## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_{a,r} \) be a binary decision variable where:
- \( x_{a,r} = 1 \) if airline \( a \) is assigned to route \( r \),
- \( x_{a,r} = 0 \) otherwise.

#### Objective Function
Minimize the total operational cost:
\[
\text{Minimize} \quad \sum_{a \in A} \sum_{r \in R} c_{a,r} \cdot x_{a,r}
\]
where:
- \( c_{a,r} \) is the cost of assigning airline \( a \) to route \( r \).

#### Constraints
1. **Route Coverage**: Each route must be assigned to exactly one airline:
\[
\sum_{a \in A} x_{a,r} = 1 \quad \forall r \in R
\]
2. **Airline Capacity**: No airline can be assigned more routes than its capacity:
\[
\sum_{r \in R} x_{a,r} \leq C_a \quad \forall a \in A
\]
where:
- \( C_a \) is the capacity of airline \( a \).

#### Data Source Verification
- \( c_{a,r} \): `cost_airline_route.cost`
- \( C_a \): `capacity_airline.capacity`

### Complete Numerical Model
Using the provided data:

#### Decision Variables
\( x_{1,1}, x_{1,2}, x_{1,3} \) (binary variables for airline 1 and routes 1, 2, 3)

#### Objective Function
\[
\text{Minimize} \quad 1200 \cdot x_{1,1} + 1800 \cdot x_{1,2} + 2500 \cdot x_{1,3}
\]

#### Constraints
1. **Route Coverage**:
\[
x_{1,1} + x_{1,2} + x_{1,3} = 1
\]
2. **Airline Capacity**:
\[
x_{1,1} + x_{1,2} + x_{1,3} \leq 8
\]

This is a complete, immediately solvable linear programming model.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def flight_assignment_optimization():
    # 1. MODEL & DATA SETUP
    model = gp.Model("flight_assignment")
    
    # Data from the problem
    airlines = [1, 2, 3]
    routes = [1, 2, 3]
    
    # Cost data
    cost_data = {
        (1, 1): 1200.0,
        (1, 2): 1800.0,
        (1, 3): 2500.0
    }
    
    # Capacity data
    capacity_data = {
        1: 8,
        2: 12,
        3: 6
    }
    
    # Validate array lengths
    assert len(cost_data) == len(airlines) * len(routes), "Cost data length mismatch"
    assert len(capacity_data) == len(airlines), "Capacity data length mismatch"
    
    # 2. VARIABLES
    x = model.addVars(airlines, routes, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(cost_data[a, r] * x[a, r] for a in airlines for r in routes), GRB.MINIMIZE)
    
    # 4. CONSTRAINTS
    # Route Coverage: Each route must be assigned to exactly one airline
    for r in routes:
        model.addConstr(gp.quicksum(x[a, r] for a in airlines) == 1, name=f"route_coverage_{r}")
    
    # Airline Capacity: No airline can be assigned more routes than its capacity
    for a in airlines:
        model.addConstr(gp.quicksum(x[a, r] for r in routes) <= capacity_data[a], name=f"capacity_{a}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for a in airlines:
            for r in routes:
                if x[a, r].x > 1e-6:
                    print(f"Airline {a} assigned to route {r}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
flight_assignment_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmph7hf41op.py", line 64, in <module>
    flight_assignment_optimization()
  File "/tmp/tmph7hf41op.py", line 29, in flight_assignment_optimization
    assert len(cost_data) == len(airlines) * len(routes), "Cost data length mismatch"
AssertionError: Cost data length mismatch

**Analysis**: Gurobipy encountered an error due to a mismatch in the length of cost data. This suggests an issue with the input data structure or preparation.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def flight_assignment_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="flight_assignment_optimization")
    
    # Data from the problem
    airlines = [1, 2, 3]
    routes = [1, 2, 3]
    
    # Cost data
    cost_data = [
        (1, 1, 1200.0),
        (1, 2, 1800.0),
        (1, 3, 2500.0)
    ]
    
    # Capacity data
    capacity_data = [
        (1, 8),
        (2, 12),
        (3, 6)
    ]
    
    # Validate array lengths
    assert len(cost_data) == len(airlines) * len(routes), "Cost data length mismatch"
    assert len(capacity_data) == len(airlines), "Capacity data length mismatch"
    
    # Create dictionaries for cost and capacity
    cost = {(a, r): c for a, r, c in cost_data}
    capacity = {a: c for a, c in capacity_data}
    
