Advertisement

Efficient contraflow algorithms for quickest evacuation planning

  • Urmila Pyakurel
  • Hari Nandan Nath
  • Tanka Nath Dhamala
Articles

Abstract

The optimization models and algorithms with their implementations on flow over time problems have been an emerging field of research because of largely increasing human-created and natural disasters worldwide. For an optimal use of transportation network to shift affected people and normalize the disastrous situation as quickly and Efficiently as possible, contraflow configuration is one of the highly applicable operations research (OR) models. It increases the outbound road capacities by reversing the direction of arcs towards the safe destinations that not only minimize the congestion and increase the flow but also decrease the evacuation time significantly. In this paper, we sketch the state of quickest flow solutions and solve the quickest contraflow problem with constant transit times on arcs proving that the problem can be solved in strongly polynomial time O(nm2(log n)2), where n and m are number of nodes and number of arcs, respectively in the network. This contraflow solution has the same computational time bound as that of the best min-cost flow solution. Moreover, we also introduce the contraflow approach with load dependent transit times on arcs and present an Efficient algorithm to solve the quickest contraflow problem approximately. Supporting the claim, our computational experiments on Kathmandu road network and on randomly generated instances perform very well matching the theoretical results. For sufficiently large number of evacuees, about double flow can be shifted with the same evacuation time and about half time is sufficient to push the given flow value with contraflow reconfiguration.

Keywords

evacuation planning contra flow flow over time quickest flow load dependent transit time 

MSC(2010)

90B10 90C27 68Q25 90B06 90B20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by Deutscher Akademischer Austauschdienst (German Academic Exchange Service) Partnership Program (with University of Kaiserslautern, Germany and Mindanao State University, Iligan Institute of Technology, Iligan, Philippines) and AvH Research Group Linkage Program (with Technische Universität Bergakademie Freiberg) in Graph Theory and Optimization at Central Department of Mathematics, Tribhuvan University, Kathmandu, Nepal. The first author was supported by the AvH Foundation for the Georg Forster Research Fellowship for post doctoral researchers at Technische Universität Bergakademic Freiberg Germany. The authors thank the anonymous referees for their valuable suggestions to improve the quality of this work.

