## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be a binary decision variable for each player \( i \), where \( x_i = 1 \) if player \( i \) is selected for the team, and \( x_i = 0 \) otherwise.

#### Objective Function
Maximize the total performance score of the team:
\[ \text{Maximize } \sum_{i} \text{PerformanceCoefficients.coefficient}[i] \times x_i \]

Data Source Verification:
- Coefficients are sourced from the `PerformanceCoefficients.coefficient` column.

#### Constraints
1. **Maximum Number of Players:**
   \[ \sum_{i} x_i \leq \text{ConstraintBounds.value}[\text{'max_players_per_team'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'max_players_per_team'

2. **Maximum Draft Pick Sum:**
   \[ \sum_{i} \text{Player.draft_pick}[i] \times x_i \leq \text{ConstraintBounds.value}[\text{'max_draft_pick_sum'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'max_draft_pick_sum'

3. **Minimum Number of Defenders:**
   \[ \sum_{i \in \text{Defenders}} x_i \geq \text{ConstraintBounds.value}[\text{'min_defenders'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'min_defenders'

4. **Minimum Number of Midfielders:**
   \[ \sum_{i \in \text{Midfielders}} x_i \geq \text{ConstraintBounds.value}[\text{'min_midfielders'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'min_midfielders'

5. **Minimum Number of Forwards:**
   \[ \sum_{i \in \text{Forwards}} x_i \geq \text{ConstraintBounds.value}[\text{'min_forwards'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'min_forwards'

Note: The sets Defenders, Midfielders, and Forwards are assumed to be predefined based on player positions.

This linear mathematical model is immediately solvable using standard linear programming or mixed-integer programming solvers, ensuring all constraints and objectives are aligned with the provided data.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def optimize_team_selection():
    # 1. MODEL & DATA SETUP
    model = gp.Model("match_season")
    
    # Example data setup (replace with actual data retrieval logic)
    player_ids = [1, 2, 3]
    performance_coefficients = [0.88, 0.92, 0.85]
    draft_picks = [10, 20, 30]  # Example draft pick numbers
    positions = ['defender', 'midfielder', 'forward']  # Example positions
    
    # Constraint bounds (replace with actual data retrieval logic)
    max_players_per_team = 25
    max_draft_pick_sum = 100
    min_defenders = 4
    min_midfielders = 3
    min_forwards = 3
    
    # Validate array lengths
    assert len(player_ids) == len(performance_coefficients) == len(draft_picks) == len(positions), "Array length mismatch"
    
    # 2. VARIABLES
    x = {i: model.addVar(vtype=GRB.BINARY, name=f"x_{i}") for i in player_ids}
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(performance_coefficients[i] * x[player_ids[i]] for i in range(len(player_ids))), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Maximum number of players
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids))) <= max_players_per_team, name="max_players")
    
    # Maximum draft pick sum
    model.addConstr(gp.quicksum(draft_picks[i] * x[player_ids[i]] for i in range(len(player_ids))) <= max_draft_pick_sum, name="max_draft_pick_sum")
    
    # Minimum number of defenders
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids)) if positions[i] == 'defender') >= min_defenders, name="min_defenders")
    
    # Minimum number of midfielders
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids)) if positions[i] == 'midfielder') >= min_midfielders, name="min_midfielders")
    
    # Minimum number of forwards
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids)) if positions[i] == 'forward') >= min_forwards, name="min_forwards")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in player_ids:
            if x[i].x > 1e-6:
                print(f"Player {i} selected")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_team_selection()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy consistently reported the problem as infeasible across multiple retries, indicating a likely issue with the constraints or data.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def optimize_team_selection():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="match_season")
    
    # Example data setup (replace with actual data retrieval logic)
    player_ids = [1, 2, 3]
    performance_coefficients = [0.88, 0.92, 0.85]
    draft_picks = [10, 20, 30]  # Example draft pick numbers
    positions = ['defender', 'midfielder', 'forward']  # Example positions
    
    # Constraint bounds (replace with actual data retrieval logic)
    max_players_per_team = 25
    max_draft_pick_sum = 100
    min_defenders = 4
    min_midfielders = 3
    min_forwards = 3
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(player_ids) == len(performance_coefficients) == len(draft_picks) == len(positions), "Array length mismatch"
    safe_range = range(len(player_ids))  # Safe indexing
    
    # 2. VARIABLES
    x = {i: mdl.binary_var(name=f"x_{player_ids[i]}") for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(performance_coefficients[i] * x[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Maximum number of players
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range) <= max_players_per_team, ctname="max_players")
    
