Sequential optimization approach for the multi-class user equilibrium problem in a continuous transportation system

We consider a city region with several facilities that are competing for customers of different classes. Within the city region, the road network is dense, and can be represented as a continuum. Customers are continuously distributed over space, and they choose a facility by considering both the transportation cost and market externalities. More importantly, the model takes into account the different transportation cost functions and market externalities to which different customer classes are subjected. A logit-type distribution of demand is specified to model the decision-making process of users' facility choice. We develop a sequential optimization approach to decompose the complex multi-class and multi-facility problem into a series of smaller single-class and single-facility sub-problems. An efficient solution algorithm is then proposed to solve the resultant problem. A numerical example is given to demonstrate the effectiveness and potential applicability of the proposed methodology.