## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be the number of police officers allocated to county \( i \).
- \( x_i \) is an integer variable for each county.

#### Objective Function
Minimize the total crime rate across all counties, which can be expressed as:
\[ \text{Minimize } \sum_{i} \text{Crime\_rate}_i \times x_i \]

#### Constraints
1. **Total Officers Constraint**: The total number of police officers allocated across all counties must not exceed the total number of officers available.
   \[
   \sum_{i} x_i \leq \text{Total\_Officers}
   \]
   - Data Source Verification: Total number of officers available is a business configuration parameter.

2. **Minimum Officers per County**: Each county must have at least a minimum number of officers assigned.
   \[
   x_i \geq \text{Minimum\_Officers}_i \quad \forall i
   \]
   - Data Source Verification: Minimum number of officers is a business configuration parameter.

3. **Residents per Officer Constraint**: The number of officers allocated to each county should not exceed the number of residents divided by the residents per officer.
   \[
   x_i \leq \frac{\text{Population}_i}{\text{Residents\_per\_officer}_i} \quad \forall i
   \]
   - Data Source Verification: \(\text{Population}_i\) and \(\text{Residents\_per\_officer}_i\) are from the `county_public_safety` table.

4. **Case Burden Constraint**: The case burden per officer must not exceed the maximum allowed.
   \[
   \frac{\text{Case\_burden}_i}{x_i} \leq \text{Max\_Case\_Burden} \quad \forall i
   \]
   - This constraint can be linearized by ensuring:
   \[
   x_i \geq \frac{\text{Case\_burden}_i}{\text{Max\_Case\_Burden}} \quad \forall i
   \]
   - Data Source Verification: \(\text{Case\_burden}_i\) is from the `county_public_safety` table, and \(\text{Max\_Case\_Burden}\) is a business configuration parameter.

#### Data Source Verification
- \(\text{Crime\_rate}_i\), \(\text{Population}_i\), \(\text{Residents\_per\_officer}_i\), and \(\text{Case\_burden}_i\) are from the `county_public_safety` table.
- \(\text{Total\_Officers}\), \(\text{Minimum\_Officers}_i\), and \(\text{Max\_Case\_Burden}\) are business configuration parameters.

This linear model is immediately solvable using standard linear programming solvers, ensuring that all constraints and the objective function are linear.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def optimize_police_allocation():
    # 1. MODEL & DATA SETUP
    model = gp.Model("county_public_safety")
    
    # Data from the database
    crime_rates = [0.15, 0.25, 0.18, 0.22, 0.2]
    populations = [1500, 2500, 1800, 2200, 2000]
    residents_per_officer = [150, 125, 180, 110, 100]
    case_burdens = [7.0, 12.0, 9.0, 11.0, 10.0]
    min_officers = [10, 20, 12, 18, 15]
    
    # Business configuration parameters
    total_officers = 75
    max_case_burden = 15.0
    
    # Validate array lengths
    n_counties = len(crime_rates)
    assert len(populations) == len(residents_per_officer) == len(case_burdens) == len(min_officers) == n_counties, "Array length mismatch"
    
    # 2. VARIABLES
    x = model.addVars(n_counties, vtype=GRB.INTEGER, name="x", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(crime_rates[i] * x[i] for i in range(n_counties)), GRB.MINIMIZE)
    
    # 4. CONSTRAINTS
    # Total Officers Constraint
    model.addConstr(gp.quicksum(x[i] for i in range(n_counties)) <= total_officers, name="total_officers")
    
    # Minimum Officers per County
    model.addConstrs((x[i] >= min_officers[i] for i in range(n_counties)), name="min_officers")
    
    # Residents per Officer Constraint
    model.addConstrs((x[i] <= populations[i] // residents_per_officer[i] for i in range(n_counties)), name="residents_per_officer")
    
    # Case Burden Constraint
    model.addConstrs((x[i] >= case_burdens[i] / max_case_burden for i in range(n_counties)), name="case_burden")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_counties):
            print(f"x[{i}] = {x[i].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
optimize_police_allocation()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy consistently reports the problem as infeasible, indicating a potential issue with the constraints or data.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def optimize_police_allocation():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="county_public_safety")
    
    # Data from the database
    crime_rates = [0.15, 0.25, 0.18, 0.22, 0.2]
    populations = [1500, 2500, 1800, 2200, 2000]
    residents_per_officer = [150, 125, 180, 110, 100]
    case_burdens = [7.0, 12.0, 9.0, 11.0, 10.0]
    min_officers = [10, 20, 12, 18, 15]
    
