Reliability-based stochastic transit assignment: Formulations and capacity paradox

Document Type

Journal Article

Publication Date


Subject Area

operations - frequency, operations - capacity, operations - reliability, ridership - demand


Frequency-based transit assignment, Reliability-based user equilibrium, Variational inequality, Braess and capacity paradox


This study develops link-based and approach-based variational inequality (VI) formulations for the frequency-based transit assignment with supply uncertainty, where link flows and flow on each outgoing link from each node are decision variables, respectively. Both the mean and variance of travel cost, including the covariance of in-vehicle travel costs, are captured in both formulations. To address the covariance of in-vehicle travel costs between different links on the same transit line, an augmented route-section network representation is developed, allowing us to apply the dynamic programming method to compute the value of the mapping function of the VI. The approach-based formulation can be solved by an extragradient method that only requires mild assumptions for convergence. It is found that the number of links carrying flow and equilibrium cost can be underestimated if supply uncertainty is not considered.

The study also introduces and examines the capacity paradox, a phenomenon in which the network maximum throughput may be reduced after new transit lines are added to a transit network or after the frequency of an existing line is increased. It is found that the capacity paradox may or may not occur simultaneously with the Braess-like paradox, a phenomenon in which providing new transit lines to a network may deteriorate the network performance in terms of the total weighted sum of the mean and variance of travel cost of all of the passengers. The demand level and the degree of risk aversion of passengers are the key factors that determine the occurrence of the capacity paradox.


Permission to publish the abstract has been given by Elsevier, copyright remains with them.


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