## 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, and \( x_i = 0 \) otherwise.

#### Objective Function
Minimize the total number of yellow cards among the selected players:
\[ \text{Minimize} \quad \sum_{i} yCard_i \times x_i \]

Data Source Verification:
- \( yCard_i \) is the number of yellow cards for player \( i \), sourced from `Tryout.yCard`.

#### Constraints

1. **Position Requirements:**
   For each position \( p \), the number of selected players must match the required number of players for that position.
   \[
   \sum_{i \in P_p} x_i = \text{required\_players}_p \quad \forall p
   \]
   where \( P_p \) is the set of players trying out for position \( p \).

   Data Source Verification:
   - \( \text{required\_players}_p \) is sourced from `PositionRequirements.required_players`.

2. **College Diversity:**
   The number of players selected from any single college must not exceed the maximum allowed.
   \[
   \sum_{i \in C_c} x_i \leq \text{max\_players}_c \quad \forall c
   \]
   where \( C_c \) is the set of players from college \( c \).

   Data Source Verification:
   - \( \text{max\_players}_c \) is sourced from `CollegeLimits.max_players`.

3. **Binary Decision Variables:**
   \[
   x_i \in \{0, 1\} \quad \forall i
   \]

Data Source Verification:
- Player data, including position and college, is sourced from `Tryout.pPos` and `Tryout.cName`.

This formulation provides a complete and immediately solvable linear mathematical model, ensuring that all positions are filled, college diversity is maintained, and the total number of yellow cards is minimized.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def soccer_team_selection():
    # 1. MODEL & DATA SETUP
    model = gp.Model("soccer_2")

    # Data from the problem statement
    players = [
        {"pID": 1, "pPos": "Goalkeeper", "cName": "College A", "yCard": 0},
        {"pID": 2, "pPos": "Defender", "cName": "College B", "yCard": 1},
        {"pID": 3, "pPos": "Midfielder", "cName": "College C", "yCard": 2},
        {"pID": 4, "pPos": "Forward", "cName": "College A", "yCard": 1},
        {"pID": 5, "pPos": "Defender", "cName": "College B", "yCard": 0}
    ]

    position_requirements = {
        "Goalkeeper": 1,
        "Defender": 4,
        "Midfielder": 4,
        "Forward": 2
    }

    college_limits = {
        "College A": 3,
        "College B": 3,
        "College C": 3
    }

    # 2. VARIABLES
    x = {player["pID"]: model.addVar(vtype=GRB.BINARY, name=f"x_{player['pID']}")
         for player in players}

    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(player["yCard"] * x[player["pID"]] for player in players), GRB.MINIMIZE)

    # 4. CONSTRAINTS

    # Position requirements
    for position, required_players in position_requirements.items():
        model.addConstr(
            gp.quicksum(x[player["pID"]] for player in players if player["pPos"] == position) == required_players,
            name=f"position_{position}"
        )

    # College diversity
    for college, max_players in college_limits.items():
        model.addConstr(
            gp.quicksum(x[player["pID"]] for player in players if player["cName"] == college) <= max_players,
            name=f"college_{college}"
        )

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for player in players:
            if x[player["pID"]].x > 1e-6:
                print(f"Player {player['pID']} selected")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

    return model

# Run the optimization
soccer_team_selection()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy consistently reports infeasibility, indicating a potential issue with the constraints or data.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def soccer_team_selection():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="soccer_team_selection")

    # Data from the problem statement
    players = [
        {"pID": 1, "pPos": "Goalkeeper", "cName": "College A", "yCard": 0},
        {"pID": 2, "pPos": "Defender", "cName": "College B", "yCard": 1},
        {"pID": 3, "pPos": "Midfielder", "cName": "College C", "yCard": 2},
        {"pID": 4, "pPos": "Forward", "cName": "College A", "yCard": 1},
        {"pID": 5, "pPos": "Defender", "cName": "College B", "yCard": 0}
    ]

    position_requirements = {
        "Goalkeeper": 1,
        "Defender": 4,
        "Midfielder": 4,
        "Forward": 2
    }

    college_limits = {
        "College A": 3,
        "College B": 3,
        "College C": 3
    }

