# Complete Optimization Problem and Solution: insurance_policies

## 1. Problem Context and Goals

### Context  
An insurance company is focused on optimizing its resource allocation to efficiently settle claims. The primary decision involves determining the amount to be settled for each claim, denoted as Amount_Settled[i], where i represents each individual claim. The company's operational objective is to minimize the total payouts across all claims. This is achieved by ensuring that the total amount settled is as low as possible while adhering to the constraints of the maximum claimable amount and the necessity to settle claims within a specified time frame. The business configuration includes a critical parameter: the maximum allowable time period for settling a claim, which is used as a constraint to ensure timely settlements. This parameter is set at 30 days, aligning with industry standards to balance efficient processing and customer satisfaction. The data presented reflects current operational information, focusing on precise decision-making that naturally leads to a linear optimization formulation. Resource limitations are expressed through linear constraints, avoiding any scenarios that would require nonlinear relationships.

### Goals  
The primary goal of the optimization process is to minimize the total amount paid out in claim settlements. This involves strategically determining the settlement amount for each claim to ensure the total payouts are minimized. Success in this optimization is measured by the reduction in total settlement costs, aligning with the company's financial objectives. The goal is articulated in natural language, emphasizing the linear nature of the optimization task without resorting to mathematical symbols or equations.

## 2. Constraints    

The optimization process is governed by two main constraints that align with linear mathematical forms:

- The amount settled for each claim must not exceed the maximum amount claimed. This ensures that settlements are fair and within the agreed-upon limits.
- Each claim must be settled within a maximum period of 30 days from the date it was received. This constraint ensures that the company adheres to industry standards for timely claim processing, enhancing customer satisfaction and operational efficiency.

These constraints are described in business terms that naturally lead to linear mathematical expressions, avoiding any need for variable products or divisions.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Added Date_Claim_Received column to Claims table to address missing data requirement for settlement period calculation. Updated business configuration logic for max_settlement_period.

CREATE TABLE Claims (
  Amount_Settled FLOAT,
  Amount_Claimed FLOAT,
  Date_Claim_Settled DATE,
  Date_Claim_Received DATE
);


```

### Data Dictionary  
The Claims table serves as the central repository for information related to insurance claims and their settlements. Each column within the table has a specific business purpose and role in the optimization process:

- **Amount_Settled**: Represents the amount settled for each claim. It plays a crucial role in the objective function, where the goal is to minimize the total payouts.
- **Amount_Claimed**: Indicates the maximum amount that can be settled for each claim. This serves as a constraint bound, ensuring settlements do not exceed the claimed amounts.
- **Date_Claim_Settled**: The date on which the claim was settled. It is used to ensure that claims are settled within the allowed period, contributing to the timely processing constraint.
- **Date_Claim_Received**: The date on which the claim was received. This information is essential for calculating the settlement period constraint, ensuring compliance with the maximum allowable time frame.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical insurance claim scenarios, ensuring that the amounts settled are realistic and within the claimed amounts, and that settlement periods are within the configured maximum period.

-- Realistic data for Claims
INSERT INTO Claims (Amount_Settled, Amount_Claimed, Date_Claim_Settled, Date_Claim_Received) VALUES (1000.0, 1200.0, 2023-01-15, 2023-01-01);
INSERT INTO Claims (Amount_Settled, Amount_Claimed, Date_Claim_Settled, Date_Claim_Received) VALUES (1500.0, 1600.0, 2023-02-20, 2023-02-01);
INSERT INTO Claims (Amount_Settled, Amount_Claimed, Date_Claim_Settled, Date_Claim_Received) VALUES (2000.0, 2500.0, 2023-03-10, 2023-03-01);


```

## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_i \): Amount settled for claim \( i \). This is a continuous decision variable representing the amount to be settled for each claim.