    # 2. VARIABLES
    x = mdl.binary_var_dict(keys=[(a, r) for a in airlines for r in routes], name="x")
    
    # 3. OBJECTIVE FUNCTION
    total_cost = mdl.sum(cost[(a, r)] * x[(a, r)] for a in airlines for r in routes)
    mdl.minimize(total_cost)
    
    # 4. CONSTRAINTS
    
    # Route Coverage: Each route must be assigned to exactly one airline
    for r in routes:
        mdl.add_constraint(mdl.sum(x[(a, r)] for a in airlines) == 1, ctname=f"route_coverage_{r}")
    
    # Airline Capacity: No airline can be assigned more routes than its capacity
    for a in airlines:
        mdl.add_constraint(mdl.sum(x[(a, r)] for r in routes) <= capacity[a], ctname=f"capacity_{a}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for a in airlines:
            for r in routes:
                if solution.get_value(x[(a, r)]) > 0:
                    print(f"Airline {a} assigned to route {r}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
flight_assignment_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpkdl4nhoy.py", line 68, in <module>
    flight_assignment_optimization()
  File "/tmp/tmpkdl4nhoy.py", line 28, in flight_assignment_optimization
    assert len(cost_data) == len(airlines) * len(routes), "Cost data length mismatch"
AssertionError: Cost data length mismatch

**Analysis**: DOCplex also failed with the same error, indicating a systemic issue with the cost data provided to the solver.

## 7. Pyomo Implementation

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

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

def optimize_flight_assignment():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Define airlines and routes
    airlines = [1, 2, 3]
    routes = [1, 2, 3]
    
    # Cost data from cost_airline_route table
    cost_data = {
        (1, 1): 1200.0,
        (1, 2): 1800.0,
        (1, 3): 2500.0
    }
    
    # Capacity data from capacity_airline table
    capacity_data = {
        1: 8,
        2: 12,
        3: 6
    }
    
    # Validate array lengths
    assert len(cost_data) == len(airlines) * len(routes), "Cost data length mismatch"
    assert len(capacity_data) == len(airlines), "Capacity data length mismatch"
    
    # 3. SETS
    model.A = pyo.Set(initialize=airlines)
    model.R = pyo.Set(initialize=routes)
    
    # 4. PARAMETERS
    model.cost = pyo.Param(model.A, model.R, initialize=cost_data)
    model.capacity = pyo.Param(model.A, initialize=capacity_data)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.A, model.R, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.cost[a, r] * model.x[a, r] for a in model.A for r in model.R)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    # Route Coverage: Each route must be assigned to exactly one airline
    def route_coverage_rule(model, r):
        return sum(model.x[a, r] for a in model.A) == 1
    model.route_coverage = pyo.Constraint(model.R, rule=route_coverage_rule)
    
    # Airline Capacity: No airline can be assigned more routes than its capacity
    def airline_capacity_rule(model, a):
        return sum(model.x[a, r] for r in model.R) <= model.capacity[a]
    model.airline_capacity = pyo.Constraint(model.A, rule=airline_capacity_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("\nVariable values:")
        for a in model.A:
            for r in model.R:
                x_val = pyo.value(model.x[a, r])
                if x_val > 1e-6:  # Only print non-zero values
                    print(f"x[{a}, {r}] = {int(x_val)}")
        
    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_flight_assignment()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpopca6jcm.py", line 95, in <module>
    optimize_flight_assignment()
  File "/tmp/tmpopca6jcm.py", line 30, in optimize_flight_assignment
    assert len(cost_data) == len(airlines) * len(routes), "Cost data length mismatch"
AssertionError: Cost data length mismatch

**Analysis**: Pyomo experienced the same error as the other solvers, reinforcing the conclusion that the problem lies in the data preparation stage.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.20s | N/A | 4 |
| Docplex | ERROR | N/A | 1.14s | N/A | 4 |
| Pyomo | ERROR | N/A | 0.70s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Cost data length mismatch
- Incorrect data preparation
- Missing or incomplete data
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: none
**Reasoning**: All solvers failed due to the same data mismatch issue. The problem must be addressed at the data preparation level before any solver can be effectively used.

### Business Interpretation
**Overall Strategy**: The optimization process cannot proceed due to data issues, preventing any meaningful business decision.
**Objective Value Meaning**: No optimal objective value is available due to the failure in the optimization process.
**Resource Allocation Summary**: Resource allocation cannot be determined until the data issues are resolved.
**Implementation Recommendations**: 1. Verify and correct the cost data to ensure it matches the expected dimensions. 2. Re-run the optimization with the corrected data. 3. Validate the results with at least two different solvers to ensure consistency.