The directed profitable rural postman problem with incompatibility constraints

In this paper, we study a variant of the directed rural postman problem (RPP) where profits are associated with arcs to be served, and incompatibility constraints may exist between nodes and profitable arcs leaving them. If convenient, some of the incompatibilities can be removed provided that penalties are paid. The problem looks for a tour starting and ending at the depot that maximizes the difference between collected profits and total cost as sum of traveling costs and paid penalties, while satisfying remaining incompatibilities. The problem finds application in the domain of road transportation service, and in particular in the context of horizontal collaboration among carriers and shippers. We call this problem the directed profitable rural postman problem with incompatibility constraints. We propose two problem formulations and introduce a matheuristic procedure exploiting the presence of a variant of the generalized independent set problem (GISP) and of the directed rural postman problem (DRPP) as subproblems. Computational results show how the matheuristic is effective outperforming in many cases the result obtained in one hour computing time by a straightforward branch-and-cut approach implemented with IBM CPLEX 12.6.2 on instances with up to 500 nodes, 1535 arcs, 1132 profitable arcs, and 10,743 incompatibilities.