# Complete Optimization Problem and Solution: battle_death

## 1. Problem Context and Goals

### Context  
In military operations, the allocation of ships to battles is a critical decision that directly impacts the outcome of engagements. The goal is to minimize the total number of casualties (killed and injured) across all battles by optimally deploying ships, considering their tonnage and the constraints of each battle. Each ship can only be deployed to one battle, and the total tonnage of ships deployed to a battle must not exceed the maximum tonnage allowed for that battle. The decision to deploy a ship to a specific battle is represented as a binary choice: a ship is either deployed or not. The total casualties in a battle are calculated as the sum of killed and injured personnel, which serves as the primary metric for optimization. The maximum tonnage allowed per battle is a key operational parameter that ensures realistic and feasible deployment scenarios.

### Goals  
The primary goal of this optimization problem is to minimize the total number of casualties across all battles. This is achieved by strategically deciding which ships to deploy to each battle, ensuring that the total tonnage of deployed ships does not exceed the maximum allowed for each battle. Success is measured by the reduction in the sum of killed and injured personnel across all battles, aligning with the operational objective of minimizing human losses.

## 2. Constraints    

1. **Tonnage Constraint**: For each battle, the total tonnage of all ships deployed to that battle must not exceed the maximum tonnage allowed for that battle. This ensures that the deployment of ships is within the operational capacity of each battle.

2. **Single Deployment Constraint**: Each ship can only be deployed to one battle. This ensures that ships are not overcommitted and are available for their assigned engagements.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for ship tonnage, battle constraints, and ship-to-battle deployment decisions. Configuration logic updates include scalar parameters for maximum tonnage and formulas for casualty calculations.

CREATE TABLE ship_tonnage (
  ship_id INTEGER,
  tonnage INTEGER
);

CREATE TABLE battle_constraints (
  battle_id INTEGER,
  max_tonnage INTEGER
);

CREATE TABLE ship_deployment (
  ship_id INTEGER,
  battle_id INTEGER,
  deployed BOOLEAN
);

CREATE TABLE death (
  battle_id INTEGER,
  ship_id INTEGER,
  killed INTEGER,
  injured INTEGER
);


```

### Data Dictionary  
- **ship_tonnage**: Contains the tonnage of each ship available for deployment.  
  - *ship_id*: Unique identifier for each ship.  
  - *tonnage*: Tonnage of the ship, used in the tonnage constraint.  

- **battle_constraints**: Specifies the maximum tonnage allowed per battle.  
  - *battle_id*: Unique identifier for each battle.  
  - *max_tonnage*: Maximum tonnage allowed in the battle, used as a constraint bound in the optimization model.  

- **ship_deployment**: Represents the binary decision variable indicating if a ship is deployed to a battle.  
  - *ship_id*: Unique identifier for each ship.  
  - *battle_id*: Unique identifier for each battle.  
  - *deployed*: Indicates if the ship is deployed to the battle, used as a decision variable in the optimization model.  

- **death**: Records the casualties in each battle.  
  - *battle_id*: Unique identifier for each battle.  
  - *ship_id*: Unique identifier for each ship.  
  - *killed*: Number of killed in the battle, used as an objective coefficient in the optimization model.  
  - *injured*: Number of injured in the battle, used as an objective coefficient in the optimization model.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic military ship tonnages, battle constraints, and casualty figures, ensuring a meaningful and solvable optimization problem.

-- Realistic data for ship_tonnage
INSERT INTO ship_tonnage (ship_id, tonnage) VALUES (1, 5000);
INSERT INTO ship_tonnage (ship_id, tonnage) VALUES (2, 7000);
INSERT INTO ship_tonnage (ship_id, tonnage) VALUES (3, 9000);

-- Realistic data for battle_constraints
INSERT INTO battle_constraints (battle_id, max_tonnage) VALUES (1, 10000);
INSERT INTO battle_constraints (battle_id, max_tonnage) VALUES (2, 12000);
INSERT INTO battle_constraints (battle_id, max_tonnage) VALUES (3, 15000);

-- Realistic data for ship_deployment
INSERT INTO ship_deployment (ship_id, battle_id, deployed) VALUES (1, 1, True);
INSERT INTO ship_deployment (ship_id, battle_id, deployed) VALUES (2, 2, True);
INSERT INTO ship_deployment (ship_id, battle_id, deployed) VALUES (3, 3, True);

-- Realistic data for death
INSERT INTO death (battle_id, ship_id, killed, injured) VALUES (1, 1, 10, 15);
INSERT INTO death (battle_id, ship_id, killed, injured) VALUES (2, 2, 20, 25);
INSERT INTO death (battle_id, ship_id, killed, injured) VALUES (3, 3, 30, 35);


```

## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{s,b} \): Binary decision variable indicating whether ship \( s \) is deployed to battle \( b \).  
  - \( x_{s,b} = 1 \) if ship \( s \) is deployed to battle \( b \), otherwise \( x_{s,b} = 0 \).

