I somewhat know how to compare two MILP formulations of a problem that both use the same set of decision variables (as in the classical MTZ vs DFJ formulations of the TSP). I was wondering how two formulations of a problem that use different sets of decision variables are compared. Can we just compare the LP-relaxation bounds?
For example, a route-based formulation for a vehicle routing problem (using an exponential number of variables) is usually considered to provide a better LP-relaxation bound. However, such a formulation employs a completely different set of decision variables. What is the right way to show that such a formulation is better? Is there a standard definition?