Side constrained traffic assignment problem for multiclass flow

Capacity constraints (or side constraints) - though representing realistic features - are largely overlooked in the traffic assignment due to the inherent mathematical complexities. To this end; we first relaxed the capacity constraints by an intuitive interpretation of their corresponding Lagrange values, that is, the amount of penalty imposed to the travel time of the oversaturated road to make them saturated. This approach is basically a subgradient method in which the penalty terms bear some resemblances to the marginal cost of the concept of system optimal traffic flow. We then circumvented the complexity of multiclass facet by adopting a bias term for each user class in the Beckmann's formulation. Hence, we arrived at an uncapacitated single-class TAP in which the penalty terms are updated iteratively. The proposed algorithm obviates any additional parameter, which is not a trivial task as shown in the past studies.