Target Patterns and Horseshoes from a Perturbed Central-Force Problem: Some Temporally Periodic Solutions to Reaction-Diffusion Equations

Existence and construction of some time-periodic solutions of a class of reaction-diffusion equations is described. These give examples of center-structures from which emanate trains of waves with outward directed group velocity, in an infinite one-dimensional spatial domain. A discrete set of distinct types of center-structures is found. Similar results are found for sufficiently large finite regions with impermeable boundaries. The existence of many other time periodic solutions corresponding to spatially infinite irregular arrays of center-structures of different types is also demonstrated for these systems. Some numerical examples are presented.     

A Note on the Use of Lagrangean Relaxation in Railway Scheduling

A previously published integer linear programming model for finding a minimum cost schedule for railway locomotives to work a given set of trains is considered. Lagrangean relaxation is suggested as a effective means of solving the model.     

Timetabling optimization of a single railway track line with sensitivity analysis

The paper deals with the timetabling problem of a single-track railway line. To solve the timetabling problem, we propose a three-stage approach combining several optimization criteria. Initially and mainly, the maximum relative travel time (ratio of travel time to minimum possible travel time) is minimized subject to a set of constraints, including departure time, train speed, minimum and maximum dwell time, and headway at track segments and stations. Since this problem has many solutions, the process is repeated for other trains, keeping the relative travel times of the critical train fixed, until all trains have been assigned their optimal relative travel times. In the second stage, the prompt allocation of trains is a secondary objective, and finally, in the third stage, the one minimizing the sum of the station dwell times of all trains, keeping the relative travel times constant, is selected to reduce fuel consumption, as a tertiary objective. To consider the user preferences in the optimization problems, the user preference departure time is used instead of the actual planned departure times. In order to guarantee that the exact or a very good approximate global optimum is attained, an algorithm based on the bisection rule is used. This method allows the computation time to be reduced in at least one order of magnitude for 42 trains. The problem of sensitivity analysis is also discussed, and closed form formulas for the sensitivities in terms of the dual variables are given. Several examples of applications are presented to illustrate the goodness of the proposed method. The results show that an adequate selection of intermediate stations and of the departure times are crucial in the good performance of the line and that inadequate spacings between consecutive trains can block the line. In addition, it is shown that, in order to improve performance, regional trains must be scheduled just ahead of or following the long distance trains, rather than having independent schedules. The sensitivities are shown to be very useful in identifying critical trains, segments, stations, departure times, and headways and in suggesting line infrastructure changes.     

货物列车的编组调度问题

研究的是货物列车的编组和调度问题.通过对问题的深入研究,设计了一种车辆编组调度方案的算法.按照这种算法,在数据处理的基础上利用VC编写每个问题的处理程序,实现了对列车的快速安全高效的调度.对每个问题进行处理,都得到符合要求的结果.问题一首先对整个车辆编组调度的问题进行分析,在尽量保证新组装列车满载的基础上,使每班的中时尽可能少.为此,本文解决了两个关键问题:一是选车问题,二是拆解重组的问题.采用梯形方案对列车车辆进行编队重组,对选车问题主要采用按照时间先后顺序的选车方案,然后通过启发式算法配合遗传算法的选车方案对按时间先后顺序的方案进行检验.从编写的VC程序的运行结果来看,两种方案都可得到满意的结果,遗传算法得到的结果更为合理.另外,为了达到中时最短,采用双推双滑的方式利用驼峰线,提高了调度效率,并在驼峰线和编组道之间加入了碰撞检验模块,保证了列车调度时的安全性.问题二的求解是在问题一的基础上对待拆列车按优先级进行分类.对优先级高的列车先进行拆解.救灾车辆最高,其次是军列和发往S1的车辆,最后是一般车辆.问题三的处理主要是在问题二的基础上,通过提前获得列车的相关信息来决定编组场的列车离开编组场的时刻,从而缩短车辆的中时.问题四在原有模型基础上对编组方案进行了修改,利用编写的VC程序重新计算了每班的中时和列车的调度方案.问题五主要分析了整个系统瓶颈所在,分析了提高资源利用率的可行性.最后,通过对站名的调整,达到了对地质灾害等对铁路系统的破坏突发情况的有效处理,并且进一步分析了如何提高车站的效率的调度方案和建议.     

