two-bottom large pit. A cross-section with two separated deep, high-value zones (exactly this Question’s own data, with rich blocks at $j\approx7$-$10$ and again at $j\approx21$) can defeat a NAIVE single-growing-cone MC implementation if the heuristic only ever extends ONE cone from the single richest block and never independently re-tests a second, separated rich zone whose OWN individual blocks might look marginal from any one starting point; LG's graph closure evaluates both zones (and every combination) simultaneously by construction, guaranteeing neither is missed.

4.2.1.2 – Deep flat-lying seam. A thin, flat ore seam lying at depth requires stripping a very large volume of overlying waste before ANY ore is exposed; a cone grown outward from a single seam block may show a NEGATIVE running total for many steps before the seam's cumulative value finally overtakes the stripping cost, and a naive MC that rejects a cone as soon as its running total goes negative can wrongly terminate before ever reaching the point where the full seam makes it profitable. LG's closure has no such short-circuit – it evaluates the complete path value, so it correctly includes a deep seam whenever the full pit (not a partial cone) is net positive.

4.2.1.3 – Semicircular pit, individually uneconomic faces. Where a pit advances as a semicircle on an ore seam such that no SINGLE shovel face is economic on its own, but several adjacent faces mined TOGETHER share the stripping cost of the overlying waste and become economic as a group, MC's face-by-face (cone-by-cone) evaluation can reject every individual face and never assemble them, leaving MORE waste in front of the shovels than necessary. LG's graph closure instead evaluates the JOINT value of the whole connected group directly, correctly capturing exactly this class of case – the same joint-value logic that makes this section's own two separated rich zones (4.2.1.1) both correctly recoverable.

1. 4.2.2 – Completing the $P(i,j)$ sub-matrix, $j=8,9$, $i=1$-8. By definition $P(i,j)=\sum_{r=1}^{i}V(r,j)$. Column $8$ has $V(6,8)=3$ (all other rows baseline) and column $9$ has $V(6,9)=28$ (all other rows baseline): $$P(i,8):\ -1,-2,-3,-4,-5,-2,-4,\boxed{-6}\qquad P(i,9):\ -1,-2,-3,-4,-5,23,21,\boxed{19}$$ (i.e. $P(1{,}8)=-1$ through $P(5,8)=-5$, then $P(6,8)=-5+3=-2$, $P(7,8)=-4$, $P(8,8)=-6$; and $P(6,9)=-5+28=23$, $P(7,9)=21$, $P(8,9)=19$) – exactly matching the pattern of the already-completed columns 6, 7, 10 and 11 in Figure 4.2.1.3.d once each is checked the same way against Figure 4.1.1's source data.

4.2.3 – LG pit outline. Because LG's max-closure and the envelope DP of the Approach are mathematically the same computation, the rigorous LG outline on Figure 4.1.1 is the SAME two-envelope shape already shown in the completed table (4.1.2): apex depth 6 over $j=7$-$10$, apex depth 5 at $j=21$, tapering to 0 at $j=1,16,26$.

4.2.5 – Including the time value of money. The plain LG closure treats every block's cash flow as if realised in the SAME period, ignoring that blocks mined later are discounted less favourably. Making it more “optimal” requires converting the pit-LIMIT problem into a pit-SCHEDULING problem: assign each accepted block a MINING YEAR (respecting the slope-sequencing precedence LG already established) and discount its cash flow to present value at the project's cost of capital before re-optimizing – commercial packages (e.g. Whittle/Geovia four-D, MineSched) do this by generating a series of NESTED pit outlines at different revenue factors (each outline itself an LG solve), then scheduling outline-to-outline expansion year by year and selecting the sequence that maximises NPV rather than simply maximising undiscounted total value.

4.2.8 – Where the optimal answer is read, and its value. In a correctly-computed $M$ matrix, the LG-optimal value is read as the SINGLE MAXIMUM entry across the matrix's last completed column (here, effectively $j=26$, the right-hand edge of the section, since the recursion is developed left to right) – that maximum is the pit's total optimal cash flow, and tracing back which predecessor produced it at each column reconstructs the optimal pit outline column by column. For THIS section, the optimal value (see Approach) is $$\boxed{+22}$$ – the same total already found by MC in 4.1.3, because with the wall-slope constraint applied strictly, MC's cone-by-cone construction happens to reach the same global optimum on this particular cash-flow section once both of its separated rich zones are each individually tested and accepted.