## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be the amount settled for claim \( i \), where \( i \) corresponds to the `Claim_ID` in the `Claims` table.  
  \( x_i \) is a continuous variable representing the settlement amount for each claim.

#### Objective Function
Minimize the total settlement cost:  
\[
\text{Minimize } \sum_{i} x_i
\]  
Here, the coefficient for each \( x_i \) is 1, as the objective is to minimize the sum of all settlement amounts.

#### Constraints
1. **Total Budget Constraint**:  
   The sum of all settlement amounts must not exceed the total budget:  
   \[
   \sum_{i} x_i \leq \text{Budgets.Total\_Budget}
   \]  
   From the data, `Budgets.Total_Budget = 500000`.

2. **Maximum Settlement per Claim**:  
   The settlement amount for any individual claim must not exceed the maximum allowed settlement per claim:  
   \[
   x_i \leq \text{Budgets.Maximum\_Settlement\_Per\_Claim} \quad \forall i
   \]  
   From the data, `Budgets.Maximum_Settlement_Per_Claim = 50000`.

3. **Minimum Settlement Percentage**:  
   The settlement amount for each claim must be at least 50% of the claimed amount:  
   \[
   x_i \geq 0.5 \times \text{Claims.Amount\_Claimed}_i \quad \forall i
   \]  
   From the data, `Claims.Amount_Claimed` values are:  
   - Claim 1: 15000  
   - Claim 2: 25000  
   - Claim 3: 35000  

#### Data Source Verification
- **Objective Function**: The coefficient 1 for \( x_i \) is derived from the problem goal of minimizing the total settlement cost.  
- **Total Budget Constraint**: `Budgets.Total_Budget = 500000` is used as the upper bound.  
- **Maximum Settlement per Claim**: `Budgets.Maximum_Settlement_Per_Claim = 50000` is used as the upper bound for each \( x_i \).  
- **Minimum Settlement Percentage**: `Claims.Amount_Claimed` values are used to calculate the lower bound for each \( x_i \).  

This formulation is 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 Insurance Policy Optimization Problem
"""

import gurobipy as gp
from gurobipy import GRB

def optimize_insurance_settlements():
    """Optimize insurance settlements to minimize total cost while adhering to constraints."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("insurance_policies")
    
    # Data from the problem
    claims = {
        1: 15000,
        2: 25000,
        3: 35000
    }
    total_budget = 500000
    max_settlement_per_claim = 50000
    min_settlement_percentage = 0.5
    
    # CRITICAL: Validate array lengths before loops
    claim_ids = list(claims.keys())
    n_claims = len(claim_ids)
    assert n_claims > 0, "No claims provided"
    
    # 2. VARIABLES
    # Decision variables: settlement amount for each claim
    x = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"x_{i}", lb=0) 
         for i in claim_ids}
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the total settlement cost
    model.setObjective(gp.quicksum(x[i] for i in claim_ids), GRB.MINIMIZE)
    
    # 4. CONSTRAINTS
    
    # Total Budget Constraint
    model.addConstr(gp.quicksum(x[i] for i in claim_ids) <= total_budget, name="total_budget")
    
    # Maximum Settlement per Claim
    for i in claim_ids:
        model.addConstr(x[i] <= max_settlement_per_claim, name=f"max_settlement_{i}")
    
    # Minimum Settlement Percentage
    for i in claim_ids:
        model.addConstr(x[i] >= min_settlement_percentage * claims[i], name=f"min_settlement_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in claim_ids:
            print(f"Settlement for claim {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_insurance_settlements()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 37500.0
**Execution Time**: 0.18 seconds
**Reliability**: high
**Analysis**: Gurobipy found the optimal solution efficiently with the lowest execution time.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCPLEX implementation for insurance_policies optimization problem
"""

from docplex.mp.model import Model

def optimize_insurance_settlements():
    """Optimize insurance settlements to minimize total cost while adhering to constraints."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="insurance_policies")
    
    # Data from the problem
    claims = [
        {"Claim_ID": 1, "Amount_Claimed": 15000},
        {"Claim_ID": 2, "Amount_Claimed": 25000},
        {"Claim_ID": 3, "Amount_Claimed": 35000}
    ]
    budgets = {"Total_Budget": 500000, "Maximum_Settlement_Per_Claim": 50000}
    
    # Extract data into lists for easier access
    claim_ids = [claim["Claim_ID"] for claim in claims]
    amounts_claimed = [claim["Amount_Claimed"] for claim in claims]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(claim_ids) == len(amounts_claimed), "Array length mismatch"
    safe_range = range(min(len(claim_ids), len(amounts_claimed)))  # Safe indexing
    
