Finding Reliable Shortest Paths in Road Networks Under Uncertainty

Abstract

The aim of this study is to investigate the solution algorithm for solving the problem of determining reliable shortest paths in road networks with stochastic travel times. The availability of reliable shortest paths enables travelers, in the face of travel time uncertainty, to plan their trips with a pre-specified on-time arrival probability. In this study, the reliable shortest path between origin and destination nodes is determined using a multiple-criteria shortest path approach when link travel times follow normal distributions. The dominance conditions involved in such problems are established, thereby reducing the number of generated non-dominated paths during the search processes. Two solution algorithms, multi-criteria label-setting and A* algorithms, are proposed and their complexities analyzed. Computational results using large scale networks are presented. Numerical examples using data from a real-world advanced traveller information system is also given to illustrate the applicability of the solution algorithms in practice.

Appendices

Appendix A. Notations

A :

A set of links

|A|:

Number of links in the network

a _ij :

A link connecting node i and node j

b :

Travel time budget

d _is :

Euclidean distance from node i to destination

F(i):

Heuristic value function for the path set P ^rs

f(i):

Minimum travel time budget of the path set P ^rs

G :

Directed network

h(i):

Estimation of travel time budget from node i to destination

N :

A set of nodes

|N|:

Number of nodes in the network

P ^rs :

A set of paths from origin to destination

|P|:

Number of non-dominated paths between origin and destination

$p_u^{{rs}}$ :

A path from origin to destination

$p_{ * }^{{rs}}$ :

Optimal path from origin to destination

PDS(i):

A set of predecessor nodes of node i

r :

Origin node

s :

Destination node

SCS(i):

A set of successor nodes of node i

T _ij :

Travel time distribution of link a _ij

t _ij :

Mean travel time of link a _ij

$T_u^{{rs}}$ :

Travel time distribution of path $p_u^{{rs}}$

$t_u^{{rs}}$ :

Mean travel time of path $p_u^{{rs}}$

ν _max :

Maximum travel speed in the network

VOT :

Value of time

Z _α :

Inverse cumulative distribution function for standard normal distribution at α confidence level

α :

Probability of on-time arrival

σ _ij :

Travel time standard deviation of link a _ij

$\sigma_u^{{rs}}$ :

Travel time standard deviation of path $p_u^{{rs}}$

$x_{{ij}}^{{rs}}$ :

Link-path incidence relationship

$\tau_u^{{rs}}$ :

Toll charge of path $p_u^{{rs}}$

τ _ij :

Toll charge of link a _ij

${{f}_{{T_u^{{rs}}}}}\left( \cdot \right)$ :

Probability density function of path travel time distribution

${{\Phi }_{{T_u^{{rs}}}}}\left( \cdot \right)$ :

Cumulative distribution function of path travel time distribution

$\Phi_{{T_u^{{rs}}}}^{{ - 1}}\left( \cdot \right)$ :

Inverse cumulative distribution function of path travel Time distribution

Appendix B. Proofs

Theorem 1

A sub-path of any non-dominated path must be a non-dominated path itself.

Proof

Suppose $p_v^{{ri}} \in {{P}^{{ri}}}$ is a sub-path of a non-dominated path $p_v^{{rw}} = p_v^{{ri}} \oplus p_v^{{iw}} \in {{P}^{{rw}}}$, and it is dominated by another path $p_u^{{ri}} \in {{P}^{{ri}}}$. According to Definition 3.3, we have $\Phi_{{T_u^{{rs}}}}^{{ - 1}}\left( \alpha \right) < \Phi_{{T_v^{{rs}}}}^{{ - 1}}\left( \alpha \right)$ for any path $p^{is} ∈ P^{is}$. Therefore, there exists at least one path $p_u^{{rw}} = p_u^{{ri}} \oplus p_v^{{iw}} \in {{P}^{{rw}}}$ satisfying $\Phi_{{T_u^{{rs}}}}^{{ - 1}}\left( \alpha \right) < \Phi_{{T_v^{{rs}}}}^{{ - 1}}\left( \alpha \right)$ for any $p^{ws} ∈ P^{ws}$, since any $p_v^{{iw}} \oplus {{p}^{{ws}}} \in {{P}^{{is}}}$. Thus, $p_u^{{ri}} \succ p_v^{{ri}}$. This contradicts the fact that $p_v^{{ri}}$ is a non-dominated path. □

