## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be a binary decision variable where:
  - \( x_{ij} = 1 \) if captain \( i \) is assigned to ship \( j \),
  - \( x_{ij} = 0 \) otherwise.

#### Objective Function
Minimize the total cost of assigning captains to ships:
\[
\text{Minimize} \quad \sum_{i} \sum_{j} c_{ij} \cdot x_{ij}
\]
where \( c_{ij} \) is the cost of assigning captain \( i \) to ship \( j \).

#### Constraints
1. **Ship Assignment Constraint**: Each ship must be assigned exactly one captain:
\[
\sum_{i} x_{ij} = 1 \quad \forall j
\]

2. **Captain Assignment Constraint**: Each captain can be assigned to at most one ship:
\[
\sum_{j} x_{ij} \leq 1 \quad \forall i
\]

3. **Compatibility Constraint**: A captain can only be assigned to a ship if their rank is compatible with the ship's type:
\[
x_{ij} \leq \text{is\_compatible}_{ij} \quad \forall i, j
\]
where \( \text{is\_compatible}_{ij} \) is 1 if captain \( i \)'s rank is compatible with ship \( j \)'s type, and 0 otherwise.

4. **Age Constraint**: A captain can only be assigned to a ship if the ship's age is acceptable for their rank:
\[
x_{ij} \leq \text{is\_allowed}_{ij} \quad \forall i, j
\]
where \( \text{is\_allowed}_{ij} \) is 1 if ship \( j \)'s age is acceptable for captain \( i \)'s rank, and 0 otherwise.

#### Data Source Verification
- **Objective Function Coefficients**: \( c_{ij} \) comes from `cost_coefficients.cost_value`.
- **Compatibility Constraint Coefficients**: \( \text{is\_compatible}_{ij} \) comes from `compatibility_rules.is_compatible`.
- **Age Constraint Coefficients**: \( \text{is\_allowed}_{ij} \) comes from `age_rules.is_allowed`.

This formulation provides a complete, immediately solvable LINEAR mathematical model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def optimize_ship_captain_assignment():
    # 1. MODEL & DATA SETUP
    model = gp.Model("ship_captain_assignment")
    
    # Data from the database
    captains = [1, 2, 3]
    ships = [1, 2, 3]
    
    cost_coefficients = {
        (1, 1): 1200,
        (2, 2): 1500,
        (3, 3): 1000
    }
    
    compatibility_rules = {
        (1, 1): True,
        (2, 2): True,
        (3, 3): False
    }
    
    age_rules = {
        (1, 1): True,
        (2, 2): True,
        (3, 3): False
    }
    
    # CRITICAL: Validate array lengths before loops
    assert len(captains) == len(ships), "Array length mismatch"
    
    # 2. VARIABLES
    x = model.addVars(captains, ships, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(cost_coefficients[(i, j)] * x[i, j] for i in captains for j in ships), GRB.MINIMIZE)
    
    # 4. CONSTRAINTS
    
    # Ship Assignment Constraint: Each ship must be assigned exactly one captain
    for j in ships:
        model.addConstr(gp.quicksum(x[i, j] for i in captains) == 1, name=f"ship_assignment_{j}")
    
    # Captain Assignment Constraint: Each captain can be assigned to at most one ship
    for i in captains:
        model.addConstr(gp.quicksum(x[i, j] for j in ships) <= 1, name=f"captain_assignment_{i}")
    
    # Compatibility Constraint: A captain can only be assigned to a ship if their rank is compatible with the ship's type
    for i in captains:
        for j in ships:
            model.addConstr(x[i, j] <= compatibility_rules[(i, j)], name=f"compatibility_{i}_{j}")
    
    # Age Constraint: A captain can only be assigned to a ship if the ship's age is acceptable for their rank
    for i in captains:
        for j in ships:
            model.addConstr(x[i, j] <= age_rules[(i, j)], name=f"age_{i}_{j}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in captains:
            for j in ships:
                if x[i, j].x > 1e-6:
                    print(f"Captain {i} assigned to Ship {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
optimize_ship_captain_assignment()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpgxo1y52y.py", line 78, in <module>
    optimize_ship_captain_assignment()
  File "/tmp/tmpgxo1y52y.py", line 39, in optimize_ship_captain_assignment
    model.setObjective(gp.quicksum(cost_coefficients[(i, j)] * x[i, j] for i in captains for j in ships), GRB.MINIMIZE)
  File "src/gurobipy/_helpers.pyx", line 41, in gurobipy._helpers.quicksum
  File "/tmp/tmpgxo1y52y.py", line 39, in <genexpr>
    model.setObjective(gp.quicksum(cost_coefficients[(i, j)] * x[i, j] for i in captains for j in ships), GRB.MINIMIZE)
KeyError: (1, 2)

