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.

我们解决了矿山的设计优化问题，在矿山中，矿石集中在长而细的矿脉中，通常使用所谓的“自上而下”的露天开采方法。在这样的矿山中，设想将地面以下的大块土地提取出来，并从概念上将其分成几层，每层为三维矩形块。矿山设计规定了哪些块作为支柱的一部分留下，以提供岩土结构的稳定性。其余部分经过提取和处理后获得矿石。我们寻求一种受岩土工程稳定性约束而使利润最大化的设计，我们将其表示为具有侧面约束的集合划分问题。由于复杂的岩土考虑，保证可行性的公式将需要成倍数量的变量和约束。我们设计了一种方法来限制需要包含的变量的数量，并开发一种启发式方法，在该方法中，将违反约束条件迭代地纳入到配方中，从而消除了会导致不稳定的绝大多数空隙（矿井中的开口）。通过有限元分析对岩土工程稳定性进行最终评估是必要的，但我们发现，系统地包含相对简单的约束条件足以使矿山设计通过此评估。在基于真实数据的案例研究中，我们的方法提供了一种满足有限元分析标准的矿山设计，其利润估计比公司采矿工程师确定的最佳解决方案高出16％，