A Method for Evaluating the Operational Reliability of a Transportation Network

In this paper, we present a method for identifying street network points (nodes) considered as strategic (for example, intersections, bridges and overpass) for operating a transportation network. These network points are defined as strategic in th e sense of, despite the problems that can occurred on them, simultaneously, it can generate a traffic jam in the streets or even interrupt the traffic flow between a specific pair of origin-dest iny in the transportation network. Thus, the identification of sets of these points allows evalu ating the vulnerability of a transportation network, and, consequently, its operational reliabi lity. IINTRODUCTION The proposed method identifies sets of nodes in the network, in such a way that, in the distribution of the traffic flows, nodes of a same set are studied as a whole for obtaining a better and more efficient circulation of the traffic flow, making the traffic operation less vulnerable. For that objective, we use the concept of Articulat ion Subset from the graph theory. The elements of these subsets are nodes of a graph, or network, that, if taken out of the graph, divide it in two parts and, consequently, interrupt the between a pa ir of nodes (each one located in each part of the graph). It is important to identify the nodes that belong to these articulation subsets, and, mainly, the smaller of these subsets, called Minimal Articu lation Subset (MAS). From a transportation network point-of-view, the smaller this subsets is, smaller will be the network operational reliability. Thus a reliability indicator can be de fined, based on the cardinality of the minimal articulation subset. In order to defining these sub sets of strategic nodes and the reliability indicator, we apply in the proposed method the conc ept of network in layers, associated to a minimum path algorithm. For a better understanding of the proposed method, we can imagine a network that has a great flow of vehicles in the rush hour between a suburba n and a central areas. If this network is dense considering the number of links, there will be many paths among them, and the paths possibly will have common nodes. If this set of common nodes ha simultaneous problems, there will be chaos in the network, because the interruption does n t allow the vehicle circulation making the transportation among these regions very difficult. For this reason, a reliability evaluation can help the decision maker when it regards in the network o perational changes, in the planning of the total flow distribution, and even the in eventual r est iction of circulation planned for the network street. This problem may be considered not only in the urba n tr nsportation but also in the regional as well, involving, in the latter, passengers or freig ht transportation in railways networks or multimodal system. 2 – METHOD DEVELOPMENT The developed method makes it possible the identifi ca on of the minimum set of strategic points, that is, the identification of the node set(s) whic h, if taken off the network, detaches it or hinders the flow between (not adjacent) two nodes, in such a way that a node defined as origin belongs to one of the parts detached and the destination node to th other part. According to the graph theory, the number of elemen ts of the minimum of these sets defines the number of disjoint paths (if they do not cross) bet ween two specific nodes (origin and destination) in a directed network or between any pairs of nodes in undirected n twork. This means that the possible paths between a pair of origin and destina tio pass obligatorily for these points. The basic principle for determination of these sets is he transformation of the initial network in a network in levels (Dinic, in Syslo, 1983), in which , in the initial level, the node source (in the directed net) or one of the nodes of any pair of ve rtic s, not adjacent (in the undirected graph). In the second level are the nodes adjacent to those. T hat is, from the second level, the layers are formed by immediately adjacent nodes to the ones of the previous layer. A network in levels is a partial network that does not present the liaisons among nodes of the same layer. The node considered as origin belongs t o the initial layer and immediate successor nodes of it, to the second layer, and the successor s of them, to the third layer, except those already that belong to the previous layers. Thus, each node of the same layer is a linking point with the next layer. The structure of a network in levels is similar to a tree formed from a node of origin (root) in level 1, and the arcs and adjacent nodes to this no de/root belonging to level 2. Regarding the developed method, it can be identifie d the nodes that belong to each one of the layers of the graph; from this point, we can define node sets that can represent articulation sets and determine, using the minimum of these sets, the s rategic points of the liaison network between a pair origin-destination. 