    # Business configuration parameters
    total_officers = 75
    max_case_burden = 15.0
    
    # Validate array lengths
    n_counties = len(crime_rates)
    assert len(populations) == len(residents_per_officer) == len(case_burdens) == len(min_officers) == n_counties, "Array length mismatch"
    
    safe_range = range(n_counties)  # Safe indexing
    
    # 2. VARIABLES
    x = {i: mdl.integer_var(name=f"x_{i}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(crime_rates[i] * x[i] for i in safe_range)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    # Total Officers Constraint
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range) <= total_officers, ctname="total_officers")
    
    # Minimum Officers per County
    for i in safe_range:
        mdl.add_constraint(x[i] >= min_officers[i], ctname=f"min_officers_{i}")
    
    # Residents per Officer Constraint
    for i in safe_range:
        mdl.add_constraint(x[i] <= populations[i] // residents_per_officer[i], ctname=f"residents_per_officer_{i}")
    
    # Case Burden Constraint
    for i in safe_range:
        mdl.add_constraint(x[i] >= case_burdens[i] / max_case_burden, ctname=f"case_burden_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(x[i])
            if value > 1e-6:
                print(f"x[{i}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Execute the optimization
optimize_police_allocation()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reports infeasibility, suggesting that the problem formulation or data constraints are not satisfiable.

## 7. Pyomo Implementation

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

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

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

    # 2. DATA SETUP
    # Data from the county_public_safety table
    crime_rates = [0.15, 0.25, 0.18, 0.22, 0.2]
    populations = [1500, 2500, 1800, 2200, 2000]
    residents_per_officer = [150, 125, 180, 110, 100]
    case_burdens = [7.0, 12.0, 9.0, 11.0, 10.0]
    min_officers = [10, 20, 12, 18, 15]
    total_officers = 75
    max_case_burden = 10.0

    # Validate array lengths
    n_counties = len(crime_rates)
    assert len(populations) == len(residents_per_officer) == len(case_burdens) == len(min_officers) == n_counties, "Array length mismatch"

    # 3. SETS
    model.I = pyo.RangeSet(1, n_counties)

    # 4. PARAMETERS
    model.crime_rate = pyo.Param(model.I, initialize={i+1: crime_rates[i] for i in range(n_counties)})
    model.population = pyo.Param(model.I, initialize={i+1: populations[i] for i in range(n_counties)})
    model.residents_per_officer = pyo.Param(model.I, initialize={i+1: residents_per_officer[i] for i in range(n_counties)})
    model.case_burden = pyo.Param(model.I, initialize={i+1: case_burdens[i] for i in range(n_counties)})
    model.min_officers = pyo.Param(model.I, initialize={i+1: min_officers[i] for i in range(n_counties)})

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

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

    # 7. CONSTRAINTS
    def total_officers_rule(model):
        return sum(model.x[i] for i in model.I) <= total_officers
    model.total_officers_constraint = pyo.Constraint(rule=total_officers_rule)

    def min_officers_rule(model, i):
        return model.x[i] >= model.min_officers[i]
    model.min_officers_constraint = pyo.Constraint(model.I, rule=min_officers_rule)

    def residents_per_officer_rule(model, i):
        return model.x[i] <= model.population[i] / model.residents_per_officer[i]
    model.residents_per_officer_constraint = pyo.Constraint(model.I, rule=residents_per_officer_rule)

    def case_burden_rule(model, i):
        return model.x[i] >= model.case_burden[i] / max_case_burden
    model.case_burden_constraint = pyo.Constraint(model.I, rule=case_burden_rule)

    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    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)}")
        for i in model.I:
            print(f"x[{i}] = {pyo.value(model.x[i])}")
    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_police_allocation()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo's infeasibility result aligns with the other solvers, reinforcing the likelihood of a formulation or data issue.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.26s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 7.19s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 2.58s | N/A | 4 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently report infeasibility, indicating a fundamental issue with the problem setup rather than solver performance.

### Business Interpretation
**Overall Strategy**: The current allocation problem cannot be solved with the given constraints and data. This suggests a need to revisit the constraints or data inputs.
**Objective Value Meaning**: The objective value cannot be interpreted as the problem is infeasible. This means the current constraints do not allow for a feasible allocation of officers.
**Resource Allocation Summary**: No feasible allocation of officers can be determined with the current constraints and data.
**Implementation Recommendations**: Review and adjust the constraints and data inputs. Consider relaxing some constraints or verifying the accuracy of the data in the `county_public_safety` table and business configuration parameters.