## 4. Mathematical Optimization Formulation

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

#### Objective Function
Minimize the total number of deaths:
\[
\text{Minimize } \sum_{i,j} \text{deaths}_{ij} \times x_{ij}
\]
where \(\text{deaths}_{ij}\) is the number of deaths caused by ship \( i \) in battle \( j \), sourced from the `deaths_by_ship_battle` table.

#### Constraints
1. **Each ship can be assigned to at most one battle:**
   \[
   \sum_{j} x_{ij} \leq 1 \quad \forall i
   \]

2. **Each battle must receive at least the minimum required number of ships:**
   \[
   \sum_{i} x_{ij} \geq \text{min\_ships}_j \quad \forall j
   \]
   where \(\text{min\_ships}_j\) is the minimum number of ships required for battle \( j \), sourced from the `battle_requirements` table.

3. **The total number of ships assigned cannot exceed the maximum number available:**
   \[
   \sum_{i,j} x_{ij} \leq \text{max\_ships}
   \]
   where \(\text{max\_ships}\) is a scalar parameter representing the maximum number of ships available for assignment.

#### Data Source Verification:
- \(\text{deaths}_{ij}\) from `deaths_by_ship_battle.deaths`
- \(\text{min\_ships}_j\) from `battle_requirements.min_ships`
- \(\text{max\_ships}\) is a business configuration parameter not explicitly provided in the data but assumed to be known from operational constraints.

This formulation provides a complete and solvable linear model for minimizing casualties by optimally assigning ships to battles under the given constraints.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def naval_optimization():
    # 1. MODEL & DATA SETUP
    model = gp.Model("battle_death_minimization")
    
    # Example data setup (replace with actual data retrieval logic)
    ships = [1, 2, 3]
    battles = [101, 102, 103]
    deaths_data = {
        (1, 101): 15,
        (2, 102): 25,
        (3, 103): 5
    }
    min_ships_required = {
        101: 3,
        102: 5,
        103: 2
    }
    max_ships = 5  # Example maximum number of ships available

    # CRITICAL: Validate array lengths before loops
    assert len(deaths_data) == len(ships) * len(battles), "Array length mismatch"

    # 2. VARIABLES
    # Binary decision variables for ship assignments
    x = model.addVars(ships, battles, vtype=GRB.BINARY, name="x")

    # 3. OBJECTIVE FUNCTION
    # Minimize the total number of deaths
    model.setObjective(gp.quicksum(deaths_data[i, j] * x[i, j] for i in ships for j in battles), GRB.MINIMIZE)

    # 4. CONSTRAINTS
    # Each ship can be assigned to at most one battle
    model.addConstrs((gp.quicksum(x[i, j] for j in battles) <= 1 for i in ships), name="ship_assignment")

    # Each battle must receive at least the minimum required number of ships
    model.addConstrs((gp.quicksum(x[i, j] for i in ships) >= min_ships_required[j] for j in battles), name="battle_requirement")

    # The total number of ships assigned cannot exceed the maximum number available
    model.addConstr(gp.quicksum(x[i, j] for i in ships for j in battles) <= max_ships, name="max_ships")

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in ships:
            for j in battles:
                if x[i, j].x > 1e-6:
                    print(f"Ship {i} assigned to Battle {j}: {x[i, j].x:.0f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

    return model

# Run the optimization
naval_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpth5ku8br.py", line 63, in <module>
    naval_optimization()
  File "/tmp/tmpth5ku8br.py", line 26, in naval_optimization
    assert len(deaths_data) == len(ships) * len(battles), "Array length mismatch"
AssertionError: Array length mismatch

**Analysis**: Gurobipy encountered an 'Array length mismatch' error, indicating a potential issue with the input data dimensions or data integrity.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def naval_optimization():
    """Optimizes ship assignments to minimize battle deaths."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="battle_death_minimization")
    
    # Data from the problem context
    ships = [1, 2, 3]
    battles = [101, 102, 103]
    deaths_data = {
        (1, 101): 15,
        (2, 102): 25,
        (3, 103): 5
    }
    min_ships_required = {
        101: 3,
        102: 5,
        103: 2
    }
    max_ships = 5  # Assumed maximum number of ships available
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(deaths_data) == len(ships), "Array length mismatch"
    
    # 2. VARIABLES
    # Binary decision variables for ship assignments
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in ships for j in battles}
    
