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Open-pit mines are surface excavations created to extract valuable material which, in most cases, is located near the surface. Pushbacks are connected regions of a mine with enough working area to support the mining operation over a defined period. Pushbacks designed without taking into consideration haulage ramps are defined here as semi-practical pushbacks. We introduce a new optimisation model for semi-practical pushbacks that accounts for operational conditions: minimum operational width and connectivity within the blocks that compose the pushbacks. Additionally, we propose an algorithm that uses a Sliding Window Heuristic, variable bounding, and other preprocessing routines to obtain solutions with less than 10% optimality gap for mining instances ranging between 30,000 to 50,000 blocks. [ABSTRACT FROM AUTHOR]

A Euclidean skeleton is a set of edges in the interior (or on the boundary) of a polygon that intersects any line segment that joins two points outside of the polygon and that intersects the polygon. In this paper we study minimum cardinality Euclidean skeletons and develop an algorithm for constructing them. We first prove a number of structural properties of minimum skeletons and use these to develop a canonical form. We then design an exact algorithm which initially generates a set of canonical skeleton edges, then executes a pruning module to reduce the set of candidate edges, and finally runs existing integer linear programming code to output an optimal solution. Finally, we perform computational testing on our algorithm to demonstrate its performance, and observe a number of experimental properties of minimum skeletons. [ABSTRACT FROM AUTHOR]

We introduce the concept of an obstacle skeleton, which is a set of line segments inside a polygonal obstacle ω that can be used in place of ω when performing intersection tests for obstacle-avoiding network problems in the plane. A skeleton can have significantly fewer line segments compared to the number of line segments in the boundary of the original obstacle, and therefore performing intersection tests on a skeleton (rather than the original obstacle) can significantly reduce the CPU time required by algorithms for computing solutions to obstacle-avoidance problems. A minimum skeleton is a skeleton with the smallest possible number of line segments. We provide an exact O (n 2) algorithm for computing minimum skeletons for rectilinear obstacles in the rectilinear plane that are rectilinearly convex. [ABSTRACT FROM AUTHOR]

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from $$120^{\circ }$$ . This notion arises in numerical approximations of minimum Steiner trees. We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It has been conjectured that this relative error is at most linear in the maximum error at the angles, independent of the number of terminals. We verify this conjecture for the two-dimensional case as long as the maximum angle error is sufficiently small in terms of the number of terminals. In the two-dimensional case we derive a lower bound for the relative error in length. This bound is linear in terms of the maximum angle error when the angle error is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error in length for larger values of the maximum angle error and calculate exact values in the plane for three and four terminals. [ABSTRACT FROM AUTHOR]

In the design of underground tunnel layout, the development cost is often dependent on the direction of the tunnel at each point due to directional ground fracturing. This paper considers the problem of finding a minimum cost curvature-constrained path between two directed points in 3-space, where the cost at every point along the path depends on the instantaneous direction. This anisotropic behaviour of the cost models the development cost of a tunnel in ground with faulting planes that are almost vertical. The main result we prove in this paper is that there exists an optimal path of the form $$\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}$$ (or a degeneracy), where $$\mathcal {C}$$ represents a segment of a helix with unit radius and $$\mathcal {S}$$ represents a straight line segment. This generalises a previous result that in the restriction of the problem to the horizontal plane there always exists a path of the form $$\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}$$ or a degeneracy which is optimal. We also prove some key structural results which are necessary for creating an algorithm which can construct an optimal path between a given pair of directed points in 3-space with a prescribed directional cost function. [ABSTRACT FROM AUTHOR]

The recent emergence of distributed generation, smart meters, and electric vehicles means that much attention is now being given to network modelling and analysis at the distribution, rather than transmission, level. Many optimisation studies, both regarding technical and economic questions, aim to satisfy the constraints posed by grid infrastructure. We explore in detail one of these network constraints, minimum required voltage, at the distribution level and demonstrate that the physical locations of individual loads in the network play a significant role in determining whether voltages throughout the network remain within required limits or not. Our simulations use real distribution network data and are run on models of two real neighbourhoods. We show that the addition of a single load at a weak point of the network can have the same impact as considerably greater numbers of loads at stronger locations of the network. This has important implications for applications such as electric vehicle charging, and suggests that spatial distribution of loads should be taken into account when analysing network stability. [ABSTRACT FROM AUTHOR]

