Okay, here it is, the optimal solution for 25% flashlight carry:

Alex -> far side, Sam + flashlight -> 3/4 across (1½ minutes)
Sam + flashlight -> 1/4 across (1 minute, total = 2½ minutes)
Sam + flashlight -> near side, Francis -> 1/4 across (2 minutes, total = 4½ minutes)
Pat + flashlight -> far side, Francis -> 3/4 across (4 minutes, total = 8½ minutes)
Alex + flashlight -> 3/4 across, Francis -> far side (2 minutes, total = 10½ minutes)
Alex + flashlight -> 1/4 across (½ minute, total = 11 minutes)
Sam -> 1/4 across, lit by Alex (½ minutes, total = 11½ minutes)
Sam + Alex + flashlight -> far side (1½ minutes, total = 13 minutes)

You can see how subtle the addition of flashlight carry to the problem really is.  In almost every case, a movement involves the flashlight holder remaining 1/4 from either side.  And notice that as Francis is crossing the bridge, first a programmer goes toward the near side (while he does the first 1/4), and then a programmer comes back from the far side (while he does the last 1/4).  It would be really hard for a human to examine all the possibilities and devise this solution, I think...

(If you have no idea what this is all about, please see the bridge of the four programmers, revisiting the bridge of the programmers, and more bridgework.  The program which found the optimal solution can be found here.)