A Configuration Model for the Line Planning Problem

We propose a novel extended formulation for the line planning problem in public transport. It is based on a new concept of frequency configurations that account for all possible options to provide a required transportation capacity on an infrastructure edge. We show that this model yields a strong LP relaxation. It implies, in particular, general classes of facet defining inequalities for the standard model. 1998 ACM Subject Classification G.2.3 Applications in Discrete Mathematics

[1]  Ralf Borndörfer,et al.  Konrad-zuse-zentrum Für Informationstechnik Berlin a Hypergraph Model for Railway Vehicle Rotation Planning a Hypergraph Model for Railway Vehicle Rotation Planning * , 2022 .

[2]  Leo Kroon,et al.  On solving multi-type line planning problems , 2002 .

[3]  Tobias Achterberg,et al.  Constraint integer programming , 2007 .

[4]  Michael R. Bussieck,et al.  Discrete optimization in public rail transport , 1997, Math. Program..

[5]  Ellis L. Johnson,et al.  A note of the knapsack problem with special ordered sets , 1981, Oper. Res. Lett..

[6]  Leo G. Kroon,et al.  A Branch-and-Cut Approach for Solving Railway Line-Planning Problems , 2004, Transp. Sci..

[7]  Andrea Lodi,et al.  MIR closures of polyhedral sets , 2009, Math. Program..

[8]  Ralf Borndörfer,et al.  Optimierung des Linienplans 2010 in Potsdam , 2012 .

[9]  Timo Berthold.  Primal Heuristics for Mixed Integer Programs , 2006 .

[10]  Michael R. Bussieck,et al.  Optimal Lines in Public Rail Transport , 1998 .

[11]  G. Nemhauser,et al.  Integer Programming , 2020 .

[12]  Ralf Borndörfer,et al.  Models for Railway Track Allocation , 2007, ATMOS.

[13]  Laurence A. Wolsey,et al.  Valid inequalities for 0-1 knapsacks and mips with generalised upper bound constraints , 1990, Discret. Appl. Math..

[14]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[15]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[16]  P. Kreuzer,et al.  Optimal lines for railway systems , 1997 .

[17]  Christian Raack,et al.  Capacitated Network Design - Multi-Commodity Flow Formulations, Cutting Planes, and Demand Uncertainty (Netzwerkdesign - Mehrgüterflussformulierungen, Schnittebenen und Bedarfsunsicherheit) , 2012 .

[18]  Anita Schöbel,et al.  Line planning in public transportation: models and methods , 2012, OR Spectr..

[19]  M. Stoer,et al.  A polyhedral approach to multicommodity survivable network design , 1994 .

[20]  Martin Grötschel,et al.  A Column-Generation Approach to Line Planning in Public Transport , 2007, Transp. Sci..

[21]  Anita Schöbel,et al.  Line Planning with Minimal Traveling Time , 2005, ATMOS.

[22]  Hanif D. Sherali,et al.  Sequential and Simultaneous Liftings of Minimal Cover Inequalities for Generalized Upper Bound Constrained Knapsack Polytopes , 1995, SIAM J. Discret. Math..

[23]  Marika Karbstein.  Line Planning and Connectivity , 2013 .

[24]  Ralf Borndörfer,et al.  A Direct Connection Approach to Integrated Line Planning and Passenger Routing , 2012, ATMOS.

[25]  Nico M. van Dijk,et al.  Cost optimal allocation of rail passenger lines , 1998, Eur. J. Oper. Res..