A fast method to develop an optimal operational sublevel stope design

Designing a sublevel stoping mine is a challenging task that requires determining the best layout in terms of economic value while respecting geomechanical constraints that limit the stopes' shape. Because of its relevance and challenging aspects, many authors have proposed methods to approximate or compute optimal stope designs or to provide raw designs that can be used as a guideline. This paper follows previous approaches by approximating the shape of the stopes using the block model as support and working in two stages: first, an enumeration algorithm generates all valid stopes, and second, an optimization model selects the set of stopes with maximum value. However, our approach adds several improvements. Firstly, we ensure that the stopes generated in the first stage are profitable and geotechnically stable. Secondly, the mathematical model of the second stage incorporates the organization of stopes into drifts and levels, which yields a more operational layout. Thirdly, the approach is suitable for algorithmic and theoretical contributions: We propose a fast heuristic algorithm for the general case of the model but also show that in some cases, the optimization problem reduces to finding the shortest path in an ad-hoc graph; thus, its optimum can be computed efficiently. Fourthly, we test our approach on 4 block models involving 84,000 to over 3 million blocks, or about 53,000 to 13.5 million valid stopes, respectively. The resulting optimal layouts are not only operationally feasible, but optimal solutions can be found in less than 1 h using freely available linear programming solvers or in less than a second using a shortest-path algorithm when it applies. Finally, we performed sensibility analyses to assess the variability of the value and geometric location of optimal stopes showing that the solutions are robust and that even with perturbations of 20% in the economic values of the stopes, the value and tonnage of the optimal solutions do not change more than 0.4% and that in most cases, 95% of the extracted blocks remain the same.