Multiple vehicles with no capacities and cost matrix

Question

I'm trying to create a VRP using two vehicles that are of the same type and start at the same location. I want the solution to create a route for each vehicle in the problem in the most cost effective way. I'm using GraphHopper to calculate the road distances and times between each service location, and creating a transport cost matrix with that information.

My problem is that all the services are put into one route and the second vehicle is never used. In the output, I can see that the problem contains two vehicles, but the solution only shows one route with one vehicle.

I can get both vehicles to be used if I set capacities on the vehicles and sizes for the services. I understand that if I leave them unspecified, they will be ignored, but will that cause all services to use the same vehicle?

Am I missing something?

I can post code samples if need.

Thanks!

Answer 1

Without capacity constraints, it will always be optimal to use a single vehicle (if the triangle inequality is satisfied, which it is for road distances). In other words, VRP without capacity constraints = TSP.

Think about it like this: Suppose you have two routes, one ends at city A (before returning to the depot) and the other begins at city B (after leaving the depot). Now imagine deleting the leg from A to depot and the leg from depot to B, and adding a leg from A to B. By the triangle inequality, c_AB <= c_A,depot + C_depot,B, so the new solution is cheaper than the old one.