The theory of Steiner trees has been extensively applied in physical network design problems to locate a Steiner point that minimizes the total length of a tree. However, maximizing the total generated cash flows of a tree has not been investigated. Such a tree has costs associated with its edges and values associated with nodes. In order to reach the nodes in the tree, the edges need to be constructed. The edges are constructed in a particular order and the costs of constructing the edges and the values at the nodes are discounted over time. These discounted costs and values generate cash flows. In this paper, we study the problem of optimally locating a single Steiner point so as to maximize the sum of all the discounted cash flows, known as the Net Present Value (NPV).An application of this problem occurs in underground mining where, we want to optimally locate a junction point in the underground access network to maximize the NPV. We propose an efficient iterative algorithm to optimally locate a single degree-3 Steiner point. We show this algorithm converges quickly and the Steiner point is unique subject to realistic design parameters.
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.
A gradient-constrained discounted Steiner tree is a network interconnecting given set of nodes in Euclidean space where the gradients of the edges are all no more than an upper bound which defines the maximum gradient. In such a tree, the costs are associated with its edges and values are associated with nodes and are discounted over time. In this paper, we study the problem of optimally locating a single Steiner point in the presence of the gradient constraint in a tree so as to maximize the sum of all the discounted cash flows, known as the net present value (NPV). An edge in the tree is labelled as a b edge, or a m edge, or an f edge if the gradient between its endpoints is greater than, or equal to, or less than the maximum gradient respectively. The set of edge labels at a discounted Steiner point is called its labelling. The optimal location of the discounted Steiner point is obtained for the labellings that can occur in a gradient-constrained discounted Steiner tree. In this paper, we propose the gradient-constrained discounted Steiner point algorithm to optimally locate the discounted Steiner point in the presence of a gradient constraint in a network. This algorithm is applied to a case study. This problem occurs in underground mining, where we focus on the optimization of underground mine access to obtain maximum NPV in the presence of a gradient constraint. The gradient constraint defines the navigability conditions for trucks along the underground tunnels.
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