## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be the amount of resources allocated to county \( i \).
- \( x_i \) is a continuous decision variable representing the resource allocation for each county.

#### Objective Function
Maximize the total expected votes:
\[ \text{Maximize } Z = 0.6x_1 + 1.5x_2 + 0.9x_3 \]

#### Constraints
1. Total budget constraint:
   \[ x_1 + x_2 + x_3 \leq \text{Total Budget} \]

2. Minimum resource allocation for each county:
   \[ x_1 \geq \text{Minimum Allocation for County 1} \]
   \[ x_2 \geq \text{Minimum Allocation for County 2} \]
   \[ x_3 \geq \text{Minimum Allocation for County 3} \]

3. Maximum resource allocation for each county:
   \[ x_1 \leq \text{Maximum Allocation for County 1} \]
   \[ x_2 \leq \text{Maximum Allocation for County 2} \]
   \[ x_3 \leq \text{Maximum Allocation for County 3} \]

Data Source Verification:
- Objective function coefficients are sourced from `ObjectiveCoefficients.votes_coefficient`.
- Decision variables \( x_i \) correspond to `DecisionVariables.resource_allocation`.
- Total budget, minimum, and maximum allocations are assumed to be provided as parameters in the business configuration, though specific values are not detailed in the provided data.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def election_optimization():
    """Optimize resource allocation for maximizing votes in an election"""

    # 1. MODEL & DATA SETUP
    model = gp.Model("election_optimization")

    # Data: Coefficients and constraints
    counties = [1, 2, 3]
    votes_coefficient = {1: 0.6, 2: 1.5, 3: 0.9}
    total_budget = 3000
    min_allocation = {1: 500, 2: 1000, 3: 600}
    max_allocation = {1: 1000, 2: 2000, 3: 1200}

    # CRITICAL: Validate array lengths before loops
    assert len(votes_coefficient) == len(min_allocation) == len(max_allocation) == len(counties), "Array length mismatch"

    # 2. VARIABLES
    # Variable dictionaries for resource allocation
    resource_allocation = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"x_{i}", lb=0) for i in counties}

    # 3. OBJECTIVE FUNCTION
    # Maximize total expected votes
    model.setObjective(gp.quicksum(votes_coefficient[i] * resource_allocation[i] for i in counties), GRB.MAXIMIZE)

    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS

    # Total budget constraint
    model.addConstr(gp.quicksum(resource_allocation[i] for i in counties) <= total_budget, name="total_budget")

    # Minimum and maximum resource allocation constraints for each county
    for i in counties:
        model.addConstr(resource_allocation[i] >= min_allocation[i], name=f"min_allocation_{i}")
        model.addConstr(resource_allocation[i] <= max_allocation[i], name=f"max_allocation_{i}")

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in counties:
            if resource_allocation[i].x > 1e-6:
                print(f"Resource allocation for county {i}: {resource_allocation[i].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
election_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3690.0
**Execution Time**: 1.51 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with a reliable status and no errors.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def election_optimization():
    """Optimize resource allocation for maximizing votes in an election"""

    # 1. MODEL & DATA SETUP
    mdl = Model(name="election_optimization")

    # Data: Coefficients and constraints
    votes_coefficients = [0.6, 1.5, 0.9]
    min_allocations = [500, 1000, 600]
    max_allocations = [1000, 2000, 1200]
    total_budget = 3000

    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(votes_coefficients) == len(min_allocations) == len(max_allocations), "Array length mismatch"
    safe_range = range(len(votes_coefficients))  # Safe indexing

    # 2. VARIABLES
    # Variable dictionaries (most common)
    resource_allocation = {i: mdl.continuous_var(name=f"x_{i}", lb=min_allocations[i], ub=max_allocations[i]) for i in safe_range}

    # 3. OBJECTIVE FUNCTION
    # Use mdl.sum() for summations (DOCplex specific)
    objective = mdl.sum(votes_coefficients[i] * resource_allocation[i] for i in safe_range)
    mdl.maximize(objective)

    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS

    # Total budget constraint
    total_usage = mdl.sum(resource_allocation[i] for i in safe_range)
    mdl.add_constraint(total_usage <= total_budget, ctname="total_budget")

    # 5. SOLVING & RESULTS
    solution = mdl.solve()

    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(resource_allocation[i])
            print(f"Resource allocation for county {i+1}: {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Run the optimization
election_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3690.0
**Execution Time**: 10.34 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, albeit with a longer execution time compared to Gurobipy.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def election_optimization():
    """Election resource allocation optimization using Pyomo"""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    counties = [1, 2, 3]
    votes_coefficients = {1: 0.6, 2: 1.5, 3: 0.9}
    min_allocation = {1: 500, 2: 1000, 3: 600}
    max_allocation = {1: 1000, 2: 2000, 3: 1200}
    total_budget = 3000
    
    # CRITICAL: Validate array lengths before indexing
    assert len(votes_coefficients) == len(min_allocation) == len(max_allocation) == len(counties), "Array length mismatch"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.Set(initialize=counties)
    
    # 4. PARAMETERS (data containers)
    model.votes_coefficient = pyo.Param(model.I, initialize=votes_coefficients)
    model.min_allocation = pyo.Param(model.I, initialize=min_allocation)
    model.max_allocation = pyo.Param(model.I, initialize=max_allocation)
    
    # 5. VARIABLES
    model.resource_allocation = pyo.Var(model.I, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.votes_coefficient[i] * model.resource_allocation[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Total budget constraint
    def budget_constraint_rule(model):
        return sum(model.resource_allocation[i] for i in model.I) <= total_budget
    model.budget_constraint = pyo.Constraint(rule=budget_constraint_rule)
    
    # Minimum and maximum allocation constraints
    def min_allocation_rule(model, i):
        return model.resource_allocation[i] >= model.min_allocation[i]
    model.min_allocation_constraint = pyo.Constraint(model.I, rule=min_allocation_rule)
    
    def max_allocation_rule(model, i):
        return model.resource_allocation[i] <= model.max_allocation[i]
    model.max_allocation_constraint = pyo.Constraint(model.I, rule=max_allocation_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):.3f}")
        
        # Extract variable values
        print("\nResource allocation:")
        for i in model.I:
            allocation_val = pyo.value(model.resource_allocation[i])
            print(f"County {i}: {allocation_val:.3f}")
        
    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
election_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3690.0
**Execution Time**: 7.29 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a moderate execution time, consistent with other solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 3690.00 | 1.51s | N/A | N/A |
| Docplex | OPTIMAL | 3690.00 | 10.34s | N/A | N/A |
| Pyomo | OPTIMAL | 3690.00 | 7.29s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 3690.0

### Final Recommendation
**Recommended Optimal Value**: 3690.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its fastest execution time and high reliability, making it suitable for time-sensitive decision-making.

### Business Interpretation
**Overall Strategy**: The optimal allocation of resources will maximize the expected votes, achieving the best possible outcome within the given constraints.
**Objective Value Meaning**: The optimal objective value of 3690.0 represents the maximum expected votes achievable with the given resource allocation.
**Resource Allocation Summary**: Resources should be allocated to maximize votes while adhering to budget and allocation constraints for each county.
**Implementation Recommendations**: Implement the solution by allocating resources as per the solver's optimal decision variables, ensuring compliance with all constraints.