Optimal Information Location for Adaptive Routing

Abstract

One strategy for addressing uncertain roadway conditions and travel times is to provide real-time travel information to drivers through variable message signs, highway advisory radio, or other means. However, providing such information is often costly, and decisions must be made about the most useful places to inform drivers about local conditions. This paper addresses this question, building on adaptive routing algorithms describing optimal traveler behavior in stochastic networks with en route information. Three specific problem contexts are formulated: routing of a single vehicle, assignment of multiple vehicles in an uncongested network, and adaptive equilibrium with congestion. A network contraction procedure is described which makes an enumerative algorithm computationally feasible for small-to-medium sized roadway networks, along with heuristics which can be applied for large-scale networks. These algorithms are demonstrated on three networks of varying size.

Appendices

Appendix

Algorithms and Pseudocode

This appendix provides a brief overview of the TD-OSP and UER2 algorithms used in this paper. TD-OSP is taken directly from Waller and Ziliaskopoulos (2002), and determines the expected cost of an optimal adaptive routing policy through a label-correcting technique involving a scan-eligible list SEL. The presentation of the algorithm below is adapted to the notation in this paper, and to allow both information nodes and non-information nodes. Input to TD-OSP is a destination v, and the output is the optimal policy π ^* with respect to the (fixed) arc costs by state, provided as input.

The version shown here is used for CIP, when the entire routing policy is needed. For UIP and IIP, all that is needed is the expected cost of the optimal routing policy, a variation for which Waller and Ziliaskopoulos (2002) provides a pseudopolynomial algorithm involving an inner reduction step of the expected cost vector.

The UER2 algorithm is used to evaluate feasible solutions for the CIP problem, finding an equilibrium among policies rather than fixed paths as in the traditional deterministic user equilibrium problem. The problem setting in this paper matches Model B of Unnikrishnan and Waller (2009), who shows that the equilibrium state-dependent link flow matrix $\mathbf{X} = [ x_{ij}^s ]$ solves the optimization problem

$$ \min \sum\limits_{(i,j) \in A} \sum\limits_{s \in S_{ij}} \int_0^{x_{ij}^s} c_{ij}^s (x) dx $$

where the $x_{ij}^s$ are generated by a feasible policy assignment with respect to the demand matrix D. Unnikrishnan (2008) uses an incidence matrix to map each policy to the proportion of its flow which uses arc (i,j) in state s, and then applies the Frank-Wolfe algorithm to solve this program (Algorithm 2).

In this paper, we adopt a different approach for mapping policies to arc flows, applying a “policy loading” algorithm (Algorithm 3) which assigns flow from all origins to a given destination v in each stage. Each node i is associated with a label T representing the number of vehicles at node i which have yet to reach the destination. Initially, T _i is equal to the demand from i to v. Nodes with positive T _i are scanned, and these vehicles assigned to adjacent nodes according to the routing policy and the probability of each message being received, thus increasing T for the adjacent nodes, and reducing T _i to zero. Since vehicles typically move from nodes with higher expected cost labels L to nodes with lower expected cost labels, a binary heap is used to keep track of the nodes with positive T, identifying the highest-cost node at each iteration. This alteration we name UER2.

The prime advantages of UER2 are (i) the ability to handle networks with cycles, through an iterative approach and a tolerance T _min  ≪ ||D|| to terminate long cycles; and (ii) a substantial (approximately O(m)) reduction in the computation time needed, since all origins corresponding to the same destination are processed simultaneously and redundant flow shifts are eliminated. The implications of this change, and proofs of convergence and correctness, are discussed more fully in Boyles (2009) and Boyles and Waller (2009).