If you want an ink-cheap printer the z3100 is a good one. It uses virtually no ink maintaining the heads. The epsons all waste large amounts of ink doing just that even when you see no head clogs. When the head does clog and you do a Power Clean you can watch the huge drop in all ink levels.
The z series also uses considerably less ink in the printing process.
It seems like a pretty far stretch to suggest purchasing a $3500 24" large format printer to replace a $1300 17" desktop printer to save ink "because of cleaning".
I would image it would take a very long time to save enough ink from the head cleaning of a 3800 to justify the z3100 (which also consumes some ink to stay clear, and eventually consumes the head because too many nozzles have failed. Yes, this may overall be lower in cost, but you can't pin the cleaning costs on the Epson without also allowing for these costs on the HP).
I'm not sure you would ever consume over $2300 worth of ink (nearly 5 full sets of ink) to just clear the heads, especially in a 3800 which is far less prone to clogging than it's bigger brothers.
It is true that the larger cartridges of the z3100, much like the larger cartridges of Epson and Canon printers will lower your ink costs, but that isn't because of head cleaning ... it's just cheaper per ml.
Hey, I don't know which is technology is better and "cheaper", using more ink to keep the head clean or designing heads with spare nozzles to use when one becomes too clogged, and replacing heads when too many nozzles become clogged.
I'm not sure it is worth discussion. I know one simple thing ...you can always get someone else to print a high quality print for less than you can print one yourself. If you choose to print it yourself, it isn't usually about saving money, it is about control. I can buy a really good 8x10 print for under two dollars, and the lab that produces it is making very good margins ... it costs them about 0.15 in paper and chemistry to make that print. No, it isn't inkjet, and to some it may not be "good enough" ... but it's pretty dang good.
If you choose to do it yourself, it probably isn't about saving money. You do it yourself because you want more control (different look, different paper, etc.) you want to know exactly what it will look like, believe you can get a higher quality print, or perhaps just because you don't want to wait a couple of days to get it. Whatever the reason, if you do your own prints, it probably isn't because you are trying to save money.
To the original poster ... if it is about money and you propose using a cheaper ink in your printer, you probably need to review your objectives, and consider if any printer is even appropriate. One of the reasons you buy an Epson K3 printer, or HPz3100 printer, is the inks themselves.