cirisiamo

Hi All,
I am developing software to solve a linear programming problem (for my team Athletic) and I can not find the mathematical representation for one of the constraints.

The problem is the following:
Maximize score of team presenting:

- Scores of at least 12 different competitions
- Maximum 18 scores
- For each competition, you can submit as 2 scores
- For every athlete can submit as 2 scores

So we have a list of scores
Score 1: High Jump, Athlete 1, Points: 500 (x1)
Score 2: High Jump, Athlete 2 Points: 600 (x2)
Score 3: Long Jump, Athlete 1, Points: 750 (x3)
Score 4: Disco, athlete 3 Points: 600 (x4)
Score 5: Disco, athlete 2 Points: 500 (x5)
Score 6: High Jump, Athlete 4 Points: 700 (x6)
etc. etc. etc.

What I did was to consider each score as a variable to obtain an objective function
max 500*x1 + 600*x2 + 750*x3 + 600*x4 + . + m*xn

and then set various constraints on the variables so that they can take as value only 0 or 1

Constraint on the maximum number of competition:
x1 + x2 + x3 + x4 + x5 + ... + xn<=18

Constraint on each athlete:
x1 + x3<=2
....
x2 + x5<=2

Constraint on each competition:
x1 + x2 + x6<=2
...
x4 + x5<=2

I can not find a mathematical representation of the constraint to take at least 12 different competitions

Can anyone help me?

Thank you and have a nice day