Minisum and minimax paths of a moving facility on a network

This paper is concerned with a problem of finding an optimal path in a network for a single service facility which moves between fixed points, and continuously examines the potential service of a set of existing facilities located on the network. The existing facilities may be discretely distributed at the nodes of the network, or may also be continuously distributed along the links of the network. Two types of objectives are examined. The first is to find a path that minimizes the sum of the weighted distances between the moving service facility and the existing facilities over all instants of time during the travel period. The second is to find a path that minimizes the sum of the farthest weighted distances between the moving service facility and the existing facilities over all instants of time during the travel period. By appropriately redefining link lengths, it is shown that the standard shortest-path algorithms can be applied to solve these problems. The methodology is illustrated via numerical examples.