The cheapest way is (unless you're using tokens) probably to craft 3 lumps of wood, upgrade 2 twice, and 1 once. It's the same fuel cost of you craft 5 lumps of wood, and only upgrade them once, but it takes more time. It's 10 fuel in total either way.
I have upgraded like 10 times(5 random lvl15 weapons as i didn´t know about the lump of wood -.-) and then the lump of wood 5 times, but it still doesn´t show as if a have upgraded any weapons. P.S: The first 5 weapons were 45 min before upgrading the lump of wood.