Control of chaotic solutions of the Hindmarsh–Rose equations

Irregular bursting and spiking solutions of the Hindmarsh–Rose model for the electrical activity of neuron cell bodies have been converted by a chaos control algorithm to periodic spike trains. A proportional feedback method is used to control both chaotic spike trains and chaotic bursting by applying controlling perturbations to membrane parameters.     

Bicriteria train scheduling for high-speed passenger railroad planning applications

This paper is concerned with a double-track train scheduling problem for planning applications with multiple objectives. Focusing on a high-speed passenger rail line in an existing network, the problem is to minimize both (1) the expected waiting times for high-speed trains and (2) the total travel times of high-speed and medium-speed trains. By applying two practical priority rules, the problem with the second criterion is decomposed and formulated as a series of multi-mode resource constrained project scheduling problems in order to explicitly model acceleration and deceleration times. A branch-and-bound algorithm with effective dominance rules is developed to generate Pareto solutions for the bicriteria scheduling problem, and a beam search algorithm with utility evaluation rules is used to construct a representative set of non-dominated solutions. A case study based on Beijing–Shanghai high-speed railroad in China illustrates the methodology and compares the performance of the proposed algorithms.     

Dynamics of an Overhead Line and Pantograph System

The dynamic interaction between pantograph and catenary is studied. The overhead electrification system for high speed trains is modelled by visco‐elastically connected double homogeneous strings system, one finite suspended on rigid concentrated supports and the second of infinite length. The pantograph is represented by one degree of freedom oscillator moving along the string at a constant speed. The model is new in that it gives possibility to consider the influence of the locomotive transverse vibrations on railway overhead contact system. The dynamic state of the investigated system is described by a nonlinear set of coupled partial differential equations with complicated boundary conditions. Dynamic transverse displacements of the connected strings are determined. General results are illustrated by numerical examples.     

Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients

A new active-set method for smooth box-constrained minimization is introduced. The algorithm combines an unconstrained method, including a new line-search which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradient) for dropping constraints from the working set. Global convergence is proved. A computer implementation is fully described and a numerical comparison assesses the reliability of the new algorithm.     

An improved local branching approach for train formation planning

The train formation plan (TFP) determines routing and frequency of trains, and assigns the demands to trains. In this paper, an improved local branching algorithm is proposed for the TFP model in Iranian railway. This solution strategy is exact in nature, although it is designed to improve the heuristic behavior of the mixed integer programming (MIP) solver at hand. In the local branching algorithm, additional constraints are built in the model for the binary variables, but in the improved local branching algorithm, the additional constraints are built in the model for integer variables. A state-of-the-art method is applied for parameter tuning using design of experiments approach. To evaluate the proposed solution method, we have simulated and solved twenty test problems. The results show the efficiency and effectiveness of the proposed approach. The proposed algorithm is implemented for Iranian Railway network as a case study.     

Algorithm for Constructing a Guaranteeing Program Package in a Control Problem with Incomplete Information

A package control problem is considered for a target set at a moment of time. The dynamic system under control is described by linear differential equations, the control area is a convex compact, and the target set is convex and closed. A version of the subsequent approximations method in extended space is proposed for constructing elements of a guaranteeing program package in the case of regular clusters.     

Method of particular solutions for linear,two-point boundary-value problems

The methods commonly employed for solving linear, two-point boundary-value problems require the use of two sets of differential equations: the original set and the derived set. This derived set is the adjoint set if the method of adjoint equations is used, the Green's functions set if the method of Green's functions is used, and the homogeneous set if the method of complementary functions is used.With particular regard to high-speed digital computing operations, this paper explores an alternate method, the method of particular solutions, in which only the original, nonhomogeneous set is used. A general theory is presented for a linear differential system ofnth order. The boundary-value problem is solved by combining linearly several particular solutions of the original, nonhomogeneous set. Both the case of an uncontrolled system and the case of a controlled system are considered.This research, supported by the NASA-Manned Spacecraft Center, Grant No. NGR-44-006-089, is a condensed version of the investigations described in Refs. 1 and 2.     

External Ellipsoidal Estimation of the Union of Two Concentric Ellipsoids and Its Applications

For an arbitrary set representable as the convex hull formed by the union of two concentric ellipsoids we propose a method to construct a family of external undominated ellipsoidal approximations and represent the estimated set as the intersection of all estimates from a given family. A sufficient condition of undominated guaranteed ellipsoidal approximation of a convex compactum is derived. A method is described that for certain classes of sets (such as the intersection of an ellipsoid or a cone with two halfspaces) constructs a family of internal undominated ellipsoidal approximations using the previous formulas for the external estimates of the union of concentric ellipsoids.     

