90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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Periodic timetabling is a challenging planning task in public transport.
As safety requirements are crucial, track allocation is indispensable for validating the practical feasibility of a railway timetable. For busy stations with limited capacities, this requires a detailed planning of turnarounds. It is therefore desirable to integrate timetabling not only with track allocation, but also with vehicle scheduling and line planning. This is captured by the Integrated Line Planning and Turn-Sensitive Periodic Timetabling Problem with Track Choice, whose MIP formulation has been demonstrated to be effective for construction site railway rescheduling, as long as a good quality initial solution is available. In this paper, we discuss how to generate such a solution by extending the SAT formulation of the Periodic Event Scheduling Problem with track choice, track occupation, and minimum service frequency components. The SAT approach is superior to pure MIP on real-world instances of the S-Bahn Berlin network.
The Periodic Event Scheduling Problem (PESP) is a notoriously hard combinatorial optimization problem, essential for the design of periodic timetables in public transportation. The coefficients of the integer variables in the standard mixed integer linear programming formulations of PESP are the period time, e.g., 60 for a horizon of one hour with a resolution of one minute. In many application scenarios, lines with different frequencies have to be scheduled, leading to period times with many divisors. It then seems natural to consider derived instances, where the period time is a divisor of the original one, thereby smaller, and bounds are scaled and rounded accordingly. To this end, we identify two rounding schemes: wide and tight. We then discuss the approximation performance of both strategies, in theory and practice.
We present incremental heuristics for the Periodic Event Scheduling Problem (PESP), the standard mathematical tool to optimize periodic timetables in public transport. The core of our method is to solve successively larger subinstances making use of previously found solutions. Introducing the technical notion of free stratifications, we formulate a general scheme for incremental heuristics for PESP. More practically, we use line and station information to create heuristics that add lines or stations one by one, and we evaluate these heuristics on instances of the benchmarking library PESPlib. This approach is indeed viable, and leads to new incumbent solutions for six PESPlib instances.
The Periodic Event Scheduling Problem (PESP) is the central mathematical tool for periodic timetable optimization in public transport. PESP can be formulated in several ways as a mixed-integer linear program with typically general integer variables.
We investigate the split closure of these formulations and show that split inequalities are identical with the recently introduced flip inequalities. While split inequalities are a general mixed-integer programming technique, flip inequalities are defined in purely combinatorial terms, namely cycles and arc sets of the digraph underlying the PESP instance. It is known that flip inequalities can be separated in pseudo-polynomial time. We prove that this is best possible unless P $=$ NP, but also observe that the complexity becomes linear-time if the cycle defining the flip inequality is fixed.
Moreover, introducing mixed-integer-compatible maps, we compare the split closures of different formulations, and show that reformulation or binarization by subdivision do not lead to stronger split closures. Finally, we estimate computationally how much of the optimality gap of the instances of the benchmark library PESPlib can be closed exclusively by split cuts, and provide better dual bounds for five instances.
The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks, and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.
Classic models to derive a timetable for public transport often face a chicken-and-egg situation: A good timetable should offer passengers routes with small travel times, but the route choice of passengers depends on the timetable. While models that fix passenger routes were frequently considered in the literature, integrated models that simultaneously optimize timetables and passenger routes have seen increasing attention lately. This creates a growing need for a set of instances that allows to test and compare new algorithmic developments for the integrated problem. Our paper addresses this requirement by presenting TimPassLib, a new benchmark library of instances for integrated periodic timetabling and passenger routing.
The Periodic Event Scheduling Problem (PESP) is the central mathematical model behind the optimization of periodic timetables in public transport. We apply Benders decomposition to the incidence-based MIP formulation of PESP. The resulting formulation exhibits particularly nice features: The subproblem is a minimum cost network flow problem, and feasibility cuts are equivalent to the well-known cycle inequalities by Odijk. We integrate the Benders approach into a branch-and-cut framework, and assess the performance of this method on instances derived from the benchmarking library PESPlib.
Periodic timetable optimization problems in public transport can be modeled as mixed-integer linear programs by means of the Periodic Event Scheduling Problem (PESP). In order to keep the branch-and-bound tree small, minimum integral cycle bases have been proven successful. We examine forward cycle bases, where no cycle is allowed to contain a backward arc. After reviewing the theory of these bases, we describe the construction of an integral forward cycle basis on a line-based event-activity network. Adding turnarounds to the instance \texttt{R1L1} of the benchmark library PESPlib, we computationally evaluate three types of forward cycle bases in the Pareto sense, and come up with significant improvements concerning dual bounds.
We investigate preprocessing for single-source shortest path queries in digraphs, where arc costs are only known to lie in an interval. More precisely, we want to decide for each arc whether it is part of some shortest path tree for some realization of costs. We show that this problem is solvable in polynomial time by giving a combinatorial algorithm, using optimal structures that we call forks. Our algorithm turns out to be very efficient in practice, and is sometimes even superior in quality to a heuristic developed for the one-to-one shortest path problem in the context of passenger routing in public transport.
We present an optimization model which is capable of routing and ordering trains on a microscopic level under a moving block regime. Based on a general timetabling definition (GTTP) that allows the plug in of arbitrarily detailed methods to compute running and headway times, we describe a layered graph approach using velocity expansion, and develop a mixed integer linear programming formulation. Finally, we present promising results for a German corridor scenario with mixed traffic, indicating that applying branch-and-cut to our model is able to solve reasonably sized instances with up to hundred trains to optimality.