Figure 1

Examples of the three representations of liner shipping service route data studied in this paper. The input data is three routes, labeled \(r_{1}\), \(r_{2}\), and \(r_{3}\), visiting the ports A, B, C, D, E. The path graph is the traditional directed network representation of the routes, where an edge exists from u to v if the edge appears in at least one route. We also add parallel edges for every route in which the edge exists (or equivalently keep a set of route labels for each edge). This graph represents how ships and cargo can move through the network. The directed co-route graph is also a directed graph, but an edge exists from port u to port v if in at least one route port v appears in any succeeding ports of call after port u. The length of the shortest path between any two pairs of nodes in the co-route graph is the minimum-route distance (distances from A shown in (b)). In the undirected co-route graph, every route is made into a clique, or fully connected undirected graph. This representation was used for service route data in previous work [11], emphasizing that cargo transportation between any two ports in a same route can be realized by one single vessel. All minimum-route paths between A and D, which require two routes and do not allow any port to appear more than once, are shown in (c)