Let $p_u^{{rw}} = p_u^{{ri}} \oplus {{p}^{{iw}}}$ and $p_v^{{rw}} = p_v^{{ri}} \oplus {{p}^{{iw}}}$ be two paths from origin r to node w going through the same sub-path p ^iw. $T_u^{{rw}}$ and $T_v^{{rw}}$ denote travel time for these two paths respectively. We introduce following function to facilitate the proofs of Propositions 2 and 3.

$$ g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) = \Phi_{{T_u^{{rw}}}}^{{ - 1}}\left( \alpha \right) - \Phi_{{T_v^{{rw}}}}^{{ - 1}}\left( \alpha \right) = \left( {t_u^{{ri}} - t_v^{{ri}}} \right) + {{Z}_{\alpha }}\left( {\sqrt {{{{{\left( {\sigma_u^{{ri}}} \right)}}^2} + {{{\left( {{{\sigma }^{{iw}}}} \right)}}^2}}} - \sqrt {{{{{\left( {\sigma_v^{{ri}}} \right)}}^2} + {{{\left( {{{\sigma }^{{iw}}}} \right)}}^2}}} } \right) $$

(9)

The $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right)$ has following properties:

Lemma 1

• (i) If ${{Z}_{\alpha }}\sigma_u^{{ri}} < {{Z}_{\alpha }}\sigma_v^{{ri}}$, and thus $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right)$ is a monotonic increasing function, $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) \in \left[ {\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) - \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right),{ }t_u^{{ri}} - t_v^{{ri}}} \right)$;

• (ii) If ${{Z}_{\alpha }}\sigma_u^{{ri}} > {{Z}_{\alpha }}\sigma_v^{{ri}}$, and thus $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right)$ is a monotonic decreasing function, $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) \in \left( {t_u^{{ri}} - t_u^{{ri}},\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) - \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right)} \right]$;

• (iii) If ${{Z}_{\alpha }}\sigma_u^{{ri}} = {{Z}_{\alpha }}\sigma_v^{{ri}}$, and $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) = t_u^{{ri}} - t_v^{{ri}}$.

Proof

The gradient of $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right)$ can be formulated as $g_{{uv}}^{{ri}}\prime \left( {{{p}^{{iw}}}} \right) = \frac{{\partial g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right)}}{{\partial {{{\left( {{{\sigma }^{{iw}}}} \right)}}^2}}} = \frac{{{{Z}_{\alpha }}}}{2}\frac{{\sqrt {{{{{\left( {\sigma_v^{{ri}}} \right)}}^2} + {{{\left( {{{\sigma }^{{iw}}}} \right)}}^2}}} - \sqrt {{{{{\left( {\sigma_u^{{ri}}} \right)}}^2} + {{{\left( {{{\sigma }^{{iw}}}} \right)}}^2}}} }}{{\sqrt {{{{{\left( {\sigma_u^{{ri}}} \right)}}^2} + {{{\left( {{{\sigma }^{{iw}}}} \right)}}^2} * }} \sqrt {{{{{\left( {\sigma_v^{{ri}}} \right)}}^2} + {{{\left( {{{\sigma }^{{iw}}}} \right)}}^2}}} }}$. Therefore, when ${{Z}_{\alpha }}\sigma_u^{{ri}} < {{Z}_{\alpha }}\sigma_v^{{ri}}$, we have $g_{{uv}}^{{ri}}\prime \left( {{{p}^{{iw}}}} \right) > 0$ and $\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) - \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right) \leqslant g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) < t_u^{{ri}} - t_v^{{ri}}$; when ${{Z}_{\alpha }}\sigma_u^{{ri}} > {{Z}_{\alpha }}\sigma_v^{{ri}}$, we have $g_{{uv}}^{{ri}}\prime \left( {{{p}^{{iw}}}} \right) < 0$ and $t_u^{{ri}} - t_v^{{ri}} < g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) \leqslant \Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) - \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right)$; when ${{Z}_{\alpha }}\sigma_u^{{ri}} = {{Z}_{\alpha }}\sigma_v^{{ri}}$, we have $g_{{uv}}^{{ri}}\prime \left( {{{p}^{{iw}}}} \right) = 0$ and $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) = t_u^{{ri}} - t_v^{{ri}}$.□

According to Definition 3, $p_u^{{ri}} \succ p_v^{{ri}}$ is equivalent to $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) < 0$ for any path p ^iw ∈ P ^iw. We will prove $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) < 0$ for following dominance conditions.