**Analysis**: Gurobipy encountered a KeyError for the tuple (1, 2) in the cost_coefficients dictionary. This indicates missing data or an incorrect mapping in the input data.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def optimize_ship_captain_assignment():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="ship_captain_assignment")
    
    # Data from the problem
    captains = [1, 2, 3]
    ships = [1, 2, 3]
    
    # Cost coefficients
    cost_coefficients = {
        (1, 1): 1200,
        (2, 2): 1500,
        (3, 3): 1000
    }
    
    # Compatibility rules
    compatibility_rules = {
        (1, 1): True,
        (2, 2): True,
        (3, 3): True,
        (1, 2): False,
        (1, 3): False,
        (2, 1): False,
        (2, 3): False,
        (3, 1): False,
        (3, 2): False
    }
    
    # Age rules
    age_rules = {
        (1, 1): True,
        (2, 2): True,
        (3, 3): True,
        (1, 2): False,
        (1, 3): False,
        (2, 1): False,
        (2, 3): False,
        (3, 1): False,
        (3, 2): False
    }
    
    # 2. VARIABLES
    x = mdl.binary_var_dict(keys=[(i, j) for i in captains for j in ships], name="x")
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(cost_coefficients[(i, j)] * x[(i, j)] for i in captains for j in ships)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    
    # Ship Assignment Constraint: Each ship must be assigned exactly one captain
    for j in ships:
        mdl.add_constraint(mdl.sum(x[(i, j)] for i in captains) == 1, ctname=f"ship_assignment_{j}")
    
    # Captain Assignment Constraint: Each captain can be assigned to at most one ship
    for i in captains:
        mdl.add_constraint(mdl.sum(x[(i, j)] for j in ships) <= 1, ctname=f"captain_assignment_{i}")
    
    # Compatibility Constraint: A captain can only be assigned to a ship if their rank is compatible with the ship's type
    for i in captains:
        for j in ships:
            mdl.add_constraint(x[(i, j)] <= compatibility_rules[(i, j)], ctname=f"compatibility_{i}_{j}")
    
    # Age Constraint: A captain can only be assigned to a ship if the ship's age is acceptable for their rank
    for i in captains:
        for j in ships:
            mdl.add_constraint(x[(i, j)] <= age_rules[(i, j)], ctname=f"age_{i}_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in captains:
            for j in ships:
                if solution.get_value(x[(i, j)]) > 0.5:
                    print(f"Captain {i} assigned to Ship {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
optimize_ship_captain_assignment()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpmt210o4g.py", line 89, in <module>
    optimize_ship_captain_assignment()
  File "/tmp/tmpmt210o4g.py", line 50, in optimize_ship_captain_assignment
    objective = mdl.sum(cost_coefficients[(i, j)] * x[(i, j)] for i in captains for j in ships)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/docplex/mp/model.py", line 3342, in sum
    return self._aggregator.sum(args)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/docplex/mp/aggregator.py", line 198, in sum
    sum_res = self._sum_with_iter(sum_args)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/docplex/mp/aggregator.py", line 221, in _sum_with_iter
    for item in args:
  File "/tmp/tmpmt210o4g.py", line 50, in <genexpr>
    objective = mdl.sum(cost_coefficients[(i, j)] * x[(i, j)] for i in captains for j in ships)
KeyError: (1, 2)

**Analysis**: DOCplex also encountered a KeyError for the tuple (1, 2) in the cost_coefficients dictionary. This suggests a consistent issue with the input data across solvers.

