## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{i,j} \) be the number of votes received by contestant \( i \) from state \( j \).
- \( x_{i,j} \) are integer decision variables, as votes must be whole numbers.

#### Objective Function
Maximize the total number of votes received by all contestants:
\[ \text{Maximize } \sum_{i} \sum_{j} x_{i,j} \]

#### Constraints
1. **Maximum Votes per Contestant**: Each contestant's total votes from all states must not exceed the maximum number of votes allowed per contestant.
   \[
   \sum_{j} x_{i,j} \leq \text{MaxVotes}_i \quad \forall i
   \]
   - Data Source Verification: MaxVotes is a business configuration parameter.

2. **Minimum Votes per State**: Each contestant must receive at least a minimum number of votes from each state.
   \[
   x_{i,j} \geq \text{MinVotes}_{i,j} \quad \forall i, j
   \]
   - Data Source Verification: MinVotes is a business configuration parameter.

3. **Non-negativity and Integer Constraints**: Votes must be non-negative integers.
   \[
   x_{i,j} \geq 0 \quad \text{and integer} \quad \forall i, j
   \]

Data Source Verification:
- The decision variables \( x_{i,j} \) correspond to the "votes" column in the `CONTESTANT_VOTES` table.
- The constraints use parameters MaxVotes and MinVotes, which are assumed to be defined in the business configuration, though not explicitly provided in the data.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_votes():
    """Optimize the distribution of votes among contestants and states."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("voter_1")
    
    # Example data setup
    contestants = [1, 2, 3]
    states = [1, 2, 3]
    
    # Parameters (these would typically come from business configuration)
    MaxVotes = {1: 300, 2: 300, 3: 300}  # Maximum votes per contestant
    MinVotes = {(1, 1): 50, (1, 2): 50, (1, 3): 50,
                (2, 1): 50, (2, 2): 50, (2, 3): 50,
                (3, 1): 50, (3, 2): 50, (3, 3): 50}  # Minimum votes per state
    
    # CRITICAL: Validate array lengths before loops
    assert len(contestants) > 0 and len(states) > 0, "Contestants or states list is empty"
    
    # 2. VARIABLES
    # Decision variables: votes received by contestant i from state j
    x = model.addVars(contestants, states, vtype=GRB.INTEGER, name="x", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of votes received by all contestants
    model.setObjective(gp.quicksum(x[i, j] for i in contestants for j in states), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Maximum Votes per Contestant
    model.addConstrs((gp.quicksum(x[i, j] for j in states) <= MaxVotes[i] for i in contestants), name="max_votes")
    
    # Minimum Votes per State
    model.addConstrs((x[i, j] >= MinVotes[i, j] for i in contestants for j in states), name="min_votes")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in contestants:
            for j in states:
                if x[i, j].x > 1e-6:
                    print(f"x[{i},{j}] = {x[i, j].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
optimize_votes()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 900.0
**Execution Time**: 0.19 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the shortest execution time, indicating efficient performance and reliability.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def voter_optimization():
    """Optimization model for maximizing votes for contestants"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="voter_optimization")
    
    # Example data setup
    contestants = [1, 2, 3]
    states = [1, 2, 3]
    
    # Maximum votes per contestant (example values)
    max_votes = {1: 300, 2: 295, 3: 305}
    
    # Minimum votes per contestant per state (example values)
    min_votes = {
        (1, 1): 50, (1, 2): 40, (1, 3): 30,
        (2, 1): 45, (2, 2): 35, (2, 3): 25,
        (3, 1): 55, (3, 2): 45, (3, 3): 35
    }
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(contestants) == len(max_votes), "Array length mismatch for contestants and max_votes"
    assert len(states) * len(contestants) == len(min_votes), "Array length mismatch for min_votes"
    
    # 2. VARIABLES
    # Variable dictionaries for votes
    votes = {(i, j): mdl.integer_var(name=f"x_{i}_{j}", lb=0) for i in contestants for j in states}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total votes
    objective = mdl.sum(votes[i, j] for i in contestants for j in states)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Maximum votes per contestant
    for i in contestants:
        mdl.add_constraint(mdl.sum(votes[i, j] for j in states) <= max_votes[i], ctname=f"max_votes_{i}")
    
    # Minimum votes per contestant per state
    for i in contestants:
        for j in states:
            mdl.add_constraint(votes[i, j] >= min_votes[i, j], ctname=f"min_votes_{i}_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in contestants:
            for j in states:
                value = solution.get_value(votes[i, j])
                if value > 1e-6:
                    print(f"Votes for contestant {i} from state {j}: {value:.0f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
voter_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 900.0
**Execution Time**: 2.15 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, though with a longer execution time compared to Gurobipy, but still reliable.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def voting_optimization():
    """Optimize the distribution of votes among contestants"""

    # 1. MODEL CREATION
    model = pyo.ConcreteModel()

    # 2. DATA SETUP
    # Example data for demonstration purposes
    contestants = [1, 2, 3]
    states = [1, 2, 3]
    max_votes = {1: 300, 2: 295, 3: 305}  # Maximum votes per contestant
    min_votes = {(1, 1): 50, (1, 2): 40, (1, 3): 60,
                 (2, 1): 45, (2, 2): 55, (2, 3): 50,
                 (3, 1): 40, (3, 2): 50, (3, 3): 55}  # Minimum votes per state

    # 3. SETS
    model.I = pyo.Set(initialize=contestants)
    model.J = pyo.Set(initialize=states)

    # 4. PARAMETERS
    model.MaxVotes = pyo.Param(model.I, initialize=max_votes)
    model.MinVotes = pyo.Param(model.I, model.J, initialize=min_votes)

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

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

    # 7. CONSTRAINTS
    # Maximum Votes per Contestant
    def max_votes_rule(model, i):
        return sum(model.x[i, j] for j in model.J) <= model.MaxVotes[i]
    model.max_votes_constraint = pyo.Constraint(model.I, rule=max_votes_rule)

    # Minimum Votes per State
    def min_votes_rule(model, i, j):
        return model.x[i, j] >= model.MinVotes[i, j]
    model.min_votes_constraint = pyo.Constraint(model.I, model.J, rule=min_votes_rule)

    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')

    # Solve the model
    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)}")

        # Extract variable values
        print("\nVariable values:")
        for i in model.I:
            for j in model.J:
                x_val = pyo.value(model.x[i, j])
                if x_val > 1e-6:  # Only print non-zero values
                    print(f"x[{i},{j}] = {x_val}")

    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
voting_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 900.0
**Execution Time**: 2.02 seconds
**Reliability**: high
**Analysis**: Pyomo achieved the optimal solution with a similar execution time to DOCplex, confirming the solution's reliability.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 900.00 | 0.19s | N/A | N/A |
| Docplex | OPTIMAL | 900.00 | 2.15s | N/A | N/A |
| Pyomo | OPTIMAL | 900.00 | 2.02s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 900.0

### Final Recommendation
**Recommended Optimal Value**: 900.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time, which is beneficial for large-scale problems or when quick decision-making is required.

### Business Interpretation
**Overall Strategy**: The optimal value of 900 indicates the maximum achievable votes under the given constraints, ensuring each contestant receives the maximum allowable votes while meeting minimum state requirements.
**Objective Value Meaning**: The optimal objective value of 900 represents the total votes distributed among contestants, maximizing their reach and engagement.
**Resource Allocation Summary**: Resources (votes) should be allocated to ensure each contestant meets the minimum state requirements while maximizing their total votes.
**Implementation Recommendations**: Implement the solution by configuring the voting system to adhere to the constraints, ensuring each contestant receives the optimal number of votes as calculated.