# How can I perform discrete optimisation of a variable over a data set

This question relates directly to a dataset I've generated for Fantasy Premier League, but I'm also curious how I can apply this to a more general case.

Data

I have a list of premier league players, and for each player p I have calculated a predicted score s for that player over a period of time.

Each player inherently has a type t, ranging from 1-4, and a team m ranging from 1-20. Each player also has an inherent cost c.

Constraints

I can construct a team of consisting of 15 of these players. I am constrained by the number of players I can choose from each type t and team m such that, the team must have:

• 2 players of type t=1
• 5 players of type t=2
• 5 players of type t=3
• 3 players of type t=4
• no more than 3 players from any one team m
• a total cost sum(m) <= some budget value B

Problem

How can I create a team which maximises the sum of predicted scores sum(s) such that all constraints have been met. The brute force approach of generating every allowed team is not viable.

Let $$P$$ be the set of players. Each player $$p$$ has a type $$t_p$$, expected score $$s_p$$, team $$m_p$$ and cost $$c_p$$.

Let $$x_p$$ be a binary decision variable equal to 1 if $$p$$ is chosen for the team, $$0$$ otherwise.

The constraints are:

• Must choose 15 players - note that in this case this constraint is redundant and can be omitted, as it's implied by the 4 constraints after it: $$\sum_{p \in P} x_p = 15$$
• 2 players of type $$t = 1$$: $$\sum_{p \in P: t_p = 1} x_p = 2$$
• 5 players of type $$t = 2$$: $$\sum_{p \in P: t_p = 2} x_p = 5$$
• 5 players of type $$t = 3$$: $$\sum_{p \in P: t_p = 3} x_p = 5$$
• 3 players of type $$t = 4$$: $$\sum_{p \in P: t_p = 4} x_p = 3$$
• no more than 3 players from any one team $$m$$: $$\sum_{p \in P: m_p = i} x_p \leq 3 ; \forall i = 1, \ldots, 20$$
• Total cost below budget value $$B$$: $$\sum_{p \in P} c_p x_p \leq B$$

Finally, the objective function of maximixing sum of predicted scores will be $$\sum_{p \in P} s_p x_p.$$

There are several alternatives to implement and solve this model, such as:

• An AML such as AMPL or GAMS connected to a mathematical programming solver
• Via integer programming or constraint programming modules through programming languages like Python.
• Reading the sets and parameters in a spreadsheet and solve using the tools like Excel's Solver (if model size and complexity allow for it).
• @RobPratt Thanks for pointing that out, answer edited to reflect this. – dhasson Nov 27 '20 at 21:42