Finding Short Integral Cycle Bases for Cyclic Timetabling
نویسنده
چکیده
Cyclic timetabling for public transportation companies is usually modeled by the periodic event scheduling problem. To deduce a mixed-integer programming formulation, artificial integer variables have to be introduced. There are many ways to define these integer variables. We show that the minimal number of integer variables required to encode an instance is achieved by introducing an integer variable for each element of some integral cycle basis. An integral cycle basis consists of |A| − |V |+ 1 oriented cycles of a directed graph D = (V, A) that enable any oriented cycle of the directed graph to be expressed as an integer linear combination. The solution times for the originating application vary extremely with different integral cycle bases. However, our computational studies show that the width of integral cycle bases is a good empirical measure for the solution time of the MIP. Clearly, integral cycle bases permit a much wider choice than the former standard approach, in which integer variables are associated with the co-tree arcs of some spanning tree. Hence, to formulate better solvable integer programs, we present algorithms that construct integral cycle bases of small width. To that end, we investigate classes of directed cycle bases that are closely related to integral cycle bases, namely (generalized) fundamental and undirected cycle bases. This gives rise to both, a compact classification of directed cycle bases and notable reductions of running times for cyclic timetabling.
منابع مشابه
Minimum Cycle Bases in Graphs Algorithms and Applications
A cycle basis of a graph is a family of cycles which spans all cycles of the graph. In an undirected graph, a cycle is simply a set of edges with respect to which every vertex has even degree. We view cycles as vectors indexed by edges. The entry for an edge is one if the edge belongs to the cycle and is zero otherwise. Addition of cycles corresponds to vector addition modulo 2 (symmetric diffe...
متن کاملComparison of Radial Consolidation Behavior of Clay under Three Types of Cyclic Loading
Vertical drains and stone columns which have been used in infrastructure construction for highways, ports, coastal regions, etc., provide significant benefits for improving soil characteristics such as reducing the drainage length and accelerating the consolidation process. So the investigation of the radial consolidation is inevitable. Soils may be subjected to cyclic loading such as silos, ta...
متن کاملA New Solution for the Cyclic Multiple-Part Type Three-Machine Robotic Cell Problem based on the Particle Swarm Meta-heuristic
In this paper, we develop a new mathematical model for a cyclic multiple-part type threemachine robotic cell problem. In this robotic cell a robot is used for material handling. The objective is finding a part sequence to minimize the cycle time (i.e.; maximize the throughput) with assumption of known robot movement. The developed model is based on Petri nets and provides a new method to calcul...
متن کاملA Numerical Approach for Solving of Two-Dimensional Linear Fredholm Integral Equations with Boubaker Polynomial Bases
In this paper, a new collocation method, which is based on Boubaker polynomials, is introduced for the approximate solutions of a class of two-dimensional linear Fredholm integral equationsof the second kind. The properties of two-dimensional Boubaker functions are presented. The fundamental matrices of integration with the collocation points are utilized to reduce the solution of the integral ...
متن کامل