Maximizing fixed circles in a fixed rectangle

Not sure if this fits here or elsewhere, but I'll try this...

I'm putting together a project where we're buying some rather expensive plastic sheets and making circles out of it.

We need to make roughly 18" diameter circles (actually, because of the router width and some errors, it works out to 18 1/4" or 18.125" per circle).

We've got a few choices of width/length sheets. I'm not sure how many we can get per sheet, obviously the more the merrier.

I know there's packing theories out there, etc, but haven't been able to adapt any of them for our purpose. I tried hexagonal packing etc, but have been failing miserably.

I was told the most we can squeeze on a 48" x 120" sheet is 14 circles, I'm pretty sure that's correct. If we only needed 18" we may be able to fit 17 circles, but, that little bit makes a big difference.

The other two sizes we can choose from are 60" x 120" and 96" x 240" (the bigger the sheet, the more expensive obviously).

Can anyone point me in the direction of an algorithm or something to calculate the number of circles we can fit? I've searched for a program online and found nothing...