Optimal paths in dynamic networks with dependent random link travel times

This paper addresses the problem of finding optimal paths in a network where all link travel times are stochastic and time-dependent, and correlated over time and space. A disutility function of travel time is defined to evaluate the paths, and those with the minimum expected disutility are defined as the optimal paths. Bellman’s Principle (Bellman, 1958) is shown to be invalid if the optimality or non-dominance of a path and its sub-paths is defined with respect to the complete set of departure times and joint realizations of link travel time. An exact label-correcting algorithm is designed to find optimal paths based on a new property for which Bellman’s Principle holds. The algorithm has exponential worst-case computational complexity. Computational tests are conducted on three types of networks. Although the average running time is exponential, the number of the optimal path candidates is polynomial on two networks and grows exponentially in the third one. Computational results in large networks and analytical results in a small network show that stochastic dependencies affect optimal path finding in a stochastic network, and that the impact is closely related to the levels of correlation and risk attitude.

► A sub-path of an optimal/non-dominated path is not necessarily optimal/non-dominated.
► For any origin and departure time, there must exist a pure optimal path.
► An exact label-correcting algorithm is designed, terminating with all pure paths in a finite number of steps.
► The computational tests show a polynomial running time for the algorithm in a typical type of network.
► The impact of dependencies on optimal path solution depends on the level of correlations and travelers risk attitude.