Using branch‐and‐price approach to solve the directed network design problem with relays

We present node‐arc and arc‐path formulations, and develop a branch‐and‐price approach for the directed network design problem with relays (DNDR). The DNDR problem can be used to model many network design problems in transportation, service, and telecommunication system, where relay points are necessary. The DNDR problem consists of introducing a subset of arcs and locating relays on a subset of nodes such that in the resulting network, the total cost (arc cost plus relay cost) is minimized, and there exists a directed path linking the origin and destination of each commodity, in which the distances between the origin and the first relay, any two consecutive relays, and the last relay and the destination do not exceed a predefined distance limit. With the node‐arc formulation, we can directly solve small DNDR instances using mixed integer programming solver. With the arc‐path formulation, we design a branch‐and‐price approach, which is a variant of branch‐and‐bound with bounds provided by solving linear programs using column generation at each node of the branch‐and‐bound tree. We design two methods to efficiently price out columns and present computational results on a set of 290 generated instances. Results demonstrate that our proposed branch‐and‐price approach is a computationally efficient procedure for solving the DNDR problem.