As many of you know I regularly run F3J events and handle all the matrix and
scoring tasks. In the past I have always generated the flight matrix using
a random assignment and rarely have any complaints. However, I am now
looking at generating the fairest possible matrix for the US F3J Team
Selections and have run a number of tests and often see anomalies in the
matrix where a pilot flies against another an inordinate number of times or
very few times. I ain't the sharpest pencil in the box, so I could use some
suggestions on how to improve the matrix from some of the smart math people
in the group. I am a programmer so if I understand the algorithm I can
program it.. :)
Here are the constraints:
1. There are 10 teams
2. Each team consists of 4 pilots (if there are less than 4 pilots a
BYE will fill the empty slots).
3. Each pilot will fly once in a round (so there are 4 groups in a
4. Pilots on the same team are protected and will never fly against
another team member.
5. We will schedule 24 total rounds in the matrix
Here is what I have attempted so far:
1. Pure random - for each round I process each team. On each team
every pilot has a fixed pilot number between 1 and 4 (including BYES). I
then compute a random sequence of the position numbers 1-4 and assign the
flight group based on that sequence. Repeat for each team. When all teams
have been computed for a round I go on to compute the next round.
2. Random with statistical preference - I do the above, but I
calculate a factor to use in determining the variance in the matrix and run
1000 trials and select the one with the least variance. To do this I
compute and record how many time each pilot flies against the other pilots.
For each pilot I compute the standard deviation of these numbers. I then
add all the pilots standard deviations together and compute the average
standard deviation for the entire matrix. I pick the matrix that generates
the lowest standard deviation in 1000 trials. Generally I see a minimum
standard deviation of about 1.28. Some individual matrix SDs are 1.9 or so.
3. Random with statistical preference and permutiations - for this
approach I calculate all the possible permutations of the sequence 1-4.
Then for each team I generate a random sequence of those permutations
(1-24). I then assign the group position for each pilot in a round based on
that teams random set of permutations. I then do the same as #2 and
calculate the minimum SD of all the matrixes and select the lowest.
Interestingly this approach was nearly identical in minimum SD as the #2
It just feels to me like there should be a better way to get a more even
distribution - I'm hoping one of you math heads out there can help me
improve the matrix.
If not - it is what it is and I did my best.
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