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 multi period 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 largescale 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 methodology for mine sequencing that combines LP relaxation via the critical multiplier algorithm, a topological sorting–based rounding heuristic, and a local search procedure to refine solutions. Lamghari et al. (2015) proposed a two-stage hybrid approach in which an initial solution is generated by decomposing the IP or mixed-integer linear programming (MILP) model into period-wise LP subproblems and then improved using a variable neighbourhood descent (VND) procedure to escape local optima. Mai et al. (2018) proposed a methodology that aggregates blocks into economically positive TopCones respecting slope and operational constraints, followed by an IP model to maximise NPV under precedence, grade control, and capacity constraints (Mariz and Soofastaei, 2022). Regardless of how the sequencing problem is modelled and solved, the transfer function that converts geological block information into economic information is usually an oversimplification of reality, based on the concept of cutoff grade proposed by Lane (1964), which would determine the minimum acceptable grade for a block to be considered as ore. The general calculation of the block value applied to mining can be described as Equation (1):
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