1.3 Cut-Off Grade Optimization & Lerchs-Grossmann Pit Limits

Pit optimization determines the physical boundaries and extraction sequence of an open-pit mine to maximize economic return. It bridges geological resource block models with financial performance by defining which mineralized blocks should be extracted, processed, or discarded as waste over time.

Cut-Off Grade Concepts & Mathematical Formulation

The cut-off grade is the threshold concentration of valuable mineral that differentiates ore (material processed for profit) from waste (material dumped or left in ground).

Break-Even Cut-Off Grade vs. Marginal Cut-Off Grade

1. Break-Even Cut-Off Grade ($g_{be}$): Applies when deciding whether an in-situ block should be mined as ore or left unmined. It must cover all costs: mining, milling, and general administration (G&A):

$g_{be} = \frac{C_{\text{mining}} + C_{\text{milling}} + C_{\text{G\&A}}}{(P - C_{\text{refining}}) \times R_{\text{met}}}$

Where:

• $C_{\text{mining}}$ = Mining cost per tonne of material ($/t)
• $C_{\text{milling}}$ = Processing/milling cost per tonne of ore ($/t)
• $C_{\text{G&A}}$ = General and administrative cost per tonne of ore ($/t)
• $P$ = Commodity selling price per unit of metal ($/unit)
• $C_{\text{refining}}$ = Refining, smelting, and marketing cost per unit of metal ($/unit)
• $R_{\text{met}}$ = Metallurgical recovery factor (decimal fraction)

1. Marginal (Internal) Cut-Off Grade ($g_m$): Applies to material that has already been excavated from the pit floor. Because the mining cost ($C_{\text{mining}}$) is already sunk, the decision is whether to send the block to the processing plant or to the waste dump:

$g_m = \frac{C_{\text{milling}} + C_{\text{G\&A}}}{(P - C_{\text{refining}}) \times R_{\text{met}}}$

If a block's grade $g$ satisfies $g_m \le g < g_{be}$, it is economically beneficial to process the block rather than dump it, provided it is already excavated.

Lane's Theory of Cut-Off Grade Optimization

Developed by Kenneth Lane (1988), Lane's theory optimizes cut-off grades over time to maximize the Net Present Value (NPV) of a mining operation under capacity constraints.

The Three Operational Capacity Constraints

Lane identified three physical bottleneck constraints in any mining system:

1. Mining Capacity Constraint ($C_m$): Maximum rate of total material (ore + waste) the mining fleet can excavate per year.
2. Milling / Processing Capacity Constraint ($C_c$): Maximum rate of ore the plant can process per year.
3. Refining / Marketing Capacity Constraint ($C_r$): Maximum rate of finished product the refinery or market can absorb per year.

Economic Mechanics of Dynamic Cut-Off Grades

To maximize NPV, Lane showed that cut-off grades should not remain static. In the early stages of mine life, discounting heavily penalizes future cash flows. Therefore, setting a high cut-off grade during early years maximizes cash flow density by feeding only premium high-grade ore to the mill while stockpiling or discarding lower-grade material. Over time, as high-grade reserves diminish and capital is recovered, the optimal cut-off grade declines progressively toward the marginal cut-off grade ($g_m$).

Ultimate Pit Limit Determination

The ultimate pit limit defines the maximum physical boundary of an open pit at the end of its economic life. A block is included in the ultimate pit shell only if its inclusion yields a positive incremental net economic value, considering the required waste blocks that must be removed above it to satisfy slope stability constraints.

Block economic value ($v_i$) for an ore block $i$ is calculated as:

$v_i = Q_i \cdot \left[ g_i \cdot (P - C_{\text{refining}}) \cdot R_{\text{met}} - (C_{\text{milling}} + C_{\text{G\&A}}) \right] - Q_i \cdot C_{\text{mining}}$

For a waste block $j$:

$v_j = - Q_j \cdot C_{\text{mining}}$

Pit Optimization Algorithms

Computers determine ultimate pit shells using 3D block models containing millions of spatial blocks.

