User-Equilibrium Route Set Analysis of a Large Road Network

By 1953 Martin Beckmann had formulated and analyzed the static, deterministic user equilibrium (UE) route choice model with variable origin-destination flows (demand) (Beckmann et al, 1956; Boyce, 2004). Twenty years later, Suzanne Evans (1973, 1976) proposed and analyzed an algorithm for solving a version of Beckmann’s model that corresponded to practitioner models with origin-destination (OD) constraints and transit as well as road modes. The effectiveness of her algorithm for large networks was studied by several researchers; see Boyce and Bar-Gera (2004) for a recent review. Other young researchers at that same time proposed and tested algorithms for solving the user-equilibrium problem with fixed OD flows (LeBlanc et al, 1975; Nguyen, 1974; Florian and Nguyen, 1976), building upon the pioneering theoretical research of Jorgensen (1963), Dafermos (1968) and Brynooghe et al (1969). More recently, Patriksson (1994) thoroughly synthesized the accumulated literature from this 40-year research effort, and proposed his own user equilibrium algorithms for fixed and variable flow problems (Larsson and Patriksson, 1992; Lundgren and Patriksson, 1998). Nagurney (1999) synthesized a related literature on the variational inequality approach to a family of problems, including user-equilibrium; Florian
and Hearn (1999) offered their review, which includes recent developments on road pricing