I have had this problem. I am using peanut-butter baited traps, which is slow if there is more than one or two.
Other things I do:
I feed the chickens at night, when I'm about to lock them up in the rat-proof section.
I move the tractor frequently so that the rats don't establish a network of tunnels into the run.
I had one coop where I diminished the problem by trimming a tree back that was allowing them on the roof.
I don't like poison, because I worry about other animals eating the dead rats. It is effective, though.
When you're raising the food, remember that rats can climb and jump, so it needs to be set up in a way that they can't do either comfortably.