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A model for multi-class road network recovery scheduling of regional road networks

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Abstract

In this paper, an optimisation model for recovery planning of road networks is presented in which both social and economic resilience is aimed to be achieved. The model is formulated as a bi-level multi-objective discrete network design problem which forms a non-convex mixed integer non-linear problem. Solved by a Branch and Bound method, the solution algorithm employs an outer approximation method to estimate the lower bound of each node in the Branch and Bound search tree. The solution algorithm exploits a unique approach for lower-bound computation dealing with a disrupted multi-class network that may not be able to satisfy the demand between all OD pairs due to damaged links. The model is assessed by applying it on the Sioux Falls network. It is also illustrated how the Pareto-optimal solutions achieved by the multi-objective optimisation can vary depending on the emphasis placed on different classes of vehicles.

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Acknowledgements

This paper is part of an ongoing research project on optimising diversion costs during road network recovery. This research is being conducted in the Centre for Disaster Management and Public Safety (CDMPS) at the Department of Infrastructure Engineering at The University of Melbourne. The authors acknowledge the kind support from the Australian Research Council’s Linkage Project, “Planning and Managing Transport Systems for Extreme Events Through Spatial Enablement” (LP140100369), VicRoads, and The Shire of Mornington Peninsula for providing us with invaluable resources.

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Correspondence to Arash Kaviani.

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Authors’ contribution

Arash Kaviani: Ideation, Literature Review, Study Design, Problem Formulation, Mathematical Programming, Data Generation, Data Analysis, Manuscript Writing and Editing. Russell G. Thompson: Literature Review, Study Design, Problem Formulation, Data Analysis, Manuscript Writing and Editing. Abbas Rajabifard: Study Design, Manuscript Editing. Majid Sarvi: Study Design, Manuscript Editing.

Appendices

Appendix 1: Obviating path set enumeration in lower-bound computation

Avoiding path set enumeration in lower-bound computation can potentially improve the run time of the algorithm when applied on the problems with a large number of OD pairs. In doing so, Eq. 40 can be replaced with the following:

$$\begin{aligned} \begin{aligned} X&= \Big \{\upsilon _{a,k,\pi }(\varepsilon ^2)|\\&\sum _{(i,j) \in A_k}\upsilon _{(i,j),k,\pi }(\varepsilon ^2)-\sum _{(j,i) \in A_k} \upsilon _{(j,i),k,\pi }(\varepsilon ^2) \le q^{N}_{ik};\forall i \in N,\forall k \in K,\forall \pi \in \varPi ;\\&\upsilon _{(i,j),k,\pi }(\varepsilon ^2)\ge 0, \forall (i,j) \in N,\forall k \in K,\forall \pi \in \varPi ; \Big \} \end{aligned} \end{aligned}$$
(52)

where i and j are representing a link’s source and target nodes respectively so that a link is defined as \(a = (i,j)\). \(A_k\) is also the set of links that serve a vehicle class \(k \in K\) in the road network. Moreover, \(q^{N}_{ik}\) is a restructured node-based demand vector that is defined for each node i and a vehicle class \(k \in K\). That said, three types of nodes are considered: source node, sink nodes and intermediate nodes. \(q^{N}_{ik}\ge 0\) should denote the amount of supply for source nodes whereas \(q^{N}_{ik}\le 0\) denote the demand size for the sink nodes. Eventually, the same figure for the intermediate nodes should be equal to zero in \(q^{N}_{ik}\). These definitions have been adapted from a relaxed form of a multi-class SO assignment problem that has no consideration for paths. Although such a relaxed formulation could result in a higher gap between lower and upper bounds in the algorithm specially for congested networks, it is still worth mentioning as a potential alternative for faster lower-bound computation.

Appendix 2: Sioux Falls network

See Table 6 for Sioux Falls network characterestics and Table 7 for demand specification.

Table 6 Link properties of the Sioux Falls network
Table 7 OD matrices for vehicle classes A, B and C

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Kaviani, A., Thompson, R.G., Rajabifard, A. et al. A model for multi-class road network recovery scheduling of regional road networks. Transportation 47, 109–143 (2020). https://doi.org/10.1007/s11116-017-9852-5

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