## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be a binary decision variable where:
  - \( x_i = 1 \) if bodybuilder \( i \) is selected for the team.
  - \( x_i = 0 \) otherwise.
  
  Here, \( i = 1, 2, 3 \) corresponds to the three bodybuilders in the database.

#### Objective Function
Maximize the total performance score:
\[
\text{Maximize } Z = 150.5x_1 + 160.0x_2 + 170.5x_3 + 200.0x_1 + 210.5x_2 + 220.0x_3
\]
Simplifying:
\[
\text{Maximize } Z = 350.5x_1 + 370.5x_2 + 390.5x_3
\]

#### Constraints
1. **Team Size Limit**:
\[
x_1 + x_2 + x_3 \leq 5
\]
2. **Minimum Average Height**:
\[
175.0x_1 + 180.0x_2 + 185.0x_3 \geq 170(x_1 + x_2 + x_3)
\]
Simplifying:
\[
5.0x_1 + 10.0x_2 + 15.0x_3 \geq 0
\]
3. **Maximum Average Weight**:
\[
90.0x_1 + 95.0x_2 + 100.0x_3 \leq 100(x_1 + x_2 + x_3)
\]
Simplifying:
\[
-10.0x_1 - 5.0x_2 + 0.0x_3 \leq 0
\]

#### Data Source Verification
- **Objective Function Coefficients**:
  - \( 150.5, 160.0, 170.5 \) from `body_builder.Snatch`.
  - \( 200.0, 210.5, 220.0 \) from `body_builder.Clean_Jerk`.
  
- **Constraint Coefficients**:
  - **Height**: \( 175.0, 180.0, 185.0 \) from `people.Height`.
  - **Weight**: \( 90.0, 95.0, 100.0 \) from `people.Weight`.
  
- **Team Size Limit**: The constant \( 5 \) is derived from the business configuration.

This formulation is a complete, immediately solvable Linear Programming (LP) model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def bodybuilder_optimization():
    """Optimization model for selecting bodybuilders to maximize performance score."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("bodybuilder_selection")
    
    # Data from the database
    snatch_scores = [150.5, 160.0, 170.5]
    clean_jerk_scores = [200.0, 210.5, 220.0]
    heights = [175.0, 180.0, 185.0]
    weights = [90.0, 95.0, 100.0]
    
    # Number of bodybuilders
    n_bodybuilders = len(snatch_scores)
    
    # CRITICAL: Validate array lengths before loops
    assert len(snatch_scores) == len(clean_jerk_scores) == len(heights) == len(weights) == n_bodybuilders, "Array length mismatch"
    
    # 2. VARIABLES
    # Binary decision variables for selecting bodybuilders
    x = {i: model.addVar(vtype=GRB.BINARY, name=f"x_{i}") for i in range(n_bodybuilders)}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total performance score
    total_score = gp.quicksum((snatch_scores[i] + clean_jerk_scores[i]) * x[i] for i in range(n_bodybuilders))
    model.setObjective(total_score, GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Team Size Limit: Total number of bodybuilders selected <= 5
    model.addConstr(gp.quicksum(x[i] for i in range(n_bodybuilders)) <= 5, name="team_size_limit")
    
    # Minimum Average Height: Average height of selected bodybuilders >= 170 cm
    model.addConstr(gp.quicksum(heights[i] * x[i] for i in range(n_bodybuilders)) >= 170 * gp.quicksum(x[i] for i in range(n_bodybuilders)), name="min_avg_height")
    
    # Maximum Average Weight: Average weight of selected bodybuilders <= 100 kg
    model.addConstr(gp.quicksum(weights[i] * x[i] for i in range(n_bodybuilders)) <= 100 * gp.quicksum(x[i] for i in range(n_bodybuilders)), name="max_avg_weight")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_bodybuilders):
            if x[i].x > 0.5:
                print(f"Bodybuilder {i} is selected.")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

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

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1111.5
**Execution Time**: 0.17 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 Bodybuilder Team Selection Problem
"""

from docplex.mp.model import Model

def bodybuilder_team_selection():
    """Optimization model to select a team of bodybuilders maximizing performance score."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="bodybuilder_team_selection")
    
    # Data from the database
    snatch_scores = [150.5, 160.0, 170.5]
    clean_jerk_scores = [200.0, 210.5, 220.0]
    heights = [175.0, 180.0, 185.0]
    weights = [90.0, 95.0, 100.0]
    
    # Business configuration
    max_team_size = 5
    min_avg_height = 170.0
    max_avg_weight = 100.0
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(snatch_scores) == len(clean_jerk_scores) == len(heights) == len(weights), "Array length mismatch"
    safe_range = range(min(len(snatch_scores), len(clean_jerk_scores), len(heights), len(weights)))  # Safe indexing
    
