A Bio-Inspired Approach to Traffic Network Equilibrium Assignment Problem

Finding an equilibrium state of the traffic assignment plays a significant role in the design of transportation networks. We adapt the path finding mathematical model of slime mold Physarum polycephalum to solve the traffic equilibrium assignment problem. We make three contributions in this paper. First, we propose a generalized Physarum model to solve the shortest path problem in directed and asymmetric graphs. Second, we extend it further to resolve the network design problem with multiple source nodes and sink nodes. At last, we demonstrate that the Physarum solver converges to the user-optimized (Wardrop) equilibrium by dynamically updating the costs of links in the network. In addition, convergence of the developed algorithm is proved. Numerical examples are used to demonstrate the efficiency of the proposed algorithm. The superiority of the proposed algorithm is demonstrated in comparison with several other algorithms, including the Frank-Wolfe algorithm, conjugate Frank-Wolfe algorithm, biconjugate Frank-Wolfe algorithm, and gradient projection algorithm.

[1]  P. D. Site,et al.  Reference-Dependent Stochastic User Equilibrium with Endogenous Reference Points , 2013 .

[2]  Yosef Sheffi,et al.  Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods , 1985 .

[3]  R. Cervero.  Road Expansion, Urban Growth, and Induced Travel: A Path Analysis , 2001 .

[4]  Robert B. Dial,et al.  A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration , 2006 .

[5]  Toshimitsu Ushio,et al.  An application of a computational ecology model to a routing method in computer networks , 2002, IEEE Trans. Syst. Man Cybern. Part B.

[6]  Michail-Antisthenis I. Tsompanas,et al.  Evolving Transport Networks With Cellular Automata Models Inspired by Slime Mould , 2015, IEEE Transactions on Cybernetics.

[7]  Y. Nishiura,et al.  Obtaining multiple separate food sources: behavioural intelligence in the Physarum plasmodium , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[8]  Paola Iovanna,et al.  Self-Adaptation in Next-Generation Internet Networks: How to React to Traffic Changes While Respecting QoS? , 2006, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[9]  Vinayak Dixit,et al.  Is equilibrium in transport pure Nash, mixed or Stochastic? , 2014 .

[10]  Sankaran Mahadevan,et al.  Solving 0-1 knapsack problems based on amoeboid organism algorithm , 2013, Appl. Math. Comput..

[11]  George L. Nemhauser,et al.  A Column Generation Algorithm for Optimal Traffic Assignment , 1973 .

[12]  Kurt Mehlhorn,et al.  Physarum can compute shortest paths , 2011, SODA.

[13]  Sankaran Mahadevan,et al.  Physarum solver: a bio-inspired method for sustainable supply chain network design problem , 2017, Ann. Oper. Res..

[14]  Yong Deng,et al.  Route selection for emergency logistics management: A bio-inspired algorithm , 2013 .

[15]  Andrew Adamatzky,et al.  Physarum Machines: Computers from Slime Mould , 2010 .

[16]  Erik T. Verhoef,et al.  SECOND-BEST CONGESTION PRICING IN GENERAL NETWORKS. HEURISTIC ALGORITHMS FOR FINDING SECOND-BEST OPTIMAL TOLL LEVELS AND TOLL POINTS , 2002 .

[17]  Andrew Adamatzky,et al.  Route 20, Autobahn 7, and Slime Mold: Approximating the Longest Roads in USA and Germany With Slime Mold on 3-D Terrains , 2012, IEEE Transactions on Cybernetics.

[18]  Shang-Hua Teng,et al.  Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems , 2003, STOC '04.

[19]  Stefano Pallottino,et al.  Equilibrium traffic assignment for large scale transit networks , 1988 .

[20]  Michael Florian,et al.  A New Look at Projected Gradient Method for Equilibrium Assignment , 2009 .

[21]  A. Tero,et al.  A mathematical model for adaptive transport network in path finding by true slime mold. , 2007, Journal of theoretical biology.