    # 2. VARIABLES
    x = {player["pID"]: mdl.binary_var(name=f"x_{player['pID']}") for player in players}

    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(player["yCard"] * x[player["pID"]] for player in players)
    mdl.minimize(objective)

    # 4. CONSTRAINTS

    # Position requirements
    for position, required_players in position_requirements.items():
        mdl.add_constraint(
            mdl.sum(x[player["pID"]] for player in players if player["pPos"] == position) == required_players,
            ctname=f"position_{position}"
        )

    # College diversity
    for college, max_players in college_limits.items():
        mdl.add_constraint(
            mdl.sum(x[player["pID"]] for player in players if player["cName"] == college) <= max_players,
            ctname=f"college_{college}"
        )

    # 5. SOLVING & RESULTS
    solution = mdl.solve()

    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for player in players:
            if solution.get_value(x[player["pID"]]) > 0.5:
                print(f"Player {player['pID']} selected")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Execute the function
soccer_team_selection()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reports infeasibility, suggesting that the problem constraints cannot be satisfied with the given data.

## 7. Pyomo Implementation

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

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

def soccer_optimization():
    """Pyomo model for minimizing yellow cards in soccer team selection"""

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

    # 2. DATA SETUP
    # Player data
    players = [1, 2, 3, 4, 5]
    positions = ['Goalkeeper', 'Defender', 'Midfielder', 'Forward']
    colleges = ['College A', 'College B', 'College C']
    
    # Player attributes
    player_positions = {1: 'Goalkeeper', 2: 'Defender', 3: 'Midfielder', 4: 'Forward', 5: 'Defender'}
    player_colleges = {1: 'College A', 2: 'College B', 3: 'College C', 4: 'College A', 5: 'College B'}
    yellow_cards = {1: 0, 2: 1, 3: 2, 4: 1, 5: 0}

    # Position requirements
    position_requirements = {'Goalkeeper': 1, 'Defender': 4, 'Midfielder': 4, 'Forward': 2}

    # College limits
    college_limits = {'College A': 3, 'College B': 3, 'College C': 3}

    # 3. SETS
    model.Players = pyo.Set(initialize=players)
    model.Positions = pyo.Set(initialize=positions)
    model.Colleges = pyo.Set(initialize=colleges)

    # 4. PARAMETERS
    model.player_positions = pyo.Param(model.Players, initialize=player_positions)
    model.player_colleges = pyo.Param(model.Players, initialize=player_colleges)
    model.yellow_cards = pyo.Param(model.Players, initialize=yellow_cards)
    model.position_requirements = pyo.Param(model.Positions, initialize=position_requirements)
    model.college_limits = pyo.Param(model.Colleges, initialize=college_limits)

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

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

    # 7. CONSTRAINTS

    # Position requirements
    def position_constraint_rule(model, p):
        return sum(model.x[i] for i in model.Players if model.player_positions[i] == p) == model.position_requirements[p]
    model.position_constraints = pyo.Constraint(model.Positions, rule=position_constraint_rule)

    # College diversity
    def college_constraint_rule(model, c):
        return sum(model.x[i] for i in model.Players if model.player_colleges[i] == c) <= model.college_limits[c]
    model.college_constraints = pyo.Constraint(model.Colleges, rule=college_constraint_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

# Run the optimization
soccer_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo confirms infeasibility, aligning with the results from the other solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.27s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 2.37s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 2.91s | 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**: HIGH
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently report infeasibility, indicating a high confidence in the result. The issue likely lies in the problem formulation or data.

### Business Interpretation
**Overall Strategy**: The current constraints and data do not allow for a feasible selection of players that meets all requirements.
**Objective Value Meaning**: The objective value represents the total number of yellow cards among selected players, which cannot be minimized due to infeasibility.
**Resource Allocation Summary**: Re-evaluate the constraints and data to ensure they accurately reflect the team's needs and available resources.
**Implementation Recommendations**: Review and adjust the position requirements and college diversity constraints. Verify the accuracy of the input data.