Several years ago I was trying to figure out a full quadrant grid for the main trolls, and ended up with a whole bunch of complicated papers and a headache. Like a devilish combination of sudoku and those puzzles they give you at school all "the person with the red house's neighbour is allergic to cats". Hate those. It'd be much easier with some kind of tool to assist. It might also help people writing stories to quickly generate a bunch of relationships between characters.
So I made some mockups. Unfortunately I have no coding knowledge, so I cannot make it myself. And the couple of coders who were interested aren't any longer. But it's a start.
This most basic version would include only the quadrants, and assume that you want every character to fill their full grid with no allowances for quadrant blurring or polyamory. The biggest thing that I am not sure about is whether the either-or system of choosing the ships is a realistic one. The best alternative that springs to mind is to generate the ships not chosen as "preferred" randomly, and to be able to click on them to keep that ship and then do random generation again kinda like how yatzee works.