Outsourcing production might be feasible, in fact you might find it's not even the highest cost, the overhead of a retail store or per piece internet advertising cost may be as much as the actual production.
One lead, when I searched: T-shirt drop shipper, an ad for vistaprints would come up, a few months ago the price was $4.50 a shirt.
All in all this is a rather saturated highly competitive business, if you could corner a small market like being the sole provider for some band or sports team you might have an easier time than trying to find space on the internet, I recall another guy here who had created a nice little web site for his original designs and had not made a sale in more than a week.
(Yesterday some news piece showed a kid in India making Happy Face printed tshirts who earns the retail cost of one shirt in a month.)
In process nothing. Silk screen printing came from the fact that screens were made of silk now that they are made from synthetic materials the name has switched to screen printing.