    # 2. VARIABLES
    # Binary decision variables for each bodybuilder
    x = {i: mdl.binary_var(name=f"x_{i}") for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total performance score
    total_score = mdl.sum((snatch_scores[i] + clean_jerk_scores[i]) * x[i] for i in safe_range)
    mdl.maximize(total_score)
    
    # 4. CONSTRAINTS
    
    # Team Size Limit
    team_size = mdl.sum(x[i] for i in safe_range)
    mdl.add_constraint(team_size <= max_team_size, ctname="team_size_limit")
    
    # Minimum Average Height
    total_height = mdl.sum(heights[i] * x[i] for i in safe_range)
    mdl.add_constraint(total_height >= min_avg_height * team_size, ctname="min_avg_height")
    
    # Maximum Average Weight
    total_weight = mdl.sum(weights[i] * x[i] for i in safe_range)
    mdl.add_constraint(total_weight <= max_avg_weight * team_size, ctname="max_avg_weight")
    
    # 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"Bodybuilder {i} selected: {value:.0f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

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

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1111.5
**Execution Time**: 1.06 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 Bodybuilder Team Selection Problem
"""

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

def bodybuilder_optimization():
    """Optimization model for selecting bodybuilders to maximize performance score."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Performance scores (Snatch + Clean & Jerk)
    performance_scores = [350.5, 370.5, 390.5]
    
    # Physical attributes
    heights = [175.0, 180.0, 185.0]
    weights = [90.0, 95.0, 100.0]
    
    # Number of bodybuilders
    n_bodybuilders = len(performance_scores)
    
    # CRITICAL: Validate array lengths before indexing
    assert len(performance_scores) == len(heights) == len(weights) == n_bodybuilders, "Array length mismatch"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.RangeSet(1, n_bodybuilders)  # 1-based indexing
    
    # 4. PARAMETERS (data containers)
    model.performance = pyo.Param(model.I, initialize={i+1: performance_scores[i] for i in range(n_bodybuilders)})
    model.height = pyo.Param(model.I, initialize={i+1: heights[i] for i in range(n_bodybuilders)})
    model.weight = pyo.Param(model.I, initialize={i+1: weights[i] for i in range(n_bodybuilders)})
    
    # 5. VARIABLES
    # Binary decision variables
    model.x = pyo.Var(model.I, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.performance[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Team Size Limit
    def team_size_rule(model):
        return sum(model.x[i] for i in model.I) <= 5
    model.team_size_constraint = pyo.Constraint(rule=team_size_rule)
    
    # Minimum Average Height
    def min_height_rule(model):
        return sum(model.height[i] * model.x[i] for i in model.I) >= 170 * sum(model.x[i] for i in model.I)
    model.min_height_constraint = pyo.Constraint(rule=min_height_rule)
    
    # Maximum Average Weight
    def max_weight_rule(model):
        return sum(model.weight[i] * model.x[i] for i in model.I) <= 100 * sum(model.x[i] for i in model.I)
    model.max_weight_constraint = pyo.Constraint(rule=max_weight_rule)
    
    # 8. SOLVING WITH GUROBI (your available solver)
    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 bodybuilders:")
        for i in model.I:
            if pyo.value(model.x[i]) > 0.5:  # Only print selected bodybuilders
                print(f"Bodybuilder {i} is 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
if __name__ == "__main__":
    bodybuilder_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1111.5
**Execution Time**: 0.78 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 | 1111.50 | 0.17s | N/A | N/A |
| Docplex | OPTIMAL | 1111.50 | 1.06s | N/A | N/A |
| Pyomo | OPTIMAL | 1111.50 | 0.78s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 1111.5

### Final Recommendation
**Recommended Optimal Value**: 1111.5
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its fastest execution time while achieving the same optimal solution as the other solvers.

### Optimal Decision Variables
- **x_1** = 1.000
  - *Business Meaning*: Bodybuilder 1 is selected for the team, contributing 350.5 to the total performance score.
- **x_2** = 1.000
  - *Business Meaning*: Bodybuilder 2 is selected for the team, contributing 370.5 to the total performance score.
- **x_3** = 1.000
  - *Business Meaning*: Bodybuilder 3 is selected for the team, contributing 390.5 to the total performance score.

### Business Interpretation
**Overall Strategy**: The optimal team includes all three bodybuilders, maximizing the total performance score while satisfying all constraints.
**Objective Value Meaning**: The total performance score of 1111.5 represents the maximum achievable performance given the constraints.
**Resource Allocation Summary**: All three bodybuilders should be selected for the team to maximize performance.
**Implementation Recommendations**: Proceed with selecting all three bodybuilders for the team. Ensure that the team size, height, and weight constraints are satisfied during implementation.