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Journal of Optimization Theory and Applications

, Volume 127, Issue 3, pp 497–513 | Cite as

Arriving on Time

  • Y. Y. Fan
  • R. E. Kalaba
  • J. E. MooreII
Article

Abstract

This research proposes a procedure for identifying dynamic routing policies in stochastic transportation networks. It addresses the problem of maximizing the probability of arriving on time. Given a current location (node), the goal is to identify the next node to visit so that the probability of arriving at the destination by time t or sooner is maximized, given the probability density functions for the link travel times. The Bellman principle of optimality is applied to formulate the mathematical model of this problem. The unknown functions describing the maximum probability of arriving on time are estimated accurately for a few sample networks by using the Picard method of successive approximations. The maximum probabilities can be evaluated without enumerating the network paths. The Laplace transform and its numerical inversion are introduced to reduce the computational cost of evaluating the convolution integrals that result from the successive approximation procedure.

Keywords

Optimal routing stochastic shortest path problems dynamic programming convolution integrals 

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References

  1. 1.
    Bellman, R.E. 1958On a Routing ProblemQuarterly of Applied Mathematics168790zbMATHMathSciNetGoogle Scholar
  2. 2.
    Dreyfus, S.E. 1969An Appraisal of Some Shortest-Path AlgorithmsOperations Research17395412zbMATHGoogle Scholar
  3. 3.
    Howard, R.A. 1971Dynamic Probabilistic SystemsJohn Wiley and SonsNew York, NYGoogle Scholar
  4. 4.
    Hall, R.W. 1986The Fastest Path through a Network with Random Time-Dependent Travel TimesTransportation Science20182188Google Scholar
  5. 5.
    Fu, L., Rilett, L.R. 1998Expected Shortest Paths in Dynamic Stochastic Traffic NetworksTransportation Research32B499516Google Scholar
  6. 6.
    Waller, S.T., Ziliaskopoulos, A.K. 2002On the Online Shortest Path Problem with Limited Arc Cost DependenciesNetworks40216227CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Miller-Hooks, E.D., Mahmassani, H.S. 2000Least Expected Time Paths in Stochastic, Time-Varying Transportation NetworksTransportation Science34198215CrossRefzbMATHGoogle Scholar
  8. 8.
    Loui, P. 1983Optimal Paths in Graphs with Stochastic or Multidimentional WeightsCommunications of the ACM26670676CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Frank, H. 1969Shortest Paths in Probabilistic GraphsOperations Research17583599zbMATHMathSciNetGoogle Scholar
  10. 10.
    Sigal, C.E., Pritsker, A.A.B., Solberg, J.J. 1980The Stochastic Shortest Route ProblemOperations Research2811221129MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bellman, R.E., Kalaba, R.E. 1965Dynamic Programming and Modern Control TheoryMcGraw-HillNew York, NYGoogle Scholar
  12. 12.
    White, D.J. 1993Minimizing a Threshold Probability in Discounted Markov Decision ProcessesJournal of Mathematical Analysis and Applications173634646CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Wu, C., Lin, Y. 1999Minimizing Risk Models in Markovian Decision Processes with Policies Depending on Target ValuesJournal of Mathematical Analysis and Applications2314767MathSciNetzbMATHGoogle Scholar
  14. 14.
    Srivastava, H.M., Buschman, R.G. 1992Theory and Applications of Convolution Integral EquationsKluwer Academic PublishersDordrecht, NetherlandszbMATHGoogle Scholar
  15. 15.
    Bellman, R.E., Kalaba, R.E. 1966Numerical Inversion of the Laplace TransformAmerican Elsevier Publishing CompanyNew York, NYzbMATHGoogle Scholar
  16. 16.
    Fan, Y.Y., Kalaba, R.E. 2003Dynamic Programming and Pseudo-InversesApplied Mathematics and Computation139323342MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Y. Y. Fan
    • 1
  • R. E. Kalaba
    • 2
  • J. E. MooreII
    • 3
  1. 1.Department of Civil and Environmental EngineeringUniversity of CaliforniaDavis
  2. 2.Departments of Biomedical, Economics, and Electrical EngineeringUniversity of Southern CaliforniaLos Angeles
  3. 3.Daniel J. Epstein Department of Industrial and System Engineering, Department of Civil and Environmental Engineering, and School of Policy, Planning, and DevelopmentUniversity of Southern CaliforniaLos Angeles

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