#### Objective Function
Minimize the total number of casualties (killed + injured) across all battles:  
\[
\text{Minimize} \quad \sum_{s} \sum_{b} (k_{s,b} + i_{s,b}) \cdot x_{s,b}
\]  
Where:  
- \( k_{s,b} \): Number of killed personnel if ship \( s \) is deployed to battle \( b \) (from `death.killed`).  
- \( i_{s,b} \): Number of injured personnel if ship \( s \) is deployed to battle \( b \) (from `death.injured`).

#### Constraints
1. **Tonnage Constraint**: For each battle \( b \), the total tonnage of deployed ships must not exceed the maximum allowed tonnage:  
\[
\sum_{s} t_{s} \cdot x_{s,b} \leq T_{b} \quad \forall b
\]  
Where:  
- \( t_{s} \): Tonnage of ship \( s \) (from `ship_tonnage.tonnage`).  
- \( T_{b} \): Maximum tonnage allowed for battle \( b \) (from `battle_constraints.max_tonnage`).  

2. **Single Deployment Constraint**: Each ship \( s \) can only be deployed to one battle:  
\[
\sum_{b} x_{s,b} \leq 1 \quad \forall s
\]

#### Data Source Verification
- \( k_{s,b} \): `death.killed`  
- \( i_{s,b} \): `death.injured`  
- \( t_{s} \): `ship_tonnage.tonnage`  
- \( T_{b} \): `battle_constraints.max_tonnage`  

This is a complete, immediately solvable Mixed-Integer Linear Programming (MILP) 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_battle_death():
    # 1. MODEL & DATA SETUP
    model = gp.Model("battle_death_optimization")
    
    # Data from the problem
    ships = [1, 2, 3]
    battles = [1, 2, 3]
    
    ship_tonnage = {
        1: 5000,
        2: 7000,
        3: 9000
    }
    
    battle_constraints = {
        1: 10000,
        2: 12000,
        3: 15000
    }
    
    death_data = {
        (1, 1): {'killed': 10, 'injured': 15},
        (2, 2): {'killed': 20, 'injured': 25},
        (3, 3): {'killed': 30, 'injured': 35}
    }
    
    # CRITICAL: Validate array lengths before loops
    assert len(ships) == len(ship_tonnage), "Ship tonnage data length mismatch"
    assert len(battles) == len(battle_constraints), "Battle constraints data length mismatch"
    assert len(death_data) == len(ships) * len(battles), "Casualties data length mismatch"
    
    # 2. VARIABLES
    x = model.addVars(ships, battles, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(
        gp.quicksum((death_data[(s, b)]['killed'] + death_data[(s, b)]['injured']) * x[s, b] 
                    for s in ships for b in battles), 
        GRB.MINIMIZE
    )
    
    # 4. CONSTRAINTS
    
    # Tonnage Constraint
    for b in battles:
        model.addConstr(
            gp.quicksum(ship_tonnage[s] * x[s, b] for s in ships) <= battle_constraints[b], 
            name=f"tonnage_{b}"
        )
    
    # Single Deployment Constraint
    for s in ships:
        model.addConstr(
            gp.quicksum(x[s, b] for b in battles) <= 1, 
            name=f"single_deployment_{s}"
        )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for s in ships:
            for b in battles:
                if x[s, b].x > 1e-6:
                    print(f"Ship {s} deployed to Battle {b}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
optimize_battle_death()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpeau32k50.py", line 80, in <module>
    optimize_battle_death()
  File "/tmp/tmpeau32k50.py", line 35, in optimize_battle_death
    assert len(death_data) == len(ships) * len(battles), "Casualties data length mismatch"
AssertionError: Casualties data length mismatch

**Analysis**: Gurobipy encountered an error due to a data mismatch in the casualties data. The error message indicates that the length of the casualties data does not match the expected length based on the number of ships and battles. This suggests a potential issue with data preparation or input.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def optimize_battle_death():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="battle_death_optimization")
    
    # Data from the database
    ships = [1, 2, 3]
    battles = [1, 2, 3]
    
    ship_tonnage = {
        1: 5000,
        2: 7000,
        3: 9000
    }
    
    battle_constraints = {
        1: 10000,
        2: 12000,
        3: 15000
    }
    
    death_data = {
        (1, 1): {'killed': 10, 'injured': 15},
        (2, 2): {'killed': 20, 'injured': 25},
        (3, 3): {'killed': 30, 'injured': 35}
    }
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(ships) == len(ship_tonnage), "Ship tonnage data length mismatch"
    assert len(battles) == len(battle_constraints), "Battle constraints data length mismatch"
    assert len(death_data) == len(ships) * len(battles), "Casualties data length mismatch"
    
    # 2. VARIABLES
    x = {(s, b): mdl.binary_var(name=f"x_{s}_{b}") for s in ships for b in battles}
    
    # 3. OBJECTIVE FUNCTION
    total_casualties = mdl.sum((death_data[(s, b)]['killed'] + death_data[(s, b)]['injured']) * x[(s, b)] for s in ships for b in battles)
    mdl.minimize(total_casualties)
    
