## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be a binary decision variable where:
  - \( x_i = 1 \) if player \( i \) is selected for tryouts.
  - \( x_i = 0 \) if player \( i \) is not selected for tryouts.

#### Objective Function
Maximize the total skill level of the selected players:
\[
\text{Maximize } Z = \sum_{i} \text{skill_level}_i \times x_i
\]
where \( \text{skill_level}_i \) is the skill level of player \( i \).

#### Constraints
1. **Total Players Constraint**:
\[
\sum_{i} x_i \leq 20
\]
2. **Forwards Constraint**:
\[
\sum_{i \in \text{Forwards}} x_i \leq 5
\]
3. **Midfielders Constraint**:
\[
\sum_{i \in \text{Midfielders}} x_i \leq 7
\]
4. **Defenders Constraint**:
\[
\sum_{i \in \text{Defenders}} x_i \leq 6
\]
5. **Goalkeepers Constraint**:
\[
\sum_{i \in \text{Goalkeepers}} x_i \leq 2
\]

#### Data Source Verification
- **skill_level_i**: The skill level of player \( i \) is obtained from the `player_skills.skill_level` column.
- **Total Players Limit**: The constant 20 is derived from the business configuration for the total number of players allowed.
- **Forwards Limit**: The constant 5 is derived from the business configuration for the maximum number of forwards allowed.
- **Midfielders Limit**: The constant 7 is derived from the business configuration for the maximum number of midfielders allowed.
- **Defenders Limit**: The constant 6 is derived from the business configuration for the maximum number of defenders allowed.
- **Goalkeepers Limit**: The constant 2 is derived from the business configuration for the maximum number of goalkeepers allowed.

This formulation provides a complete, immediately solvable LINEAR mathematical model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy 12.0.2 Implementation for Soccer Team Selection Optimization
"""

import gurobipy as gp
from gurobipy import GRB

def optimize_soccer_selection():
    """Optimize soccer team selection based on skill levels and position constraints."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("soccer_selection")
    
    # Example data (replace with actual data from database)
    player_ids = [1, 2, 3, 4, 5]
    skill_levels = [8.5, 7.2, 9.0, 6.8, 7.9]
    positions = ['Forward', 'Midfielder', 'Forward', 'Defender', 'Goalkeeper']
    
    # Position limits
    total_players_limit = 20
    forwards_limit = 5
    midfielders_limit = 7
    defenders_limit = 6
    goalkeepers_limit = 2
    
    # CRITICAL: Validate array lengths before loops
    assert len(player_ids) == len(skill_levels) == len(positions), "Array length mismatch"
    
    # 2. VARIABLES
    # Binary decision variables for each player
    x = {i: model.addVar(vtype=GRB.BINARY, name=f"x_{player_ids[i]}") 
         for i in range(len(player_ids))}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total skill level of selected players
    model.setObjective(gp.quicksum(skill_levels[i] * x[i] for i in range(len(player_ids))), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Total Players Constraint
    model.addConstr(gp.quicksum(x[i] for i in range(len(player_ids))) <= total_players_limit, name="total_players_limit")
    
    # Forwards Constraint
    model.addConstr(gp.quicksum(x[i] for i in range(len(player_ids)) if positions[i] == 'Forward') <= forwards_limit, name="forwards_limit")
    
    # Midfielders Constraint
    model.addConstr(gp.quicksum(x[i] for i in range(len(player_ids)) if positions[i] == 'Midfielder') <= midfielders_limit, name="midfielders_limit")
    
    # Defenders Constraint
    model.addConstr(gp.quicksum(x[i] for i in range(len(player_ids)) if positions[i] == 'Defender') <= defenders_limit, name="defenders_limit")
    
    # Goalkeepers Constraint
    model.addConstr(gp.quicksum(x[i] for i in range(len(player_ids)) if positions[i] == 'Goalkeeper') <= goalkeepers_limit, name="goalkeepers_limit")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(len(player_ids)):
            if x[i].x > 0.5:
                print(f"Player {player_ids[i]} selected with skill level {skill_levels[i]}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
if __name__ == "__main__":
    optimize_soccer_selection()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 39.4
**Execution Time**: 0.19 seconds
**Reliability**: high
**Analysis**: Gurobipy achieved the optimal solution efficiently with the fastest execution time.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCPLEX implementation for soccer team selection optimization problem.
"""

from docplex.mp.model import Model

def soccer_team_selection():
    """Optimize soccer team selection to maximize total skill level."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="soccer_team_selection")
    
    # Player data
    player_ids = [1, 2, 3, 4, 5]
    skill_levels = [8.5, 7.2, 9.0, 6.8, 7.9]
    positions = ['Forward', 'Midfielder', 'Defender', 'Goalkeeper', 'Midfielder']  # Example positions
    
    # Position limits
    total_players_limit = 20
    forwards_limit = 5
    midfielders_limit = 7
    defenders_limit = 6
    goalkeepers_limit = 2
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(player_ids) == len(skill_levels) == len(positions), "Array length mismatch"
    safe_range = range(min(len(player_ids), len(skill_levels), len(positions)))  # Safe indexing
    
