Stubborn has many names, Max, Talon, Sophie, and Cosmo

For those with LGD (livestock guard dogs) how do you keep them in with the animals?  Ours roams all night.  I have been notified that they have been spotted on a trail cam over two miles away!  Before any of you say, put up a fence.  I have a fence!  They dig under the fence!  I have patched the fence and they find a new hole!  I have put down chicken wire, but seriously do I need to do it to the entire perimeter of 15 acres?  Just when I thought I had time to work on my crafts!
These dogs are about to put me in the looney bin!
Top is Max with my daughter Olivia, the sleeping bear is Cosmo and the yellow dog is Talon.  Max would get out but he would stay in the yard.  Cosmo did not get out but once he did he would roam.  Neither will come back when you are yelling at them to, “Stop”, “Leave It”, or “mine”.  There is nothing more hard headed than a Great Pyrenees!  I am concerned for their safety as well as them being a nuisance to the neighbors!
Talon is a rehomed dog from a family that had medical issues.  He is a good dog but he will not be contained and he chases cars!  Ok, I hear it now, “How is he a good dog?”  Well he is good when he is in with the animals.  He is not food aggressive with them and does not bark at them when they come by his food bowl.  He appears to not bark as much as the Max or Cosmo.  I swear I think they can hear a gnat fart two counties away!
Then there are Percy and Sophie.  These are the girls.  Both have been spayed so that is not an issue but they also don’t like to be contained.  Sophie has an issue with the knees on her hind legs.  The ligaments are stiff and she does not walk normally, more like crabs moves around.  But she has amazing upper body strength. When she gets out she like to lay right by the road.  We have had countless people stop by to let us know our dog has been hit.  Once she was laying in red clay and it got on her coat.  We had three people stop to let us know our dog was bleeding and having trouble walking and should probably see a vet.  Nope it is just Sophie laying by the dang road!
The bottom of our fencing looks like Stanford and Sons truck unloaded along it.  We have pipes, logs, bricks, cinder blocks, concrete, rocks, you name it we have probably used it to chink up a hole!  On Sunday we again fixed the fence!  We not only worked on the holes in the bottom of the fence but we worked on the over the top places as well. 
All is well! NOT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  They were in the pastures on Monday, they were in the pastures on Monday night.  They were not in the pastures on Tuesday morning!  ARGGGGGGGGGGGGGGGGHHHHHHHHHHHHHHHHHHH!  Before heading off to work on that gray but warmish morning, I was once again out in the barn yard chinking up holes in the fence! 
I priced an invisible fence system for the dogs, for what we will need it will be close to $1000.  We have an electric fence charger but not the fence or the insulators.   What to do, What to do?  I refuse to chain them up, doesn’t matter as they can snap a chain.  Well the ones I had on a dog chain. 
After Christmas I will put up the electric wire and see where that gets me.  I will get batteries for the training collar for Talon to make him stop chasing cars.  I did read a blog from a LGD trainer that called most LGD owners lazy, stupid, and ignorant.  I don’t think I fall under that heading but I do think I totally under estimated the stubbornness of the Great Pyrenees!

Left: Talon, Right: Max with daughter Olivia, Far Right: Cosmo