"the"

'the glass and a half' is the name of the company so therefore the rules of grammar doesn't really apply to it. It isn't a sentence just a noun.

Is this anything to do with http://www.aglassandahalffullproductions.com/ ?

Yes it is the production compant of Cadbury's. I believed the poster on the name using 'the' rather than 'a'. You have proved that the poster was right in his belief of the language, but mistaken in the title of the company.

Ok am confused.

Cadbury has more than one production so how would I refer to one of it's productions?

"A" production not "the" production as there is more than one. Is the name of the company not "a glass and a half full productions"?

I'm confused (not that it really matters). The OP asked whether it should be "the glass full and a half production" or "a glass and a half full production", and I didn't understand your reply. I'm wondering now if he or she actually meant to ask whether it should be "the glass and a half full production" or "a glass and a half full production".

I think the poster was trying to find out what was correct in English. Given that it is a name it doesn't really matter. Anyway as you have shown it is 'A glass and a half productions'.