    # 3. OBJECTIVE FUNCTION
    # Minimize total deaths
    objective = mdl.sum(deaths_data.get((i, j), 0) * x[i, j] for i in ships for j in battles)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    
    # Each ship can be assigned to at most one battle
    for i in ships:
        mdl.add_constraint(mdl.sum(x[i, j] for j in battles) <= 1, ctname=f"ship_{i}_assignment")
    
    # Each battle must receive at least the minimum required number of ships
    for j in battles:
        mdl.add_constraint(mdl.sum(x[i, j] for i in ships) >= min_ships_required[j], ctname=f"battle_{j}_min_ships")
    
    # The total number of ships assigned cannot exceed the maximum number available
    mdl.add_constraint(mdl.sum(x[i, j] for i in ships for j in battles) <= max_ships, ctname="max_ships_limit")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in ships:
            for j in battles:
                if solution.get_value(x[i, j]) > 0.5:  # Binary variable threshold
                    print(f"Ship {i} assigned to Battle {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

# Execute the optimization
naval_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex reported the problem as infeasible, suggesting that the constraints may be too restrictive or conflicting.

## 7. Pyomo Implementation

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

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

def naval_optimization():
    """Optimize ship assignments to minimize battle deaths."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the problem description
    ships = [1, 2, 3]
    battles = [101, 102, 103]
    deaths_data = {
        (1, 101): 15,
        (2, 102): 25,
        (3, 103): 5
    }
    min_ships_data = {
        101: 3,
        102: 5,
        103: 2
    }
    max_ships = 5  # Example maximum number of ships available
    
    # CRITICAL: Validate array lengths before indexing
    assert len(deaths_data) == len(ships) * len(battles), "Array length mismatch"
    
    # 3. SETS
    model.Ships = pyo.Set(initialize=ships)
    model.Battles = pyo.Set(initialize=battles)
    
    # 4. PARAMETERS
    model.deaths = pyo.Param(model.Ships, model.Battles, initialize=deaths_data, default=0)
    model.min_ships = pyo.Param(model.Battles, initialize=min_ships_data)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.Ships, model.Battles, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.deaths[i, j] * model.x[i, j] for i in model.Ships for j in model.Battles)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    
    # Each ship can be assigned to at most one battle
    def ship_assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.Battles) <= 1
    model.ship_assignment_constraint = pyo.Constraint(model.Ships, rule=ship_assignment_rule)
    
    # Each battle must receive at least the minimum required number of ships
    def battle_requirement_rule(model, j):
        return sum(model.x[i, j] for i in model.Ships) >= model.min_ships[j]
    model.battle_requirement_constraint = pyo.Constraint(model.Battles, rule=battle_requirement_rule)
    
    # The total number of ships assigned cannot exceed the maximum number available
    def max_ships_rule(model):
        return sum(model.x[i, j] for i in model.Ships for j in model.Battles) <= max_ships
    model.max_ships_constraint = pyo.Constraint(rule=max_ships_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
    # 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("\nAssignment values:")
        for i in model.Ships:
            for j in model.Battles:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0.5:  # Only print assignments
                    print(f"Ship {i} assigned to Battle {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
naval_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmp3zx6t6c7.py", line 98, in <module>
    naval_optimization()
  File "/tmp/tmp3zx6t6c7.py", line 29, in naval_optimization
    assert len(deaths_data) == len(ships) * len(battles), "Array length mismatch"
AssertionError: Array length mismatch

**Analysis**: Pyomo also encountered an 'Array length mismatch' error, similar to Gurobipy, indicating a consistent issue with 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 | 0.77s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 8.08s | N/A | 4 |
| Pyomo | ERROR | N/A | 4.36s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Data integrity issues leading to array length mismatches
- Overly restrictive or conflicting constraints causing infeasibility
- Incorrect or incomplete data input
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: Given the errors and infeasibility, no single solver can be recommended. The focus should be on resolving data and model issues.

### Business Interpretation
**Overall Strategy**: The current model and data inputs are not yielding a feasible or error-free solution, indicating a need for data verification and model adjustment.
**Objective Value Meaning**: The optimal objective value represents the minimized total number of deaths across all battles.
**Resource Allocation Summary**: Currently, no feasible allocation of ships to battles can be determined due to data and model issues.
**Implementation Recommendations**: Verify and correct the input data dimensions and integrity. Re-evaluate the constraints to ensure they are not overly restrictive or conflicting. Once resolved, re-run the optimization using multiple solvers to ensure consistency.