Application of an improved Lagrangian relaxation approach in the constrained long-term production scheduling problem under grade uncertainty

In open-pit mines, the long-term production scheduling (LTPS) problem is a mixed-integer programming problem and is considered as a class of NP-hard problems that has to be solved in a reasonably small time owing to the operational requirements. The LTPS problem cannot be considered a well-solved problem. The current article presents hybrid models to elucidate the LTPS problem regarding grade uncertainty with the involvement of Lagrangian relaxation and augmented Lagrangian relaxation (ALR) with metaheuristic methods, firefly algorithm (FA) and bat algorithm. The results demonstrate that the ALR-FA has the best results in terms of net present value, average ore grade and computational time, and it is significantly better than the conventional method. Finally, analysis of the results shows that the proposed method generates a near-optimal solution within a reasonable time; thus, it could be a good proposition for use in the industry.