=== Problem Context ===
# Complete Optimization Problem and Solution: riding_club

## 1. Problem Context and Goals

### Context  
The riding club operates with the goal of maximizing the total points earned across all clubs in a season. This is achieved by optimally assigning players to coaches based on two key factors: the votes received by each player and the rank of each coach. The assignment of players to coaches is a binary decision, meaning each player is either assigned to a specific coach or not. 

The club has established a business configuration that limits the number of players each coach can handle to a maximum of four. This ensures that coaches can provide adequate attention and support to their assigned players. The votes received by players and the ranks of coaches are used as coefficients in the optimization model to determine the total points earned. 

The optimization problem is designed to be linear, ensuring computational efficiency and scalability. This means that all relationships and constraints are expressed in a straightforward, additive manner without any complex interactions such as variable products or divisions.

### Goals  
The primary goal of the optimization is to maximize the total points earned by all clubs. This is achieved by strategically assigning players to coaches in a way that leverages the coach's rank and the player's votes. The success of the optimization is measured by the total points calculated, which is directly influenced by the combination of player votes and coach ranks for each assignment. 

The optimization process ensures that each player is assigned to exactly one coach and that no coach exceeds their maximum capacity of four players. The goal is to find the optimal assignment that maximizes the overall points while adhering to these constraints.

## 2. Constraints    

The optimization problem is subject to two main constraints:

1. **Single Assignment Constraint**: Each player must be assigned to exactly one coach. This ensures that no player is left without a coach or assigned to multiple coaches simultaneously.

2. **Coach Capacity Constraint**: Each coach can handle a maximum of four players. This ensures that coaches are not overwhelmed and can provide adequate attention to each player assigned to them.

These constraints are designed to be linear, meaning they are expressed in a straightforward, additive manner without any complex interactions. This ensures that the optimization problem remains computationally efficient and scalable.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for missing optimization requirements and updating business configuration logic to handle scalar parameters and formulas.

CREATE TABLE player (
  Player_ID INTEGER,
  Votes INTEGER
);

CREATE TABLE coach (
  Coach_ID INTEGER,
  Rank INTEGER
);

CREATE TABLE assignment (
  Player_ID INTEGER,
  Coach_ID INTEGER,
  Assignment BOOLEAN
);
```

### Data Dictionary  
- **Player Table**: Represents the players in the riding club. Each player has a unique identifier (`Player_ID`) and a number of votes (`Votes`) they have received. The votes are used as coefficients in the objective function to calculate the total points.
  
- **Coach Table**: Represents the coaches in the riding club. Each coach has a unique identifier (`Coach_ID`) and a rank (`Rank`). The rank is used as a coefficient in the objective function to calculate the total points.
  
- **Assignment Table**: Represents the assignment of players to coaches. Each record indicates whether a specific player (`Player_ID`) is assigned to a specific coach (`Coach_ID`). The assignment is a binary decision variable in the optimization model.


=== Schema ===
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for missing optimization requirements and updating business configuration logic to handle scalar parameters and formulas.

CREATE TABLE player (
  Player_ID INTEGER,
  Votes INTEGER
);

CREATE TABLE coach (
  Coach_ID INTEGER,
  Rank INTEGER
);

CREATE TABLE assignment (
  Player_ID INTEGER,
  Coach_ID INTEGER,
  Assignment BOOLEAN
);



=== LLM Response ===
```sql
-- Query Description: Retrieve all players with their votes, which are coefficients in the objective function.
-- This data is essential for calculating the total points based on player votes.
SELECT Player_ID, Votes
FROM player;

-- Query Description: Retrieve all coaches with their ranks, which are coefficients in the objective function.
-- This data is essential for calculating the total points based on coach ranks.
SELECT Coach_ID, Rank
FROM coach;

-- Query Description: Retrieve all current assignments of players to coaches.
-- This data is crucial for understanding the current state of assignments and ensuring constraints are met.
SELECT Player_ID, Coach_ID, Assignment
FROM assignment
WHERE Assignment = TRUE;

-- Query Description: Count the number of players currently assigned to each coach.
-- This data is important for enforcing the coach capacity constraint (max 4 players per coach).
SELECT Coach_ID, COUNT(Player_ID) AS Player_Count
FROM assignment
WHERE Assignment = TRUE
GROUP BY Coach_ID;

-- Query Description: Retrieve all possible player-coach pairs that are not currently assigned.
-- This data is useful for identifying potential new assignments that could improve the total points.
SELECT p.Player_ID, c.Coach_ID
FROM player p
CROSS JOIN coach c
LEFT JOIN assignment a ON p.Player_ID = a.Player_ID AND c.Coach_ID = a.Coach_ID
WHERE a.Assignment IS NULL OR a.Assignment = FALSE;

-- Query Description: Calculate the potential points for each unassigned player-coach pair.
-- This data helps in evaluating the impact of assigning a player to a coach on the total points.
SELECT p.Player_ID, c.Coach_ID, (p.Votes * c.Rank) AS Potential_Points
FROM player p
CROSS JOIN coach c
LEFT JOIN assignment a ON p.Player_ID = a.Player_ID AND c.Coach_ID = a.Coach_ID
WHERE a.Assignment IS NULL OR a.Assignment = FALSE;

-- Query Description: Retrieve the total points for all current assignments.
-- This data provides the baseline total points that the optimization aims to improve.
SELECT SUM(p.Votes * c.Rank) AS Total_Points
FROM assignment a
JOIN player p ON a.Player_ID = p.Player_ID
JOIN coach c ON a.Coach_ID = c.Coach_ID
WHERE a.Assignment = TRUE;

-- Query Description: Retrieve players who are not currently assigned to any coach.
-- This data is important for ensuring the single assignment constraint is met.
SELECT p.Player_ID
FROM player p
LEFT JOIN assignment a ON p.Player_ID = a.Player_ID
WHERE a.Assignment IS NULL OR a.Assignment = FALSE;

-- Query Description: Retrieve coaches who have not reached their maximum capacity.
-- This data is useful for identifying coaches who can still take on more players.
SELECT c.Coach_ID
FROM coach c
LEFT JOIN (
    SELECT Coach_ID, COUNT(Player_ID) AS Player_Count
    FROM assignment
    WHERE Assignment = TRUE
    GROUP BY Coach_ID
) AS assigned_counts ON c.Coach_ID = assigned_counts.Coach_ID
WHERE assigned_counts.Player_Count IS NULL OR assigned_counts.Player_Count < 4;
```

These queries are designed to retrieve the necessary data for the optimization problem, including the coefficients for the objective function, the current state of assignments, and the constraints that need to be enforced. The data retrieved will be used to make informed decisions on how to assign players to coaches to maximize the total points while adhering to the constraints.
