Combined Gravity Model Trip Distribution and Paired Combinatorial Logit Stochastic User Equilibrium Problem

The equivalent mathematical formulation of the combined doubly-constrained gravity-based trip distribution and paired-combinatorial-logit stochastic user equilibrium assignment problem (CDA-PCL-SUE) is proposed. Its first order conditions are shown to be equal to the gravity equations and PCL formula. The proposed solution method is a path-based partial linearization algorithm to approximately solve the restricted CDA-PCL-SUE. The proposed algorithm is a three-phase iterative process. Phase 1 is an entropy maximization problem on O-D flow space that can be solved by Bregman's balancing algorithm. Phase 2 is a PCL SUE problem that can be solved by PCL formula. Phase 3 is line search. CDA-PCL-SUE is solved on a small network and a real network, the city of Winnipeg network. The proposed algorithms with the six line search methods, namely, golden section (GS), bisection (BS), Armijo's rule (AR), method of successive averages (MSA), self-regulated averaging (SRA) scheme, and quadratic interpolation (QI) scheme, are compared in terms of various convergence characteristics: root mean square error, step size, KKT-based mean square error and objective function. In terms of computational efficiency, under different path set sizes, dispersion parameters, impedance parameters and demand levels, the following line search methods are ordered from best to worst: SRA, GS, AR, QI, BS and MSA. The performances of Armijo's rule and QI have greater variances. The performance of QI is worse with the increase of the path set size. Given all other factors being the same, the increase of dispersion parameter, path set size or demand level yields the increase of CPU time, whereas the change of impedance parameter does not influence CPU time. In addition, CDA-PCL-SUE is compared with its multinomial-logit counterpart (CDA-MNL-SUE).