extract, in which period, and to which destinations to send them is the most important aspect of mine planning (once the project's production rate has been determined), a decision directly responsible for determining the economic return of the enterprise (Caccetta and Hill, 2003). From a computational point of view, the mine sequencing problem is associated with the precedence constrained knapsack problem (PCKP), a notorious NP-hard problem, meaning that there are no algorithms capable of solving it exactly in an acceptable computational time (Espinoza et al., 2013). When modelling this problem, the vertical precedence relationships between blocks represent a large computational burden, in addition to other necessary constraints such as mining equipment capacity, processing plant capacity, feed grades, and stockpiling, among others. For this reason, various approaches have been used since the 1960s to subdivide the intractable main problem into manageable subproblems. Among these, the ultimate pit limit (UPL) problem should be highlighted, which aims to determine which blocks will be extracted and which will not be extracted throughout the mine's lifetime, significantly reducing the size of the optimisation problem by eliminating blocks that would never be extracted (Caccetta and Hill, 2003). Several approaches have been proposed to model and solve this problem, including the floating cone heuristic (Pana, 1965), the exact graph-based algorithm of Lerchs and Grossmann (1965), and the pseudoflow algorithm (Hochbaum, 2001). Even after solving the UPL problem beforehand, depending on the size of the block model under study, the complexity of the problem would generally still be considerably bigger for the mine sequencing problem to be solved, considering the remaining blocks. For this reason, mine planning practitioners employ the parametric nested pit algorithm (Lerchs and Grossmann, 1965) and then select some of these pits, make operational adjustments (such as inserting a minimum pit width), and name these intermediate pits as phases (Meagher et al., 2014). Finally, it would be possible to solve the mine sequencing problem for each phase, and yet it is common for sequencing to be performed on clusters of blocks or aggregates called bench-phases (Mariz et al., 2024), as this further reduces the complexity of the strategic mine sequencing problem. Figure 1 shows a block model (left) with an ultimate pit limit (red lines) and four phases in red, yellow, magenta, and blue colors (black lines), whose blocks have been aggregated into bench-phase units (right). The reason why the outlines of the final pit are generally larger than the outlines of the sequenced phases is that the UPL problem disregards the effect of time on the economic value of the project, while the mine sequencing problem applies a discount rate to the periods. Figure 1 – Traditional mine planning approach: block model (left) with an ultimate pit limit (red lines) and four phases in red, yellow, magenta, and blue colors (black lines),
RkJQdWJsaXNoZXIy MTM0Mzk2