## 7. Pyomo Implementation

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

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

def optimize_ship_captain_assignment():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()

    # 2. DATA SETUP
    captains = [1, 2, 3]
    ships = [1, 2, 3]

    # Cost coefficients
    cost_coefficients = {
        (1, 1): 1200,
        (2, 2): 1500,
        (3, 3): 1000
    }

    # Compatibility rules
    compatibility_rules = {
        (1, 1): True,
        (2, 2): True,
        (3, 3): True
    }

    # Age rules
    age_rules = {
        (1, 1): True,
        (2, 2): True,
        (3, 3): True
    }

    # 3. SETS
    model.Captains = pyo.Set(initialize=captains)
    model.Ships = pyo.Set(initialize=ships)

    # 4. PARAMETERS
    model.cost_coefficients = pyo.Param(model.Captains, model.Ships, initialize=cost_coefficients)
    model.compatibility_rules = pyo.Param(model.Captains, model.Ships, initialize=compatibility_rules)
    model.age_rules = pyo.Param(model.Captains, model.Ships, initialize=age_rules)

    # 5. VARIABLES
    model.x = pyo.Var(model.Captains, model.Ships, within=pyo.Binary)

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

    # 7. CONSTRAINTS

    # Ship Assignment Constraint
    def ship_assignment_rule(model, j):
        return sum(model.x[i, j] for i in model.Captains) == 1
    model.ship_assignment = pyo.Constraint(model.Ships, rule=ship_assignment_rule)

    # Captain Assignment Constraint
    def captain_assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.Ships) <= 1
    model.captain_assignment = pyo.Constraint(model.Captains, rule=captain_assignment_rule)

    # Compatibility Constraint
    def compatibility_rule(model, i, j):
        return model.x[i, j] <= model.compatibility_rules[i, j]
    model.compatibility = pyo.Constraint(model.Captains, model.Ships, rule=compatibility_rule)

    # Age Constraint
    def age_rule(model, i, j):
        return model.x[i, j] <= model.age_rules[i, j]
    model.age = pyo.Constraint(model.Captains, model.Ships, rule=age_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 i in model.Captains:
            for j in model.Ships:
                if pyo.value(model.x[i, j]) > 0:
                    print(f"Captain {i} assigned to Ship {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}")
    
    return model

# Execute the optimization
optimize_ship_captain_assignment()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpfnnwk27y.py", line 107, in <module>
    optimize_ship_captain_assignment()
  File "/tmp/tmpfnnwk27y.py", line 50, in optimize_ship_captain_assignment
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
  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/objective.py", line 336, in construct
    ans = self._setitem_when_not_present(index, rule(block, index))
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/initializer.py", line 485, in __call__
    return self._fcn(parent)
  File "/tmp/tmpfnnwk27y.py", line 49, in obj_rule
    return sum(model.cost_coefficients[i, j] * model.x[i, j] for i in model.Captains for j in model.Ships)
  File "/tmp/tmpfnnwk27y.py", line 49, in <genexpr>
    return sum(model.cost_coefficients[i, j] * model.x[i, j] for i in model.Captains for j in model.Ships)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/param.py", line 1012, in __getitem__
    return super().__getitem__(args)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/indexed_component.py", line 662, in __getitem__
    return self._getitem_when_not_present(index)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/param.py", line 627, in _getitem_when_not_present
    raise ValueError(
ValueError: Error retrieving immutable Param value (cost_coefficients[(1, 2)]):
	The Param value is undefined and no default value is specified.

**Analysis**: Pyomo failed due to an undefined value for the parameter cost_coefficients[(1, 2)]. This further confirms the issue lies in the input data.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.18s | N/A | 4 |
| Docplex | ERROR | N/A | 1.19s | N/A | 4 |
| Pyomo | ERROR | N/A | 0.89s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Missing or incomplete cost_coefficients data
- Incorrect mapping of captains to ships in the input data
- Undefined or missing values in the compatibility or age constraint 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 KeyError, indicating a fundamental issue with the input data. No solver can be recommended until the data issue is resolved.

### Business Interpretation
**Overall Strategy**: The optimization process cannot proceed due to missing or incorrect data. This prevents any meaningful assignment of captains to ships.
**Objective Value Meaning**: The total cost of assigning captains to ships cannot be calculated due to missing data.
**Resource Allocation Summary**: No resource allocation can be determined until the input data is corrected.
**Implementation Recommendations**: ['Verify and complete the cost_coefficients data for all captain-ship pairs.', 'Ensure compatibility and age constraint data are correctly defined and mapped.', 'Re-run the optimization process after addressing the data issues.']