Multi-commodity location-routing: Flow intercepting formulation and branch-and-cut algorithm

- We propose a new ILP formulation for the multi-commodity location-routing problem.
- The new formulation models the location decisions by a flow intercepting approach.
- The usage of the proposed formulation for the design of a single-tier freight distribution system is discussed.
- A branch-and-cut algorithm is proposed and experienced on test networks with different topologies.

Research on the location-routing problem (LRP) is very active, producing a good number of effective exact and approximated solution approaches. It is noteworthy that most of the contributions present in the literature address the single-commodity LRP, whereas the multi-commodity case has been scarcely investigated. Yet, this issue assumes an important role in many LRP applications, particularly in the context of designing single-tier freight distribution City Logistics systems. To fill this gap, we define a new multi-commodity LRP, proposing an original integer linear programming model for it. The proposed formulation takes into account the multi-commodity feature of the problem, modeling the strategic location and the tactical routing decisions using the flow intercepting approach. We therefore name this problem the flow intercepting facility location-routing problem. It is solved by a branch-and-cut algorithm which exploits cuts derived and adapted from literature. The proposed method is successfully experienced and validated on test instances reproducing different network topologies and problem settings.