## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{f,a,c} \): Binary decision variable indicating whether flight \( f \) is assigned to airport \( a \) and company \( c \).  
  - \( x_{f,a,c} \in \{0, 1\} \) for all \( f \in F \), \( a \in A \), \( c \in C \).

#### Objective Function
Minimize the total fuel consumption:  
\[
\text{Minimize } Z = \sum_{f \in F} \sum_{a \in A} \sum_{c \in C} \text{coefficient}_f \cdot x_{f,a,c}
\]  
- **Data Source Verification**:  
  - \( \text{coefficient}_f \) comes from `fuel_consumption_coefficients.coefficient`.

#### Constraints
1. **Flight Assignment Constraint**: Each flight must be assigned to exactly one airport and one company.  
\[
\sum_{a \in A} \sum_{c \in C} x_{f,a,c} = 1 \quad \forall f \in F
\]

2. **Airport Capacity Constraint**: The total number of flights assigned to an airport cannot exceed its capacity.  
\[
\sum_{f \in F} \sum_{c \in C} x_{f,a,c} \leq \text{capacity}_a \quad \forall a \in A
\]  
- **Data Source Verification**:  
  - \( \text{capacity}_a \) comes from `airport_capacities.capacity`.

3. **Company Capacity Constraint**: The total number of flights assigned to a company cannot exceed its capacity.  
\[
\sum_{f \in F} \sum_{a \in A} x_{f,a,c} \leq \text{capacity}_c \quad \forall c \in C
\]  
- **Data Source Verification**:  
  - \( \text{capacity}_c \) comes from `company_capacities.capacity`.

4. **Velocity Constraint**: Each flight must operate within its predefined minimum and maximum velocity limits.  
\[
\text{velocity_min}_f \leq \text{velocity}_f \leq \text{velocity_max}_f \quad \forall f \in F
\]  
- **Data Source Verification**:  
  - \( \text{velocity_min}_f \) and \( \text{velocity_max}_f \) come from `velocity_constraints.velocity_min` and `velocity_constraints.velocity_max`.

5. **Altitude Constraint**: Each flight must operate within its predefined minimum and maximum altitude limits.  
\[
\text{altitude_min}_f \leq \text{altitude}_f \leq \text{altitude_max}_f \quad \forall f \in F
\]  
- **Data Source Verification**:  
  - \( \text{altitude_min}_f \) and \( \text{altitude_max}_f \) come from `altitude_constraints.altitude_min` and `altitude_constraints.altitude_max`.

#### Summary
This is a Mixed-Integer Programming (MIP) formulation with a linear objective function and linear constraints. The decision variables \( x_{f,a,c} \) are binary, ensuring that each flight is assigned to exactly one airport and one company. The constraints enforce operational limits on airport and company capacities, as well as velocity and altitude ranges for each flight. The objective function minimizes total fuel consumption based on predefined coefficients.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy Implementation for Flight Scheduling Optimization Problem
"""

import gurobipy as gp
from gurobipy import GRB

def flight_scheduling_optimization():
    # 1. MODEL & DATA SETUP
    model = gp.Model("flight_scheduling")
    
    # Example data (replace with actual data loading from database)
    flights = [1, 2, 3]
    airports = [1, 2, 3]
    companies = [1, 2, 3]
    
    fuel_coefficients = {1: 0.5, 2: 0.6, 3: 0.7}
    airport_capacities = {1: 10, 2: 15, 3: 20}
    company_capacities = {1: 5, 2: 10, 3: 15}
    
    # CRITICAL: Validate array lengths before loops
    assert len(flights) == len(fuel_coefficients), "Flight data length mismatch"
    assert len(airports) == len(airport_capacities), "Airport data length mismatch"
    assert len(companies) == len(company_capacities), "Company data length mismatch"
    
    # 2. VARIABLES
    # Binary decision variable: x[f, a, c] = 1 if flight f is assigned to airport a and company c
    x = model.addVars(flights, airports, companies, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Minimize total fuel consumption
    model.setObjective(gp.quicksum(fuel_coefficients[f] * x[f, a, c] for f in flights for a in airports for c in companies), GRB.MINIMIZE)
    
    # 4. CONSTRAINTS
    
    # Flight Assignment Constraint: Each flight must be assigned to exactly one airport and one company
    for f in flights:
        model.addConstr(gp.quicksum(x[f, a, c] for a in airports for c in companies) == 1, name=f"flight_assignment_{f}")
    
    # Airport Capacity Constraint: Total flights assigned to an airport cannot exceed its capacity
    for a in airports:
        model.addConstr(gp.quicksum(x[f, a, c] for f in flights for c in companies) <= airport_capacities[a], name=f"airport_capacity_{a}")
    
    # Company Capacity Constraint: Total flights assigned to a company cannot exceed its capacity
    for c in companies:
        model.addConstr(gp.quicksum(x[f, a, c] for f in flights for a in airports) <= company_capacities[c], name=f"company_capacity_{c}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for f in flights:
            for a in airports:
                for c in companies:
                    if x[f, a, c].x > 0.5:
                        print(f"Flight {f} assigned to Airport {a} and Company {c}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
if __name__ == "__main__":
    flight_scheduling_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.8
**Execution Time**: 0.17 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution with the lowest execution time, indicating high efficiency and reliability.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCplex 2.29.245 Implementation for Flight Scheduling Optimization
"""

from docplex.mp.model import Model

def flight_scheduling_optimization():
    """Optimize flight assignments to minimize total fuel consumption."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="flight_scheduling")
    
