Effective hybrid branch-and-cut algorithm for the inventory routing problem with open vehicle routing constraints

This study considers a kind of inventory routing problem, which jointly optimizes the open vehicle routing decision and the inventory replenishment in a real-world distribution scenario. That is, the considered problem is an integrated optimization problem with two coupled subproblems (IOP_TCSP), i.e., the open vehicle routing problem (OVRP) and the inventory replenishment problem. The criterion is to minimize the total logistics and inventory costs under multiple periods. The IOP_TCSP is modelled as a mixed integer programming problem, and then a hybrid branch-and-cut algorithm combining novel Lagrangian heuristic approach and valid inequalities (HB&C_NLHAVI) is devised to deal with it. Test results on 72 instances with different scales demonstrate that the devised HB&C is more effective than the commercial solver Gurobi. Specifically, the HB&C can obviously reduce optimality gaps for many instances within the similar or less running time, and it can reduce the optimality gaps by 12–24% within only 60–70% of Gurobi’s running time for almost all large-scale instances.