## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_i \) represent the amount of money allocated to entrepreneur \( i \), where \( i \) corresponds to the `entrepreneur_id` in the `entrepreneurs` table.  
\( x_i \geq 0 \) for all \( i \).  

#### Objective Function
Maximize the total expected return on investment:  
\[
\text{Maximize } Z = \sum_{i} (\text{return_rate}_i \times x_i)
\]  
Where \( \text{return_rate}_i \) is the expected return rate for entrepreneur \( i \).  

#### Constraints
1. **Total Budget Constraint**: The sum of all investments must not exceed the available budget of one million dollars:  
\[
\sum_{i} x_i \leq 1,000,000
\]  
2. **Individual Investment Limit**: The amount allocated to any single entrepreneur must not exceed two hundred thousand dollars:  
\[
x_i \leq 200,000 \quad \forall i
\]  

#### Data Source Verification
- \( \text{return_rate}_i \): `entrepreneurs.return_rate`  
- Budget limit: Business configuration parameter (1,000,000)  
- Individual investment limit: Business configuration parameter (200,000)  

This is a complete, immediately solvable LINEAR programming model.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy Implementation for Entrepreneur Investment Optimization
"""

import gurobipy as gp
from gurobipy import GRB

def optimize_investment():
    # 1. MODEL & DATA SETUP
    model = gp.Model("entrepreneur_investment")
    
    # Data from the entrepreneurs table
    entrepreneur_ids = [1, 2, 3]
    return_rates = [0.12, 0.18, 0.1]
    initial_investments = [150000, 200000, 100000]
    
    # Business configuration parameters
    total_budget = 1000000
    individual_investment_limit = 200000
    
    # CRITICAL: Validate array lengths before loops
    assert len(entrepreneur_ids) == len(return_rates) == len(initial_investments), "Array length mismatch"
    
    # 2. VARIABLES
    # Decision variables: amount allocated to each entrepreneur
    x = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"x_{i}", lb=0) 
         for i in entrepreneur_ids}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total expected return on investment
    model.setObjective(gp.quicksum(return_rates[i-1] * x[i] for i in entrepreneur_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Total Budget Constraint
    model.addConstr(gp.quicksum(x[i] for i in entrepreneur_ids) <= total_budget, name="total_budget")
    
    # Individual Investment Limit
    for i in entrepreneur_ids:
        model.addConstr(x[i] <= individual_investment_limit, name=f"individual_limit_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in entrepreneur_ids:
            if x[i].x > 1e-6:
                print(f"Allocation to entrepreneur {i}: {x[i].x:.2f}")
    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__":
    optimize_investment()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 80000.0
**Execution Time**: 1.02 seconds
**Reliability**: high
**Analysis**: Gurobipy successfully found the optimal solution with the highest efficiency in terms of execution time.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCplex 2.29.245 Implementation for Entrepreneur Investment Optimization
"""

from docplex.mp.model import Model

def optimize_investment():
    """Optimize investment allocation across entrepreneurs to maximize return."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="entrepreneur_investment")
    
    # Data from the entrepreneurs table
    entrepreneur_ids = [1, 2, 3]
    return_rates = [0.12, 0.18, 0.1]
    initial_investments = [150000, 200000, 100000]
    
    # Business configuration parameters
    total_budget = 1000000
    individual_investment_limit = 200000
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(entrepreneur_ids) == len(return_rates) == len(initial_investments), "Array length mismatch"
    safe_range = range(min(len(entrepreneur_ids), len(return_rates), len(initial_investments)))  # Safe indexing
    
    # 2. VARIABLES
    # Decision variables: amount allocated to each entrepreneur
    x = {i: mdl.continuous_var(name=f"x_{entrepreneur_ids[i]}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total expected return on investment
    objective = mdl.sum(return_rates[i] * x[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Total Budget Constraint
    total_investment = mdl.sum(x[i] for i in safe_range)
    mdl.add_constraint(total_investment <= total_budget, ctname="total_budget")
    
    # Individual Investment Limit
    for i in safe_range:
        mdl.add_constraint(x[i] <= individual_investment_limit, ctname=f"individual_limit_{entrepreneur_ids[i]}")
    
    # 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])
            print(f"Allocated to entrepreneur {entrepreneur_ids[i]}: ${value:.2f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

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

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 80000.0
**Execution Time**: 5.33 seconds
**Reliability**: high
**Analysis**: DOCplex also found the optimal solution but took longer compared to Gurobipy.

## 7. Pyomo Implementation

```python
#!/usr/bin/env python3
"""
Pyomo 6.9.2 Implementation for Entrepreneur Investment Optimization
"""

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

def optimize_investment():
    """Optimize investment allocation across entrepreneurs to maximize return"""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Realistic data for entrepreneurs
    entrepreneurs_data = {
        1: {'return_rate': 0.12},
        2: {'return_rate': 0.18},
        3: {'return_rate': 0.1}
    }
    
    # Business configuration parameters
    total_budget = 1000000  # $1,000,000
    max_individual_investment = 200000  # $200,000
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.Set(initialize=entrepreneurs_data.keys())  # Set of entrepreneurs
    
    # 4. PARAMETERS (data containers)
    model.return_rate = pyo.Param(model.I, initialize={i: entrepreneurs_data[i]['return_rate'] for i in model.I})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeReals, bounds=(0, max_individual_investment))
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.return_rate[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Total Budget Constraint
    def budget_rule(model):
        return sum(model.x[i] for i in model.I) <= total_budget
    model.budget_constraint = pyo.Constraint(rule=budget_rule)
    
    # Individual Investment Limit (already handled by variable bounds)
    
    # 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("\nInvestment allocations:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"Entrepreneur {i}: ${x_val:.2f}")
        
    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__":
    optimize_investment()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 80000.0
**Execution Time**: 3.10 seconds
**Reliability**: high
**Analysis**: Pyomo found 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 | 80000.00 | 1.02s | N/A | N/A |
| Docplex | OPTIMAL | 80000.00 | 5.33s | N/A | N/A |
| Pyomo | OPTIMAL | 80000.00 | 3.10s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 80000.0

### Final Recommendation
**Recommended Optimal Value**: 80000.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its faster execution time while still providing the same optimal solution as the other solvers.

### Business Interpretation
**Overall Strategy**: The optimal total expected return on investment is $80,000, which is the maximum achievable given the constraints.
**Objective Value Meaning**: The optimal objective value of $80,000 represents the maximum expected return on investment achievable with the given budget and individual investment limits.
**Resource Allocation Summary**: Resources should be allocated to entrepreneurs in a way that maximizes the total expected return, ensuring no single entrepreneur receives more than $200,000 and the total investment does not exceed $1,000,000.
**Implementation Recommendations**: Allocate funds to entrepreneurs based on their return rates, ensuring compliance with individual and total budget constraints. Use Gurobipy for future optimizations due to its efficiency.