Analytical method for deriving distance distributions in continuous networks in which travelers visit exactly one facility between origin and destination

In this paper, a method is developed for evaluating the locations of facilities in a network in which users visit exactly one facility on their way from origin to destination, such as the daily commute to work. The focus is on a continuous network in which the origins and destinations of trips are distributed uniformly and independently along edges of the network, and an analytical method is proposed for deriving the distance distributions. It is assumed that every traveler selects a route that minimizes the sum of the distance from origin to facility and the distance from facility to destination. Compared to a single summary index such as average distance, distance distributions contain rich information about the overall accessibility of facilities and so are useful for analyzing actual facility configurations and evaluating several planning alternatives. The proposed method can also be used to evaluate solutions obtained from facility location models that assume demands represented as flows traveling over a network. For the case of only one facility, a method is presented that uses an extended shortest-path tree rooted at the facility node. For the case of two or more facilities, an existing method is extended to obtain the shortest travel-length distribution in a continuous network in the case in which travelers visit exactly one facility on their way from origin to destination. By applying the proposed framework to an actual road network, it is found that the shapes of distributions differ greatly depending on the location of facilities and hence they are much more useful compared to using a single index such as the average distance.