I second
@rayra suggestion of steel roof panels. It is much cheaper and can go up pretty quickly, especially if you can order them cut to size like you can at Menards.
I put steel roofing on the underside of my deck last year by first ripping 2x material into a long wedge and then attaching it to the underside of the deck joists. I ran them perpendicular to the joists and spaced them to be the distance between the ridges on the steel roofing. This 2x material gave me the mounting and slope I needed for my steel roofing panels to be attached and drain.
I then attached the roofing to the ripped 2x material using steel roof screws. The roofing was attached such that the ridges faced the deck and screwed directly into the ripped 2x material. This left the larger area of the roof panel to act as a channel to drain water and minimized steel contact with the wood. I left openings at both ends to help ensure there would be additional air movement to dry things out.
I don't recall specifics but I believe I have 3" of slope over 10' and it drains really well. Winter hasn't been bad. I do clear my deck of snow and that likely helps but the roof of the house drains onto the deck so I did have quite a bit of water freezing and melting on the underside of the deck but because of the drainage and the number of screws I put in it seems to be holding up well without any signs of ice weight pulling it down.