## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{ij} \): Binary decision variable indicating whether flight \( i \) is assigned to airport \( j \). \( x_{ij} = 1 \) if flight \( i \) is assigned to airport \( j \), otherwise \( x_{ij} = 0 \).

#### Objective Function
Minimize the total operational cost of assigning flights to airports:

\[
\text{Minimize } \sum_{i,j} \text{distance}_{ij} \times \text{fee\_rate}_j \times x_{ij}
\]

Where:
- \(\text{distance}_{ij}\) is the distance between flight \( i \) and airport \( j \) from the `Distances` table.
- \(\text{fee\_rate}_j\) is the fee rate for airport \( j \) from the `FeeRates` table.

#### Constraints
1. **Each flight must be assigned to exactly one airport:**

\[
\sum_{j} x_{ij} = 1 \quad \forall i
\]

2. **The total number of flights assigned to an airport must not exceed its capacity:**

\[
\sum_{i} x_{ij} \leq \text{capacity}_j \quad \forall j
\]

Where:
- \(\text{capacity}_j\) is the capacity of airport \( j \) from the `Airports` table.

#### Data Source Verification:
- \(\text{distance}_{ij}\) is sourced from `Distances.distance`.
- \(\text{fee\_rate}_j\) is sourced from `FeeRates.fee_rate`.
- \(\text{capacity}_j\) is sourced from `Airports.capacity`.

This formulation provides a complete linear programming model that can be solved using standard optimization software to determine the optimal assignment of flights to airports, minimizing the total operational cost while respecting all constraints.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 4

import gurobipy as gp
from gurobipy import GRB

def flight_company_optimization():
    """Optimize flight assignments to minimize operational costs."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("flight_company")
    
    # Example data setup (replace with actual data retrieval logic)
    flight_ids = [1, 2, 3]
    airport_ids = [101, 102, 103]
    
    distances = {
        (1, 101): 550.0,
        (2, 102): 800.0,
        (3, 103): 950.0
    }
    
    fee_rates = {
        101: 12.0,
        102: 18.0,
        103: 22.0
    }
    
    capacities = {
        101: 120,
        102: 180,
        103: 150
    }
    
    # CRITICAL: Validate data lengths
    assert len(flight_ids) * len(airport_ids) == len(distances), "Data length mismatch"
    
    # 2. VARIABLES
    x = model.addVars(flight_ids, airport_ids, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(
        gp.quicksum(distances[i, j] * fee_rates[j] * x[i, j] for i in flight_ids for j in airport_ids),
        GRB.MINIMIZE
    )
    
    # 4. CONSTRAINTS
    # Each flight must be assigned to exactly one airport
    model.addConstrs(
        (gp.quicksum(x[i, j] for j in airport_ids) == 1 for i in flight_ids),
        name="assign_flight"
    )
    
    # The total number of flights assigned to an airport must not exceed its capacity
    model.addConstrs(
        (gp.quicksum(x[i, j] for i in flight_ids) <= capacities[j] for j in airport_ids),
        name="capacity_limit"
    )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in flight_ids:
            for j in airport_ids:
                if x[i, j].x > 1e-6:
                    print(f"Flight {i} assigned to Airport {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    else:
        print("Optimization ended with status", model.status)

# Run the optimization
flight_company_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpx906n5r7.py", line 76, in <module>
    flight_company_optimization()
  File "/tmp/tmpx906n5r7.py", line 35, in flight_company_optimization
    assert len(flight_ids) * len(airport_ids) == len(distances), "Data length mismatch"
AssertionError: Data length mismatch

**Analysis**: Gurobipy encountered a data length mismatch error, indicating a potential issue with the input data structure or data integrity. The solver could not proceed to find a solution.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 4

from docplex.mp.model import Model

def flight_company_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="flight_company")
    
    # Data from the problem statement
    flight_ids = [1, 2, 3]
    airport_ids = [101, 102, 103]
    distances = {
        (1, 101): 550.0,
        (2, 102): 800.0,
        (3, 103): 950.0
    }
    fee_rates = {
        101: 12.0,
        102: 18.0,
        103: 22.0
    }
    capacities = {
        101: 120,
        102: 180,
        103: 150
    }
    
    # CRITICAL: Validate data consistency
    assert len(flight_ids) * len(airport_ids) == len(distances), "Data length mismatch"
    
    # 2. VARIABLES
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in flight_ids for j in airport_ids}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(distances[i, j] * fee_rates[j] * x[i, j] for i in flight_ids for j in airport_ids)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    # Each flight must be assigned to exactly one airport
    for i in flight_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for j in airport_ids) == 1, ctname=f"assign_flight_{i}")
    
