# Complete Optimization Problem and Solution: sports_competition

## 1. Problem Context and Goals

### Context  
The sports league is focused on optimizing the allocation of players to clubs to maximize the total points scored across all clubs. This decision-making process involves assigning players to clubs in a way that respects the operational constraints of each club, including the maximum and minimum number of players allowed and the total budget available for player salaries. The league uses a binary decision variable to determine whether a player is assigned to a specific club, ensuring that each assignment is clear and definitive. The points scored by each player in a club are used as the primary metric to evaluate the success of the allocation. The league’s operational parameters are designed to align with a linear optimization approach, avoiding complex scenarios that would require nonlinear relationships, such as variable products or divisions. The business configuration includes scalar parameters like the points scored by players in clubs and the binary decision variable for player assignments, ensuring that the optimization problem remains straightforward and mathematically consistent.

### Goals  
The primary goal of the optimization is to maximize the total points scored by all clubs through the optimal assignment of players. This is achieved by considering the points each player is expected to contribute to a club and ensuring that the assignments respect the operational constraints of each club. Success is measured by the total points accumulated across all clubs, which directly reflects the effectiveness of the player allocation strategy. The optimization process focuses on linear relationships, ensuring that the objective and constraints are straightforward and aligned with the league’s operational requirements.

## 2. Constraints  

The optimization problem is subject to several key constraints that ensure the feasibility and fairness of the player allocations:  
1. **Maximum Players per Club**: Each club has a limit on the number of players it can accommodate. The total number of players assigned to a club must not exceed this limit.  
2. **Minimum Players per Club**: Each club must have a minimum number of players to ensure competitive viability. The total number of players assigned to a club must meet or exceed this requirement.  
3. **Budget Limit**: Each club has a total budget available for player salaries. The combined salaries of all players assigned to a club must not exceed this budget.  

These constraints are designed to align with the league’s operational requirements and ensure that the optimization problem remains linear and mathematically consistent.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating new tables for missing optimization requirements (player_points and player_club_assignments) and updating business configuration logic with scalar parameters and formulas.

CREATE TABLE club_constraints (
  club_id INTEGER,
  max_players INTEGER,
  min_players INTEGER,
  budget FLOAT
);

CREATE TABLE player_salaries (
  player_id INTEGER,
  salary FLOAT
);

CREATE TABLE player_points (
  player_id INTEGER,
  club_id INTEGER,
  points INTEGER
);

CREATE TABLE player_club_assignments (
  player_id INTEGER,
  club_id INTEGER,
  assignment INTEGER
);
```

### Data Dictionary  
- **club_constraints**: This table defines the operational constraints for each club, including the maximum and minimum number of players allowed and the total budget available for player salaries. These constraints are used to ensure that the player allocations are feasible and fair.  
  - **club_id**: Unique identifier for the club.  
  - **max_players**: Maximum number of players allowed per club.  
  - **min_players**: Minimum number of players required per club.  
  - **budget**: Total budget available for player salaries per club.  

- **player_salaries**: This table contains the salary information for each player, which is used to ensure that the total salaries of players assigned to a club do not exceed the club’s budget.  
  - **player_id**: Unique identifier for the player.  
  - **salary**: Salary of the player.  

- **player_points**: This table records the points scored by each player in a club, which are used as the primary metric for evaluating the success of the player allocations.  
  - **player_id**: Unique identifier for the player.  
  - **club_id**: Unique identifier for the club.  
  - **points**: Points scored by the player in the club.  

- **player_club_assignments**: This table represents the binary decision variable indicating whether a player is assigned to a club.  
  - **player_id**: Unique identifier for the player.  
  - **club_id**: Unique identifier for the club.  
  - **assignment**: Binary decision variable indicating whether the player is assigned to the club.  

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic sports league scenarios, considering player performance, club budgets, and typical team sizes. Data was generated to ensure a balanced and solvable optimization problem.

-- Realistic data for club_constraints
INSERT INTO club_constraints (club_id, max_players, min_players, budget) VALUES (1, 25, 18, 1500000);
INSERT INTO club_constraints (club_id, max_players, min_players, budget) VALUES (2, 30, 20, 2000000);
INSERT INTO club_constraints (club_id, max_players, min_players, budget) VALUES (3, 22, 15, 1000000);

-- Realistic data for player_salaries
INSERT INTO player_salaries (player_id, salary) VALUES (1, 75000);
INSERT INTO player_salaries (player_id, salary) VALUES (2, 100000);
INSERT INTO player_salaries (player_id, salary) VALUES (3, 50000);

-- Realistic data for player_points
INSERT INTO player_points (player_id, club_id, points) VALUES (1, 1, 15);
INSERT INTO player_points (player_id, club_id, points) VALUES (2, 2, 20);
INSERT INTO player_points (player_id, club_id, points) VALUES (3, 3, 10);

-- Realistic data for player_club_assignments
INSERT INTO player_club_assignments (player_id, club_id, assignment) VALUES (1, 1, 1);
INSERT INTO player_club_assignments (player_id, club_id, assignment) VALUES (2, 2, 1);
INSERT INTO player_club_assignments (player_id, club_id, assignment) VALUES (3, 3, 1);
```

## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{p,c} \) be a binary decision variable indicating whether player \( p \) is assigned to club \( c \).  
  \( x_{p,c} \in \{0, 1\} \) for all players \( p \) and clubs \( c \).

#### Objective Function
Maximize the total points scored across all clubs:  
\[
\text{Maximize } \sum_{p} \sum_{c} \text{player_points.points}_{p,c} \times x_{p,c}
\]

#### Constraints
1. **Maximum Players per Club**:  
   The total number of players assigned to each club \( c \) must not exceed the club’s maximum player limit:  
   \[
   \sum_{p} x_{p,c} \leq \text{club_constraints.max_players}_c \quad \forall c
   \]

2. **Minimum Players per Club**:  
   The total number of players assigned to each club \( c \) must meet or exceed the club’s minimum player requirement:  
   \[
   \sum_{p} x_{p,c} \geq \text{club_constraints.min_players}_c \quad \forall c
   \]

3. **Budget Limit**:  
   The total salaries of players assigned to each club \( c \) must not exceed the club’s budget:  
   \[
   \sum_{p} \text{player_salaries.salary}_p \times x_{p,c} \leq \text{club_constraints.budget}_c \quad \forall c
   \]

4. **Player Assignment**:  
   Each player \( p \) can be assigned to at most one club:  
   \[
   \sum_{c} x_{p,c} \leq 1 \quad \forall p
   \]

#### Data Source Verification
- **Objective Function**:  
  - \( \text{player_points.points}_{p,c} \) comes from the `player_points.points` column.  
- **Maximum Players per Club**:  
  - \( \text{club_constraints.max_players}_c \) comes from the `club_constraints.max_players` column.  
- **Minimum Players per Club**:  
  - \( \text{club_constraints.min_players}_c \) comes from the `club_constraints.min_players` column.  
- **Budget Limit**:  
  - \( \text{player_salaries.salary}_p \) comes from the `player_salaries.salary` column.  
  - \( \text{club_constraints.budget}_c \) comes from the `club_constraints.budget` column.  

This formulation provides a complete, immediately solvable linear mathematical model using the provided data.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 4

import gurobipy as gp
from gurobipy import GRB

def sports_competition_optimization():
    # 1. MODEL & DATA SETUP
    model = gp.Model("sports_competition")

    # Data from the database schema
    clubs = [1, 2, 3]
    players = [1, 2, 3]

    # Club constraints
    club_constraints = {
        1: {'max_players': 25, 'min_players': 18, 'budget': 1500000},
        2: {'max_players': 30, 'min_players': 20, 'budget': 2000000},
        3: {'max_players': 22, 'min_players': 15, 'budget': 1000000}
    }

    # Player salaries
    player_salaries = {
        1: 75000,
        2: 100000,
        3: 50000
    }

    # Player points
    player_points = {
        (1, 1): 15,
        (2, 2): 20,
        (3, 3): 10
    }

    # Validate array lengths before loops
    assert len(player_points) == len(players) * len(clubs), "Player points length mismatch"

    # 2. VARIABLES
    # Binary decision variable indicating whether player p is assigned to club c
    x = model.addVars(players, clubs, vtype=GRB.BINARY, name="x")

    # 3. OBJECTIVE FUNCTION
    # Maximize the total points scored across all clubs
    model.setObjective(gp.quicksum(player_points[p, c] * x[p, c] for p in players for c in clubs), GRB.MAXIMIZE)

    # 4. CONSTRAINTS

    # Maximum Players per Club
    for c in clubs:
        model.addConstr(gp.quicksum(x[p, c] for p in players) <= club_constraints[c]['max_players'], name=f"max_players_{c}")