    # Example data (replace with actual data loading from database)
    flights = [1, 2, 3]
    airports = [1, 2, 3]
    companies = [1, 2, 3]
    
    fuel_coefficients = {1: 0.5, 2: 0.6, 3: 0.7}
    airport_capacities = {1: 10, 2: 15, 3: 20}
    company_capacities = {1: 5, 2: 10, 3: 15}
    
    # Validate array lengths
    assert len(flights) == len(fuel_coefficients), "Flight data mismatch"
    assert len(airports) == len(airport_capacities), "Airport data mismatch"
    assert len(companies) == len(company_capacities), "Company data mismatch"
    
    # 2. VARIABLES
    # Binary decision variable: x[f, a, c] = 1 if flight f is assigned to airport a and company c
    x = mdl.binary_var_cube(flights, airports, companies, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Minimize total fuel consumption
    objective = mdl.sum(fuel_coefficients[f] * x[f, a, c] for f in flights for a in airports for c in companies)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    
    # Flight Assignment Constraint: Each flight must be assigned to exactly one airport and one company
    for f in flights:
        mdl.add_constraint(mdl.sum(x[f, a, c] for a in airports for c in companies) == 1, ctname=f"flight_assignment_{f}")
    
    # Airport Capacity Constraint: Total flights assigned to an airport cannot exceed its capacity
    for a in airports:
        mdl.add_constraint(mdl.sum(x[f, a, c] for f in flights for c in companies) <= airport_capacities[a], ctname=f"airport_capacity_{a}")
    
    # Company Capacity Constraint: Total flights assigned to a company cannot exceed its capacity
    for c in companies:
        mdl.add_constraint(mdl.sum(x[f, a, c] for f in flights for a in airports) <= company_capacities[c], ctname=f"company_capacity_{c}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for f in flights:
            for a in airports:
                for c in companies:
                    if solution.get_value(x[f, a, c]) > 0.5:  # Check if assigned
                        print(f"Flight {f} assigned to Airport {a} and Company {c}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
if __name__ == "__main__":
    flight_scheduling_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.7999999999999998
**Execution Time**: 0.95 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, but with a slightly longer execution time compared to Gurobipy.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def flight_scheduling_optimization():
    """Optimize flight scheduling to minimize total fuel consumption."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data (replace with actual data loading from database)
    flights = [1, 2, 3]
    airports = [1, 2, 3]
    companies = [1, 2, 3]
    
    fuel_coefficients = {1: 0.5, 2: 0.6, 3: 0.7}
    airport_capacities = {1: 10, 2: 15, 3: 20}
    company_capacities = {1: 5, 2: 10, 3: 15}
    
    # Validate array lengths
    assert len(flights) == len(fuel_coefficients), "Fuel coefficients length mismatch"
    assert len(airports) == len(airport_capacities), "Airport capacities length mismatch"
    assert len(companies) == len(company_capacities), "Company capacities length mismatch"
    
    # 3. SETS
    model.F = pyo.Set(initialize=flights)  # Flights
    model.A = pyo.Set(initialize=airports)  # Airports
    model.C = pyo.Set(initialize=companies)  # Companies
    
    # 4. PARAMETERS
    model.fuel_coefficient = pyo.Param(model.F, initialize=fuel_coefficients)
    model.airport_capacity = pyo.Param(model.A, initialize=airport_capacities)
    model.company_capacity = pyo.Param(model.C, initialize=company_capacities)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.F, model.A, model.C, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.fuel_coefficient[f] * model.x[f, a, c] for f in model.F for a in model.A for c in model.C)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    
    # Flight Assignment Constraint
    def flight_assignment_rule(model, f):
        return sum(model.x[f, a, c] for a in model.A for c in model.C) == 1
    model.flight_assignment = pyo.Constraint(model.F, rule=flight_assignment_rule)
    
    # Airport Capacity Constraint
    def airport_capacity_rule(model, a):
        return sum(model.x[f, a, c] for f in model.F for c in model.C) <= model.airport_capacity[a]
    model.airport_capacity_constraint = pyo.Constraint(model.A, rule=airport_capacity_rule)
    
    # Company Capacity Constraint
    def company_capacity_rule(model, c):
        return sum(model.x[f, a, c] for f in model.F for a in model.A) <= model.company_capacity[c]
    model.company_capacity_constraint = pyo.Constraint(model.C, rule=company_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("\nFlight assignments:")
        for f in model.F:
            for a in model.A:
                for c in model.C:
                    if pyo.value(model.x[f, a, c]) > 0.5:  # Only print assigned flights
                        print(f"Flight {f} assigned to Airport {a} and Company {c}")
        
    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
if __name__ == "__main__":
    flight_scheduling_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.8
**Execution Time**: 0.96 seconds
**Reliability**: high
**Analysis**: Pyomo found an optimal solution, but with the longest 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 | 1.80 | 0.17s | N/A | N/A |
| Docplex | OPTIMAL | 1.80 | 0.95s | N/A | N/A |
| Pyomo | OPTIMAL | 1.80 | 0.96s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 1.8

### Final Recommendation
**Recommended Optimal Value**: 1.8
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its optimal solution and the fastest execution time, indicating superior performance.

### Business Interpretation
**Overall Strategy**: The optimal solution minimizes total fuel consumption, leading to cost savings and improved operational efficiency.
**Objective Value Meaning**: The optimal objective value of 1.8 represents the minimized total fuel consumption across all flights.
**Resource Allocation Summary**: Flights should be assigned to airports and companies in a way that respects capacity constraints and minimizes fuel usage.
**Implementation Recommendations**: Implement the flight assignments as per the optimal solution, ensuring compliance with airport and company capacities, and monitor fuel consumption for continuous improvement.