A solver puzzle (solver ignoring conditions?)

[kalil]

So, I've been trying to solve a puzzle. I've got a series of conitionals - X can be present in the group so long as neither Y or Z is present, or so long as W is present and only one of Y and Z are. I want to maximize the size of the group.

I've set this up in OO.o using a set of formulas, so that in a given row J is positive or 0 so long as the conditions are met, and F is either 1 or 0 depending on wether that member is present. I then set 'solver' to maximize the sum of column F by varying the values in column F, with the conditions that values in column F must be binary and that values in column J must be greater then or equal to 0. Solver then spits out '1' for all values in column F, ignoring that this makes all the values in J negative.

I'm not sure that there is only one 'best' solution to this problem, but there are valid solutions (the simplest one being to leave all values at 0, and the largest I've found manually giving a sum of 13). I don't know why Solver is giving me invalid solutions. I downloaded the non-linear solver extension, and those did not perform any better. You can find my document here. Thanks for any assistance.