Application of the Pseudoflow Algorithm in Ultimate Pit Limit Optimization via Precedence Computation

The determination of the Ultimate Pit Limit (UPIT) — the optimal contour of an open-pit mine that maximizes the economic value of the deposit while respecting slope angles — is a central problem in mine planning. Its exact formulation as a Maximum Closure Problem in oriented graphs, solved by max-flow/min-cut algorithms, produces the exact and globally optimal solution, but requires efficient computational structures for real instances with millions of blocks.

This work presents a pipeline in Python composed of three steps. First, the parallel calculation of precedences is performed via KDTree with adaptive radius and a geomechanical cone filter parameterizable by the slope angle α, followed by transitive reduction to eliminate redundant arcs. Then, the maximum flow graph is constructed. Finally, exact resolution is applied using Hochbaum's pseudoflow algorithm. The method is generic: α and the number of precedence levels are input parameters, allowing direct application to any block model and enabling systematic geomechanical sensitivity analyses — for example, α ∈ {35°, 40°, 45°, 50°, 55°} — without any code changes.

The McLaughlin instance (MineLib) — 2,140,342 blocks of 25 × 25 × 20 ft, with α = 45° and 8 levels of precedence — was used as a benchmark. The precedence calculation generated 73,143,782 arcs in 136.1 s (15,723 blocks/s), with an average of 34.17 arcs per block. The maximum flow resolution was completed in 34.2 s, totalizing less than 3 minutes from start to the optimal result. The obtained UPIT value was 1,495,886,962, corresponding to the difference between the sum of the positive values of the selected blocks (1,632,471,928) and the value of the minimum cut (136,584,966). The difference relative to the solution published in MineLib is exactly zero, confirming the correctness of the implementation.

According to the results obtained, it is suggested that the pipeline maintains scalability and practical efficiency for large-scale real instances, being applicable to any angle α and block dimension through simple reconfiguration of input parameters.