Bi-Objective Evolutionary Optimization for Large-Scale Open Pit Mine Scheduling Problem under Uncertainty with Chance Constraints

TL;DR

The paper tackles large-scale OPMSP under geological uncertainty by formulating a bi-objective optimization that maximizes expected discounted NPV while minimizing its risk, without fixing a confidence level. It adopts an integer-vector schedule encoding and introduces domain-specific initialization and a PeriodSwapMutation, integrated into GSEMO, MOEA/D, and NSGA-II to generate Pareto fronts capturing risk–return trade-offs. Experimental results on MineLib instances show the bi-objective approach yields higher mean chance-constrained NPV and substantially lower variability than a fixed-alpha single-objective baseline, with MOEA/D-OPMSP often delivering the best performance. This approach provides a flexible, robust planning tool for open-pit mining under uncertainty, enabling decision-makers to select schedules across a range of confidence levels without re-optimizing for each $\alpha$.

Abstract

The open-pit mine scheduling problem (OPMSP) is a complex, computationally expensive process in long-term mine planning, constrained by operational and geological dependencies. Traditional deterministic approaches often ignore geological uncertainty, leading to suboptimal and potentially infeasible production schedules. Chance constraints allow modeling of stochastic components by ensuring probabilistic constraints are satisfied with high probability. This paper presents a bi-objective formulation of the OPMSP that simultaneously maximizes expected net present value and minimizes scheduling risk, independent of the confidence level required for the constraint. Solutions are represented using integer encoding, inherently satisfying reserve constraints. We introduce a domain-specific greedy randomized initialization and a precedence-aware period-swap mutation operator. We integrate these operators into three multi-objective evolutionary algorithms: the global simple evolutionary multi-objective optimizer (GSEMO), a mutation-only variant of multi-objective evolutionary algorithm based on decomposition (MOEA/D), and non-dominated sorting genetic algorithm II (NSGA-II). We compare our bi-objective formulation against the single-objective approach, which depends on a specific confidence level, by analyzing mine deposits consisting of up to 112 687 blocks. Results demonstrate that the proposed bi-objective formulation yields more robust and balanced trade-offs between economic value and risk compared to single-objective, confidence-dependent approach.

Figures (13)

• Figure 1: Flowchart for block value calculation for ore and waste blocks.
• Figure 2: Cross-sections along the Y-plane showing the mineral grade distributions.
• Figure 3: Histogram with KDE and QQ plot for the distribution of total economic value from 50 ensemble realizations.
• Figure 4: Box plot of chance constrained discounted NPV ($ in millions) for (1+1) EA-OPMSP, GSEMO-OPMSP, MOEA/D-OPMSP, and NSGA-II-OPMSP across three benchmark instances and confidence levels $\alpha = {0.60, 0.90, 0.99}$.
• Figure 5: Yearly discounted expected profit ($\mu_t$) as bar charts (left) and standard deviation ($\sigma_t$) as box plots (right) for confidence level $\alpha = 0.6$, for (1+1) EA-OPMSP, GSEMO-OPMSP, MOEA/D-OPMSP, and NSGA-II-OPMSP across the Newman1, Marvin, and Mclaughlin Limit instances.