2.2Method for identifying the sets of Strategic P oints of the Network Given any network between an origin and a destinati o defined previously, the following steps must be followed: Step 1 Set value 1 for all nodes of the network. Step 2 Using an algorithm of minimum path, to fin d the minimum path from the origin node to all nodes of the network. After that, to identify t he layer to which each one of the nodes belongs, based on the length of the path between t h origin node and each one of nodes of the network. Step 3 – As soon as identified the nodes belonging to each level, the following checks must be made for the identification of articulation subsets : iIt is verified, in each layer, if each one of the nodes that are in the same level have paths to the destination node. In this in case, two situations can occur: a) does not exist path between some of these node s and the destination; in this case, the node that does not belongs to a path is n ot part of a minimal articulation subset (MAS). b) there is a path between (among) the node (s) a nd the destination. In this case, at least one of the successors of these nodes is in th e ext layer; if not, the node does not belong to a minimal articulation subset; on the contrary, it belongs to a minimal articulation subset. ii For the node subsets that are in a superior or in the same layer to the destination, it must be observed that: a) the subgroups of minimal articulation subsets t o which these nodes belong will have a number of elements equivalents to the last l ayer before the destination node. b) thus, the sets formed by the layers that belong t the destination (C d), or after it, will be defined substituting its respective predece ssors in the set of nodes of the last layer before the destination node, except in c ase that the node in the previous layer has other successors nodes in the final layer. Step 4 To verify the different minimal articulati on subset Formed the sets in the previous step, another verif ication must be done for ratifying the cardinality of the minimal subsets, that is: a) given a set, to verify, for each one of the nod es, if the same ones are connected to two or more nodes in the next layer. If positive, to ve rify if none of the other node of this layer is connected to these same nodes or to one of these . If not, to substitute this node by its successors in the next layer, forming a new subset. b) given a set, to verify if two or more nodes hav e the same unique successor node in the next layer and they are not linked to any other nod e. To substitute, then, in this set, these nodes by the successor node. Step 5 To identify the minimum set(s) of the sets determined in the previous step. In step 2 of this procedure, we can use any algorit hm of minimum path, however the algorithms in matrix as Floyd and Dantzig’s, (Minieka, 1990), have as final result the necessary answers for steps 2 and 3, since it supplies the length of the paths among all pairs of the network. 2.3Defining a Network Reliability Index (NRI) The Network Reliability Index (NRI) is defined in function of the cardinality of the mi nimal articulation set. We consider that, if greater this set is, better are the chances of routes to the destination and, fewer the possibilities of a colla pse in the network due to the impediment of flux in some nodes of the network. For this reason, the Network Reliability Index is d efined as: NRI = 1-1/N where N is the number of elements of the minimal ar ticulation subset identified in Step 5. As more greater the value of RI, greater is the reliab ility of the network. This means that there are fewer possibilities for making unpracticable the fl ow between the region of origin and the region of destination 3 PRACTICAL EXAMPLE We consider the highway network of Figure 3 that re presents a road system establishing connection among 5 areas of a city (A1,A2,A3,A4 and A5) with the central zone (C). For evaluating the reliability of this network among th e areas and the central region, and for identifying the sets of strategic points of it, we apply the proposed algorithm, as followed: Step 1: As it exists several origins, we create a f ictitious origin node linking it to the nodes of th e areas. We set values V ij =1 for all arcs, including those connecting the fi ctitious origin, and we apply an algorithm of minimum path to determine the length of the paths from the origin to one of nodes. The results are in Table 1. Table 1 Value of the minimum pa ths for each node node A1 = 1 Node A4 = 1 Node 7 = 2 Node 10 = 3 N ode 13 = 4 Node 16= 5 A2 = 1 A5 = 1 8 = 2 11 = 3 14 = 3 17= 4 A3 = 1 6 = 2 9 = 2 12 = 4 15 = 4 18=5 e C = 5 Step 2: We identify the nodes which belong to each one of the layers using the length of the paths, obtaining the following node subgroups by le vels. c1={A1,A2,A3,A4,A5} c2={6,7,8,9 } c3={10,11,14} c4 ={12,13,15,17} c5 ={18, 16,C) Fig. 3 – Network for the practical example Step 3: We verify, in each set, if each one of