A Comparison of Dynamic User Optimal States with Zero, Fixed and Variable Tolerances

摘要

The notion of dynamic user equilibrium (DUE) and dynamic user optimum (DUO) were still treated interchangeably in the dynamic traffic assignment (DTA) literature until tolerance-based DUO was recently proposed as alternative definitions of DUO. The latter distinguishes DUO from DUE, and the tolerances can be prescribed a priori (i.e. DUO with fixed tolerance) or endogenously (i.e. DUO with variable tolerance). These tolerances reflect the differences in travel costs between used paths of an origin-destination (OD) pair. Since DUE requires zero difference in travel costs between used paths of an OD pair, it can be regarded as DUO with zero tolerance. This paper first compares the three alternative definitions of DUO and discusses their relationships. Secondly, the existence conditions of these DUO solutions are discussed. A key conclusion is that DUO with variable tolerance always exists even when the other two types of DUO solutions may not. The second conclusion is that, when the tolerances are equal to zero, the solution must be a DUE solution. Furthermore, DUE is a special case of DUO with variable tolerance; since the tolerance in the concept of DUO with variable tolerance is an endogenous variable, if a DUE solution exists, an efficient method for solving for DUO with variable tolerance should directly find such a DUE solution. All these are illustrated by means of three examples. Finally, it is acknowledged that the analysis of DUO with variable tolerance in the same rigor as that of DUE is a challenging problem and worthy of further research.