A binary integer linear programming model for optimizing underground stope layout

Designing stope layouts in underground mines is challenging. Traditional software tools often rely on heuristics, which can lead to less-than-optimal or impractical results, ultimately reducing profitability [1]. This study introduces a novel two-dimensional binary integer linear programming (BILP) model to address this, integrating efficient shape constraints — adapted from Queyranne and Wolsey’s production planning formulations — alongside geotechnical and grade constraints [2]. Applied to a numerical example of a sublevel gold deposit with 774 blocks, the BILP model achieved a maximum economic value of $34.4 million by optimally selecting blocks that met all constraints. This solution was reached in 1.83 h with a 0.00 percent gap tolerance. The results confirm that LP-based approaches can be both computationally efficient and robust for solving complex stope layout problems, offering a rigorous, optimal and profitable alternative for strategic mine planning. Introduction The strategic planning of underground mines is paramount for the safe and profitable extraction of deeply seated ore deposits. A primary challenge in this process is the design of an optimal stope layout that maximizes economic value while adhering to a multitude of geotechnical and geometric constraints. Traditional stope layout optimization (SLO) methods, often relying on heuristic algorithms within commercial mining software, are limited in their capacity to guarantee a globally optimal solution. This suboptimality can result in significant financial losses over the life of the mine. While mathematical programming techniques, such as mixed-integer programming (MIP), offer the advantage of optimality, they have historically struggled with the nonlinear nature of complex shape constraints [3]. This has often led to the generation of solutions that are not practically feasible for a mining operation. This research aims to bridge the gap between theoretical optimality and practical feasibility by developing and validating a rigorous algorithm that can create a two-dimensional stope layout that maximizes the value of a deposit subject to grade, geotechnical and geometric constraints. The BILP model’s ability to generate practical, optimal and profitable stope designs provides a significant improvement over traditional methods for strategic mine planning. Methodology This study proposes a two-dimensional BILP model to optimize stope layouts, with the objective of maximizing the total economic value of a deposit. The core of this methodology is its innovative approach to formulating geometric and geotechnical constraints, which are crucial for generating practical and feasible stope shapes. The model draws from Queyranne and Wolsey’s formulations for tight constraints in production planning to enforce block contiguity and rectangular shapes within the stope layout optimization problem [2]. Fig. 1 Framework of the BILP model. The model’s framework is built upon several key components, as seen in Fig. 1. First, a regularized block model of a gold deposit is converted into an economic block model, where the value of each block is calculated based on metal prices, costs and recovery rates. Second, the model’s objective is to maximize the sum of the economic values of all blocks selected for mining, subject to a comprehensive set of linear constraints. These constraints ensure practicality by defining stope dimensions, preventing overlaps, and enforcing minimum stope grade and geotechnical pillar sizes. Results The BILP model was successfully verified using a two-dimensional section of a gold deposit. The numerical example, which contained 774 blocks with 144 classified as valuable, served as the primary test case. The model’s optimization process successfully selected a final stope layout that maximized economic value while adhering to all defined constraints. The results are shown in Table 1. Fig. 2 Optimal stope layout for the base-case scenario. The case study showed that the model identified stope layouts that maximized value at $34.4 million. Results were produced within two hours of computing time, demonstrating both accuracy and efficiency as well as confirming a globally optimal solution in a reasonable computational time. The model demonstrated its power to adapt stope shapes to the geological and geotechnical characteristics of the deposit for maximum recovery, as the generated stopes had variable heights and lengths. It preferentially selected blocks with higher grades, with the majority of stopes located in the central and right portions of the deposit, where grades are highest. The model also validated the enforcement of geotechnical pillar constraints, as shown in Fig. 2. Conclusions The results demonstrate that optimization models can produce practical stope layouts that enhance recovery while ensuring stability, providing mine planners with a rigorous alternative to traditional design methods. The proposed model successfully found a globally optimal layout that maximized the undiscounted profit from the deposit while adhering to operational, technical and economic constraints.For the base case, the results show that the model can generate stopes with variable heights and lengths, allowing mining operations to adapt to irregular mineral deposit peripheries and minimize dilution. The model also successfully incorporates geotechnical pillar requirements, which are critical for a stable operating environment. Regarding computational experiments, the model was found to be highly sensitive to key input parameters, including cutoff grade, stope dimensions and pillar sizes. This sensitivity underscores its utility as a powerful tool for strategic mine planning and scenario analysis. Despite its strengths, the BILP model has a key limitation: the model does not include diagonal pillar constraints, which are important for stability and would need to be considered in practice. This will require future post-processing. Additionally, long solution times due to the use of binary variables suggest a need for preprocessing techniques in future work to improve computational efficiency. Selected references Janiszewski M, Pontow S, Rinne M (2022) Industry survey on the current state of stope design methods in the underground mining sector. Energies (Basel) 15(1):240 Queyranne M, Wolsey LA (2017) Tight MIP formulations for bounded up/down times and interval-dependent start-ups. Math Program 164(1–2):129–155 Nhleko AS, Tholana T, Neingo PN (2018) A review of underground stope boundary optimization algorithms. Resour Policy 56:59–69

A binary integer linear programming model for optimizing underground stope layout

Authored by: Theophilus Mensah and Kwame Awuah-Offei

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