The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within O( n) time. In this paper we generalise their approach in order to solve the k-Steiner tree problem, in which the Steiner minimum tree may contain up to k Steiner points for a given constant k. We also extend their approach further to encompass other normed planes, and to solve a much wider class of problems, including the k-bottleneck Steiner tree problem and other generalised k-Steiner tree problems. We show that, for any fixed k, such problems can be solved in O( n) time. [ABSTRACT FROM AUTHOR]

The history of the Euclidean Steiner tree problem, which is the problem of constructing a shortest possible network interconnecting a set of given points in the Euclidean plane, goes back to Gergonne in the early nineteenth century. We present a detailed account of the mathematical contributions of some of the earliest papers on the Euclidean Steiner tree problem. Furthermore, we link these initial contributions with results from the recent literature on the problem. [ABSTRACT FROM AUTHOR]

The Steiner tree problem in Euclidean space $$E^3$$ asks for a minimum length network $$T$$ , called a Euclidean Steiner Minimum Tree (ESMT), spanning a given set of points. This problem is NP-hard and the hardness is inherently due to the number of feasible topologies (underlying graph structure of $$T$$ ) which increases exponentially as the number of given points increases. Planarity is a very strong condition that gives a big difference between the ESMT problem in the Euclidean plane $$E^2$$ and in Euclidean $$d$$ -space $$E^d (d\ge 3)$$ : the ESMT problem in the plane is practically solvable whereas the ESMT problem in $$d$$ -space is really intractable. The simplest tree rearrangement technique is to repeatedly replace a subtree spanning four nodes in $$T$$ with another subtree spanning the same four nodes. This technique is referred to as the Swapping 4-point Topology/ Tree technique in this paper. An indicator (or quasi-indicator) of $$T$$ plays a similar role in the optimization of the length $$L(T)$$ of $$T$$ in the discrete topology space (the underlying graph structure of $$T$$ ) to the derivative of a differentiable function which indicates a fastest direction of descent. $$T$$ will be called S4pT-optimal if it is optimal with respect to swapping 4-point subtrees. In this paper we first make a complete analysis of 4-point trees in Euclidean space exploring all possible types of 4-point trees and their properties. We review some known indicators of 4-point ESMTs in $$E^2$$ , and give some simple geometric proofs of these indicators. Then, we translate these indicators to $$E^3$$ , producing eight quasi-indicators in $$E^3$$ using computational experiments, the best quasi-indicator $$\rho _\mathrm{osr}$$ is sifted with an effectiveness for randomly generated 4-point sets as high as 98.62 %. Finally we show how $$\rho _\mathrm{osr}$$ is used to find an S4pT-optimal ESMT on 14 probability vectors in $$4d$$ -space with a detailed example. [ABSTRACT FROM AUTHOR]

This paper studies the problem of finding the minimum cost curvature-constrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvature-constrained paths of minimum length, commonly referred to as Dubins paths. We show that there always exists a path of the form $\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}$ or a degeneracy which is optimal, where $\mathcal {C}$ represents an arc of maximum curvature, and $\mathcal {S}$ represents a straight line. This result is also extended to the case where there is not only a directional-cost, but the cost of curved sections are scaled up by a factor w≥1. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint, the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground, and curved sections may incur more cost due to additional support and ventilation. [ABSTRACT FROM AUTHOR]

This paper looks at the problem of finding the minimum cost curvature-constrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvature-constrained paths of minimum length, commonly referred to as Dubins paths. We conclude that if the reciprocal of the directional-cost function is strictly polarly convex, then the forms of the optimal paths are of the same forms as Dubins paths. If we relax the strict polar convexity to weak polar convexity, then we show that there exists a Dubins path which is optimal. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint and the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground. [ABSTRACT FROM AUTHOR]

