User heterogeneity and bi-criteria system optimum

For a traffic network with fixed demand of heterogeneous users in terms of their different values of time (VOT), the system performance can be measured either in time unit by the total system travel time (in short, system time), or in monetary unit by the total system travel cost (in short, system cost). Thus we have two different objectives for network optimization, i.e. to minimize system time and to minimize system cost, which naturally gives rise to a bi-objective minimization problem. A Pareto optimum of this bi-objective optimization problem represents a bi-criteria system optimum for network optimization in the sense that, at each Pareto optimum, neither system time nor system cost can be further reduced without increasing the other one. In this paper we prove that any Pareto optimum can be decentralized into multi-class user equilibrium by positive anonymous link tolls. We then bound the system performance gap when optimized by the two different criteria. Specifically, we provide answers to the following questions: When system time is minimized, how far could the corresponding system cost deviate from its minimum value? Conversely, when system cost is minimized, how far could the corresponding system time deviate from its minimum value? More generally, how far can the system time and system cost at a given bi-criteria Pareto optimum deviate from their respective single-criterion based system optimum?