Separation of Cycle Inequalities for the Periodic Timetabling Problem

Cycle inequalities play an important role in the polyhedral study of the periodic timetabling problem. We give the first pseudo-polynomial time separation algorithm for cycle inequalities, and we give a rigorous proof for the pseudo-polynomial time separability of the change-cycle inequalities. The efficiency of these cutting planes is demonstrated on real-world instances of the periodic timetabling problem

Determining All Integer Vertices of the PESP Polytope by Flipping Arcs

We investigate polyhedral aspects of the Periodic Event Scheduling Problem (PESP), the mathematical basis for periodic timetabling problems in public transport. Flipping the orientation of arcs, we obtain a new class of valid inequalities, the flip inequalities, comprising both the known cycle and change-cycle inequalities. For a point of the LP relaxation, a violated flip inequality can be found in pseudo-polynomial time, and even in linear time for a spanning tree solution. Our main result is that the integer vertices of the polytope described by the flip inequalities are exactly the vertices of the PESP polytope, i.e., the convex hull of all feasible periodic slacks with corresponding modulo parameters. Moreover, we show that this flip polytope equals the PESP polytope in some special cases. On the computational side, we devise several heuristic approaches concerning the separation of cutting planes from flip inequalities. We finally present better dual bounds for the smallest and largest instance of the benchmarking library PESPlib

The Second Chvatal Closure Can Yield Better Railway Timetables

We investigate the polyhedral structure of the Periodic Event Scheduling Problem (PESP), which is commonly used in periodic railway timetable optimization. This is the first investigation of Chvatal closures and of the Chvatal rank of PESP instances. In most detail, we first provide a PESP instance on only two events, whose Chvatal rank is very large. Second, we identify an instance for which we prove that it is feasible over the first Chvatal closure, and also feasible for another prominent class of known valid inequalities, which we reveal to live in much larger Chvatal closures. In contrast, this instance turns out to be infeasible already over the second Chvatal closure. We obtain the latter result by introducing new valid inequalities for the PESP, the multi-circuit cuts. In the past, for other classes of valid inequalities for the PESP, it had been observed that these do not have any effect in practical computations. In contrast, the new multi-circuit cuts that we are introducing here indeed show some effect in the computations that we perform on several real-world instances - a positive effect, in most of the cases

A MILP model for quasi-periodic strategic train timetabling

In railways, the long-term strategic planning is the process of evaluating improvements to the railway network (e.g., upgrading a single track line to a double track line) and changes to the composition/frequency of train services (e.g., adding 1 train per hour along a certain route). The effects of different combinations of infrastructure upgrades and updated train services (also called scenarios), are usually evaluated by creating new feasible timetables followed by extensive simulation. Strategic Train Timetabling (STT) is indeed the task of producing new tentative timetables for these what-if scenarios. Unlike the more classic train timetabling, STT can often overlook (or at least give less importance to) some complementary aspects, such as crew and rolling stock scheduling. On the other hand, the different scenarios are likely to lead to very different timetables, hindering the common and effective practice of using existing timetables to warm start the solution process. We introduce the concept of quasi-periodic timetables, that are timetables where certain subsets of trains need to start at almost (rather than precisely) the same minute of every period. The additional flexibility offered by quasi-periodic timetables turned out to be crucial in real-life scenarios characterized by elevated train traffic. We describe a MILP based approach for strategic quasi-periodic train timetabling and we test it on 4 different realistic what-if scenarios for an important line in Norway. The timetables produced by our algorithm were ultimately used by the Norwegian Railway Directorate to select 3 out of the 4 scenarios for phasing the progressive expansion of the JȪren line.publishedVersio

