# Complete Optimization Problem and Solution: insurance_policies

## 1. Problem Context and Goals

### Context  
The insurance company is focused on minimizing the total cost of settling claims while ensuring fairness and adherence to budget constraints. The company must allocate settlement amounts across multiple claims in a way that reduces the overall payout. This allocation is subject to several operational constraints:  
- The total amount settled across all claims must not exceed the total budget available for settlements.  
- The amount settled for each individual claim must not exceed the maximum settlement amount allowed per claim.  
- The amount settled for each claim must be at least a minimum percentage of the claimed amount, ensuring fairness to policyholders.  

The total budget available for settlements is currently set at $500,000, and the maximum amount that can be settled per claim is $50,000. Additionally, the company ensures that each settlement is at least 50% of the claimed amount. These parameters are critical for maintaining financial stability and fairness in the settlement process.  

### Goals  
The primary goal of this optimization problem is to minimize the total amount settled across all claims. This is achieved by determining the optimal settlement amount for each claim while adhering to the constraints mentioned above. Success is measured by achieving the lowest possible total settlement cost without violating any of the operational or fairness constraints.  

## 2. Constraints  

The optimization problem is subject to the following constraints, which ensure that the settlement process remains within operational and financial limits:  
1. **Total Budget Constraint**: The sum of all settlement amounts across claims must not exceed the total budget available for settlements, which is $500,000.  
2. **Maximum Settlement per Claim**: The settlement amount for any individual claim must not exceed the maximum allowed settlement per claim, which is $50,000.  
3. **Minimum Settlement Percentage**: The settlement amount for each claim must be at least 50% of the claimed amount, ensuring a fair minimum payout to policyholders.  

These constraints are designed to align with the company’s financial and operational policies while maintaining fairness in the settlement process.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding a Budgets table for Total_Budget and Maximum_Settlement_Per_Claim. Configuration logic updated with scalar parameters for Total_Budget and Maximum_Settlement_Per_Claim, and a business logic formula for minimum settlement percentage.

CREATE TABLE Claims (
  Claim_ID INTEGER,
  Amount_Claimed FLOAT
);

CREATE TABLE Settlements (
  Claim_ID INTEGER,
  Amount_Settled FLOAT
);

CREATE TABLE Budgets (
  Total_Budget FLOAT,
  Maximum_Settlement_Per_Claim FLOAT
);
```

### Data Dictionary  
- **Claims Table**:  
  - **Claim_ID**: A unique identifier for each insurance claim.  
  - **Amount_Claimed**: The amount requested by the policyholder for each claim. This value is used to determine the minimum settlement amount per claim.  

- **Settlements Table**:  
  - **Claim_ID**: A unique identifier for each claim, linking it to the corresponding settlement.  
  - **Amount_Settled**: The amount allocated to settle each claim. This is the decision variable to be optimized.  

- **Budgets Table**:  
  - **Total_Budget**: The total amount of money available for settling all claims. This value is used as an upper bound for the sum of all settlements.  
  - **Maximum_Settlement_Per_Claim**: The maximum amount that can be allocated to settle any single claim. This value is used as an upper bound for individual settlements.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic insurance claim scenarios, ensuring that the total budget and maximum settlement per claim are sufficient to cover the claims while maintaining business profitability. The minimum settlement percentage ensures fairness to policyholders.

-- Realistic data for Claims
INSERT INTO Claims (Claim_ID, Amount_Claimed) VALUES (1, 15000);
INSERT INTO Claims (Claim_ID, Amount_Claimed) VALUES (2, 25000);
INSERT INTO Claims (Claim_ID, Amount_Claimed) VALUES (3, 35000);

-- Realistic data for Settlements
INSERT INTO Settlements (Claim_ID, Amount_Settled) VALUES (1, 7500);
INSERT INTO Settlements (Claim_ID, Amount_Settled) VALUES (2, 12500);
INSERT INTO Settlements (Claim_ID, Amount_Settled) VALUES (3, 17500);

-- Realistic data for Budgets
INSERT INTO Budgets (Total_Budget, Maximum_Settlement_Per_Claim) VALUES (500000, 50000);
```

## 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.