Introduction
I: DC J: Depot
8.1 Introduction and Problem Definition
16 5
8
a rc s elecTion anD r ouTing for r esToraTion
of n eT work c onnecTiviT y af Ter a D isasTer
AY ŞE N U R A S A LY A N D F. SI BE L S A L M A N
16 6 aYŞe nur asalY and f. sibel salman
and major gateways in the transportation system, such as airports and ports, should be accessible.
One of the outcomes of a high-impact disaster is the disruption of transportation systems, which cripples postdisaster emergency and relief activities. In the 2013 Bohol earthquake and Typhoon Haiyan, rescue workers struggled to reach ravaged towns and villages in the central Philippines (Mogato and Ng 2013). Relief operations were hampered because roads, airports, and bridges had been destroyed or were covered in wreckage. After the 2011 devastating earth- quake and the resulting tsunami in northeast Japan, almost 4000 road segments, 78 bridges, and 29 railway locations were reported to be damaged (BBC News and National Police Agency of Japan 2012). Accumulated debris in the downtown of Kamaishi City, Iwate Prefecture, and a damaged arterial road (National Highway 45) vir- tually isolated the community from rescue efforts. About 76% of the highways in the area were closed due to damage.
This study focuses on logistics planning to ensure connectivity of road networks in the immediate disaster response stage. As experi- enced in many cases worldwide, roads can be severely damaged in a natural disaster. For instance, in a high-magnitude earthquake, (1) some parts of the roads may be affected as follows: blocked by building, lamppost, tree, and car debris, and deformed, distorted, and ruptured due to ground failure and liquefaction; and (2) vulner- able structures such as bridges and viaducts may collapse. Damage to other infrastructure networks, such as natural gas or drainage sys- tems, may also cause dysfunctionality in the roads. As a result, traffic is blocked at various links of the road network, and some nodes may become unreachable.
Some of the damaged roads can be cleared or restored in a short time, whereas it may take many hours, days, or months to eliminate other types of damage. For example, after the 2011 earthquake and tsunami in Japan, Japanese road administrators immediately launched an emergency road restoration operation with the cooperation of local construction companies. The efforts concentrated on 16 routes, to establish first the vertical artery, followed by east–west routes. The operation was completed after 9 days. In general, the emergency res- toration goal is to ensure connectivity of the road network and pro- vide accessibility between people in different areas as fast as possible.
16 7
arC seleC tion and routing
For this purpose, first, the road conditions are assessed, and time to clear/open the roads is estimated. The tasks that take too long are postponed to later stages. Then, among the remaining tasks, a subset that enables connectivity should be selected, and a fleet of machinery or vehicles routed to conduct them in the shortest time. Since some people will want to evacuate the disaster area, while others will be coming in for help, strong connectivity of the network is required.
Recently, several studies focused on upgrading a road network or improving accessibility after a disaster situation. These studies are reviewed in Section 8.2. To the best of our knowledge, the restora- tion of the roads after a disaster by routing a fleet of vehicles in order to ensure strong connectivity of a network has not been addressed in the literature. In this study, we define a new network optimization problem to address this topic. Since the problem combines arc routing and network design elements, it is called Arc Routing for Connectivity Problem (ARCP).
Before we define ARCP formally, some definitions may be useful.
A connected graph contains a directed path from a node i to another node j or a directed path from j to i for every pair of nodes i and j.
Otherwise, the graph is disconnected. A graph is strongly connected if it contains a directed path from i to j and a directed path from j to i for every pair of nodes i and j. Otherwise, the graph is disconnected in the strong sense. We define ARCP on a directed, strongly connected, and simple graph G = (V, A) with nonnegative arc costs. After a natu- ral disaster, speed of transportation is highly dependent on road and extraordinary traffic conditions, as also stated in Nolz et al. (2011).
Therefore, costs are calculated in terms of estimated time instead of distance. Traversal time on an unblocked (i.e., not blocked initially) or a blocked arc after it has been unblocked (i.e., opened) is equal to cij, where (i, j) represents the arc. We refer to the fleet of emergency response machineries (including possibly lighting, drainage pump, and satellite communication vehicles) that move together as a single vehicle, which is located initially at a node d, for example, its depot or an emergency response facility. Moreover, a subset B of arcs, which are determined to be blocked according to postdisaster information on road conditions, are given such that GB = (V, A\B) is disconnected in the strong sense. The set B consists of all blocked arcs, and the set R, a subset of B, represents the arcs that will be traversed and cleared
16 8 aYŞe nur asalY and f. sibel salman
by the vehicle in order to restore strong connectivity of the graph.
The set R is not known in advance, and its selection is a decision in the problem. The solution identifies R and constructs a walk for the vehicle that starts at its depot. We want the walk in the solution to cover arcs in R. In other words, the arcs in the set A\B∪R should induce a connected graph, GR, on the set V.
We assume that there are |Q| disconnected components in GB, where Q is the set of disconnected components, in the strong sense.
Each component in Q consists of strongly connected nodes. We parti- tion Q into three classes: (1) components within which the nodes are strongly connected and which require at least one incoming and one outgoing arc in order to be strongly connected to the remaining graph, (2) components that require at least one outgoing but no incoming arc to be unblocked in order to be strongly connected to the remaining graph, and (3) components that require at least one incoming but no outgoing arc to be unblocked in order to be strongly connected to the remaining network. Moreover, unblocking, that is, passing through a blocked arc for the first time, results in work time in addition to its traversal time. More formally, we define the additional time of unblocking arc (i, j) as bij where bij ≥ 0. In a walk, cij time units elapse each time an arc is traversed, and in addition, bij units elapse once for each blocked arc that is unblocked during the walk. In other words, a blocked arc is unblocked by a vehicle in its first traversal of that arc.
We assume that traffic cannot flow in both directions after a blocked road is unblocked in one direction by a vehicle. Considering that allowing traffic in the reverse direction would slow down response activities, this is a reasonable assumption.
The objective is to minimize the time at which the graph becomes strongly connected. That is, by definition, there must be a path from each vertex to every other vertex in the network. In order to connect all the disconnected components, at least two arcs in opposite direc- tions within the cutset of a component must be unblocked. Otherwise, the network cannot be strongly connected. Since we are interested in minimizing the time when the graph becomes connected, return of the vehicle to its depot is not considered. Therefore, the walk is open. We can define the objective function as min c(W) + b(W), where W is walk of the vehicle; c(W) is traversal time, and c(W) is calculated by summing up the traversal time of arcs (in terms of cij) that are
16 9
arC seleC tion and routing
traversed by the vehicle; b(W) is the total additional time (in terms of bij) of unblocking for the vehicle.
The aim of this study is to develop a solution method to the con- nectivity problem that generates a solution in a short time. We formu- late ARCP and observe for which cases it can be solved in reasonably short time by numerical tests. Our tests are performed on instances generated considering Istanbul road network at a macro level and its vulnerability to a potential earthquake. Our analysis of the solutions over a set of scenarios provides some insights for preparedness.
The organization of this study is as follows: Section 8.2 reviews relevant studies in the literature. Section 8.3 gives computational complexity proof of the ARCP. In Section 8.4, a mixed integer pro- gramming (MIP) model for ARCP is given. Section 8.5 presents the data related to Istanbul highway network, and Section 8.6 gives the computational results. Finally, in Section 8.7, we conclude the study with a summary, some comments, and directions for future research.