Solving the step-tolled bottleneck model with general user heterogeneity

Abstract

Two new numerical methods, a semi-analytical method and an exact method, are proposed in this paper for solving the step-tolled user equilibrium problem for the bottleneck model with general user heterogeneity. The semi-analytical method transforms the step-tolled user equilibrium problem into a static, asymmetric traffic assignment problem with side constraints, which is formulated and solved as a variational inequality problem (VIP). Existence and uniqueness results are obtained by examining the properties of the VIP. The exact method, on the other hand, locates the equilibrium solution by enumerating all possible combinations of user departure orders. To improve the computational efficiency of the exact method, a simple heuristic is developed to reduce the number of solutions to be enumerated. Examples of up to 1000 classes are tested. The results confirm that ignoring user heterogeneity may lead to sub-optimal design of step tolls. In all experiments, the semi-analytical method is found to be more efficient than the heuristic version of the exact method by an order of magnitude, albeit they consistently produce nearly identical equilibrium solutions.

Temporal-spatial allocation of bottleneck capacity for managing morning commute with carpool

“…Secondly, we present analytically the UE solutions of a temporal-only capacity allocation scheme (a time-limited carpooling scheme) for each of the above three cases. We show that, the impact of the reserved time window on the commute patterns for both Case 1(a) and Case 3(a) is similar to that of the single-step coarse toll scheme (Arnott et al, 1990;Laih, 1994;Laih, 2004;Lindsey et al, 2012;Xiao et al, 2011Xiao et al, , 2012Nie and Yin, 2013;Liu et al, 2015;Li et al, 2017;Xu et al, 2019), and that two important variables are the constant component of extra carpool cost and the queuing time cost for the solo drivers just passing the bottleneck at the starting/ending time of the time window: if the latter is larger, it leads to excess queuing delay; whilst if the former is larger, the scheme will lead to excess capacity waste.…”

“…Furthermore, the sign and magnitude of extra carpool cost may vary considerably across individuals. Accordingly, it is of interest to integrate user heterogeneity into the temporal-spatial capacity allocation problem for managing morning commute with carpool (Chen et al, 2015;Wu and Huang, 2015;Liu and Nie, 2017;Yu et al, 2019).…”

On the morning commute problem with carpooling behavior under parking space constraint

“…In addition, carpooling is more common in reality among commuters with lower incomes, longer trips to work, and more restricted access to auto. Accordingly, it is of interest to integrate user heterogeneity into the multi-mode morning commuting problem with carpooling behavior under parking space constraint ( Wu and Huang, 2015;Chen et al, 2015 ).…”

Carpooling with heterogeneous users in the bottleneck model

“…In particular when heterogeneity concerns both heterogeneity in value of time and in values of schedule delay, different possibilities arise. We distinguish three ideal types: ratio heterogeneity ( de Palma and Lindsey (2002) , van den Berg and Verhoef (2011a,b) ), proportional heterogeneity ( Vickrey (1973) , van den Berg and Verhoef (2011b) ), and more general heterogeneity ( Newell (1987) , Lindsey (2004) , Börjesson and Kristoffersson (2014) , Wu and Huang (2015) , Liu et al (2015) , Chen et al (2015) , , Takayama and Kuwahara (2017) ). 2 "Ratio heterogeneity" refers to heterogeneity in the ratio of the value of time over the value of schedule delay, or α i / β i , in the conventional notation.…”