Flow expansion on transportation networks with budget constraints

This study considers the Budgeted Flow Expansion (BFE) problem on transportation network. The problem input includes a given budget and a transportation network, i.e. a directed graph with edges’ capacities. In addition, each edge is associated with possible expansion capacity and the expansion cost. For instance, given a transportation network connecting two cities and possible options of expanding existing roads and/or constructing new roads, the objective is to efficiently utilize a given budget to maximize the flow between the cities. The {BFE} problem is NP-hard for the general case where the expansion options are all-or-nothing, i.e., expand by utilizing the entire expansion capacity or do not expand at all. Nonetheless, in this study we consider a special case in which any integral amount of expansion capacity can be utilize; for this case a polynomial algorithm is proposed. The algorithm iteratively expands the flow by one unit by one unit as long as the resultant cost is within the budget constraint. In each iteration, the maximum flow is found using the known Ford-Fulkerson algorithm. Based on the residual network, combined with the possible expansion of edges, the cheapest path for expanding the flow is selected. The method described can be used as an efficient tool for decision makers to attain the best improvements of transportation networks when a limited budget is available. The methodology and algorithms can be applied to a real-world road network including the exhibition and interpretation of the unique features used and the benefits expected.