    # The total number of flights assigned to an airport must not exceed its capacity
    for j in airport_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for i in flight_ids) <= capacities[j], ctname=f"capacity_airport_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in flight_ids:
            for j in airport_ids:
                if solution.get_value(x[i, j]) > 0.5:  # Binary variable, check if assigned
                    print(f"Flight {i} assigned to Airport {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

# Run the optimization
flight_company_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpth908u56.py", line 61, in <module>
    flight_company_optimization()
  File "/tmp/tmpth908u56.py", line 29, in flight_company_optimization
    assert len(flight_ids) * len(airport_ids) == len(distances), "Data length mismatch"
AssertionError: Data length mismatch

**Analysis**: DOCplex also faced a data length mismatch error, similar to Gurobipy, suggesting a consistent issue across solvers related to the input data.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 4

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

def flight_company_optimization():
    """Optimize flight assignments to minimize operational costs."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Define the data based on the provided database schema and current stored values
    flight_ids = [1, 2, 3]
    airport_ids = [101, 102, 103]
    
    distances = {
        (1, 101): 550.0,
        (2, 102): 800.0,
        (3, 103): 950.0
    }
    
    fee_rates = {
        101: 12.0,
        102: 18.0,
        103: 22.0
    }
    
    capacities = {
        101: 120,
        102: 180,
        103: 150
    }
    
    # CRITICAL: Validate data consistency
    assert all((i, j) in distances for i in flight_ids for j in airport_ids), "Data length mismatch"
    
    # 3. SETS
    model.F = pyo.Set(initialize=flight_ids)
    model.A = pyo.Set(initialize=airport_ids)
    
    # 4. PARAMETERS
    model.distance = pyo.Param(model.F * model.A, initialize=distances, within=pyo.NonNegativeReals)
    model.fee_rate = pyo.Param(model.A, initialize=fee_rates, within=pyo.NonNegativeReals)
    model.capacity = pyo.Param(model.A, initialize=capacities, within=pyo.NonNegativeIntegers)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.F, model.A, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.distance[i, j] * model.fee_rate[j] * model.x[i, j] for i in model.F for j in model.A)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    # Each flight must be assigned to exactly one airport
    def flight_assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.A) == 1
    model.flight_assignment_constraint = pyo.Constraint(model.F, rule=flight_assignment_rule)
    
    # The total number of flights assigned to an airport must not exceed its capacity
    def airport_capacity_rule(model, j):
        return sum(model.x[i, j] for i in model.F) <= model.capacity[j]
    model.airport_capacity_constraint = pyo.Constraint(model.A, rule=airport_capacity_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)}")
        
        # Extract variable values
        print("\nFlight assignments:")
        for i in model.F:
            for j in model.A:
                if pyo.value(model.x[i, j]) > 0.5:  # Binary variable, check if assigned
                    print(f"Flight {i} assigned to Airport {j}")
        
    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}")

# Run the optimization
flight_company_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmp_wb0m3_x.py", line 92, in <module>
    flight_company_optimization()
  File "/tmp/tmp_wb0m3_x.py", line 36, in flight_company_optimization
    assert all((i, j) in distances for i in flight_ids for j in airport_ids), "Data length mismatch"
AssertionError: Data length mismatch

**Analysis**: Pyomo reported a data length mismatch error, indicating that the input data does not align with the expected format or content, preventing the solver from executing successfully.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.45s | N/A | 4 |
| Docplex | ERROR | N/A | 3.84s | N/A | 4 |
| Pyomo | ERROR | N/A | 2.46s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Data length mismatch suggests that the input data tables (Distances, FeeRates, Airports) may not be correctly populated or aligned.
- There may be missing or extra entries in the Distances table compared to the expected number of flight-airport pairs.
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers encountered the same data issue, indicating that the problem lies with the input data rather than the solvers themselves. Therefore, no solver preference can be made until the data issue is resolved.

### Business Interpretation
**Overall Strategy**: The current optimization attempt failed due to data issues, and no valid assignment of flights to airports could be determined.
**Objective Value Meaning**: The optimal objective value could not be calculated due to errors in the input data, preventing any meaningful business interpretation.
**Resource Allocation Summary**: Resource allocation could not be determined due to the failure in solving the optimization problem.
**Implementation Recommendations**: Verify and correct the input data tables to ensure that all expected flight-airport pairs are present and correctly formatted. Once data integrity is ensured, re-run the optimization using the solvers.