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I need to produce a formula that takes the following parameters:

  • T = time of game in minutes
  • p = number of players on field at one time
  • s = number of substitute players

Each of these is variable on a game-to-game basis but once the game has begun they remain constant.

The goal is to ensure that each player receives an equal number of minutes on the field during the game. There is no limit on the amount of changes that can be made from substitutes (s) onto the field (p) and a player that has left the field can return later as a substitute.

The simple solution to this problem is to swap the players with the least playing time onto the field every T/(p+s) minutes. However, this often results in making changes more regularly than is necessary. Ideally, this task should be achieved with the minimum number of changes to reduce stoppages in play.

There is a very good example of the result on the internet at http://soccerslava.com/fcalc/en/fcalc.php, however I need to be able to run these calculations offline, hence needing the formula.

I'm hoping that this must be easier than it appears to be!

Matt Oakley
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  • There is no simple mathematical formula. It is a dynamic programming question, and you will have to learn how to write such code. – ghosts_in_the_code Dec 02 '14 at 14:38
  • Try fixing $T$ and $p$ as constants, and then tampering with $s$. What sort of patterns do you see? Can you find the most efficient pattern for small values, at least? – shardulc Dec 02 '14 at 15:23

1 Answers1

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I know this is an old question but I have recently had the same problem. This is my solution. It might look complicated but there are only 2 simple formula to remember

The total playing time for each player is calculated as p/s * T
assuming s is the total number of players.

If s is the total of subs then the calculation would be p/(p+s) * T

eg assuming a 1 hour 7-a-side game and a team with 9 players 7/9 * 60 = 46.67 minutes per player.

The frequency of the substitutions = T/(p+s) * assuming s = subs only

60/9 = 6.67 minutes

So every 6.67 minutes you would make 2 substitutions

The order of the substitutions would be something like this, naming the players p1, p2, p3 etc

1) p1, p2, p3, p4, p5, p6, p7

2) p8, p9, p1, p2, p3, p4, p5,

3) p6, p6, p8, p9, p1, p2, p3

4) p4, p5, p6, p7, p8, p9, p1

5) p2, p3, p4, p5, p6, p7, p8

6) p9, p1, p2, p3, p4, p5, p6

7) p7, p8, p9, p1, p2, p3, p4,

8) p5, p6, p7, p8, p9, p1, p2

9) p3, p4, p5, p6, p7, p8, p9

Each player should play 7 times and be substitute twice. Unfortunately you will need a stopwatch and a good memory to keep tabs on each pair of substitutions.

The formula should work for all combinations. I hope this helps

Jon
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