Scientific, Computing and Applied & Industrial Mathematics: Maurice Queyranne

In open pit mining, one must dig a pit, that is, excavate upper layers of ground to reach valuable minerals. The walls of the pit must satisfy some geomechanical constraints (maximum slope constraints) so as not to collapse. The ultimate pit limits problem is to determine an optimal pit, the total volume to be extracted so as to maximize total net profits. We set up the problem in a continuous space framework (as opposed to discretized space, such as with block models), and we show, under weak assumptions, the existence of an optimum pit. For this, we formulate an infinite-dimensional, optimal transportation problem of the Kantorovich type, where the cost function is lower semi-continuous and is allowed to take the value +infinity. We show that this transportation problem is a strong dual to the optimum pit problem, and also yields optimality (complementary slackness) conditions. This approach has the potential of leading to novel algorithmic approaches, yet to be explored, to the ultimate pit limits and related mine planning problems.

This is joint work with Ivar Ekeland (CEREMADE, Université Paris-Dauphine)