1. Floating Cone Method (Heuristic)

The Floating Cone method moves an inverted cone (matching geotechnical slope angles) block by block through the model. For each target ore block with positive value, the cone defines all overlying waste blocks that must be stripped. If the summed value of all blocks inside the cone is positive, the cone is cleared for removal.

Limitations: Floating Cone is a heuristic algorithm. It can fail to find the global economic optimum in complex deposits, occasionally missing profitable combinations of overlapping cones or over-excavating non-profitable zones.

2. Lerchs-Grossmann (LG) 3D Graph Theory Algorithm

Formulated by Helmut Lerchs and Ingo Grossmann (1965), the LG algorithm models the 3D block environment as a directed graph:

• Nodes: Represent individual blocks with assigned net economic values ($v_i$).
• Directed Arcs: Represent geotechnical slope precedence constraints (a directed arrow points from an underlying block to all overlying blocks that must be removed first).

The LG algorithm solves the Maximum Closure Problem by converting the graph into a normalized tree structure and applying graph transformations (rooting, strong branch identification, and weak branch pruning).

Advantage: For the supplied block values and precedence arcs, a correctly implemented Lerchs–Grossmann maximum-closure algorithm returns the optimal closure value. This is a model optimum, not the unknowable “true” project value: errors or omissions in geology, costs, prices, slopes, time, capacity, access, and constraints remain outside that guarantee.

Revenue Factors, Parameterization & Nested Pit Shells

By varying the commodity price through a multiplier called the Revenue Factor ($RF$) (e.g., $RF = 0.3, 0.4, \dots, 1.0, \dots, 1.5$), optimization software generates a set of nested pit shells:

• Starter Pit Shells ($RF = 0.3 - 0.5$): Small, high-grade pit shells with minimal stripping ratios.
• Ultimate Pit Shell ($RF = 1.0$): Optimal pit shell at the base forecast price.
• Expanded Shells ($RF > 1.0$): Pit boundaries under higher commodity price scenarios.

Mine Scheduling & DCF Pit Sequencing

Nested pit shells serve as structural guides for designing operational pushbacks (mining phases). Production scheduling software applies Discounted Cash Flow (DCF) analysis to schedule pushbacks from high-value inner shells outward, maximizing project NPV while ensuring smooth equipment movement and constant mill feed.

Comparative Overview of Pit Optimization Techniques

Cutoff Decisions Are Constrained Routing Decisions

A cutoff grade is not one permanent property of a deposit. It is a decision rule for a defined material, destination, time, and capacity. State whether the decision is to mine or leave an in-situ block, process or dump material already exposed, stockpile for later, send ore to one of several plants, or blend to satisfy a product constraint. Each decision has a different relevant-cost boundary. Mining cost may be avoidable for an in-situ block but sunk for material already loaded; haulage to the alternative destination, rehandle, stockpile loss, closure, and incremental selling costs may still matter.

When plant capacity is limiting, opportunity cost is central. A low-grade tonne that pays its direct processing cost can still destroy value if it displaces a higher-margin tonne. Lane-style optimization therefore considers mining, concentrating, and refining or market constraints together with the time value of money. High early cutoffs are not an automatic rule: grade distribution, stripping access, stockpile capacity and recovery, blending, price, development, and bottleneck shadow prices determine the schedule.

Use consistent units. For a metal grade expressed as a fraction, net value per tonne can be written as grade × recovery × payable fraction × net metal price, after converting tonnes, kilograms, pounds, or troy ounces correctly. Deduct only costs relevant to the stated routing decision. Royalties and taxes may be based on gross output, income, margin, or contract terms and cannot be inserted as one generic percent.

Validate a cutoff policy through the physical schedule and reconciled destination model. Check ore and waste movement, plant feed, recovery, product, stockpile opening and closing balances, capacity, cash flow, and Mineral Reserve basis. Run sensitivity to price, recovery, cost, exchange rate, throughput and geologic uncertainty. A mathematically optimized shell or cutoff remains conditional on the block values, slope arcs, resource model, modifying assumptions, and time basis supplied to it; it does not replace detailed phase design, access, water management, permits, or an executable schedule.