    # 4. CONSTRAINTS
    
    # Tonnage Constraint
    for b in battles:
        mdl.add_constraint(mdl.sum(ship_tonnage[s] * x[(s, b)] for s in ships) <= battle_constraints[b], ctname=f"tonnage_{b}")
    
    # Single Deployment Constraint
    for s in ships:
        mdl.add_constraint(mdl.sum(x[(s, b)] for b in battles) <= 1, ctname=f"single_deployment_{s}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for s in ships:
            for b in battles:
                if solution.get_value(x[(s, b)]) > 0.5:
                    print(f"Ship {s} deployed to Battle {b}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
optimize_battle_death()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpnqac0bv5.py", line 69, in <module>
    optimize_battle_death()
  File "/tmp/tmpnqac0bv5.py", line 34, in optimize_battle_death
    assert len(death_data) == len(ships) * len(battles), "Casualties data length mismatch"
AssertionError: Casualties data length mismatch

**Analysis**: DOCplex also encountered the same error as Gurobipy, indicating a data mismatch in the casualties data. This consistency in errors across solvers points to a systemic issue with the data rather than a solver-specific problem.

## 7. Pyomo Implementation

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

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

def optimize_battle_death():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    ships = [1, 2, 3]
    battles = [1, 2, 3]
    
    # Ship tonnage data
    ship_tonnage_data = {
        1: 5000,
        2: 7000,
        3: 9000
    }
    
    # Battle constraints data
    battle_constraints_data = {
        1: 10000,
        2: 12000,
        3: 15000
    }
    
    # Casualties data
    casualties_data = {
        (1, 1): (10, 15),
        (2, 2): (20, 25),
        (3, 3): (30, 35)
    }
    
    # Validate array lengths
    assert len(ship_tonnage_data) == len(ships), "Ship tonnage data length mismatch"
    assert len(battle_constraints_data) == len(battles), "Battle constraints data length mismatch"
    assert len(casualties_data) == len(ships) * len(battles), "Casualties data length mismatch"
    
    # 3. SETS
    model.S = pyo.Set(initialize=ships)
    model.B = pyo.Set(initialize=battles)
    
    # 4. PARAMETERS
    model.tonnage = pyo.Param(model.S, initialize=ship_tonnage_data)
    model.max_tonnage = pyo.Param(model.B, initialize=battle_constraints_data)
    model.killed = pyo.Param(model.S, model.B, initialize=lambda model, s, b: casualties_data[(s, b)][0])
    model.injured = pyo.Param(model.S, model.B, initialize=lambda model, s, b: casualties_data[(s, b)][1])
    
    # 5. VARIABLES
    model.x = pyo.Var(model.S, model.B, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum((model.killed[s, b] + model.injured[s, b]) * model.x[s, b] for s in model.S for b in model.B)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    # Tonnage Constraint
    def tonnage_rule(model, b):
        return sum(model.tonnage[s] * model.x[s, b] for s in model.S) <= model.max_tonnage[b]
    model.tonnage_constraint = pyo.Constraint(model.B, rule=tonnage_rule)
    
    # Single Deployment Constraint
    def deployment_rule(model, s):
        return sum(model.x[s, b] for b in model.B) <= 1
    model.deployment_constraint = pyo.Constraint(model.S, rule=deployment_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("\nDeployment decisions:")
        for s in model.S:
            for b in model.B:
                if pyo.value(model.x[s, b]) > 0.5:  # Only print deployed ships
                    print(f"Ship {s} deployed to Battle {b}")
        
    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_battle_death()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpke0ccvns.py", line 103, in <module>
    optimize_battle_death()
  File "/tmp/tmpke0ccvns.py", line 38, in optimize_battle_death
    assert len(casualties_data) == len(ships) * len(battles), "Casualties data length mismatch"
AssertionError: Casualties data length mismatch

**Analysis**: Pyomo failed with the same error as the other solvers, further confirming that the issue lies in the data preparation. The error message explicitly mentions a mismatch in the length of the casualties data, which needs to be addressed before reattempting the optimization.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.17s | N/A | 4 |
| Docplex | ERROR | N/A | 1.06s | 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**:
- Data mismatch in casualties data
- 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 encountered the same error, indicating a fundamental issue with the data rather than the solvers themselves. The data must be corrected before any solver can be effectively used.

### Business Interpretation
**Overall Strategy**: The optimization cannot proceed due to data inconsistencies, meaning no actionable deployment strategy can be derived at this time.
**Objective Value Meaning**: The objective value represents the total number of casualties (killed + injured) across all battles. However, it cannot be computed due to data issues.
**Resource Allocation Summary**: No resource allocation can be recommended until the data issues are resolved.
**Implementation Recommendations**: 1. Verify and correct the casualties data to ensure it matches the expected length based on the number of ships and battles. 2. Re-run the optimization with the corrected data. 3. Validate the results with at least two solvers to ensure consistency.