I was working on some VRP solvers and realized that tractability deproved when I added Fixed Cost for each vehicle (in an attempt to reduce number of vehicles used).
1- Due to the traingle inequality, it is always shorter(or equal to) to remove an intermediate node. Adding more vehicles would mean more visits to the central depot/start. Thus, using additional vehicles to fulfill a set of nodes would always result in a longer tour. If this were true,that would imply that mTSPs will naturally try to minimize number of vehicles used? If so, it feels like mTSP would always result in the same solution as TSP which does not seem to make sense.
2- What about in the case if there are capacity limits to the vehicle? I have been trying to generate examples to disprove this but have not been able to.
Is my reasoning flawed? An example or simple logical proof would be greatly appreciated