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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.
Line planning in public transport involves determining vehicle routes and assigning frequencies of service such that travel demands are satisfied. We evaluate how line plans, which are optimal with respect to in-motion costs (IMC), the objective function depending purely on arc-lengths for both user and operator costs, performs with respect to the value of resources consumed (VRC). The latter is an elaborate, socio-economic cost function which includes discomfort caused by delay, boarding and alighting times, and transfers. Even though discomfort is a large contributing factor to VRC and is entirely disregarded in IMC, we observe that the two cost functions are qualitatively comparable.
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 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.
We consider the line planning problem in public transport in the Parametric City, an idealized model that captures typical scenarios by a (small) number of parameters. The Parametric City is rotation symmetric, but optimal line plans are not always symmetric. This raises the question to quantify the symmetry gap between the best symmetric and the overall best solution. For our analysis, we formulate the line planning problem as a mixed integer linear program, that can be solved in polynomial time if the solutions are forced to be symmetric. The symmetry gap is provably small when a specific Parametric City parameter is fixed, and we give an approximation algorithm for line planning in the Parametric City in this case. While the symmetry gap can be arbitrarily large in general, we show that symmetric line plans are a good choice in most practical situations.
We consider the price-optimal earliest arrival problem in public transit (POEAP) in which we aim to calculate the Pareto-front of journeys with respect to ticket price and arrival time in a public transportation network. Public transit fare structures are often a combination of various fare strategies such as, e.g., distance-based fares, zone-based fares or flat fares. The rules that determine the actual ticket price are often very complex. Accordingly, fare structures are notoriously difficult to model as it is in general not sufficient to simply assign costs to arcs in a routing graph. Research into POEAP is scarce and usually either relies on heuristics or only considers restrictive fare models that are too limited to cover the full scope of most real-world applications. We therefore introduce conditional fare networks (CFNs), the first framework for representing a large number of real-world fare structures. We show that by relaxing label domination criteria, CFNs can be used as a building block in label-setting multi-objective shortest path algorithms. By the nature of their extensive modeling capabilities, optimizing over CFNs is NP-hard. However, we demonstrate that adapting the multi-criteria RAPTOR (MCRAP) algorithm for CFNs yields an algorithm capable of solving POEAP to optimality in less than 400 ms on average on a real-world data set. By restricting the size of the Pareto-set, running times are further reduced to below 10 ms.
Periodic timetabling is a central aspect of both the long-term organization and the day-to-day operations of a public transportation system. The Periodic Event Scheduling Problem (PESP), the combinatorial optimization problem that forms the mathematical basis of periodic timetabling, is an extremely hard problem, for which optimal solutions are hardly ever found in practice. The
most prominent solving strategies today are based on mixed-integer programming, and there is a concurrent PESP solver employing a wide range of heuristics [3]. We present tropical neighborhood search (tns), a novel PESP heuristic. The method is based on the relations between periodic timetabling and tropical geometry [4]. We implement tns into the concurrent solver, and test it on instances of the benchmarking library PESPlib. The inclusion of tns turns out to be quite beneficial to the solver: tns is able to escape local optima for the modulo network simplex algorithm, and the overall share of improvement coming from tns is substantial compared to the other methods
available in the solver. Finally, we provide better primal bounds for five PESPlib instances.
The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetabling problems in public transport. A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. To conclude we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis.
Public transportation networks are typically operated with a periodic timetable. The periodic event scheduling problem (PESP) is the standard mathematical modeling tool for periodic timetabling. PESP is a computationally very challenging problem: For example, solving the instances of the benchmarking library PESPlib to optimality seems out of reach. Since PESP can be solved in linear time on trees, and the treewidth is a rather small graph parameter in the networks of the PESPlib, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth, or even better, fixed-parameter tractable algorithms. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. Moreover, we show W[1]-hardness of the general feasibility problem with respect to treewidth, which means that we can most likely only accomplish pseudo-polynomial-time algorithms on input networks with bounded tree- or branchwidth. We present two such algorithms based on dynamic programming. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special—but standard—structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.