#### Objective Function
Minimize the total amount settled across all claims:
\[ \text{Minimize } \sum_{i} x_i \]

#### Constraints
1. **Maximum Amount Claimed Constraint**: The amount settled for each claim must not exceed the maximum amount claimed.
   \[
   x_i \leq \text{Amount_Claimed}_i \quad \forall i
   \]

2. **Settlement Period Constraint**: Each claim must be settled within 30 days from the date it was received.
   \[
   \text{Date_Claim_Settled}_i - \text{Date_Claim_Received}_i \leq 30 \quad \forall i
   \]

Data Source Verification:
- The coefficients for the maximum amount claimed constraint come from the `Claims.Amount_Claimed` column.
- The settlement period constraint uses the `Claims.Date_Claim_Settled` and `Claims.Date_Claim_Received` columns to ensure compliance with the 30-day maximum settlement period.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB
from datetime import datetime

def optimize_claim_settlements():
    """Optimize insurance claim settlements using Gurobi"""

    # 1. MODEL & DATA SETUP
    model = gp.Model("insurance_policies")

    # Sample data based on the provided SQL schema and data
    claims_data = [
        {"Amount_Claimed": 1200.0, "Date_Claim_Received": "2023-01-01", "Date_Claim_Settled": "2023-01-15"},
        {"Amount_Claimed": 1600.0, "Date_Claim_Received": "2023-02-01", "Date_Claim_Settled": "2023-02-20"},
        {"Amount_Claimed": 2500.0, "Date_Claim_Received": "2023-03-01", "Date_Claim_Settled": "2023-03-10"}
    ]

    n_claims = len(claims_data)

    # CRITICAL: Validate array lengths before loops
    assert n_claims > 0, "No claims data available"

    # 2. VARIABLES
    # Variable dictionaries for Amount_Settled
    Amount_Settled = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"Amount_Settled_{i}", lb=0)
                      for i in range(n_claims)}

    # 3. OBJECTIVE FUNCTION
    # Minimize the total amount settled
    model.setObjective(gp.quicksum(Amount_Settled[i] for i in range(n_claims)), GRB.MINIMIZE)

    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS

    # Maximum Amount Claimed Constraint
    for i in range(n_claims):
        model.addConstr(Amount_Settled[i] <= claims_data[i]["Amount_Claimed"], name=f"max_claimed_{i}")

    # Settlement Period Constraint
    for i in range(n_claims):
        date_received = datetime.strptime(claims_data[i]["Date_Claim_Received"], "%Y-%m-%d")
        date_settled = datetime.strptime(claims_data[i]["Date_Claim_Settled"], "%Y-%m-%d")
        settlement_period = (date_settled - date_received).days
        model.addConstr(settlement_period <= 30, name=f"settlement_period_{i}")

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_claims):
            if Amount_Settled[i].x > 1e-6:
                print(f"Amount_Settled[{i}] = {Amount_Settled[i].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

    return model

# Run the optimization
optimize_claim_settlements()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 0.0
**Execution Time**: 2.05 seconds
**Reliability**: high
**Analysis**: Gurobipy successfully found an optimal solution with an objective value of 0.0, indicating that all claims can be settled at zero cost, which might suggest a data or model issue.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model
from datetime import datetime

def optimize_claim_settlements():
    """Optimize insurance claim settlements to minimize total payouts."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="insurance_policies")
    
    # Sample data from the Claims table
    amount_claimed = [1200.0, 1600.0, 2500.0]
    date_claim_received = [
        datetime(2023, 1, 1),
        datetime(2023, 2, 1),
        datetime(2023, 3, 1)
    ]
    date_claim_settled = [
        datetime(2023, 1, 15),
        datetime(2023, 2, 20),
        datetime(2023, 3, 10)
    ]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(amount_claimed) == len(date_claim_received) == len(date_claim_settled), "Array length mismatch"
    safe_range = range(min(len(amount_claimed), len(date_claim_received), len(date_claim_settled)))  # Safe indexing
    
    # 2. VARIABLES
    # Variable dictionaries (most common)
    x = {i: mdl.continuous_var(name=f"x_{i}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the total amount settled across all claims
    objective = mdl.sum(x[i] for i in safe_range)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Pattern A: Maximum Amount Claimed Constraint
    for i in safe_range:
        mdl.add_constraint(x[i] <= amount_claimed[i], ctname=f"max_claim_{i}")
    
