## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be the number of flights assigned to pilot \( i \) for aircraft \( j \).
  - \( x_{ij} \) is an integer decision variable representing the number of flights.

#### Objective Function
- Minimize the total number of flights assigned:
  \[
  \text{Minimize } \sum_{i,j} x_{ij}
  \]

#### Constraints
1. **Pilot Minimum Flight Requirement:**
   - Each pilot must be assigned at least their minimum required flights:
   \[
   \sum_{j} x_{ij} \geq \text{min\_flights}_i \quad \forall i
   \]
   - Data Source Verification: \( \text{min\_flights}_i \) from `pilot_constraints.min_flights`

2. **Aircraft Maximum Flight Capacity:**
   - Each aircraft must not exceed its maximum flight capacity:
   \[
   \sum_{i} x_{ij} \leq \text{max\_flights}_j \quad \forall j
   \]
   - Data Source Verification: \( \text{max\_flights}_j \) from `aircraft_constraints.max_flights`

#### Data Source Verification
- Decision Variables \( x_{ij} \) are derived from `flights_assigned.number_of_flights`.
- Coefficients for constraints are sourced from:
  - `pilot_constraints.min_flights` for minimum flight requirements.
  - `aircraft_constraints.max_flights` for maximum flight capacities.

This linear model is immediately solvable using standard linear programming solvers, ensuring that all pilots meet their minimum flight requirements while no aircraft exceeds its maximum capacity, thereby minimizing the total number of flights assigned.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_pilot_assignment():
    """Optimize the assignment of pilots to aircraft to minimize total flights."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("pilot_record")
    
    # Data from the database
    pilots = [1, 2, 3]
    aircrafts = [101, 102, 103]
    
    # Minimum flights required for each pilot
    min_flights = {1: 10, 2: 15, 3: 12}
    
    # Maximum flights allowed for each aircraft
    max_flights = {101: 120, 102: 100, 103: 110}
    
    # Validate data lengths
    assert len(min_flights) == len(pilots), "Mismatch in pilots and min_flights data"
    assert len(max_flights) == len(aircrafts), "Mismatch in aircrafts and max_flights data"
    
    # 2. VARIABLES
    # Decision variables: number of flights assigned to each pilot for each aircraft
    x = model.addVars(pilots, aircrafts, vtype=GRB.INTEGER, name="x", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the total number of flights assigned
    model.setObjective(gp.quicksum(x[i, j] for i in pilots for j in aircrafts), GRB.MINIMIZE)
    
    # 4. CONSTRAINTS
    # Pilot minimum flight requirement
    model.addConstrs((gp.quicksum(x[i, j] for j in aircrafts) >= min_flights[i] for i in pilots), name="pilot_min_flights")
    
    # Aircraft maximum flight capacity
    model.addConstrs((gp.quicksum(x[i, j] for i in pilots) <= max_flights[j] for j in aircrafts), name="aircraft_max_flights")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in pilots:
            for j in aircrafts:
                if x[i, j].x > 1e-6:
                    print(f"x[{i},{j}] = {x[i, j].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_pilot_assignment()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 37.0
**Execution Time**: 2.55 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the shortest execution time, indicating efficient performance and reliability.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_pilot_assignments():
    """Optimize the assignment of pilots to aircraft to minimize total flights."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="pilot_record")
    
    # Data from the database schema
    pilots = [1, 2, 3]
    aircrafts = [101, 102, 103]
    
    # Minimum flights required for each pilot
    min_flights = {1: 10, 2: 15, 3: 12}
    
    # Maximum flights allowed for each aircraft
    max_flights = {101: 120, 102: 100, 103: 110}
    
    # Validate array lengths to prevent IndexError
    assert len(pilots) == len(min_flights), "Mismatch in pilots and min_flights"
    assert len(aircrafts) == len(max_flights), "Mismatch in aircrafts and max_flights"
    
