The fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network. This paper shows how transit fare can be optimized so as to balance the passenger flow on the transit network and to reduce the overload delays of passengers at transit stops. A bi‐level programming method is developed to optimize the transit fare under line capacity constraints. The upper‐level problem seeks to minimize the total network travel time, while the lower‐level problem is a stochastic user equilibrium transit assignment model with line capacity constraints. A heuristic solution algorithm based on sensitivity analysis is proposed. Numerical example is used to illustrate the application of the proposed model and solution algorithm.

Since regret theory has been introduced to model travellers’ route choices, various random regret minimization (RRM) models have been developed for the choice situation of multiple alternatives and multiple attributes. There are two approaches dealing with multi-objective optimisations: one is to combine different attributes into a single additive one and the other is to consider each attribute separately. The existing RRM models adopt the first approach to measure regret. However, travellers might not always trade off attributes in such a compensatory way. In this paper, under the assumption that travellers might consider attributes separately, we develop two new regret-based stochastic user equilibrium (SUE) models by incorporating the RRM model and the hybrid RUM-RRM model into a non-compensatory multi-objective framework. The majority of the second approaches dealing with multiple objectives generally provide a feasible solution set caused by conflicts among objectives. Different from that, the two new models provide probabilistic choice to each route, based on which a single SUE path flow pattern would be attained. Meanwhile, the compromise effect which is widely seen in consumer behaviour can be explained by the two new models. The equivalent variational inequality problems for the proposed models and a path-based algorithm using the method of successive averages have been given. Numerical examples are further conducted to illustrate the properties of the proposed models and the effectiveness of the algorithm.

In this paper, we first propose the concepts of $(\zeta ,\eta ,\lambda ,\pi )$-generalized hybrid multi-valued mappings, the set of all the common attractive points ($C{A}_{f,g}$) and the set of all the common strongly attractive points ($C_s{A}_{f,g}$), respectively for the multi-valued mappings f and g in a CAT(0) space. Moreover, we give some elementary properties in regard to the sets $A_f$, $F_f$ and $C{A}_{f,g}$ for the multi-valued mappings f and g in a complete CAT(0) space. Furthermore, we present a weak convergence theorem of common attractive points for two $(\zeta ,\eta ,\lambda ,\pi )$-generalized hybrid multi-valued mappings in the above space by virtue of Banach limits technique and Ishikawa iteration respectively. Finally, we prove strong convergence of a new viscosity approximation method for two $(\zeta ,\eta ,\lambda ,\pi )$-generalized hybrid multi-valued mappings in CAT(0) spaces, which also solves a kind of variational inequality problem.