A multi-modal network equilibrium model considering captive travelers and mode correlation

Abstract

In making daily commuting trips, a part of travelers, which are called captive travelers, rely on one transport mode due to a lack of access or affordability to other transport modes. To account for the effect of such captive travelers on network equilibrium performances, this paper proposes a multi-modal network equilibrium (MMNE) model that accounts for the captive travelers and the correlations between modes and between routes. First, a hybrid mode choice model is developed by integrating the dogit and nested logit (NL) models. The hybrid dogit–NL (DNL) model has smaller direct and cross elasticity than the NL model, it alleviates the property of irrelevant from independent alternatives and takes the dogit and NL modal splits as bounds. Second, the path-size logit (PSL) model is adopted for predicting travelers’ route choices with overlapping routes. The DNL–PSL MMNE model is formulated as a mathematical programming problem that admits an equivalent and unique solution. Then, a partial linearization algorithm with the Barzilai–Borwein (BB) step sizes is developed. The numerical results reveal that captive travelers lead to lower sensitivity toward transport policies and may cause higher network total travel time; while the perception of mode similarity may impair the overall attractiveness of modes with a high degree of similarity. The observations indicate that to promote green transportation, policy efforts should be made to make use of or adjust the captivity structure and produce diversified perceptions of and preferences for different green transport modes. The BB step sizes are suggested for low travel demand cases when solving the combined travel choice problems. Further, extensions of the DNL model with bundle captivities are discussed. The results of the paper help improve the network equilibrium prediction and support transport policymaking.

Appendix

Given Assumptions 1-4 and the assumption of independently distributed Weibull-type mode disutilities, we may define the DNW–PSL MMNE model as,

$$\begin{aligned} \begin{array}{l} \begin{aligned} \min Z &{}= {Z_1}\!+\!{Z_2}\!+\!{Z_3} \!+\!{Z_4}\!+\! Z_{5}\!+\! Z_{6}\!+\!Z_{7}\\ &{}= \sum \limits _{m \in M}\sum \limits _{a \in A^{m}} {\int _{0}^{v^{ma}} {\tau ^{ma}(v)dv}} \!+\!\sum \limits _{ij\in IJ}\sum \limits _{u \in {U_{ij}}}\sum \limits _{m \in M_{ij}^u}\frac{1}{\theta _{ij}^{um}}\sum \limits _{r\in R_{ij}^{um}}f_{ij}^{umr}\left( \textrm{ln}f_{ij}^{umr}-1\right) \\ &{}\quad - \sum \nolimits _{ij\in IJ}\sum \nolimits _{u \in {U_{ij}}}\sum \limits _{m \in M_{ij}^u}\frac{1}{\theta _{ij}^{um}}\sum \limits _{r\in R_{ij}^{um}}f_{ij}^{umr}\textrm{ln}\varpi _{ij}^{umr} \\ &{}\quad +\sum \limits _{ij \in IJ} {\sum \limits _{u \in {U_{ij}}} {\sum \limits _{m \in M_{ij}^u} {\frac{1}{\theta _{ij}^{um}}\left( q_{ij}^{um} \!-\! \frac{{{d_{ij}}\eta _{ij}^{m}}}{{1 \!+\! {\sum \nolimits _{n \in M_{ij}} {\eta _{ij}^{n}} } }}\right) \left[ {\ln \left( {q_{ij}^{um} \!-\! \frac{{{d_{ij}}\eta _{ij}^{m}}}{{1 \!+\! {\sum \nolimits _{n \in M_{ij}} {\eta _{ij}^{n}} } }}} \right) \!-\! 1} \right] } } } \\ &{}\quad +\sum \limits _{ij \in IJ} {\sum \limits _{u \in {U_{ij}}} \left({\frac{1}{\varphi _{ij}^u} \!-\! \frac{1}{\theta _{ij}^{um}}}\right)\left[ {\sum \limits _{m \in M_{ij}^u} {\left( {q_{ij}^{um} \!-\! \frac{{{d_{ij}}\eta _{ij}^{m}}}{{1 \!+\! {\sum \nolimits _{n \in M_{ij}} {\eta _{ij}^{n}} } }}} \right) } } \right] } \\ &{}\quad * \left[ {\ln \sum \limits _{m \in M_{ij}^u} {\left( {q_{ij}^{um} \!-\! \frac{{{d_{ij}}\eta _{ij}^{m}}}{{1 \!+\! {\sum \limits _{n \in M_{ij}} {\eta _{ij}^{n}} } }}} \right) } \!-\! 1} \right] \\ &{}\quad -\sum \limits _{ij\in IJ}\sum \limits _{u \in {U_{ij}}}\sum \limits _{m \in M_{ij}^u}\frac{1}{\theta _{ij}^{um}}q_{ij}^{um}\left( \textrm{ln}q_{ij}^{um}-1\right) \!-\! \sum \limits _{ij \in IJ} {\sum \limits _{u \in {U_{ij}}} {\sum \limits _{m \in M_{ij}^u} {q_{ij}^{um} \ln \Psi _{ij}^{um}} } } \end{aligned} \end{array} \end{aligned}$$

(64)

$$\begin{aligned} s.t.\ \ \sum \limits _{u \in {U_{ij}}} {\sum \limits _{m \in M_{ij}^u} {q_{ij}^{um}} }= & {} {d_{ij}},\forall ij\in IJ, \end{aligned}$$

(65)

$$\begin{aligned} \sum \limits _{r \in {R_{ij}^{um}}}f_{ij}^{umr}= & {} q_{ij}^{um}, \forall m\in M_{ij}^{u}, u\in U_{ij}, ij\in IJ, \end{aligned}$$

(66)

$$\begin{aligned} q_{ij}^{um}\ge & {} 0, \forall m\in M_{ij}^{u}, u\in U_{ij}, ij\in IJ, \end{aligned}$$

(67)

$$\begin{aligned} f_{ij}^{umr}\ge & {} 0, \forall r\in R_{ij}^{um}, m\in M_{ij}^{u}, u\in U_{ij}, ij\in IJ, \end{aligned}$$

(68)

where $Z_{1}-Z_{3}$ captures the PSL route choice problem, $Z_{4}-Z_{7}$ captures the DNW mode choice problem. The related variables and parameters are the same as those in the DNL–PSL MMNE model. Note that, Eqs. (64)–(68) admits a unique solution that is equivalent to the DNW mode choice and PSL route choice models, the detailed proofs are similar to those for the DNL–PSL MMNE model, thus omitted here.