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Tournament Schedule Optimization with Excel

R

Robapottamus

New Member
Hi Everyone


I have a problem regarding a sports tournament schedule that I was hoping someone would be able to help with.


My constraints are:

-8 teams

-Each team needs to play all other teams once (and only once), i.e. 7 games each = 28 games for the total tournament.

-7 rounds (i.e. 4 games each round).

-7 venues

-Each team needs to play once (and only once) at each venue. i.e. Each venue is a different game to be played, for instance Chess at Venue 1, Draughts at Venue 2, Poker at Venue 3.

-Each venue can only be used once each round.


The difficulty I am facing is the last constraint. The best I can get to using trial and error is 2 rounds where there is a double up on one venue.


This would normally be simple if I did not have to satisfy this last constraint, but I am assuming it is still mathematically possible. However it seems a bit too complex for Solver (for my abilities anyhow!!). Would anyone have an idea of a possible solution for this?


Your help is appreciated

Thanks

Rob
 
Hi Rob ,


Thanks for a fascinating problem.


Can you let me know whether my understanding of the constraints is correct ?


1. No team can play more than 1 match in a round.


2. No venue can be used more than once in each round.


3. No game can be played more than once in each round.


4. No game can be played at the same venue twice in the entire tournament.


Are these correct , and are there any more that I've missed ?


Narayan
 
Hi Narayan


Thanks for having a look. All are correct except the 4th constraint which I would restate as "No team can play the same game (i.e. at the same venue) more than once per tournament". I want each team to play each game once, but not more than once. And each game has a fixed venue of its own. e.g. Chess always at venue 1. So each round where they play a new game, it will be at a different venue against a different competitor.


I hope that makes sense!


There is lots of tournament scheduling literature floating around but i havent come across anything with this slight twist!


cheers


Rob
 
I'm having a bit of trouble following the terms used here. I think of "game" in the sense of chess, poker, draughts. A round would be a set of the same game played against all other teams and there are eight teams competing across the set of seven games.


Just to make sure, do you want each team to play one game against all the other teams at each venue? Each team would play a different venue in each of the rounds? So you'd have 4 chess matches, then 4 poker games, followed by 4 draughts games, etc., through seven different games?
 
No sorry, I will try and explain it a different way.


What I didn't mention at the start is that there are limited resources, i.e. only one Chess Board, only one Draught board, only one Poker Set, so you can only have two teams at this venue at one time.


So round one would look something like this:


Round 1 (for example):

Chess - Team A vs Team B

Draughts - Team C vs Team D

Poker - Team E vs Team F

BackGammon - Team G vs Team H


Now we need to remember that there are 7 different activities, which will become evident as soon as we move into Round 2:


Round 2 (for example):

Chess - Team C vs Team E

500 - Team A vs Team G

Settlers of Catan - Team B vs Team D

Canasta - Team F vs Team H


To extend this out from Team A's perspective, they will meet against all other seven teams only once in this tournament, and each of these meetings needs to be at a different game. i.e. Team A will play 7 different games once and once only. I know that seems a little wierd not to play every team at every game, but i am trying to apply this to a fun sports afternoon for a social group, where the activities are actually things like Egg and Spoon race or Sack race, where it is your time that counts against all other teams, not just whether you beat the team that happens to be at that station when your team is there.


Does that make it a little clearer?
 
Hi Rob ,


Thanks for clarifying.


If you say that each game is tied to a fixed venue , then we need not consider both the game and the venue in the constraints ; considering any one of these two , will automatically take care of the other. Is this correct ?


If yes , then there are only 3 constraints :


1. No team can play more than 1 match in a round.


2. No venue can be used more than once in each round.


3. No team can play at a venue more than once in the entire tournament.


Narayan
 
So it'd look like this?


Round Game1 Game2 Game3 Game4 Game5 Game6 Game7

1______1-2____________3-4__________ 5-6__________ 7-8

2____________ 1-3__________ 5-8__________ 6-7____2-4

3______3-5______6-8__________2-7____1-4____________

4______2-6______4-7__________1-5__________________3-8

5__________________ 2-8___________3-7____4-5____1-6

6______4-8______2-5__________3-6__________ 1-7______

7_____________________4-6____1-8____2-3__________ 5-7


This is just brute forcing it.
 
Hi


Yes Narayan that is correct. As mentioned initially it is your 2nd constraint that seems the most difficult part.


Mike - yes it would look something very similar to that. Although if i text-to-columns correctly there are 13 issues with that solution.


My closest solution so far is as below, where the X marks are issues where there will have to be a double up of some sort. Teams running down the left, round number along the top, and the letter indicating which game (or venue) two teams are at for that round:


1 2 3 4 5 6 7

1 a b c d e f g

2 a c d b f g e

3 b c a f d e g

4 b g d e c f a

5 c g a b e d f

6 c X g d f e a

7 X X c e d g f

8 X b g f c d e


thanks for looking at this for me guys.


cheers

rob
 
Sorry I will repost the above soultion so the columns line up.....


0 1 2 3 4 5 6 7

1 a b c d e f g

2 a c d b f g e

3 b c a f d e g

4 b g d e c f a

5 c g a b e d f

6 c X g d f e a

7 X X c e d g f

8 X b g f c d e
 
Hi Rob,


Have you tried evaluating if it is possible at all? Like it is not possible when you try with 4 teams, 2 rounds and 2 venues (i.e. 2 games).


If we ignore the "7 rounds" constraint, then there is at least one solutions. If this is a real life problem, I think you may add another round and give some teams some rest in some rounds.


I am not negating the possibility of a solution, just presenting my view.
 
Hi Rob ,


Anup is right. I think it is impossible with 7 venues / games. You need 8.


With 8 venues / games , the following is one solution :


Round 1 : A vs. B ( Game 1 ) , CD ( Game 4 ) , EF ( Game 2 ) , GH ( Game 3 )

Round 2 : A vs. C ( Game 2 ) , BD ( Game 5 ) , EG ( Game 1 ) , FH ( Game 4 )

Round 3 : A vs. D ( Game 3 ) , BC ( Game 6 ) , EH ( Game 5 ) , FG ( Game 7 )

Round 4 : A vs. E ( Game 4 ) , BF ( Game 3 ) , CG ( Game 5 ) , DH ( Game 1 )

Round 5 : A vs. F ( Game 5 ) , BE ( Game 7 ) , CH ( Game 8 ) , DG ( Game 2 )

Round 6 : A vs. G ( Game 6 ) , BH ( Game 2 ) , CE ( Game 3 ) , DF ( Game 8 )

Round 7 : A vs. H ( Game 7 ) , BG ( Game 4 ) , CF ( Game 1 ) , DE ( Game 6 )


Narayan
 
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