This paper proposes a new objective function and corresponding formulations for the vehicle routing problem. The new cost function defined as the product of the distance of the arc and the flow on that arc. We call a vehicle routing problem with this new objective function as the Cumulative Vehicle Routing Problem (CumVRP). Integer programming formulations with O(n2) binary variables and O(n2) constraints are developed for both collection and delivery cases. We show that the CumVRP is a generalization of the mTraveling Repairman and related problems in the literature; as an additional finding, we propose an integer programming formulation with O(n2) constraints and decision variables for the m-Traveling Repairman Problem. We discuss two additional applications of the CumVRP: the Energy-Minimizing VRP and the Average Distance-Minimizing School-Bus Routing Problem. The collection case of the proposed models for Energy Minimizing case of the CumVRP are tested and demonstrated by using CPLEX 8.1 on some problems from Turkey's 31 and 24