    # Pattern B: Settlement Period Constraint
    for i in safe_range:
        settlement_period = (date_claim_settled[i] - date_claim_received[i]).days
        mdl.add_constraint(settlement_period <= 30, ctname=f"settlement_period_{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])
            if value > 1e-6:
                print(f"x[{i}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
optimize_claim_settlements()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 0.0
**Execution Time**: 5.97 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution with an objective value of 0.0, consistent with Gurobipy, suggesting the same potential data or model issue.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def insurance_policies_optimization():
    """Optimize insurance claim settlements to minimize total payouts."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the provided SQL data
    amount_claimed = [1200.0, 1600.0, 2500.0]
    date_claim_received = ["2023-01-01", "2023-02-01", "2023-03-01"]
    date_claim_settled = ["2023-01-15", "2023-02-20", "2023-03-10"]
    
    # Convert date strings to datetime objects for calculation
    date_claim_received = [datetime.strptime(date, "%Y-%m-%d") for date in date_claim_received]
    date_claim_settled = [datetime.strptime(date, "%Y-%m-%d") for date in date_claim_settled]
    
    # Calculate settlement periods
    settlement_periods = [(settled - received).days for settled, received in zip(date_claim_settled, date_claim_received)]
    
    # CRITICAL: Validate array lengths before indexing
    n_claims = len(amount_claimed)
    assert len(settlement_periods) == n_claims, "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.RangeSet(1, n_claims)  # 1-based indexing
    
    # 4. PARAMETERS
    model.amount_claimed = pyo.Param(model.I, initialize={i+1: amount_claimed[i] for i in range(n_claims)})
    model.settlement_periods = pyo.Param(model.I, initialize={i+1: settlement_periods[i] for i in range(n_claims)})
    
    # 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
    # Maximum Amount Claimed Constraint
    def max_claimed_rule(model, i):
        return model.x[i] <= model.amount_claimed[i]
    model.max_claimed_constraint = pyo.Constraint(model.I, rule=max_claimed_rule)
    
    # Settlement Period Constraint
    def settlement_period_rule(model, i):
        return model.settlement_periods[i] <= 30
    model.settlement_period_constraint = pyo.Constraint(model.I, rule=settlement_period_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):.2f}")
        
        # Extract variable values
        print("\nVariable values:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"x[{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

# Run the optimization
insurance_policies_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpdz6cg6qa.py", line 84, in <module>
    insurance_policies_optimization()
  File "/tmp/tmpdz6cg6qa.py", line 54, in insurance_policies_optimization
    model.settlement_period_constraint = pyo.Constraint(model.I, rule=settlement_period_rule)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 571, in __setattr__
    self.add_component(name, val)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 1101, in add_component
    val.construct(data)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/constraint.py", line 722, in construct
    self._setitem_when_not_present(index, rule(block, index))
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/indexed_component.py", line 1111, in _setitem_when_not_present
    obj.set_value(value)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/constraint.py", line 469, in set_value
    raise ValueError(
ValueError: Invalid constraint expression. The constraint expression resolved to a trivial Boolean (True) instead of a Pyomo object. Please modify your rule to return Constraint.Feasible instead of True.

Error thrown for Constraint 'settlement_period_constraint[1]'

**Analysis**: Pyomo encountered an error due to a constraint expression issue, indicating a problem with the model formulation in Pyomo.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 0.00 | 2.05s | N/A | N/A |
| Docplex | OPTIMAL | 0.00 | 5.97s | N/A | N/A |
| Pyomo | ERROR | N/A | 5.16s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex
**Majority Vote Optimal Value**: 0.0

### Final Recommendation
**Recommended Optimal Value**: 0.0
**Confidence Level**: MEDIUM
**Preferred Solver(s)**: multiple
**Reasoning**: Both Gurobipy and DOCplex provided consistent results, but the zero objective value suggests a deeper issue that needs investigation.

### Business Interpretation
**Overall Strategy**: The results suggest that claims can be settled without any cost, which is likely incorrect. This indicates a need to review the model and data.
**Objective Value Meaning**: An objective value of 0.0 implies no cost for settling claims, which is unrealistic and suggests a model or data issue.
**Resource Allocation Summary**: Resources should be allocated to review and correct the model and data to ensure realistic claim settlements.
**Implementation Recommendations**: Verify the data integrity and model formulation, especially the constraints and objective function, to ensure they reflect the real-world scenario accurately.