P. A. Grossman

Decline design in underground mines using constrained path optimisation

Abstract This paper focuses on the problem of optimising the design of an underground mine decline, so as to minimise the costs associated with infrastructure development and haulage over the lifetime of the mine. A key design consideration is that the decline must be navigable by trucks and mining equipment, hence must satisfy both gradient and turning circle constraints. The decline is modelled as a mathematical network that captures the operational constraints and costs of a real mine, and is optimised using geometric techniques for constrained path optimisation. A deep understanding of the geometric properties of gradient and turning circle constrained paths has led to a very efficient procedure for designing optimal declines. This procedure has been automated in a new version of a software tool, decline optimisation tool. A case study is described indicating the substantial improvements of the new version of the decline optimisation tool over the earlier one.

Strategic Underground Mine Access Design to Maximise the Net Present Value

 In our case study, the connector that links two underground ore bodies is designed to obtain the maximum NPV.  The 2-face discounted junction point algorithm improves the NPV compared with the underground mine operation with a single face. Also, the improvement of the NPV increases with the discount rate.  In the optimisation, several breakout points are considered and then the best location of the breakout point is identified to obtain the maximum NPV.  In future research, a new algorithm will be developed to locate a single junction point with the gradient constraint. 2. 1FACE AND 2FACE DISCOUNTED JUNCTION POINT ALGORITHMS (1FDJPA & 2FDJPA)

Minimal curvature-constrained networks

This paper introduces an exact algorithm for the construction of a shortest curvature-constrained network interconnecting a given set of directed points in the plane and a gradient descent method for doing so in 3D space. Such a network will be referred to as a minimum Dubins tree, since its edges are Dubins paths (or slight variants thereof). The problem of constructing a minimum Dubins tree appears in the context of underground mining optimisation, where the objective is to construct a least-cost network of tunnels navigable by trucks with a minimum turning radius. The Dubins tree problem is similar to the Steiner tree problem, except the terminals are directed and there is a curvature constraint. We propose the minimum curvature-constrained Steiner point algorithm for determining the optimal location of the Steiner point in a 3-terminal network. We show that when two terminals are fixed and the third varied in the planar version of the problem, the Steiner point traces out a limaçon.

Combined optimisation of an open-pit mine outline and the transition depth to underground mining

Abstract Miners harvest minerals from ore-bodies in the ground by a variety of specialised mining methods, with most falling into the categories of open-pit and underground. Some ore-bodies are harvested by a combination of open-pit and underground methods. In these cases there is often material that could be mined by either method, and an economic choice has to be made. This is referred to as the transition problem and it has received some attention in the mining literature since the 1980s and more recently has had attention in the mathematics literature. The transition problem is complicated by the need in many cases to leave a crown pillar (un-mined rock above the underground mine) and for this crown pillar to have a prescribed shape. We have developed a method to optimise the design of an open-pit mine, while solving the transition problem and taking into account the need for a crown pillar with a prescribed shape. We base it on an existing method to optimise the design of an open-pit mine, framed as a maximum graph closure problem. Our method introduces non-trivial strongly connected sub-graphs (NSCSs) of the graph, a complication that previous authors on maximum graph closure problems do not appear to have covered. To obviate the need to check every method for compatibility with NSCSs, we reduce the problem to an equivalent problem without them. This has the added advantage of reducing overall processing time in cases where the number of NSCSs is large.

MINIMAL CURVATURE-CONSTRAINED PATHS IN THE PLANE WITH A CONSTRAINT ON ARCS WITH OPPOSITE ORIENTATIONS

The declines that provide vehicle access in an underground mine are typically designed as paths formed by concatenating line segments and circular arcs. In order to reduce wear on the ore trucks and the road surfaces and to enhance driver safety, such paths may be subject to a further constraint: each pair of consecutive arcs with opposite orientations must be separated by a straight line segment of at least a certain specified length. In order to reduce the construction and operational costs of the mine, it is desirable to minimize the lengths of such paths between any given pair of directed points. Some necessary and sufficient conditions are obtained for paths of this form to be locally or globally minimal with respect to length. In particular, it is shown that there is always a globally minimal path that contains at most four circular arcs.