[22]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .

[23]  Per Olov Lindberg,et al.  The Stiff Is Moving - Conjugate Direction Frank-Wolfe Methods with Applications to Traffic Assignment , 2013, Transp. Sci..

[24]  Hai-Jun Huang,et al.  A multiclass, multicriteria logit-based traffic equilibrium assignment model under ATIS , 2007, Eur. J. Oper. Res..

[25]  Soondal Park,et al.  A label-setting algorithm for finding a quickest path , 2004, Comput. Oper. Res..

[26]  Matthias Ehrgott,et al.  A framework for and empirical study of algorithms for traffic assignment , 2015, Comput. Oper. Res..

[27]  A. Tero,et al.  Minimum-risk path finding by an adaptive amoebal network. , 2007, Physical review letters.

[28]  Dongjoo Park,et al.  Solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs , 2010, Appl. Math. Comput..

[29]  Yu Nie,et al.  A Note on Bar-Gera's Algorithm for the Origin-Based Traffic Assignment Problem , 2012, Transp. Sci..

[30]  G. D'Aguí,et al.  Three Solutions for a Discrete Nonlinear Neumann Problem Involving the -Laplacian , 2010 .

[31]  Yingyan Lou,et al.  Optimal Dynamic Pricing Strategies for High-Occupancy/Toll Lanes , 2011 .

[32]  A. Tero,et al.  Rules for Biologically Inspired Adaptive Network Design , 2010, Science.

[33]  Sankaran Mahadevan,et al.  A Biologically Inspired Network Design Model , 2015, Scientific Reports.

[34]  T. Nakagaki,et al.  Smart network solutions in an amoeboid organism. , 2004, Biophysical chemistry.

[35]  Amit Kumar,et al.  An Improved Social Pressure Algorithm for Static Deterministic User Equilibrium Traffic Assignment Problem , 2011 .

[36]  Toshiyuki Nakagaki,et al.  Physarum solver: A biologically inspired method of road-network navigation , 2006 .

[37]  Xiangdong Xu,et al.  A self-adaptive gradient projection algorithm for the nonadditive traffic equilibrium problem , 2012, Comput. Oper. Res..

[38]  Vincenzo Bonifaci,et al.  Physarum can compute shortest paths: A short proof , 2013, Inf. Process. Lett..

[39]  Michael Patriksson,et al.  The Traffic Assignment Problem: Models and Methods , 2015 .

[40]  Athanasios V. Vasilakos,et al.  Physarum Optimization: A Biology-Inspired Algorithm for the Steiner Tree Problem in Networks , 2015, IEEE Transactions on Computers.

[41]  Nisheeth K. Vishnoi,et al.  Natural Algorithms for Flow Problems , 2016, SODA.

[42]  Marino Lupi,et al.  Convergence of the Frank—Wolfe algorithm in transportation networks , 1986 .

[43]  J. A. Ventura,et al.  Finiteness in restricted simplicial decomposition , 1985 .

[44]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[45]  Hillel Bar-Gera,et al.  Origin-Based Algorithm for the Traffic Assignment Problem , 2002, Transp. Sci..

[46]  R. Jayakrishnan,et al.  A FASTER PATH-BASED ALGORITHM FOR TRAFFIC ASSIGNMENT , 1994 .

[47]  Zhong Zhou,et al.  Computational Study of Alternative Methods for Static Traffic Equilibrium Assignment , 2010 .

[48]  Z. Zhang,et al.  Solving NP-Hard Problems with Physarum-Based Ant Colony System , 2017, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[49]  T. Nakagaki,et al.  Intelligence: Maze-solving by an amoeboid organism , 2000, Nature.

[50]  Larry J. LeBlanc,et al.  AN EFFICIENT APPROACH TO SOLVING THE ROAD NETWORK EQUILIBRIUM TRAFFIC ASSIGNMENT PROBLEM. IN: THE AUTOMOBILE , 1975 .