# Gradient-constrained discounted Steiner trees I: optimal tree configurations

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## Abstract

A *gradient-constrained discounted Steiner tree* *T* is a maximum Net Present Value (NPV) tree, spanning a given set N of nodes in space with edges whose gradients are all no more than an upper bound *m* which is the maximum gradient. The nodes in *T* but not in *N* are referred to as *discounted Steiner points*. Such a tree has costs associated with its edges and values associated with its 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. In this paper, we study the optimal tree configurations so as to maximize the sum of all the discounted cash flows, known as the 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 in the presence of the gradient constraint. This constraint defines the navigability conditions on mining vehicles along the underground tunnels. Labellings are essential for defining a tree configuration and indicate gradients on the edges of the network. An edge in a gradient-constrained discounted Steiner tree is labelled as an *f* edge, an *m* edge or a *b* edge, if the gradient is less, equal or greater than *m* respectively. Each tree configuration is identified by the labellings of its edges. In this paper the non-optimal sets of labellings of edges that are incident with the discounted Steiner point in a gradient-constrained discounted Steiner network are classified. This reduces the number of configurations that need to be considered when optimizing. In addition, the gradient-constrained discounted Steiner point algorithm is outlined.

## Keywords

Gradient constraint Network optimisation Optimal mine design Net Present Value Steiner points## References

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