Spike-timing-dependent plasticity (STDP) determines the evolution of the synaptic weights according to their pre- and post-synaptic activity, which in turn changes the neuronal activity on a (much) slower time scale. This paper examines the effect of STDP in a recurrently connected network stimulated by external pools of input spike trains, where both input and recurrent synapses are plastic. Our previously developed theoretical framework is extended to incorporate weight-dependent STDP and dendritic delays. The weight dynamics is determined by an interplay between the neuronal activation mechanisms, the input spike-time correlations, and the learning parameters. For the case of two external input pools, the resulting learning scheme can exhibit a symmetry breaking of the input connections such that two neuronal groups emerge, each specialized to one input pool only. In addition, we show how the recurrent connections within each neuronal group can be strengthened by STDP at the expense of those between the two groups. This neuronal self-organization can be seen as a basic dynamical ingredient for the emergence of neuronal maps induced by activity-dependent plasticity. [ABSTRACT FROM AUTHOR]

In contrast to a feed-forward architecture, the weight dynamics induced by spike-timing-dependent plasticity (STDP) in a recurrent neuronal network is not yet well understood. In this article, we extend a previous study of the impact of additive STDP in a recurrent network that is driven by spontaneous activity (no external stimulating inputs) from a fully connected network to one that is only partially connected. The asymptotic state of the network is analyzed, and it is found that the equilibrium and stability conditions for the firing rates are similar for both full and partial connectivity: STDP causes the firing rates to converge toward the same value and remain quasi-homogeneous. However, when STDP induces strong weight competition, the connectivity affects the weight dynamics in that the distribution of the weights disperses more quickly for lower density than for higher density. The asymptotic weight distribution strongly depends upon that at the beginning of the learning epoch; consequently, homogeneous connectivity alone is not sufficient to obtain homogeneous neuronal activity. In the absence of external inputs, STDP can nevertheless generate structure in the network through autocorrelation effects, for example, by introducing asymmetry in network topology. [ABSTRACT FROM AUTHOR]

Spike-timing-dependent plasticity (STDP) determines the evolution of the synaptic weights according to their pre- and post-synaptic activity, which in turn changes the neuronal activity. In this paper, we extend previous studies of input selectivity induced by (STDP) for single neurons to the biologically interesting case of a neuronal network with fixed recurrent connections and plastic connections from external pools of input neurons. We use a theoretical framework based on the Poisson neuron model to analytically describe the network dynamics (firing rates and spike-time correlations) and thus the evolution of the synaptic weights. This framework incorporates the time course of the post-synaptic potentials and synaptic delays. Our analysis focuses on the asymptotic states of a network stimulated by two homogeneous pools of “steady” inputs, namely Poisson spike trains which have fixed firing rates and spike-time correlations. The (STDP) model extends rate-based learning in that it can implement, at the same time, both a stabilization of the individual neuron firing rates and a slower weight specialization depending on the input spike-time correlations. When one input pathway has stronger within-pool correlations, the resulting synaptic dynamics induced by (STDP) are shown to be similar to those arising in the case of a purely feed-forward network: the weights from the more correlated inputs are potentiated at the expense of the remaining input connections. [ABSTRACT FROM AUTHOR]

Spike-timing-dependent plasticity (STDP) is believed to structure neuronal networks by slowly changing the strengths (or weights) of the synaptic connections between neurons depending upon their spiking activity, which in turn modifies the neuronal firing dynamics. In this paper, we investigate the change in synaptic weights induced by STDP in a recurrently connected network in which the input weights are plastic but the recurrent weights are fixed. The inputs are divided into two pools with identical constant firing rates and equal within-pool spike-time correlations, but with no between-pool correlations. Our analysis uses the Poisson neuron model in order to predict the evolution of the input synaptic weights and focuses on the asymptotic weight distribution that emerges due to STDP. The learning dynamics induces a symmetry breaking for the individual neurons, namely for sufficiently strong within-pool spike-time correlation each neuron specializes to one of the input pools. We show that the presence of fixed excitatory recurrent connections between neurons induces a group symmetry-breaking effect, in which neurons tend to specialize to the same input pool. Consequently STDP generates a functional structure on the input connections of the network. [ABSTRACT FROM AUTHOR]