Least Expected Time Paths in Stochastic, Time-Varying Transportation Networks

We consider stochastic, time-varying transportation networks, where the arc weights (arc travel times) are random variables with probability distribution functions that vary with time. Efficient procedures are widely available for determining least time paths in deterministic networks. In stochastic but time-invariant networks, least expected time paths can be determined by setting each random arc weight to its expected value and solving an equivalent deterministic problem. This paper addresses the problem of determining least expected time paths in stochastic, time-varying networks. Two procedures are presented. The first procedure determines the a priori least expected time paths from all origins to a single destination for each departure time in the peak period. The second procedure determines lower bounds on the expected times of these a priori least expected time paths. This procedure determines an exact solution for the problem where the driver is permitted to react to revealed travel times on traveled links en route, i.e., in a time-adaptive route choice framework. Modifications to each of these procedures for determining least expected cost (where cost is not necessarily travel time) paths and lower bounds on the expected costs of these paths are given. Extensive numerical tests are conducted to illustrate the algorithms' computational performance as well as the properties of the solution.

[1]  Hani S. Mahmassani,et al.  Time dependent, shortest-path algorithm for real-time intelligent vehicle highway system applications , 1993 .

[2]  V. G. Kulkarni,et al.  Shortest paths in networks with exponentially distributed arc lengths , 1986, Networks.

[3]  Randolph W. Hall,et al.  The Fastest Path through a Network with Random Time-Dependent Travel Times , 1986, Transp. Sci..

[4]  Athanasios K. Ziliaskopoulos.  Optimum path algorithms on multidimensional networks: Analysis, design, implementation and computational experience. , 1996 .

[5]  G. Gallo,et al.  SHORTEST PATH METHODS: A UNIFYING APPROACH , 1986 .

[6]  James J. Solberg,et al.  The Stochastic Shortest Route Problem , 1980, Oper. Res..

[7]  D. Malcolm,et al.  Application of a Technique for Research and Development Program Evaluation , 1959 .

[8]  John N. Tsitsiklis,et al.  Dynamic Shortest Paths in Acyclic Networks with Markovian Arc Costs , 1993, Oper. Res..

[9]  G. Thompson,et al.  Critical Path Analyses Via Chance Constrained and Stochastic Programming , 1964 .

[10]  David Connolly,et al.  Handbooks in Operations Research and Management Science, Volume 1 — Optimization , 1990 .

[11]  A. Madansky.  Inequalities for Stochastic Linear Programming Problems , 1960 .

[12]  Hossein Soroush,et al.  Path Preferences and Optimal Paths in Probabilistic Networks , 1985, Transp. Sci..

[13]  Stefano Pallottino,et al.  Shortest-path methods: Complexity, interrelations and new propositions , 1984, Networks.

[14]  S. Pallottino,et al.  Hyperpaths and shortest hyperpaths , 1989 .

[15]  Van Slyke,et al.  MONTE CARLO METHODS AND THE PERT PROBLEM , 1963 .

[16]  Elise Miller-Hooks,et al.  Least possible time paths in stochastic, time-varying networks , 1998, Comput. Oper. Res..

[17]  F. Glover,et al.  New Polynomial Shortest Path Algorithms and Their Computational Attributes , 1985 .

[18]  D. Shier,et al.  An empirical investigation of some bicriterion shortest path algorithms , 1989 .

[19]  Hossein Soroush,et al.  Optimal paths in probabilistic networks: A case with temporary preferences , 1985, Comput. Oper. Res..

[20]  U. Pape,et al.  Implementation and efficiency of Moore-algorithms for the shortest route problem , 1974, Math. Program..

[21]  Pierre Hansen,et al.  Bicriterion Path Problems , 1980 .

[22]  Elise Miller-Hooks,et al.  Optimal routing in time-varying, stochastic networks: algorithms and implementations , 1997 .

[23]  Ronald Prescott Loui,et al.  Optimal paths in graphs with stochastic or multidimensional weights , 1983, Commun. ACM.

[24]  Vidyadhar G. Kulkarni,et al.  Shortest paths in stochastic networks with ARC lengths having discrete distributions , 1993, Networks.

[25]  H. Frank,et al.  Shortest Paths in Probabilistic Graphs , 1969, Oper. Res..

[26]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[27]  Richard Bellman,et al.  ON A ROUTING PROBLEM , 1958 .

[28]  Sumit Sarkar,et al.  A Relaxation-Based Pruning Technique for a Class of Stochastic Shortest Path Problems , 1996, Transp. Sci..

[29]  Ariel Orda,et al.  Minimum weight paths in time-dependent networks , 1991, Networks.

[30]  Robert L. Smith,et al.  Fastest Paths in Time-dependent Networks for Intelligent Vehicle-Highway Systems Application , 1993, J. Intell. Transp. Syst..