First the obvious question... Has anyone considered raising rats as a food source? They eat everything (not sure how efficient the meat conversion is, but I suspect it would be really good), they breed very, very, very quickly, they are very easy to raise, and can be very tasty. Some experts say we are quickly approaching a human population where we will not be able to feed everyone even if all available land is dedicated to producing food ( 9 Billion is that magic number). I suspect that when that time comes (12 to 20 years from now), that there will be bug farms and rat farms all over the world. Some food is better than none. Bugs being 40 to 60% protein is good and rat can be tasty as any survivalist can attest.
But onto rat control...
I'm a big fan of King Snakes, when they are small they will eat young rats, when they are big they will eat adult rats. I have seen king snakes in excess of 9 feet long and very capable of eating a NY subway rat. King snakes also eat other snakes, including venomous snakes like rattle snakes and anything else. They are on my list of future projects... I would like to start breeding them and then release the young all over the farm. Granted I would build them some winter habitats, possibly even a space in the barn and greenhouse. If they know it's there, they might come back to it, however King snakes will also eat other king snakes, so once the other prey is gone, it might become a balance of nature thing or it might cause them to wander off in search of more food. Regardless, I really like them and as long as they are around, you don't have to worry about dangerous snakes as they all become king snake food. And I would only consider this with a local king snake variety (which are all over north america and other parts of the world), not an imported variety.
I'm not a fan of cats, but I have had cats when I was growing up, a few of which singlehandedly killed rats larger then they were (adult cat takes out rat that is very big, like something you would see in NY). The cat leaves the head, feet, spine and tail and eats everything else (but you can still figure out the size based on what is left). I've seen this a few times. I have also seen cats that will watch a mouse run over them and not even care, so it all depends on the cat. I'm considering calling my local animal shelter/dog pound and asking if they have any feral cats that I can adopt. I would pay them to spay/fix the cat, and then take them home and keep them in a large cage in the barn for a few days, possibly try to feed them a couple live trapped rats during that acclimation time. I would give them cat food and water as well and would prefer they be well fed before offering them the live rat as that would tell me how interested they are in eating live rats versus commercial food. After a few days of acclimation, open the cage and hope for the best. But if the cats run off, then no big deal, hopefully they will hunt the local area for a bit before getting too far. I might even keep trying that approach to see if I can find a couple good mouser/barn cats that decide to stick around the farm (as long as they don't try to mess with my birds, it's all good).
If you are currently having problems with rats then I recommend an all out, multi pronged, full on war / assault. And if you live in an area where they replenish or migrate in, then you will need to have a rat war from time to time, especially if you compost (which I hope everyone does), and/or have a garden, fruit trees, nut trees, etc. One breeding pair of rats can overrun your home in less than 90 days. I don't like poison (I have a lot of owls and hawks in my area), if I had a really bad infestation I might consider adding some for the initial battle, but not after that, and I would be very hard pressed to do so. However, if you have rats in the attic, then those I would poison, smear some nut butter on some rat poison cubes and leave on a paper plate somewhere in the attic. For the rest, get many, many, many snap traps, I like to smear a dab of nut butter on the trigger plate (sometimes a rat sized snap trap will take out a squirrel which is a serious bonus if you eat them or consider them a pest). Get some rat zapper traps, get some sticky traps, get a lot of traps, get different types as some will work better than others and some rats may avoid a certain type of trap, think big numbers and big variety (you can also vary the bait). You can even set mini snares made out of thin copper wire along fences and areas you see them sprint along. The key is you want to put all of it out at once, and kill as many as possible as fast as possible. Don't set it and forget it, stay on it and get rid of the dead rats and re-bait those traps asap (when you hear one go off, jump on it immediately, clean, re-bait, reset). If you can kill mom and dad and the babies are tiny, then they will not survive to breed more. Rats are exponential breeders, and the more you kill quickly, the better. Once you have gone a couple weeks with no kills in any of your traps, take almost all of them down, leave a couple in places that would be very hard for anything else to get to, but where you can still check once a week (those will let you know if a new batch of rats migrated in and/or they are breeding again). Once you have an infestation under control (which is easy as long as you attack aggressively), then you might consider the cat or dog or snake or just have a rat war once you notice a rat in the area. (if you see one, then there are 20 to 30 you didn't see). Save the traps you are not using in a non humid place as they rust pretty easy, but even when rusty, they can still work and kill, so don't toss them until they stop working. If you hammer them hard with a ton of traps, all at once, then you should get almost all of them within a couple days. Chickens will eat the dead ones and they appreciate the protein and bugs, or bury or compost, but choose a smart place as the dead rats may attract other predators some of which may be just as unwanted as the rats.