Underground mine plan optimisation
Abstract© 2019 David WhittleThis thesis addresses several topics relating to the planning of underground mines, with a focus on underlying mathematical models.
Some mineral resources are mined by a combination of open-pit and underground mining methods and a decision must be made as to which methods to apply to different parts of the resource. This is called the transition problem, to which Chapter 3 is dedicated. My contribution, is a graph theory-based optimization model that solves the transition problem efficiently for large data sets with various geometric constraints.
The remainder of this thesis focuses on the optimization of underground mine plans. My major contribution concerns a sub-problem that is framed as a Prize collecting Euclidean Steiner tree problem. This is a generalization of the Euclidean Steiner tree problem. A problem instance is a set of points in the plane, each with a point weight. Of interest are networks on some subset of these points. Networks can include additional vertices called Steiner points if their inclusion yields a shorter network. The value of the network is calculated as the sum of the point weights in a selected subset, less the sum of the lengths of the edges in the network connecting these points. The question is: What selection of points and connected network has the highest value?
There is a great deal of literature on the Euclidean Steiner tree problem and efficient solutions are available. In contrast, there are no solutions to the prize collecting generalization, only an approximation scheme. I have developed an algorithmic framework for the problem (Chapter 6). Included are efficient methods to determine a subset of points that must be in every solution (ruled in) and a subset of points that cannot be in any solution (ruled out). Also included are methods to generate and concatenate full Steiner trees. My generation and concatenation approaches are elaborations on existing equivalent functions for the simpler Euclidean Steiner tree problem.
Two of the ruling out methods require new universal geometric constants. For one of these, I have been able to prove an infimum. This is a strong result. The proof for this infimum is long, and a chapter is dedicated to it (Chapter 7: A universal constant for replacement argument A). For the other universal constant, I have a proof for a lower bound, and a conjecture for an infimum. This second universal constant also has its own chapter (Chapter 8: A universal constant for replacement argument B).
Finally, I have also developed two new decompositions of the underground mine planning problem (Chapters 4 and 5). These decompositions are at an early stage and I plan to apply myself to their further development in the future