2075 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 benchphase 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), whose blocks have been aggregated into bench-phase units (right) (Modified from Mariz et al., 2024). All these subjective interventions and simplifications of the main problem cause the viable solutions found (mining sequences) to be suboptimal. In contrast, the direct block scheduling (DBS) approach has become popular with the increase of computational power, in which the blocks are addressed as integer variables (blocks can only be mined in a single period, and it is not allowed to split them) and the mine sequencing problem is addressed directly (Morales et al., 2015). A large number of studies have been released on this topic in the last 20 years. Boland et al. (2007) proposed a strengthened integer programming (IP) formulation that combines precedence and production constraints into 0–1 knapsack inequalities, reducing computational effort while achieving optimal solutions. Bienstock and Zuckerberg (2010) proposed an iterative algorithm (BZ) that exploits the underlying combinatorial structure of precedence-constrained and capacitated optimisation problems to efficiently solve linear programming (LP) relaxations. By updating primal structures instead of dual information, the method accelerates convergence and achieves optimality in very large-scale instances within a small number of iterations. Moreno et al. (2010) proposed the critical multiplier algorithm to solve LP relaxations of multi-period IP-based mine sequencing problems, along with an LP-based topological sort heuristic to generate feasible solutions. Their approach decomposes a multiperiod PCKP into ordered single-period problems, which can be represented as topologically sorted partial solutions (Mariz and Soofastaei, 2022). Cullenbine et al. (2011) proposed a sliding time window heuristic to solve large-scale IP mine sequencing models, achieving solutions within about 2% of optimality when exact methods failed. The approach combines repaired, rigorous, and relaxed sub-models across time windows, with Lagrangian relaxation playing a key role in computational efficiency. Chicoisne et al. (2012) proposed a three-stage
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