This paper deals with transport user equilibrium. A modified version of the ant colony system is proposed where the ant colony heuristic is adapted in order to take into account all aspects characterizing the transport problem: multiple ODs (Origin-Destination pairs), link congestion, non-separable cost link functions, elasticity of demand, multi classes in demand. Categories and Subject Descriptors I.2.8 [Problem Solving, Control Methods, and Search]: Heuristic methods General Terms Algorithms, Performance, Experimentation. Keywords Ant colony system, User equilibrium assignment, Transportation networks, Non separable cost.