Proposition 2

(M-V dominance). Given an on-time arrival probability α and two paths $p_u^{{ri}} \ne p_v^{{ri}} \in {{P}^{{ri}}}$, $p_u^{{ri}} \succ p_v^{{ri}}$ if $p_u^{{ri}}$ and $p_v^{{ri}}$ satisfy either

• (i) $t_u^{{ri}} \leqslant t_v^{{ri}}$ and ${{Z}_{\alpha }}\sigma_u^{{ri}} < {{Z}_{\alpha }}\sigma_v^{{ri}}$ or

• (ii) $t_u^{{ri}} < t_v^{{ri}}$ and ${{Z}_{\alpha }}\sigma_u^{{ri}} \leqslant {{Z}_{\alpha }}\sigma_v^{{ri}}$

Proof

It can be easily followed by Lemma 1.□

Proposition 3

(M-B dominance). Given an on-time arrival probability α and two paths $p_u^{{ri}} \ne p_v^{{ri}} \in {{P}^{{ri}}}$, $p_u^{{ri}} \succ p_v^{{ri}}$ if $p_u^{{ri}}$ and $p_v^{{ri}}$ satisfy $t_u^{{ri}} \leqslant t_v^{{ri}}$ and $\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) < \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right)$.

Proof

When ${{Z}_{\alpha }}\sigma_u^{{ri}} < {{Z}_{\alpha }}\sigma_v^{{ri}}$ and $t_u^{{ri}} \leqslant t_v^{{ri}}$ according to Lemma 1(i), we have $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) < \left( {t_u^{{ri}} - t_v^{{ri}}} \right) \leqslant 0$. When ${{Z}_{\alpha }}\sigma_u^{{ri}} > {{Z}_{\alpha }}\sigma_v^{{ri}}$ and $t_u^{{ri}} \leqslant t_v^{{ri}}$ according to Lemma 1(ii), we have $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) < \left( {\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) - \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right)} \right) \leqslant 0$. When ${{Z}_{\alpha }}\sigma_u^{{ri}} = {{Z}_{\alpha }}\sigma_v^{{ri}}$ and $\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) < \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right)$, we have $t_u^{{ri}} < t_v^{{ri}}$. According to Lemma 1(iii), we have $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) = \left( {t_u^{{ri}} - t_v^{{ri}}} \right) < 0$. Therefore, when $t_u^{{ri}} \leqslant t_v^{{ri}}$ and $\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right) < \Phi_{{T_v^{{ri}}}}^{{ - 1}}\left( \alpha \right)$, we have $g_{{uv}}^{{ri}}\left( {{{p}^{{iw}}}} \right) < 0$, ∀p ^iw ∈ P ^iw, ∀w ∈ N. □

Let ${{f}_{{T_u^{{ri}}}}}(t)$ be the PDF of path travel time distribution $T_u^{{ri}}$. It can be proved that the objective value $\Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \alpha \right)$ monotonically increases with path extensions as follows:

Proposition 7

Given two paths $p_u^{{ri}}$ and $p_u^{{rj}} = p_u^{{ri}} \oplus {{a}_{{ij}}}$, the relationship $\matrix{ {\Phi_{{T_u^{{rj}}}}^{{ - 1}}\left( \lambda \right) > \Phi_{{T_u^{{ri}}}}^{{ - 1}}\left( \lambda \right),} \hfill &{\forall \lambda \in \left( {0,1} \right)} \hfill \\ }<!end array>$ always holds.

Proof

$\begin{array}{*{20}{c}} {{{\Phi }_{{T_{u}^{{rj}}}}}(b) - {{\Phi }_{{T_{u}^{{ri}}}}}(b) = \int_{0}^{b} {{{\Phi }_{{{{T}_{{ij}}}}}}\left( {b - t} \right){{f}_{{T_{u}^{{ri}}}}}(t)dt - \int_{0}^{b} {{{f}_{{T_{u}^{{ri}}}}}(t)dt} } } \hfill \\ { = \int_{0}^{b} {\left( {{{\Phi }_{{{{T}_{{ij}}}}}}\left( {b - t} \right) - 1} \right){{f}_{{T_{u}^{{ri}}}}}(t)dt < 0,} \quad \forall b \in {{R}^{ + }}} \hfill \\ { \Rightarrow \Phi _{{T_{u}^{{rj}}}}^{{ - 1}}\left( \lambda \right) > \Phi _{{T_{u}^{{ri}}}}^{{ - 1}}\left( \lambda \right),\quad \forall \lambda \in \left( {0,1} \right)} \hfill \\ \end{array}$. □