Traffic equilibrium with a continuously distributed bound on travel weights: the rise of range anxiety and mental account

Abstract

A new traffic network equilibrium problem with continuously distributed bounds on path weights is introduced in this paper, as an emerging modeling tool for evaluating traffic networks in which the route choice behavior of individual motorists is subject to some physical or psychological upper limit of a travel weight. Such a problem may arise from at least two traffic network instances. First, in a traffic network serving electric vehicles, the driving range of these vehicles is subject to a distance constraint formed by onboard battery capacities and electricity consumption rates as well as network-wide battery-recharging opportunities, which cause the range anxiety issue in the driving population. Second, in a tolled traffic network, while drivers take into account both travel time and road toll in their route choice decisions, many of them implicitly or explicitly set a budget constraint in their mental account for toll expense, subject to their own income levels and other personal and household socio-economic factors. In both cases, we model the upper limit of the path travel weight (i.e., distance or toll) as a continuously distributed stochastic parameter across the driving population, to reflect the diverse heterogeneity of vehicle- and/or motorist-related travel characteristics. For characterizing this weight-constrained network equilibrium problem, we proposed a convex programming model with a finite number of constraints, on the basis of a newly introduced path flow variable named interval path flow rate. We also analyzed the problem’s optimality conditions for the case of path distance limits, and studied the existence of optimal tolls for the case of path toll limits. A linear approximation algorithm was further developed for this complex network equilibrium problem, which encapsulates an efficient weight-constrained k-minimum time path search procedure to perform the network loading. Numerical results obtained from conducting quantitative analyses on example networks clearly illustrate the applicability of the modeling and solution methods for the proposed problem and reveal the mechanism of stochastic weight limits reshaping the network equilibrium.

Appendices

Appendix A

Following Florian et al. (1987), we know the following relationship between x ^{( n+1)} _a and x ^{( n)} _a , x ^{( n−1)} _a , and y ^{( n)} _a ,

$$ \begin{aligned} x_{a}^{{\left( {n + 1} \right)}} & = x_{a}^{{\left( {n - 1} \right)}} + \beta^{\left( n \right)} \left( {z_{a}^{\left( n \right)} - x_{a}^{{\left( {n - 1} \right)}} } \right) \\ & = \beta^{\left( n \right)} \left( {x_{a}^{\left( n \right)} + \alpha^{\left( n \right)} \left( {y_{a}^{\left( n \right)} - x_{a}^{\left( n \right)} } \right)} \right) + \left( {1 - \beta^{\left( n \right)} } \right)x_{a}^{{\left( {n - 1} \right)}} \\ & = \beta^{\left( n \right)} \left( {\alpha^{\left( n \right)} y_{a}^{\left( n \right)} + \left( {1 - \alpha^{\left( n \right)} } \right)x_{a}^{\left( n \right)} } \right) + \left( {1 - \beta^{\left( n \right)} } \right)x_{a}^{{\left( {n - 1} \right)}} \\ \end{aligned} $$

which can be further written as a function of only y ^{( i)} _a , i =  0, 1, …, n

$$ x_{a}^{{\left( {n + 1} \right)}} = \mathop \sum \limits_{i = 0}^{n} \rho^{{\left( {n + 1, i} \right)}} y_{a}^{\left( i \right)} $$

where $\rho^{{\left( {n + 1, i} \right)}}$ is a function of α⁽ⁱ⁾ and β⁽ⁱ⁾, where i =  0, 1, …, n, such as,

$$ \begin{aligned} \rho^{{\left( {n + 1, i} \right)}} & = \beta^{\left( n \right)} \alpha^{\left( n \right)} \quad i = n \\ \rho^{{\left( {n + 1, i} \right)}} & = \beta^{\left( n \right)} \left( {1 - \alpha^{\left( n \right)} } \right)\rho^{{\left( {n, n - 1} \right)}} \quad i = n - 1 \\ \rho^{{\left( {n + 1, i} \right)}} & = \beta^{\left( n \right)} \left( {1 - \alpha^{\left( n \right)} } \right)\rho^{{\left( {n, i} \right)}} + \left( {1 - \beta^{\left( n \right)} } \right)\rho^{{\left( {n - 1, i} \right)}} \quad i = 0,1, \ldots ,n - 2 \\ \end{aligned} $$

Then, $\beta_{ \hbox{max} }^{\left( n \right)}$ can be determined by the following formula:

$$ \beta_{ \hbox{max} }^{\left( n \right)} = \mathop {\hbox{min} }\limits_{0 \le i \le n - 2} \left\{ {\frac{{\rho^{{\left( {n - 1, i} \right)}} }}{{\rho^{{\left( {n - 1, i} \right)}} - \left( {1 - \alpha^{\left( n \right)} } \right)\rho^{{\left( {n, i} \right)}} }}: \rho^{{\left( {n - 1, i} \right)}} - \left( {1 - \alpha^{\left( n \right)} } \right)\rho^{{\left( {n, i} \right)}} > 0} \right\} $$

Appendix B

The pseudo-code of the label-correcting algorithm for finding a Pareto-optimal path set connecting an origin node r and all destination nodes in terms of two travel impedances (i.e., travel time and travel weight in our case) is given below:

In the above pseudo-code, I and J, respectively, represent the current set and the next set of nodes with updated label pairs; L(i) is the set of label pairs for node i; A(i) is the set of emanating links from node i; P(i) is the set of precedent nodes of node i; i^u denotes the precedent node that results in label pair $\left( {t\left( i \right), d\left( i \right)} \right)$ through link $\left( {i^{u} , j} \right)$.