    # Safe indexing range
    safe_pilot_range = range(len(pilots))
    safe_aircraft_range = range(len(aircrafts))
    
    # 2. VARIABLES
    # Decision variables: number of flights assigned to each pilot for each aircraft
    x = {(i, j): mdl.integer_var(name=f"x_{i}_{j}", lb=0) for i in pilots for j in aircrafts}
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the total number of flights assigned
    objective = mdl.sum(x[i, j] for i in pilots for j in aircrafts)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    
    # Pilot minimum flight requirement
    for i in pilots:
        mdl.add_constraint(mdl.sum(x[i, j] for j in aircrafts) >= min_flights[i], ctname=f"min_flights_{i}")
    
    # Aircraft maximum flight capacity
    for j in aircrafts:
        mdl.add_constraint(mdl.sum(x[i, j] for i in pilots) <= max_flights[j], ctname=f"max_flights_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in pilots:
            for j in aircrafts:
                value = solution.get_value(x[i, j])
                if value > 1e-6:
                    print(f"x[{i},{j}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Run the optimization
optimize_pilot_assignments()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 37.0
**Execution Time**: 10.02 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, albeit with a longer execution time compared to Gurobipy, but still reliable.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def pilot_record_optimization():
    """Optimize pilot flight assignments to minimize total flights"""

    # 1. MODEL CREATION
    model = pyo.ConcreteModel()

    # 2. DATA SETUP
    pilots = [1, 2, 3]
    aircrafts = [101, 102, 103]
    min_flights = {1: 10, 2: 15, 3: 12}
    max_flights = {101: 120, 102: 100, 103: 110}

    # CRITICAL: Validate data lengths before indexing
    assert len(min_flights) == len(pilots), "Mismatch in pilots and min_flights data"
    assert len(max_flights) == len(aircrafts), "Mismatch in aircrafts and max_flights data"

    # 3. SETS
    model.P = pyo.Set(initialize=pilots)
    model.A = pyo.Set(initialize=aircrafts)

    # 4. PARAMETERS
    model.min_flights = pyo.Param(model.P, initialize=min_flights)
    model.max_flights = pyo.Param(model.A, initialize=max_flights)

    # 5. VARIABLES
    model.x = pyo.Var(model.P, model.A, within=pyo.NonNegativeIntegers)

    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.x[i, j] for i in model.P for j in model.A)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)

    # 7. CONSTRAINTS

    # Pilot Minimum Flight Requirement
    def pilot_min_flights_rule(model, i):
        return sum(model.x[i, j] for j in model.A) >= model.min_flights[i]
    model.pilot_min_flights_constraint = pyo.Constraint(model.P, rule=pilot_min_flights_rule)

    # Aircraft Maximum Flight Capacity
    def aircraft_max_flights_rule(model, j):
        return sum(model.x[i, j] for i in model.P) <= model.max_flights[j]
    model.aircraft_max_flights_constraint = pyo.Constraint(model.A, rule=aircraft_max_flights_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)

    # 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.P:
            for j in model.A:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0:  # Only print non-zero values
                    print(f"Flights assigned to pilot {i} for aircraft {j}: {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

# Run the optimization
pilot_record_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 37.0
**Execution Time**: 8.30 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a moderate execution time, demonstrating reliability similar to the other solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 37.00 | 2.55s | N/A | N/A |
| Docplex | OPTIMAL | 37.00 | 10.02s | N/A | N/A |
| Pyomo | OPTIMAL | 37.00 | 8.30s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 37.0

### Final Recommendation
**Recommended Optimal Value**: 37.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time, which is beneficial for operational efficiency in real-time decision-making scenarios.

### Business Interpretation
**Overall Strategy**: The optimal assignment of flights minimizes the total number of flights while meeting all pilot and aircraft constraints.
**Objective Value Meaning**: The optimal objective value of 37 indicates the minimum total number of flights assigned across all pilots and aircraft, ensuring efficient resource utilization.
**Resource Allocation Summary**: Resources should be allocated such that each pilot meets their minimum flight requirements without exceeding aircraft capacities.
**Implementation Recommendations**: Implement the solution by assigning flights according to the optimal decision variables, ensuring compliance with all constraints and monitoring for any changes in pilot or aircraft availability.