Network optimization in railway transport planning

This work is dealing with train timetabling problem. In the first chapter, one can find an introduction to network flows which is needed for understanding deeper concepts later on. Namely, basic graph theory definitions are stated as well as core problems like the minimum cost flow and shortest path problem. Furthermore, two equivalent representations of network flows are described, including some useful properties connected to each of them. At the end of the chapter, linear programming and simplex method are introduced into some detail. In the second chapter more complex theory is introduced. At the beginning, multi-commodity flow problem is stated and few solutions approaches are briefly described. Once we settled for one of them, the rest of the chapter is dealing with Lagrangian relaxation and column generation techniques. Since column generation is the main result needed for solving our problem, some finer results, like determining lower and upper bounds, are stated. In the last, third chapter, one can find a model for representing train timetabling problem for a single line network. That model was introduced by Valentina Cacchiani in her Ph.D. thesis. In this work, periodicity of timetable is assumed because it makes computations way quicker, as well as it has some other benefits. At the end, one can find an algorithm based on column generation technique for solving introduced model. That algorithm is based on 6 steps, and after reading this work, one should be able to fully understand each of them.Ovaj rad bavi se problemom rasporeda vožnje u željezničkom prometu. U prvom poglavlju nalazi se uvod u mrežne tokove koji je potreban za razumijevanje naprednijih koncepata. Konkretno, iskazane su osnovne definicije teorije grafova kao i neki temeljni problemi poput problema najjeftinijeg toka i problema najkraćeg puta. Nadalje, opisana su dva ekvivalenta prikaza mrežnih tokova, uključujući neka korisna svojstva za svaki od njih. Na kraju poglavlja, linearno programiranje i simpleks metoda, objašnjeni su na razini razumijevanja. U drugom poglavlju nalazi se naprednija teorija koja se nadovezuje na prvo poglavlje. Na početku poglavlja prikazan je problem više dobara, kao i nekoliko pristupa rješavanju navedenog problema. Nakon što smo se odlučili za jedan od pristupa, ostatak poglavlja bavi se Lagrangeovom relaksacijom i metodom generacije stupaca. Kako je upravo metoda generacije stupaca najvažniji rezultat za rješavanje našega problema, napredniji rezultati vezani uz određivanje donjih i gornjih granica su detaljno objasnjeni. U posljednjem, trećem poglavlju, nalazi se model za prikazivanje problema rasporeda vožnje za mreže s jednom tračnicom. Navedeni model prvi puta je predstavljen u doktorskom radu Valentine Cacchiani. U ovom radu također pretpostavljamo periodičnost rasporeda vožnje kako bismo, između ostalih, ostvarili prednost poput bržeg vremena računanja. Na kraju rada nalazi se algoritam, temeljen na metodi generacije stupaca, za rješavanje predstavljenog modela. Navedeni algoritam sastoji se od 6 koraka, od kojih je svaki detaljno opisan u ovome radu

Integrated Periodic Timetabling and Vehicle Circulation Scheduling

Periodic timetabling is one of the most well researched problems in the public transport optimization literature. However, the impact timetabling has on the number of required vehicles, which directly translates to operator costs, is rarely considered. Therefore, in this paper, we consider the problem of jointly optimizing the timetable and the vehicle circulation schedule, which specifies the cyclic sequences of trips vehicles perform. In order to be able to solve realistic instances, we improve an earlier proposed formulation by contraction techniques, valid inequalities and symmetry-breaking constraints. Ultimately, this allows us to explore the trade-off between the number of vehicles and the attractiveness of the timetable from the passengers' perspective. An extensive computational stu

On Cyclic Timetabling and Cycles in Graphs

The cyclic railway timetabling problem (CRTP) essentially is defined by some constraint graph together with a cycle period time. We point out the relevance of cycles of the constraint graph for the CRTP. This covers valid inequalities for a Branch and Cut approach and special cases in that CRTP becomes easy. But emphasis will be put on the problem formulation. The most intuitive model for cyclic timetabling involves node potentials. Modelling the cyclicity appropriately immediately results in an integer variable for every restriction, or arc of the constraint graph. A more sophisticated model regards the corresponding (periodic) tension. This well-established approach only requires an integer variable for every co-tree arc. The latter may be interpreted as representants for the elements of (strictly) fundamental cycle bases. We introduce the more general concept of integral cycle bases for characterizing periodic tensions. Whereas the number of integer variables is still limited to the cyclomatic number of the constraint graph, there is a much wider choice for the cycle basis. One can profit immediately from this, because there are box constraints known for every integer variable that could ever appear