    # Minimum Players per Club
    for c in clubs:
        model.addConstr(gp.quicksum(x[p, c] for p in players) >= club_constraints[c]['min_players'], name=f"min_players_{c}")

    # Budget Limit
    for c in clubs:
        model.addConstr(gp.quicksum(player_salaries[p] * x[p, c] for p in players) <= club_constraints[c]['budget'], name=f"budget_{c}")

    # Player Assignment
    for p in players:
        model.addConstr(gp.quicksum(x[p, c] for c in clubs) <= 1, name=f"player_assignment_{p}")

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for p in players:
            for c in clubs:
                if x[p, c].x > 1e-6:
                    print(f"Player {p} assigned to Club {c}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

    return model

# Execute the optimization
sports_competition_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmp8nu8muq2.py", line 81, in <module>
    sports_competition_optimization()
  File "/tmp/tmp8nu8muq2.py", line 36, in sports_competition_optimization
    assert len(player_points) == len(players) * len(clubs), "Player points length mismatch"
AssertionError: Player points length mismatch

**Analysis**: Gurobipy failed due to a data mismatch error. The error message indicates that the length of the player_points data does not match the expected length based on the number of players and clubs. This suggests a potential issue with data preparation or input.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 4

from docplex.mp.model import Model

def sports_competition_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="sports_competition_optimization")
    
    # Data from the database schema
    clubs = [1, 2, 3]
    players = [1, 2, 3]
    
    # Club constraints
    club_constraints = {
        1: {'max_players': 25, 'min_players': 18, 'budget': 1500000},
        2: {'max_players': 30, 'min_players': 20, 'budget': 2000000},
        3: {'max_players': 22, 'min_players': 15, 'budget': 1000000}
    }
    
    # Player salaries
    player_salaries = {
        1: 75000,
        2: 100000,
        3: 50000
    }
    
    # Player points
    player_points = {
        (1, 1): 15,
        (2, 2): 20,
        (3, 3): 10
    }
    
    # Validate array lengths
    assert len(player_points) == len(players) * len(clubs), "Player points length mismatch"
    
    # 2. VARIABLES
    # Binary decision variable indicating whether player p is assigned to club c
    x = {(p, c): mdl.binary_var(name=f"x_{p}_{c}") for p in players for c in clubs}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total points scored across all clubs
    objective = mdl.sum(player_points[(p, c)] * x[(p, c)] for p in players for c in clubs)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Maximum Players per Club
    for c in clubs:
        mdl.add_constraint(mdl.sum(x[(p, c)] for p in players) <= club_constraints[c]['max_players'], ctname=f"max_players_{c}")
    
    # Minimum Players per Club
    for c in clubs:
        mdl.add_constraint(mdl.sum(x[(p, c)] for p in players) >= club_constraints[c]['min_players'], ctname=f"min_players_{c}")
    
    # Budget Limit
    for c in clubs:
        mdl.add_constraint(mdl.sum(player_salaries[p] * x[(p, c)] for p in players) <= club_constraints[c]['budget'], ctname=f"budget_{c}")
    
    # Player Assignment
    for p in players:
        mdl.add_constraint(mdl.sum(x[(p, c)] for c in clubs) <= 1, ctname=f"player_assignment_{p}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for p in players:
            for c in clubs:
                if solution.get_value(x[(p, c)]) > 0.5:
                    print(f"Player {p} is assigned to Club {c}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
sports_competition_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpvd8p5_7u.py", line 80, in <module>
    sports_competition_optimization()
  File "/tmp/tmpvd8p5_7u.py", line 35, in sports_competition_optimization
    assert len(player_points) == len(players) * len(clubs), "Player points length mismatch"
AssertionError: Player points length mismatch

**Analysis**: DOCplex also failed with the same error as Gurobipy, indicating a consistent issue with the player_points data. This reinforces the likelihood of a data preparation problem rather than a solver-specific issue.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 4

import pyomo.environ as pyo
from pyomo.opt import SolverFactory

def sports_competition_optimization():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the provided schema and realistic values
    clubs = [1, 2, 3]
    players = [1, 2, 3]
    
    # Club constraints
    club_constraints = {
        1: {'max_players': 25, 'min_players': 18, 'budget': 1500000},
        2: {'max_players': 30, 'min_players': 20, 'budget': 2000000},
        3: {'max_players': 22, 'min_players': 15, 'budget': 1000000}
    }
    
