## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{i,j} \): Binary decision variable indicating whether player \( i \) is assigned to team \( j \). \( x_{i,j} = 1 \) if player \( i \) is assigned to team \( j \), otherwise \( x_{i,j} = 0 \).

#### Objective Function
Maximize the total performance score of all teams:
\[
\text{Maximize } \sum_{i} \sum_{j} \text{PlayerPerformance.performance\_coefficient}[i] \times x_{i,j}
\]

Data Source Verification:
- Coefficients for the objective function are sourced from `PlayerPerformance.performance_coefficient`.

#### Constraints

1. **Team Size Constraints**: Each team cannot exceed its maximum number of players.
   \[
   \sum_{i} x_{i,j} \leq \text{TeamConstraints.max\_team\_size}[j] \quad \forall j
   \]

   Data Source Verification:
   - Maximum team size values are sourced from `TeamConstraints.max_team_size`.

2. **Age Limit Constraints**: The total age of players assigned to each team must not exceed the specified limit.
   \[
   \sum_{i} \text{PlayerPerformance.age}[i] \times x_{i,j} \leq \text{TeamConstraints.max\_age\_limit}[j] \quad \forall j
   \]

   Data Source Verification:
   - Player ages are sourced from `PlayerPerformance.age`.
   - Maximum age limit values are sourced from `TeamConstraints.max_age_limit`.

3. **Player Assignment Constraints**: Each player can be assigned to at most one team.
   \[
   \sum_{j} x_{i,j} \leq 1 \quad \forall i
   \]

   Data Source Verification:
   - This constraint ensures that each player is assigned to at most one team, aligning with the binary nature of the decision variable.

This linear mathematical model is immediately solvable using linear programming or mixed-integer programming techniques, ensuring that the solution adheres to the constraints and maximizes the total performance score of the school teams.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_school_player_allocation():
    """Optimize player allocation to school teams to maximize performance score."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("school_player")
    
    # Data from the database schema
    player_ids = [1, 2, 3]
    team_ids = [1, 2, 3]
    
    performance_coefficients = {1: 1.8, 2: 2.3, 3: 2.7}
    player_ages = {1: 16, 2: 17, 3: 18}
    
    max_team_sizes = {1: 15, 2: 18, 3: 20}
    max_age_limits = {1: 100, 2: 110, 3: 120}
    
    # CRITICAL: Validate array lengths before loops
    assert len(player_ids) == len(performance_coefficients) == len(player_ages), "Player data length mismatch"
    assert len(team_ids) == len(max_team_sizes) == len(max_age_limits), "Team data length mismatch"
    
    # 2. VARIABLES
    # Decision variables: x[i, j] = 1 if player i is assigned to team j, else 0
    x = model.addVars(player_ids, team_ids, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total performance score of all teams
    model.setObjective(gp.quicksum(performance_coefficients[i] * x[i, j] for i in player_ids for j in team_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Team Size Constraints: Each team cannot exceed its maximum number of players
    model.addConstrs((gp.quicksum(x[i, j] for i in player_ids) <= max_team_sizes[j] for j in team_ids), name="team_size")
    
    # Age Limit Constraints: The total age of players assigned to each team must not exceed the specified limit
    model.addConstrs((gp.quicksum(player_ages[i] * x[i, j] for i in player_ids) <= max_age_limits[j] for j in team_ids), name="age_limit")
    
    # Player Assignment Constraints: Each player can be assigned to at most one team
    model.addConstrs((gp.quicksum(x[i, j] for j in team_ids) <= 1 for i in player_ids), name="player_assignment")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in player_ids:
            for j in team_ids:
                if x[i, j].x > 1e-6:
                    print(f"Player {i} assigned to Team {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_school_player_allocation()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 6.8
**Execution Time**: 0.19 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the shortest execution time, indicating efficient performance.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_school_teams():
    """Optimize player assignments to school teams to maximize performance score."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="school_player")
    
    # Data from the database schema
    player_ids = [1, 2, 3]
    team_ids = [1, 2, 3]
    
    performance_coefficients = [1.8, 2.3, 2.7]
    player_ages = [16, 17, 18]
    
    max_team_sizes = [15, 18, 20]
    max_age_limits = [100, 110, 120]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(player_ids) == len(performance_coefficients) == len(player_ages), "Player data length mismatch"
    assert len(team_ids) == len(max_team_sizes) == len(max_age_limits), "Team data length mismatch"
    
    safe_player_range = range(len(player_ids))
    safe_team_range = range(len(team_ids))
    