    # 2. VARIABLES
    # Binary decision variables for player selection
    x = {i: mdl.binary_var(name=f"x_{player_ids[i]}") for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total skill level of selected players
    objective = mdl.sum(skill_levels[i] * x[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Total Players Constraint
    total_players = mdl.sum(x[i] for i in safe_range)
    mdl.add_constraint(total_players <= total_players_limit, ctname="total_players_limit")
    
    # Position Constraints
    forwards = mdl.sum(x[i] for i in safe_range if positions[i] == 'Forward')
    mdl.add_constraint(forwards <= forwards_limit, ctname="forwards_limit")
    
    midfielders = mdl.sum(x[i] for i in safe_range if positions[i] == 'Midfielder')
    mdl.add_constraint(midfielders <= midfielders_limit, ctname="midfielders_limit")
    
    defenders = mdl.sum(x[i] for i in safe_range if positions[i] == 'Defender')
    mdl.add_constraint(defenders <= defenders_limit, ctname="defenders_limit")
    
    goalkeepers = mdl.sum(x[i] for i in safe_range if positions[i] == 'Goalkeeper')
    mdl.add_constraint(goalkeepers <= goalkeepers_limit, ctname="goalkeepers_limit")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            if solution.get_value(x[i]) > 0.5:  # Selected players
                print(f"Player {player_ids[i]} (Position: {positions[i]}, Skill: {skill_levels[i]}) selected")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

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

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 39.4
**Execution Time**: 1.45 seconds
**Reliability**: high
**Analysis**: DOCplex also achieved the optimal solution but with a longer execution time compared to Gurobipy.

## 7. Pyomo Implementation

```python
#!/usr/bin/env python3
"""
Pyomo 6.9.2 Implementation for Soccer Team Selection Optimization
"""

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

def soccer_team_selection():
    """Optimization model for selecting soccer players for tryouts"""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data for player skills and positions
    player_ids = [1, 2, 3, 4, 5]
    skill_levels = [8.5, 7.2, 9.0, 6.8, 7.9]
    positions = ['Forward', 'Midfielder', 'Defender', 'Goalkeeper', 'Midfielder']
    
    # Validate array lengths before indexing
    assert len(player_ids) == len(skill_levels) == len(positions), "Array length mismatch"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.Set(initialize=player_ids)  # Set of players
    
    # 4. PARAMETERS (data containers)
    model.skill_level = pyo.Param(model.I, initialize={i: skill_levels[idx] for idx, i in enumerate(player_ids)})
    model.position = pyo.Param(model.I, initialize={i: positions[idx] for idx, i in enumerate(player_ids)})
    
    # 5. VARIABLES
    # Binary decision variable for player selection
    model.x = pyo.Var(model.I, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    # Maximize the total skill level of selected players
    def obj_rule(model):
        return sum(model.skill_level[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Total Players Constraint
    def total_players_rule(model):
        return sum(model.x[i] for i in model.I) <= 20
    model.total_players_constraint = pyo.Constraint(rule=total_players_rule)
    
    # Forwards Constraint
    def forwards_rule(model):
        return sum(model.x[i] for i in model.I if model.position[i] == 'Forward') <= 5
    model.forwards_constraint = pyo.Constraint(rule=forwards_rule)
    
    # Midfielders Constraint
    def midfielders_rule(model):
        return sum(model.x[i] for i in model.I if model.position[i] == 'Midfielder') <= 7
    model.midfielders_constraint = pyo.Constraint(rule=midfielders_rule)
    
    # Defenders Constraint
    def defenders_rule(model):
        return sum(model.x[i] for i in model.I if model.position[i] == 'Defender') <= 6
    model.defenders_constraint = pyo.Constraint(rule=defenders_rule)
    
    # Goalkeepers Constraint
    def goalkeepers_rule(model):
        return sum(model.x[i] for i in model.I if model.position[i] == 'Goalkeeper') <= 2
    model.goalkeepers_constraint = pyo.Constraint(rule=goalkeepers_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("\nSelected Players:")
        for i in model.I:
            if pyo.value(model.x[i]) > 0.5:  # Only print selected players
                print(f"Player {i} (Position: {model.position[i]}, Skill Level: {model.skill_level[i]})")
        
    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
if __name__ == "__main__":
    soccer_team_selection()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 39.4
**Execution Time**: 1.14 seconds
**Reliability**: high
**Analysis**: Pyomo achieved the optimal solution with an execution time between Gurobipy and DOCplex.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 39.40 | 0.19s | N/A | N/A |
| Docplex | OPTIMAL | 39.40 | 1.45s | N/A | N/A |
| Pyomo | OPTIMAL | 39.40 | 1.14s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 39.4

### Final Recommendation
**Recommended Optimal Value**: 39.4
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its fastest execution time while still achieving the optimal solution.

### Business Interpretation
**Overall Strategy**: The optimal solution suggests selecting a combination of players that maximizes the total skill level while adhering to the constraints on the number of players in each position.
**Objective Value Meaning**: The optimal total skill level of 39.4 indicates the highest possible combined skill level achievable under the given constraints.
**Resource Allocation Summary**: Allocate resources to select up to 20 players, with no more than 5 forwards, 7 midfielders, 6 defenders, and 2 goalkeepers.
**Implementation Recommendations**: Proceed with the player selection as per the optimal solution, ensuring that the constraints on the number of players in each position are strictly followed.