Large-scale road network vulnerability analysis: a sensitivity analysis based approach

Abstract

Traditionally, an assessment of transport network vulnerability is a computationally intensive operation. This article proposes a sensitivity analysis-based approach to improve computational efficiency and allow for large-scale applications of road network vulnerability analysis. Various vulnerability measures can be used with the proposed method. For illustrative purposes, this article adopts the relative accessibility index (AI), which follows the Hansen integral index, as the network vulnerability measure for evaluating the socio-economic effects of link (or road segment) capacity degradation or closure. Critical links are ranked according to the differences in the AIs between normal and degraded networks. The proposed method only requires a single computation of the network equilibrium problem. The proposed technique significantly reduces computational burden and memory storage requirements compared with the traditional approach. The road networks of the Sioux Falls city and the Bangkok metropolitan area are used to demonstrate the applicability and efficiency of the proposed method. Network manager(s) or transport planner(s) can use this approach as a decision support tool for identifying critical links in road networks. By improving these critical links or constructing new bypass roads (or parallel paths) to increase capacity redundancy, the overall vulnerability of the networks can be reduced.

Appendix A

Since the Jacobian of the probit path choice probability vector with respect to the mean path cost vector ($\nabla_{{{\mathbf{c}}^{w} }} {\mathbf{P}}^{w}$) is symmetric, it is only necessary to determine the upper triangular elements of $\nabla_{{{\mathbf{c}}^{w} }} {\mathbf{P}}^{w}$ by using an approximation method (Clark and Watling, 2002).

For route k connecting the OD movement w, the derivative of the path choice probability ($P_{k}^{w}$) with respect to the cost on an alternative route j ($C_{j}^{w}$), at a given mean path cost vector ${\mathbf{c}}^{w}$ with a variance–covariance matrix $\varvec{\Upsigma}^{w}$, is formulated as

$$ \left. {\frac{{\partial P_{k}^{w} }}{{\partial C_{j}^{w} }}} \right|_{{C = {\mathbf{c}}^{w} }} = \sqrt {\frac{{\left| {\varvec{\Upsigma}^{w} \left( {k,j} \right)} \right|}}{{2\pi \left| {\varvec{\Upsigma}^{w} } \right|}}} \exp \left( {\frac{{A\left( {k,j} \right)}}{2}} \right){\text{MNP}}_{k} \left( {{\mathbf{c}}^{w} \left( {k,j} \right),\varvec{\Upsigma}^{w} \left( {k,j} \right)} \right)\quad k < j;\;k,j \in {\mathbf{K}}^{w}, $$

where ${\mathbf{K}}^{w}$ is the subset of paths connecting the OD movement w; ${\text{MNP}}_{k} \left( {{\mathbf{c}}^{w} \left( {k,j} \right),\varvec{\Upsigma}^{w} \left( {k,j} \right)} \right)$ is the multinomial probit (MNP) choice probability for alternative $k \in {\mathbf{K}}^{w}$ corresponding to a model with N ^w − 1 alternatives (where the original model had $N^{w} = \left| {{\mathbf{K}}^{w} } \right|$ alternatives), mean cost vector ${\mathbf{c}}^{w} \left( {k,j} \right)$ and covariance matrix $\varvec{\Upsigma}^{w} \left( {k,j} \right)$; and the matrix $\varvec{\Upsigma}^{w} \left( {k,j} \right)$, row-vector ${\mathbf{c}}^{w} \left( {k,j} \right)$, and scalar $A\left( {k,j} \right)$ are computed from the following steps (with ${\mathbf{c}}^{w}$ assumed to be in row-vector notation):

Step 1: :

Calculate the inverse covariance matrix $\left( {\varvec{\Upsigma}^{w} } \right)^{ - 1}$ and the row vector ${\mathbf{c}}^{w} \cdot \left( {\varvec{\Upsigma}^{w} } \right)^{ - 1}$.

Step 2::

Form the (N ^w − 1)-square matrix ${\mathbf{D}}\left( {k,j} \right)$ from $\left( {\varvec{\Upsigma}^{w} } \right)^{ - 1}$ by:

(2.1) adding row j of $\left( {\varvec{\Upsigma}^{w} } \right)^{ - 1}$ to row k;

(2.2) adding column j of the resultant matrix from (2.1) to its column k;

(2.3) deleting row j and column j of the resultant matrix from (2.2).

Step 3::

Form the (N ^w − 1) dimensional row vector ${\mathbf{d}}\left( {k,j} \right)$ from ${\mathbf{c}}^{w} \cdot \left( {\varvec{\Upsigma}^{w} } \right)^{ - 1}$, by adding the jth element to the kth element, and then deleting the jth element.

Step 4::

Compute:

$$ \begin{aligned}\varvec{\Upsigma}^{w} \left( {k,j} \right) = \left( {{\mathbf{D}}\left( {k,j} \right)} \right)^{ - 1} , \\ {\mathbf{c}}^{w} \left( {k,j} \right) = {\mathbf{d}}\left( {k,j} \right) \cdot\varvec{\Upsigma}^{w} \left( {k,j} \right),\;{\text{and}} \\ A\left( {k,j} \right) = {\mathbf{d}}\left( {k,j} \right) \cdot\varvec{\Upsigma}^{w} \left( {k,j} \right) \cdot \left( {{\mathbf{d}}\left( {k,j} \right)} \right)^{T} - {\mathbf{c}}^{w} \cdot \left( {\varvec{\Upsigma}^{w} } \right)^{ - 1} \cdot\,\left( {{\mathbf{c}}^{w} } \right)^{T} \\ \end{aligned} $$

Since the Jacobian is symmetry and the choice probabilities must sum to 1, once all the off-diagonal elements have been determined the diagonal terms can be calculated from:

$$ \frac{{\partial P_{k}^{w} }}{{\partial C_{k}^{w} }} = - \sum\limits_{{j \in {\mathbf{K}}^{w} ,j \ne k}} {\frac{{\partial P_{k}^{w} }}{{\partial C_{j}^{w} }}} $$

This approximation method is feasible to apply for large scale networks, where ${\text{MNP}}_{k} \left( {{\mathbf{c}}^{w} \left( {k,j} \right),\varvec{\Upsigma}^{w} \left( {k,j} \right)} \right)$ can be calculated by Monte Carlo methods.