A mathematical analysis of the lerchs and grossmann algorithm and the nested lerchs and grossmann algorithm

The Lerchs and Grossmann algorithms are used in the surface mining industry to solve the Pit Limit Problem and to generate an approximate solution to the Extraction Sequence Problem. We revisit the theoretical proof of convergence of the algorithm and provide considerable examples to explain the algorithm. We demonstrate that the Lerchs and Grossmann (LG) Algorithm is a graph theoretic implementation of the dual simplex algorithm which solves the Pit Limit Problem. We also prove that the LG Algorithm converges, in a finite number of iterations, to the smallest maximum valued pit limit, when multiple maximum valued pit limits exist, and through a simple change in definition the algorithm, it can be forced to converge to the largest maximum valued pit limit. We also show that the Nested Lerchs and Grossmann (NLG) Algorithm uses Lagrangian relaxation on the volume constrained Pit Limit Problem to generate a sequence of nested pit limits whose average values decrease monotonically. Using the fact that the nested pit limits average values decrease monotonically, we prove they maximize the integral of the global cash flow function. These nested pit limits generate an approximate solution to the Extraction Sequence Problem. Finally, we formulate the Extraction Sequence Problem, which we argue must be a zero-one integer programming problem. Thus, it is classified as an NP-hard problem and $\epsilon$-approximate algorithms to the problem are also NP-hard. Thus, it is impossible to prove that the NLG Algorithm's approximate solution is within some bound of the optimal solution.