A Lagrangian relaxation approach for expansion of a highway network

This paper deals with the problem of improving an existing road network in the context of strategic planning through the creation of new highway corridors. To address this problem we analyse three mixed integer programming models. The so-called [P1] is the classical capacitated multicommodity network design model. The model named [P2] imposes on [P1] the location of a single (main path) highway corridor in the road network and [P3] adds to [P2] a set of sub-tour breaking constraints. The stated goal is to minimize the total travel time for a known origin-destination demand matrix with a given budget. In this paper we propose an efficient method for [P3], based on a Lagrangian relaxation, to obtain easily-solved sub-problems. A cutting-plane method for solving the Lagrangian sub-problems is proposed. This method generates valid cuts until an optimal solution is found. The Lagrangian dual problem is solved using the sub-gradient optimization method. A case study has been carried out for the region of Castilla-La Mancha (Spain). Computational comparisons between the proposed method and a state-of-the-art mixed-integer code are presented. The Lagrangian relaxation approach is found to be capable of generating good feasible solutions to the case study within a reasonable computational time.

[1]  Gilbert Laporte,et al.  An Integrated Methodology for the Rapid Transit Network Design Problem , 2004, ATMOS.

[2]  Erik Rolland,et al.  Efficient Algorithms for Solving the Shortest Covering Path Problem , 1994, Transp. Sci..

[3]  Michel Gendreau,et al.  A Simplex-Based Tabu Search Method for Capacitated Network Design , 2000, INFORMS J. Comput..

[4]  Iván A. Contreras,et al.  Lagrangean relaxation for the capacitated hub location problem with single assignment , 2009, OR Spectr..

[5]  Gilbert Laporte,et al.  Maximizing Trip Coverage in the Location of a Single Rapid Transit Alignment , 2005, Ann. Oper. Res..

[6]  Michel Gendreau,et al.  A heuristic for the location of a rapid transit line , 2002, Comput. Oper. Res..

[7]  Gilbert Laporte,et al.  Optimization methods for the planning of rapid transit systems , 2000, Eur. J. Oper. Res..

[8]  Ángel Marín,et al.  Integrating Robust Railway Network Design and Line Planning under Failures , 2009, Robust and Online Large-Scale Optimization.

[9]  Laureano F. Escudero,et al.  An approach for solving a modification of the extended rapid transit network design problem , 2009 .

[10]  Hasan Pirkul,et al.  The Hierarchical Network Design Problem: A New Formulation and Solution Procedures , 1991, Transp. Sci..

[11]  Timothy John Niblett.  The Maximal Covering/Shortest Path Problem Revisited: An Examination and Reformulation of the Problem to Allow the Elimination or Attachment of Sub-Tours , 2013 .

[12]  Gilbert Laporte,et al.  Locating stations on rapid transit lines , 2002, Comput. Oper. Res..

[13]  G. Laporte,et al.  Planning rapid transit networks , 2010 .

[14]  Peter A. Rogerson,et al.  Location coverage models with demand originating from nodes and paths: Application to cellular network design , 2008, European Journal of Operational Research.

[15]  Abdullah Konak,et al.  Network design problem with relays: A genetic algorithm with a path-based crossover and a set covering formulation , 2012, Eur. J. Oper. Res..

[16]  Blas Pelegrín,et al.  Applying Lagrangian relaxation to the resolution of two‐stage location problems , 1999, Ann. Oper. Res..

[17]  Teodor Gabriel Crainic,et al.  Bundle-based relaxation methods for multicommodity capacitated fixed charge network design , 2001, Discret. Appl. Math..

[18]  J. Current,et al.  The maximum covering/shortest path problem: A multiobjective network design and routing formulation , 1985 .

[19]  Paul Schonfeld,et al.  Using GIS, Genetic Algorithms, and Visualization in Highway Development , 2001 .

[20]  Gilbert Laporte,et al.  Designing Robust Rapid Transit Networks with Alternative Routes , 2008 .

[21]  Juan A. Mesa,et al.  A review of extensive facility location in networks , 1996 .

[22]  John R. Current,et al.  Multiobjective transportation network design and routing problems: Taxonomy and annotation , 1993 .

[23]  Ángel Marín.  An extension to rapid transit network design problem , 2007 .

[24]  N. Sancho.  The hierarchical network design problem with multiple primary paths , 1997 .

[25]  Rajendra S. Solanki,et al.  Using decomposition in large-scale highway network design with a quasi-optimization heuristic , 1998 .

[26]  Di Yuan,et al.  A Lagrangian Heuristic Based Branch-and-Bound Approach for the Capacitated Network Design Problem , 2000, Oper. Res..

[27]  Ricardo García-Ródenas,et al.  Determining Highway Corridors , 2012 .

[28]  Hai Yang,et al.  Models and algorithms for road network design: a review and some new developments , 1998 .

[29]  Paul Schonfeld,et al.  Integrating Genetic Algorithms and Geographic Information System to Optimize Highway Alignments , 2000 .

[30]  A. Volgenant,et al.  Reducing the hierarchical network design problem , 1989 .

[31]  Manoj K. Jha.  Using a Geographic Information System for Automated Decision Making in Highway Cost Analysis , 2001 .

[32]  Iván A. Contreras,et al.  General network design: A unified view of combined location and network design problems , 2012, Eur. J. Oper. Res..

[33]  Brian Boffey,et al.  Models for multi-path covering-routing problems , 1998, Ann. Oper. Res..

[34]  Gabriel Gutiérrez-Jarpa,et al.  Minimum cost path location for maximum traffic capture , 2010, Comput. Ind. Eng..

[35]  Ricardo García-Ródenas,et al.  Location of infrastructure in urban railway networks , 2009, Comput. Oper. Res..

[36]  Ricardo García-Ródenas,et al.  A continuous bi-level model for the expansion of highway networks , 2014, Comput. Oper. Res..

[37]  G. Ghiani,et al.  A multi-modal approach to the location of a rapid transit line , 1998 .

[38]  Min-Wook Kang.  An Alignment Optimization Model for a Simple Highway Network , 2008 .

[39]  Michel Gendreau,et al.  LOCATING A TRANSIT LINE USING TABU SEARCH , 1996 .

[40]  Boris Polyak.  Minimization of unsmooth functionals , 1969 .

[41]  J. C. Jong,et al.  Optimizing Highway Alignments with Genetic Algorithms , 1998 .

[42]  Charles ReVelle,et al.  SYMPOSIUM ON LOCATION PROBLEMS: IN MEMORY OF LEON COOPER.: THE SHORTEST COVERING PATH PROBLEM: AN APPLICATION OF LOCATIONAL CONSTRAINTS TO NETWORK DESIGN , 1984 .

[43]  Pascual Fernández,et al.  A Heuristic Procedure For Path Location With Multisource Demand , 2003 .

[44]  J. Current,et al.  The hierarchical network design problem , 1986 .