Modulational Stability of Two-Phase Sine-Gordon Wavetrains

A modulational stability analysis is presented for real, two-phase sine-Gordon wavetrains. Using recent results on the geometry of these real solutions, an invariant representation in terms of Abelian differentials is derived for the sine-Gordon modulation equations. The theory thus attains the same integrable features of the previously completed KdV and sinh-Gordon modulations. The twophase results are as follows: kink-kink trains are stable, while the breather trains and kink-radiation trains are unstable, to modulations.     

Nominal and robust train timetabling problems

In this paper we survey the main studies dealing with the train timetabling problem in its nominal and robust versions. Roughly speaking, the nominal version of the problem amounts of determining “good” timetables for a set of trains (on a railway network or on a single one-way line), satisfying the so-called track capacity constraints, with the aim of optimizing an objective function that can have different meanings according to the requests of the railway company (e.g. one can be asked to schedule the trains according to the timetables preferred by the train operators or to maximize the passenger satisfaction). Two are the main variants of the nominal problem: one is to consider a cyclic (or periodic) schedule of the trains that is repeated every given time period (for example every hour), and the other one is to consider a more congested network where only a non-cyclic schedule can be performed. In the recent years, many works have been dedicated to the robust version of the problem. In this case, the aim is to determine robust timetables for the trains, i.e. to find a schedule that avoids, in case of disruptions in the railway network, delay propagation as much as possible. We present an overview of the main works on train timetabling, underlining the differences between models and methods that have been developed to tackle the nominal and the robust versions of the problem.     

Using dual network bounds in algorithms for solving generalized set packing partitioning problems

This article deals with a method to compute bounds in algorithms for solving the generalized set packing/partitioning problems. The problems under investigation can be solved by the branch and bound method. Linear bounds computed by the simplex method are usually used. It is well known that this method breaks down on some occasions because the corresponding linear programming problems are degenerate. However, it is possible to use the dual (Lagrange) bounds instead of the linear bounds. A partial realization of this approach is described that uses a network relaxation of the initial problem. The possibilities for using the dual network bounds in the approximation techniques to solve the problems under investigation are described.     

Improvement of extreme controls and the steepest ascent method in the norm maximization problem on the reachable set

The maximization of the terminal state norm of a linear system is considered in the sense of searching for and improving extreme points of the reachable set. A sufficient optimality condition is formulated in terms of a special maximum function. A steepest ascent method for level surfaces of the objective function is constructed, and related procedures for improving extreme controls are described.     

Zur minimierung konvexer subdifferenzierbarer funktionen Über dem R n

A minimization method is described for convex subdifferentiable functions and the convergence proof is given. The method is based on an extension of the subdifferential in such a way that this point to set map is continuous. Different methods follow for a special choice of the used norm and the used extension of the subdifferential. For certain classes of functions the method is explained in detail and corresponding realisations of the extensions of the subdiffential are given.     

A Lagrangian heuristic algorithm for a real-world train timetabling problem

The train timetabling problem (TTP) aims at determining an optimal timetable for a set of trains which does not violate track capacities and satisfies some operational constraints.In this paper, we describe the design of a train timetabling system that takes into account several additional constraints that arise in real-world applications. In particular, we address the following issues:

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Manual block signaling for managing a train on a track segment between two consecutive stations.

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Station capacities, i.e., maximum number of trains that can be present in a station at the same time.

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Prescribed timetable for a subset of the trains, which is imposed when some of the trains are already scheduled on the railway line and additional trains are to be inserted.

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Maintenance operations that keep a track segment occupied for a given period.

We show how to incorporate these additional constraints into a mathematical model for a basic version of the problem, and into the resulting Lagrangian heuristic. Computational results on real-world instances from Rete Ferroviaria Italiana (RFI), the Italian railway infrastructure management company, are presented.     

Nonlinear Chebyshev fitting from the solution of ordinary differential equations

The nonlinear Chebyshev approximation of real-valued data is considered where the approximating functions are generated from the solution of parameter dependent initial value problems in ordinary differential equations. A theory for this process applied to the approximation of continuous functions on a continuum is developed by the authors in [17]. This is briefly described and extended to approximation on a discrete set. A much simplified proof of the local Haar condition is given. Some algorithmic details are described along with numerical examples of best approximations computed by the Exchange algorithm and a Gauss-Newton type method.