    # 2. VARIABLES
    # Decision variables: settlement amount for each claim
    x = {i: mdl.continuous_var(name=f"x_{claim_ids[i]}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the total settlement cost
    objective = mdl.sum(x[i] for i in safe_range)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    
    # Total Budget Constraint
    total_settlement = mdl.sum(x[i] for i in safe_range)
    mdl.add_constraint(total_settlement <= budgets["Total_Budget"], ctname="total_budget")
    
    # Maximum Settlement per Claim
    for i in safe_range:
        mdl.add_constraint(x[i] <= budgets["Maximum_Settlement_Per_Claim"], ctname=f"max_settlement_{claim_ids[i]}")
    
    # Minimum Settlement Percentage (50% of claimed amount)
    for i in safe_range:
        mdl.add_constraint(x[i] >= 0.5 * amounts_claimed[i], ctname=f"min_settlement_{claim_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"Settlement for Claim {claim_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_insurance_settlements()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 37500.0
**Execution Time**: 0.90 seconds
**Reliability**: high
**Analysis**: DOCplex also found the optimal solution but with a slightly higher execution time compared to Gurobipy.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def optimize_insurance_settlements():
    """Optimize insurance claim settlements to minimize total cost while adhering to constraints."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Claims data
    claims_data = {
        1: 15000,
        2: 25000,
        3: 35000
    }
    
    # Budget data
    total_budget = 500000
    max_settlement_per_claim = 50000
    min_settlement_percentage = 0.5
    
    # Validate array lengths before indexing
    assert len(claims_data) > 0, "Claims data is empty"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.Set(initialize=claims_data.keys())  # Set of claim IDs
    
    # 4. PARAMETERS (data containers)
    model.amount_claimed = pyo.Param(model.I, initialize=claims_data)
    model.total_budget = pyo.Param(initialize=total_budget)
    model.max_settlement_per_claim = pyo.Param(initialize=max_settlement_per_claim)
    model.min_settlement_percentage = pyo.Param(initialize=min_settlement_percentage)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    # Total Budget Constraint
    def total_budget_rule(model):
        return sum(model.x[i] for i in model.I) <= model.total_budget
    model.total_budget_constraint = pyo.Constraint(rule=total_budget_rule)
    
    # Maximum Settlement per Claim
    def max_settlement_rule(model, i):
        return model.x[i] <= model.max_settlement_per_claim
    model.max_settlement_constraint = pyo.Constraint(model.I, rule=max_settlement_rule)
    
    # Minimum Settlement Percentage
    def min_settlement_rule(model, i):
        return model.x[i] >= model.min_settlement_percentage * model.amount_claimed[i]
    model.min_settlement_constraint = pyo.Constraint(model.I, rule=min_settlement_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("\nSettlement amounts:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"Claim {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_insurance_settlements()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 37500.0
**Execution Time**: 0.96 seconds
**Reliability**: high
**Analysis**: Pyomo found the optimal solution but had the highest execution time among the three solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 37500.00 | 0.18s | N/A | N/A |
| Docplex | OPTIMAL | 37500.00 | 0.90s | N/A | N/A |
| Pyomo | OPTIMAL | 37500.00 | 0.96s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 37500.0

### Final Recommendation
**Recommended Optimal Value**: 37500.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its efficiency in finding the optimal solution with the lowest execution time.

### Optimal Decision Variables
- **x_1** = 7500.000
  - *Business Meaning*: Settlement amount for Claim 1, set to $7,500, which is 50% of the claimed amount of $15,000.
- **x_2** = 12500.000
  - *Business Meaning*: Settlement amount for Claim 2, set to $12,500, which is 50% of the claimed amount of $25,000.
- **x_3** = 17500.000
  - *Business Meaning*: Settlement amount for Claim 3, set to $17,500, which is 50% of the claimed amount of $35,000.

### Business Interpretation
**Overall Strategy**: The optimal total settlement cost is $37,500, which is the minimum possible while satisfying all constraints. This ensures that the budget is not exceeded, and each claim is settled fairly according to the minimum percentage requirement.
**Objective Value Meaning**: The optimal objective value of $37,500 represents the minimum total settlement cost that satisfies all constraints, ensuring efficient use of the budget.
**Resource Allocation Summary**: Resources should be allocated as follows: $7,500 to Claim 1, $12,500 to Claim 2, and $17,500 to Claim 3. This allocation ensures that each claim is settled at the minimum required percentage without exceeding the total budget.
**Implementation Recommendations**: 1. Verify the settlement amounts with the claimants. 2. Ensure that the payments are processed promptly. 3. Monitor the budget to ensure no further claims exceed the remaining budget.