References

  1. 1.
    Ahuja R K, Orlin J B. A capacity scaling algorithm for the constrained maximum flow problem. Networks, 1995, 25: 89–98CrossRefMATHGoogle Scholar
  2. 2.
    Anderson E J, Philpott A B. Optimisation of flows in networks over time. In: Probability, Statistics and Optimisation, vol. 27. New York: Wiley, 1994, 369–382MathSciNetMATHGoogle Scholar
  3. 3.
    Arulselvan A. Network model for disaster management. PhD Thesis. Gainesville: University of Florida, 2009Google Scholar
  4. 4.
    Baumann N, Köhler E M. Approximating earliest arrival flows with flow-dependent transit times. Discrete Appl Math, 2007, 155: 161–171MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berlin G N. The Use of Directed Routes for Assessing Escape Potential. Boston: National Fire Protection Association, 1979Google Scholar
  6. 6.
    Burkard R E, Dlaska K, Kellerer H. The quickest disjoint flow problem. Technical Report. Graz: Institute of Mathematics, Graz University of Technology, 1991, 189–191MATHGoogle Scholar
  7. 7.
    Burkard R E, Dlaska K, Klinz B. The quickest flow problem. ZOR Methods Models Oper Res, 1993, 37: 31–58MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cai X, Sha D, Wong C K. Time-varying minimum cost flow problems. Eur J Oper Res, 2001, 131: 352–374MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Carey M. Dynamic traffic assignment with more exible modelling within links. Netw Spat Econ, 2001, 1: 349–375CrossRefGoogle Scholar
  10. 10.
    Carey M, Subrahmanian E. An approach to modelling time-varying flows on congested networks. Transp Res B, 2002, 34: 157–183CrossRefGoogle Scholar
  11. 11.
    Chalmet L G, Francis R L, Saunders P B. Network models for building evacuation. Manag Sci, 1982, 28: 86–105CrossRefGoogle Scholar
  12. 12.
    Chen Y L, Chin Y H. The quickest path problem. Comput Oper Res, 1990, 17: 153–161MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dhamala T N. A survey on models and algorithms for discrete evacuation planning network problems. J Ind Manag Optim, 2015, 11: 265–289MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dhamala T N, Pyakurel U. Earliest arrival contra flow problem on series-parallel graphs. Int J Oper Res, 2013, 10: 1–13MathSciNetGoogle Scholar
  15. 15.
    Fleischer L K, Tardos E. Efficient continuous-time dynamic network flow algorithms. Oper Res Lett, 1998, 23: 71–80MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ford L R, Fulkerson D R. Flows in Networks. Princeton: Princeton University Press, 1962MATHGoogle Scholar
  17. 17.
    Goldberg A V, Tarjan R E. Finding minimum-cost circulations by successive approximations. Math Oper Res, 1990, 15: 430–466MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hamacher H W, Heller S, Rupp B. Flow location (FlowLoc) problems: Dynamic network flows and location models for evacuation planning. Ann Oper Res, 2013, 207: 161–180MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hamacher H W, Tjandra S A. Mathematical modeling of evacuation problems: A state of the art. In: Pedestrian and Evacuation Dynamics. New York: Springer, 2002, 227–266Google Scholar
  20. 20.
    Hoppe B, Tardos E. The quickest transshipment problem. Math Oper Res, 2000, 25: 36–62MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hung Y C, Chen G H. On the quickest path problem. In: Proceedings of International Conference on Computing and Information. Lecture Notes in Computer Science, vol. 497. Berlin: Springer-Verlag, 1991, 44–46Google Scholar
  22. 22.
    Kagaris D, Pantziou G E, Tragoudas S, et al. On the computation of fast data transmissions in networks with capacities and delays. In: Workshop on Algorithms and Data Structures. Lecture Notes in Computer Science, vol. 955. New York: Springer-Verlag, 1995, 291–302MathSciNetGoogle Scholar
  23. 23.
    Kaufman D E, Nonis J, Smith R L. A mixed integer linear programming model for dynamic route guidance. Transp Res B, 1998, 32: 431–440CrossRefGoogle Scholar
  24. 24.
    Kim S, Shekhar S. Contra flow network reconfiguration for evacuation planning: A summary of results. In: Proceedings of the 13th International ACM Symposium on Advances in Geographic Information Systems. New York: ACM, 2005, 250–259Google Scholar
  25. 25.
    Kim S, Shekhar S, Min M. Contra flow transportation network reconfiguration for evacuation route planning. IEEE Trans Knflowl Data Eng, 2008, 20: 1115–1129CrossRefGoogle Scholar
  26. 26.
    Köhler E, Langkau K, Skutella M. Time expanded graphs for flow-depended transit times. In: Proceedings of the 10th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 2461. London: Springer-Verlag, 2002, 599–611MATHGoogle Scholar
  27. 27.
    Köhler E, Skutella M. Flows over time with load-dependent transit times. SIAM J Optim, 2005, 15: 1185–1202MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Langkau K. Flows over time with flow-dependent transit times. PhD Thesis. Berlin: Technical University, 2003Google Scholar
  29. 29.
    Lin M, Jaillet P. On the quickest flow problem in dynamic networks: A parametric min-cost flow approach. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia: Society for Industrial and Applied Mathematics, 2015, 1343–1356CrossRefGoogle Scholar
  30. 30.
    Megiddo N. Combinatorial optimization with rational objective functions. Math Oper Res, 1979, 4: 414–424MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Miller-Hooks E, Patterson S S. On solving quickest time problems in time-dependent dynamic networks. J Math Model Algorithms, 2004, 3: 39–71MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Nasrabadi E, Hashemi S M. Minimum cost time-varying network flow problems. Optim Methods Softw, 2010, 25: 429–447MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Pel A J, Bliemer M C, Hoogendoorn S P. A review on travel behaviour modelling in dynamic traffic simulation models for evacuations. Transportation, 2012, 39: 97–123CrossRefGoogle Scholar
  34. 34.
    Pyakurel U. Evacuation planning problem with contra flow approach. PhD Thesis. Kathmandu: Tribhuvan University, 2016Google Scholar
  35. 35.
    Pyakurel U, Dhamala T N. Models and algorithms on contra flow evacuation planning network problems. Int J Oper Res, 2015, 12: 36–46MathSciNetGoogle Scholar
  36. 36.
    Pyakurel U, Dhamala T N. Continuous time dynamic contra flow models and algorithms. Adv Oper Res, 2016, 2016: Article ID 7902460Google Scholar
  37. 37.
    Pyakurel U, Dhamala T N. Evacuation planning by earliest arrival contra flow. J Ind Manag Optim, 2017, 13: 487–501MathSciNetMATHGoogle Scholar
  38. 38.
    Pyakurel U, Dhamala T N. Continuous dynamic contra flow approach for evacuation planning. Ann Oper Res, 2017, 253: 573–598MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Pyakurel U, Dhamala T N, Dempe S. Efficient continuous contra flow algorithms for evacuation planning problems. Ann Oper Res, 2017, 254: 335–364MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Pyakurel U, Hamacher H W, Dhamala T N. Generalized maximum dynamic contra flow on lossy network. Internat J Oper Res Nepal, 2014, 3: 27–44Google Scholar
  41. 41.
    Rebennack S, Arulselvan A, Elefteriadou L, et al. Complexity analysis for maximum flow problems with arc reversals. J Comb Optim, 2010, 19: 200–216MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Rosena J B, Sun S Z, Xue G L. Algorithms for the quickest path problem and the enumeration of quickest paths. Comput Oper Res, 1991, 18: 579–584MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Saho M, Shigeno M. Cancel-and-tighten algorithm for quickest flow problems. Networks, 2017, 69: 179–188MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Tjandra S A. Dynamic network flow optimization with application to the evacuation problem. PhD Thesis. Kaiserslautern: University of Kaiserslautern, 2003Google Scholar
  45. 45.
    Wang J W, Ip W H, Zhang W J. An integrated road construction and resource planning approach to the evacuation of victims from single source to multiple destinations. IEEE Trans Intell Trans Syst, 2010, 11: 277–289CrossRefGoogle Scholar
  46. 46.
    Wang J W, Wang H F, Zhang W J, et al. Evacuation planning based on the contra flow technique with consideration of evacuation priorities and traffic setup time. IEEE Trans Intell Trans Syst, 2013, 14: 480–485CrossRefGoogle Scholar
  47. 47.
    Wang J W, Wang H F, Zhou Y M, et al. On an integrated approach to resilient transportation systems in emergency situations. Nat Comput, 2017, https://doi.org/10.1007/s11047-016-9605-y Google Scholar
  48. 48.
    Zhao X, Feng Z Y, Li Y, et al. Evacuation network optimization model with lane-based reversal and routing. Math Probl Eng, 2016, 2016: 1–12Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Urmila Pyakurel
    • 1
  • Hari Nandan Nath
    • 2
  • Tanka Nath Dhamala
    • 1
  1. 1.Central Department of MathematicsTribhuvan UniversityKathmanduNepal
  2. 2.Department of Mathematics, Bhaktapur Multiple CampusTribhuvan UniversityKathmanduNepal

Personalised recommendations