    # Player salaries
    player_salaries = {
        1: 75000,
        2: 100000,
        3: 50000
    }
    
    # Player points
    player_points = {
        (1, 1): 15,
        (2, 2): 20,
        (3, 3): 10
    }
    
    # Validate array lengths
    assert len(player_points) == len(players) * len(clubs), "Player points length mismatch"
    
    # 3. SETS
    model.P = pyo.Set(initialize=players)  # Players
    model.C = pyo.Set(initialize=clubs)    # Clubs
    
    # 4. PARAMETERS
    model.points = pyo.Param(model.P, model.C, initialize=player_points)
    model.salary = pyo.Param(model.P, initialize=player_salaries)
    model.max_players = pyo.Param(model.C, initialize={c: club_constraints[c]['max_players'] for c in clubs})
    model.min_players = pyo.Param(model.C, initialize={c: club_constraints[c]['min_players'] for c in clubs})
    model.budget = pyo.Param(model.C, initialize={c: club_constraints[c]['budget'] for c in clubs})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.P, model.C, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.points[p, c] * model.x[p, c] for p in model.P for c in model.C)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    # Maximum Players per Club
    def max_players_rule(model, c):
        return sum(model.x[p, c] for p in model.P) <= model.max_players[c]
    model.max_players_constraint = pyo.Constraint(model.C, rule=max_players_rule)
    
    # Minimum Players per Club
    def min_players_rule(model, c):
        return sum(model.x[p, c] for p in model.P) >= model.min_players[c]
    model.min_players_constraint = pyo.Constraint(model.C, rule=min_players_rule)
    
    # Budget Limit
    def budget_rule(model, c):
        return sum(model.salary[p] * model.x[p, c] for p in model.P) <= model.budget[c]
    model.budget_constraint = pyo.Constraint(model.C, rule=budget_rule)
    
    # Player Assignment
    def player_assignment_rule(model, p):
        return sum(model.x[p, c] for c in model.C) <= 1
    model.player_assignment_constraint = pyo.Constraint(model.P, rule=player_assignment_rule)
    
    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    
    # Optional: Set solver options
    solver.options['TimeLimit'] = 300  # 5 minutes
    solver.options['MIPGap'] = 0.01    # 1% gap
    
    # Solve the model
    results = solver.solve(model, tee=True)  # tee=True shows solver output
    
    # 9. RESULT PROCESSING
    # Check solver status
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nPlayer assignments:")
        for p in model.P:
            for c in model.C:
                if pyo.value(model.x[p, c]) > 0.5:  # Only print assignments
                    print(f"Player {p} assigned to Club {c}")
        
    elif results.solver.termination_condition == pyo.TerminationCondition.infeasible:
        print("Problem is infeasible")
    elif results.solver.termination_condition == pyo.TerminationCondition.unbounded:
        print("Problem is unbounded")
    else:
        print(f"Solver terminated with condition: {results.solver.termination_condition}")
    
    return model

# Execute the optimization
if __name__ == "__main__":
    sports_competition_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmphesn55uk.py", line 113, in <module>
    sports_competition_optimization()
  File "/tmp/tmphesn55uk.py", line 37, in sports_competition_optimization
    assert len(player_points) == len(players) * len(clubs), "Player points length mismatch"
AssertionError: Player points length mismatch

**Analysis**: Pyomo encountered the same error as the other solvers, further confirming that the issue lies in the data rather than the solvers themselves. All solvers are consistent in their failure due to the data mismatch.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.18s | N/A | 4 |
| Docplex | ERROR | N/A | 1.12s | N/A | 4 |
| Pyomo | ERROR | N/A | 0.84s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Data preparation error: The player_points data length does not match the expected length based on the number of players and clubs.
- Incorrect data input: The player_points data may be incomplete or incorrectly formatted.
- Logical error in the model setup: The assertion check may be incorrectly implemented or based on flawed assumptions.
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: none
**Reasoning**: All solvers failed with the same error, indicating a fundamental issue with the data or model setup. Until the data mismatch is resolved, no solver can provide a valid solution.

### Business Interpretation
**Overall Strategy**: The optimization model cannot be solved due to a data mismatch error. This prevents any meaningful allocation of players to clubs or calculation of total points.
**Objective Value Meaning**: Not applicable due to solver errors.
**Resource Allocation Summary**: Not applicable due to solver errors.
**Implementation Recommendations**: 1. Verify and correct the player_points data to ensure it matches the expected length based on the number of players and clubs. 2. Re-run the optimization model after resolving the data issue. 3. Double-check the model setup and constraints to ensure they align with the problem requirements.