    # 2. VARIABLES
    # Binary decision variables for player assignments
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in safe_player_range for j in safe_team_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total performance score
    objective = mdl.sum(performance_coefficients[i] * x[i, j] for i in safe_player_range for j in safe_team_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Team Size Constraints
    for j in safe_team_range:
        mdl.add_constraint(mdl.sum(x[i, j] for i in safe_player_range) <= max_team_sizes[j], ctname=f"team_size_{j}")
    
    # Age Limit Constraints
    for j in safe_team_range:
        mdl.add_constraint(mdl.sum(player_ages[i] * x[i, j] for i in safe_player_range) <= max_age_limits[j], ctname=f"age_limit_{j}")
    
    # Player Assignment Constraints
    for i in safe_player_range:
        mdl.add_constraint(mdl.sum(x[i, j] for j in safe_team_range) <= 1, ctname=f"player_assignment_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_player_range:
            for j in safe_team_range:
                value = solution.get_value(x[i, j])
                if value > 1e-6:
                    print(f"Player {player_ids[i]} assigned to Team {team_ids[j]}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
optimize_school_teams()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 6.8
**Execution Time**: 1.70 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, albeit with a longer execution time compared to Gurobipy.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def school_player_optimization():
    """Optimize player assignments to school teams to maximize performance score."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Data from the database schema
    player_ids = [1, 2, 3]
    team_ids = [1, 2, 3]
    performance_coefficients = {1: 1.8, 2: 2.3, 3: 2.7}
    ages = {1: 16, 2: 17, 3: 18}
    max_team_sizes = {1: 15, 2: 18, 3: 20}
    max_age_limits = {1: 100, 2: 110, 3: 120}
    
    # CRITICAL: Validate array lengths before indexing
    assert len(player_ids) == len(performance_coefficients) == len(ages), "Array length mismatch"
    assert len(team_ids) == len(max_team_sizes) == len(max_age_limits), "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.Set(initialize=player_ids)
    model.J = pyo.Set(initialize=team_ids)
    
    # 4. PARAMETERS
    model.performance_coefficient = pyo.Param(model.I, initialize=performance_coefficients)
    model.age = pyo.Param(model.I, initialize=ages)
    model.max_team_size = pyo.Param(model.J, initialize=max_team_sizes)
    model.max_age_limit = pyo.Param(model.J, initialize=max_age_limits)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, model.J, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.performance_coefficient[i] * model.x[i, j] for i in model.I for j in model.J)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Team Size Constraints
    def team_size_rule(model, j):
        return sum(model.x[i, j] for i in model.I) <= model.max_team_size[j]
    model.team_size_constraint = pyo.Constraint(model.J, rule=team_size_rule)
    
    # Age Limit Constraints
    def age_limit_rule(model, j):
        return sum(model.age[i] * model.x[i, j] for i in model.I) <= model.max_age_limit[j]
    model.age_limit_constraint = pyo.Constraint(model.J, rule=age_limit_rule)
    
    # Player Assignment Constraints
    def player_assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.J) <= 1
    model.player_assignment_constraint = pyo.Constraint(model.I, 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 i in model.I:
            for j in model.J:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0.5:  # Only print assignments
                    print(f"Player {i} assigned to Team {j}")
        
    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

# Run the optimization
school_player_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 6.8
**Execution Time**: 1.01 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a moderate execution time, demonstrating reliable performance.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 6.80 | 0.19s | N/A | N/A |
| Docplex | OPTIMAL | 6.80 | 1.70s | N/A | N/A |
| Pyomo | OPTIMAL | 6.80 | 1.01s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 6.8

### Final Recommendation
**Recommended Optimal Value**: 6.8
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its efficient execution time and high reliability, making it suitable for quick decision-making.

### Business Interpretation
**Overall Strategy**: The optimal value of 6.8 indicates the maximum achievable performance score for the teams given the constraints.
**Objective Value Meaning**: The optimal objective value of 6.8 represents the highest possible performance score for the teams, ensuring the best allocation of players.
**Resource Allocation Summary**: Players should be allocated to teams in a manner that maximizes the total performance score while adhering to team size and age constraints.
**Implementation Recommendations**: Implement the solution by assigning players to teams as per the optimal decision variables, ensuring compliance with all constraints.