# Choosing better objective function for vehicle routing problem

I have a graph $$G$$ and the following variables.

$$b_{i,j}$$ is $$(i,j)$$ edge is taken or not.

$$t_{i,j}$$ is time to travel $$(i,j)$$

$$A_{i}$$, $$D_{i}$$ are arrival and departure time at node $$i$$.

My first objective is :

$$\min \sum_{i,j} b_{i,j}\cdot t_{i,j} + \sum_{i} D_{i} - A_{i}$$

My second objective is :

$$\min A_{e}$$

where $$e$$ is the end location of the path.

My question is why the solver takes a long time to solve the second objective as compared to the first objective.

As an analogy, the $$p$$-median problem is a minisum facility location problem, while the $$p$$-center problem is a minimax problem with nearly the same constraints. I once ran a test on both problems using a classic 88-node benchmark instance. CPLEX solved the $$p$$-median problem in 0.5 seconds but took more than 1600 seconds to solve the $$p$$-center problem.