Selecting support pillars in underground mines with ore veins

We address the design optimization problem for a mine in which the ore is concentrated in a system of long, thin veins and for which the so-called top-down open-stope mining method is customarily used. In such a mine, a large volume of earth below the surface is envisioned for extraction and is conceptually divided into three-dimensional rectangular blocks on each of several layers. The mine design specifies which blocks are left behind as part of a pillar to provide geotechnical structural stability; the remainder are extracted and processed to obtain ore. We seek a design that maximizes profit subject to geotechnical stability constraints, which we represent as a set partitioning problem with side constraints. Due to the complex geotechnical considerations, a formulation that guarantees feasibility would require exponentially large numbers of variables and constraints. We devise a method to limit the number of variables that need to be included and develop a heuristic in which violated constraints are iteratively incorporated into the formulation, thereby eliminating the vast majority of voids (openings in the mine) that would cause instability. A final evaluation of geotechnical stability via finite element analysis is necessary, but we have found that systematic inclusion of relatively simple constraints is adequate for the mine design to pass this evaluation. In a case study based on real data, our approach provided a mine design that satisfied the finite element analysis standards, with an estimated profit 16% higher than that of the best solution identified by the company's mining engineers, leading to tens of millions of dollars in profit enhancement.