Day-to-Day Evolution Model Based on Dynamic Reference Point with Heterogeneous Travelers

Abstract

This paper investigates the implementation of a dynamic reference point scheme to capture traveler’s mental characteristics, with their day-to-day route choice behavior and heterogeneity. The traveler’s heterogeneity focuses on their different risk attitudes. On each day, travelers choose the routes based on their estimated travel costs, which can be affected by the reference point structure and its update. Most existing studies on the day-to-day traffic assignment models are proposed to capture day-to-day flow fluctuations through a learning model based on traveler’s past experience and information, but did not work on the consideration of gain and loss, which is described by the reference point scheme, comparing with traveler’s previous experience. This study aims to develop a day-to-day dynamic evolution model, in which travelers take on a tendency to refer to their previous travel experience as a reference when coping with different travel scenarios. First, the multi-class dynamic system is proposed to model traveler’s route choice behavior in a transportation network with two traveler classes. Then, the equilibrium state and stability of the evolution model is examined. We further investigate the class-specified update structure of the reference point. Finally, numerical experiments are presented to illustrate the application of our method.

Appendix

The process of calculating the eigenvalues of the matrix ${J}_C^{\prime }{J}_P$ can be described as follows:

$$ {\displaystyle \begin{array}{l}{\left.{J}_c\right|}_{z^{\ast }}={\left.\left[\begin{array}{cc}\frac{\partial C\left({f}_1\right)}{\partial {f}_1}& \frac{\partial C\left({f}_1\right)}{\partial {f}_2}\\ {}\frac{\partial C\left({f}_2\right)}{\partial {f}_1}& \frac{\partial C\left({f}_2\right)}{\partial {f}_2}\end{array}\right]\right|}_{z^{\ast }}\\ {}={\left.\left[\begin{array}{cc}{c}_{0,1} ab{\left(\frac{f_1^{\ast }}{Q_1}\right)}^{b-1}\frac{1}{Q_1}& 0\\ {}0& {c}_{0,2} ab{\left(\frac{f_2^{\ast }}{Q_2}\right)}^{b-1}\frac{1}{Q_2}\end{array}\right]\right|}_{z^{\ast }}\end{array}} $$

(42)

$$ {\displaystyle \begin{array}{l}{\left.{J}_P\right|}_{z^{\ast }}={\left.\left[\begin{array}{cc}\frac{\partial {P}_1}{\partial {c}_1}& \frac{\partial {P}_1}{\partial {c}_2}\\ {}\frac{\partial {P}_2}{\partial {c}_1}& \frac{\partial {P}_2}{\partial {c}_2}\end{array}\right]\right|}_{z^{\ast }}\\ {}={\left.\left[\begin{array}{cc}\frac{\left(-\theta \right)\exp \left(-\theta {c}_1^{\ast}\right)\exp \left(-\theta {c}_2^{\ast}\right)}{{\left[\exp \left(-\theta {c}_1^{\ast}\right)+\exp \left(-\theta {c}_2^{\ast}\right)\right]}^2}& \frac{\theta \exp \left(-\theta {c}_1^{\ast}\right)\exp \left(-\theta {c}_2^{\ast}\right)}{{\left[\exp \left(-\theta {c}_1^{\ast}\right)+\exp \left(-\theta {c}_2^{\ast}\right)\right]}^2}\\ {}\frac{\theta \exp \left(-\theta {c}_1^{\ast}\right)\exp \left(-\theta {c}_2^{\ast}\right)}{{\left[\exp \left(-\theta {c}_1^{\ast}\right)+\exp \left(-\theta {c}_2^{\ast}\right)\right]}^2}& \frac{\left(-\theta \right)\exp \left(-\theta {c}_1^{\ast}\right)\exp \left(-\theta {c}_2^{\ast}\right)}{{\left[\exp \left(-\theta {c}_1^{\ast}\right)+\exp \left(-\theta {c}_2^{\ast}\right)\right]}^2}\end{array}\right]\right|}_{z^{\ast }}\end{array}} $$

(43)