PURPOSES : This study aims to develop an algorithm to solve the user equilibrium traffic assignment problem using soft link capacity constraints. This model is used to relax the hard capacity constraints model. METHODS : In the traffic assignment model that imposes the hard capacity constraints, the well-known solution algorithms used are the augmented Lagrange multiplier method and the inner penalty function method. The major drawback of using the hard-capacity constraint model is the feasible solution issue. If the capacities in the network are not sufficient to absorb the flow from the diverged flows through the hard capacity constraints, it might result in no solution; whereas, using a soft capacity constraint model guarantees a feasible solution because the soft capacity constraint model uses the penalization of constraint violation in the objective function. In this study, the gradient projection (GP) algorithm was adapted. RESULTS : Two numerical experiments were conducted to demonstrate the features of the soft capacity constraint model and the computational performance of the solution algorithm. The results revealed that imposing the soft link capacity constraints can ensure convergence. CONCLUSIONS : The proposed model can be easily extended by considering other traffic assignment models, for e.g., non-additive traffic equilibrium problem, stochastic traffic equilibrium model, and, elastic demand traffic equilibrium problem. Furthermore, the model can exist regardless of the sufficient capacity for each O-D pair to cater to their demands.

Even if there are no causing factors such as car crash and road works, traffic congestion come from traffic growth on the road. In this case, estimation of traffic flow helps find the solution of traffic congestion problem. In this paper, we present a optimization model which used on traffic equilibrium problem and studied the problem of inverting shortest path sets for complex traffic system. And we also develop pivotal decomposition algorithm for reliability function of complex traffic system. Several examples are illustrated.