Lerchs-Grossmann Algorithm
Also known as: Lerchs-Grossmann Method, LG Algorithm
The Lerchs-Grossmann Algorithm is a graph-theoretic method for determining the ultimate pit limit in open-pit mining operations. Introduced by Helmut Lerchs and Israel Grossmann in 1965, it maximizes the net present value of extracted ore while respecting slope stability constraints. This algorithm forms the theoretical foundation for most modern pit optimization software.
Key highlights
- Mathematically proven optimal solution to the pit limit problem under specified constraints
- Handles complex slope angle geometries and variable extraction costs across depth zones
- Efficient computational complexity (O(n log n) with modern implementations) despite combinatorial problem size
- Results are reproducible and verifiable, supporting regulatory approval and stakeholder confidence
- Flexible framework accommodates multi-commodity deposits and variable commodity prices
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Apply the Lerchs-Grossmann Algorithm when planning the ultimate pit limit for open-pit mining projects, especially when precise optimization against economic and geotechnical constraints is critical. Assume block model data is reliable and slope stability angles are well-defined. Prefer this over simpler parametric methods when pit geometry is complex or when small improvements in ore recovery justify computational cost. Consider alternatives like parametric ultimate pit or dynamic programming variants for real-time optimization in production planning.
Strengths & limitations
- Mathematically proven optimal solution to the pit limit problem under specified constraints
- Handles complex slope angle geometries and variable extraction costs across depth zones
- Efficient computational complexity (O(n log n) with modern implementations) despite combinatorial problem size
- Results are reproducible and verifiable, supporting regulatory approval and stakeholder confidence
- Flexible framework accommodates multi-commodity deposits and variable commodity prices
- Requires high-quality, dense block models which may be expensive to obtain for large deposits
- Assumes deterministic ore grades; does not natively model geological uncertainty or price volatility
- Slope angle constraints are fixed inputs; it cannot optimize angles themselves
- Computational time scales with deposit size; very large models (>10M blocks) may require decomposition or approximation
- Result is sensitive to block value assumptions; small errors in cost estimates can shift pit boundaries significantly
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
How does the algorithm handle variable slope angles in different rock types?
The algorithm uses a cone angle or slope angle as input for each direction and depth. Rock type variations are pre-processed into the block model structure: blocks above steeper walls become precedents for blocks below them. Different slope angles in different zones require different precedence relationships in the graph.
Can the algorithm handle multiple ore types or commodity prices that change over time?
The basic algorithm assigns a fixed value to each block, so it solves for a single-period optimal pit. Extensions like staged pit design use the algorithm iteratively on different discount rates or commodity prices to generate a sequence of pit shells. More advanced models directly incorporate time-based block values and solve for multi-period optimization.
What is the computational complexity, and how large a block model can I optimize efficiently?
Modern implementations using dynamic programming or push-relabel algorithms achieve O(n log n) complexity for n blocks. Deposits with 1-5 million blocks are routinely optimized in minutes on standard computers. Larger models may require block aggregation, spatial decomposition, or approximation techniques.
How sensitive is the pit limit to block value assumptions?
Very sensitive. A 5-10% change in commodity price or extraction cost can shift pit boundaries by millions of tonnes. Sensitivity analysis and scenario planning are essential. Many practitioners generate multiple pit shells at different price points (e.g., $800/oz, $1200/oz, $1600/oz gold) to support strategic planning.
Does the algorithm account for geological uncertainty or ore grade variability?
The classical algorithm uses point estimates of block grades. Probabilistic extensions use conditional simulation to generate multiple equiprobable block models and optimize each one, generating a distribution of pit limits. This captures uncertainty in pit boundary decisions.
Sources
- 1.Lerchs, H., & Grossmann, I. F. (1965). Optimum design of open-pit mines. Canadian Mining and Metallurgical Bulletin, 58(633), 47-54.
- 2.Johnson, T. B. (2014). Optimum pit limits - definition and computational procedures. Journal of the Australasian Institute of Mining and Metallurgy, 249, 21-28.
You have read it. What now?
Cite this page
ScholarGate. (2026, June 3). Lerchs-Grossmann Algorithm. ScholarGate. https://scholargate.app/mining-engineering/lerchs-grossmann-algorithm