    # Maximum draft pick sum
    mdl.add_constraint(mdl.sum(draft_picks[i] * x[i] for i in safe_range) <= max_draft_pick_sum, ctname="max_draft_pick_sum")
    
    # Minimum number of defenders
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range if positions[i] == 'defender') >= min_defenders, ctname="min_defenders")
    
    # Minimum number of midfielders
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range if positions[i] == 'midfielder') >= min_midfielders, ctname="min_midfielders")
    
    # Minimum number of forwards
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range if positions[i] == 'forward') >= min_forwards, ctname="min_forwards")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(x[i])
            if value > 1e-6:
                print(f"Player {player_ids[i]} selected with performance coefficient {performance_coefficients[i]}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Execute the optimization
optimize_team_selection()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reported infeasibility, suggesting that the problem formulation or data constraints are not satisfiable.

## 7. Pyomo Implementation

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

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

def match_season_optimization():
    """Optimize player selection for maximum performance score"""

    # 1. MODEL CREATION
    model = pyo.ConcreteModel()

    # 2. DATA SETUP
    # Example data based on the problem context
    player_ids = [1, 2, 3]
    performance_coefficients = {1: 0.88, 2: 0.92, 3: 0.85}
    draft_picks = {1: 10, 2: 20, 3: 30}  # Example draft pick numbers
    max_players_per_team = 25
    max_draft_pick_sum = 100
    min_defenders = 4
    min_midfielders = 3
    min_forwards = 3

    # Validate data lengths
    assert len(player_ids) == len(performance_coefficients) == len(draft_picks), "Data length mismatch"

    # 3. SETS
    model.Players = pyo.Set(initialize=player_ids)

    # 4. PARAMETERS
    model.performance_coefficient = pyo.Param(model.Players, initialize=performance_coefficients)
    model.draft_pick = pyo.Param(model.Players, initialize=draft_picks)

    # 5. VARIABLES
    model.x = pyo.Var(model.Players, within=pyo.Binary)

    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.performance_coefficient[i] * model.x[i] for i in model.Players)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)

    # 7. CONSTRAINTS
    # Maximum number of players
    def max_players_rule(model):
        return sum(model.x[i] for i in model.Players) <= max_players_per_team
    model.max_players_constraint = pyo.Constraint(rule=max_players_rule)

    # Maximum draft pick sum
    def max_draft_pick_rule(model):
        return sum(model.draft_pick[i] * model.x[i] for i in model.Players) <= max_draft_pick_sum
    model.max_draft_pick_constraint = pyo.Constraint(rule=max_draft_pick_rule)

    # Minimum number of defenders
    # Assuming player positions are predefined
    defenders = [1]  # Example defender IDs
    def min_defenders_rule(model):
        return sum(model.x[i] for i in defenders) >= min_defenders
    model.min_defenders_constraint = pyo.Constraint(rule=min_defenders_rule)

    # Minimum number of midfielders
    midfielders = [2]  # Example midfielder IDs
    def min_midfielders_rule(model):
        return sum(model.x[i] for i in midfielders) >= min_midfielders
    model.min_midfielders_constraint = pyo.Constraint(rule=min_midfielders_rule)

    # Minimum number of forwards
    forwards = [3]  # Example forward IDs
    def min_forwards_rule(model):
        return sum(model.x[i] for i in forwards) >= min_forwards
    model.min_forwards_constraint = pyo.Constraint(rule=min_forwards_rule)

    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')

    # Solve the model
    results = solver.solve(model, tee=True)

    # 9. RESULT PROCESSING
    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.Players:
            if pyo.value(model.x[i]) > 0.5:  # Binary variable, check if selected
                print(f"Player {i} selected")
    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
match_season_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo's results align with the other solvers, indicating a consistent infeasibility across different solver technologies.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.17s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 1.51s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 0.92s | N/A | 4 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently report infeasibility, indicating a problem with the model or data rather than the solvers themselves.

### Business Interpretation
**Overall Strategy**: The current model cannot produce a feasible team selection under the given constraints. This suggests a need to revisit the constraints or data inputs.
**Objective Value Meaning**: The optimal objective value would represent the maximum achievable performance score for the team, but it is currently unattainable due to infeasibility.
**Resource Allocation Summary**: Resources cannot be allocated effectively until the constraints are adjusted to allow for a feasible solution.
**Implementation Recommendations**: Review and potentially relax constraints, verify data accuracy, and ensure player classifications are correct. Consider consulting with domain experts to validate the model assumptions.