Analysing groups of cells to form new groups

HI
I have been directed here by someone who tells me this is the griendliest and most helpful Excel there is so here goes

What I have is a group of 16 people who have to be arranged into 4 groups of 4 on 5 different occasions.

My task is to organise the final (5th) groupings. The previous 4 sessions groups have been pretty random, and they will not always follow the same patterns on subsequent courses.

What I want/need to do is try to ensure that everyone, after my session, has been in a group with as many of the other people as possible while at the same time minimising the number of "double ups".

I am hopeful that someone can come up with a solution, otherwise I am back to my previous method that involves a pad and pencil and a whole lot of hope

I attach the data I am trying to work with.

Thanks in advance.