## 4. Mathematical Optimization Formulation

#### Decision Variables
Let:
- \( x_{i,j} \) be a binary decision variable where \( i \) represents the student (1, 2, or 3) and \( j \) represents the position (President, Vice President, Secretary, Treasurer, Class President, Class Senator). \( x_{i,j} = 1 \) if student \( i \) is elected to position \( j \), and \( x_{i,j} = 0 \) otherwise.

#### Objective Function
Maximize the total number of votes received by the elected representatives:

\[
\begin{align*}
\text{Maximize } & \sum_{i=1}^{3} \left( 25x_{i,\text{President}} + 18x_{i,\text{Vice President}} + 22x_{i,\text{Secretary}} + 15x_{i,\text{Treasurer}} + 30x_{i,\text{Class President}} + 20x_{i,\text{Class Senator}} \right) \\
& + \sum_{i=1}^{3} \left( 30x_{i,\text{President}} + 25x_{i,\text{Vice President}} + 20x_{i,\text{Secretary}} + 18x_{i,\text{Treasurer}} + 28x_{i,\text{Class President}} + 22x_{i,\text{Class Senator}} \right) \\
& + \sum_{i=1}^{3} \left( 20x_{i,\text{President}} + 30x_{i,\text{Vice President}} + 25x_{i,\text{Secretary}} + 22x_{i,\text{Treasurer}} + 18x_{i,\text{Class President}} + 25x_{i,\text{Class Senator}} \right)
\end{align*}
\]

#### Constraints
1. Each position must be filled by exactly one student:

\[
\begin{align*}
\sum_{i=1}^{3} x_{i,\text{President}} &= 1 \\
\sum_{i=1}^{3} x_{i,\text{Vice President}} &= 1 \\
\sum_{i=1}^{3} x_{i,\text{Secretary}} &= 1 \\
\sum_{i=1}^{3} x_{i,\text{Treasurer}} &= 1 \\
\sum_{i=1}^{3} x_{i,\text{Class President}} &= 1 \\
\sum_{i=1}^{3} x_{i,\text{Class Senator}} &= 1
\end{align*}
\]

2. Each student can hold at most one position:

\[
\begin{align*}
\sum_{j} x_{1,j} &\leq 1 \\
\sum_{j} x_{2,j} &\leq 1 \\
\sum_{j} x_{3,j} &\leq 1
\end{align*}
\]

Where \( j \) iterates over all positions: President, Vice President, Secretary, Treasurer, Class President, Class Senator.

Data Source Verification:
- Coefficients for the objective function are derived from the `Voting_record` table, specifically from the columns `President_Vote`, `Vice_President_Vote`, `Secretary_Vote`, `Treasurer_Vote`, `Class_President_Vote`, and `Class_Senator_Vote`.
- Constraints are based on the business logic described in the `Constraint_Bounds` table, ensuring each position is filled by exactly one student and each student holds at most one position.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def optimize_student_elections():
    # 1. MODEL & DATA SETUP
    model = gp.Model("student_elections")
    
    # Votes data from the Voting_record table
    votes = [
        [25, 18, 22, 15, 30, 20],  # Student 1
        [30, 25, 20, 18, 28, 22],  # Student 2
        [20, 30, 25, 22, 18, 25]   # Student 3
    ]
    
    n_students = len(votes)
    n_positions = len(votes[0])
    
    # Validate array lengths
    assert all(len(vote) == n_positions for vote in votes), "Array length mismatch"
    
    # 2. VARIABLES
    # Binary decision variables: x[i, j] = 1 if student i is elected to position j
    x = model.addVars(n_students, n_positions, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of votes received by the elected representatives
    model.setObjective(
        gp.quicksum(votes[i][j] * x[i, j] for i in range(n_students) for j in range(n_positions)),
        GRB.MAXIMIZE
    )
    
    # 4. CONSTRAINTS
    # Each position must be filled by exactly one student
    model.addConstrs(
        (gp.quicksum(x[i, j] for i in range(n_students)) == 1 for j in range(n_positions)),
        name="position_filled"
    )
    
    # Each student can hold at most one position
    model.addConstrs(
        (gp.quicksum(x[i, j] for j in range(n_positions)) <= 1 for i in range(n_students)),
        name="student_position_limit"
    )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_students):
            for j in range(n_positions):
                if x[i, j].x > 0.5:  # Binary variable, so check if it's 1
                    print(f"Student {i+1} is elected to position {j+1}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_student_elections()
```

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

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def optimize_student_elections():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="student_elections")
    
    # Voting data for each student and position
    votes = [
        [25, 18, 22, 15, 30, 20],  # Student 1
        [30, 25, 20, 18, 28, 22],  # Student 2
        [20, 30, 25, 22, 18, 25]   # Student 3
    ]
    
    n_students = len(votes)
    n_positions = len(votes[0])
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert all(len(v) == n_positions for v in votes), "Array length mismatch"
    safe_range_students = range(n_students)
    safe_range_positions = range(n_positions)
    
    # 2. VARIABLES
    # Binary decision variables for each student and position
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in safe_range_students for j in safe_range_positions}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of votes received by the elected representatives
    objective = mdl.sum(votes[i][j] * x[i, j] for i in safe_range_students for j in safe_range_positions)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Each position must be filled by exactly one student
    for j in safe_range_positions:
        mdl.add_constraint(mdl.sum(x[i, j] for i in safe_range_students) == 1, ctname=f"position_{j}_filled")
    
    # Each student can hold at most one position
    for i in safe_range_students:
        mdl.add_constraint(mdl.sum(x[i, j] for j in safe_range_positions) <= 1, ctname=f"student_{i}_limit")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range_students:
            for j in safe_range_positions:
                if solution.get_value(x[i, j]) > 0.5:  # Binary variable, so check > 0.5
                    print(f"Student {i+1} is elected to position {j+1}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Run the optimization
optimize_student_elections()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reports infeasibility, aligning with Gurobipy's results, suggesting a consistent issue across solvers.

## 7. Pyomo Implementation

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

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

def voter_2_optimization():
    """Pyomo model for optimizing student representative elections"""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    n_students = 3
    positions = ['President', 'Vice President', 'Secretary', 'Treasurer', 'Class President', 'Class Senator']
    
    # Votes received by each student for each position
    votes = {
        (1, 'President'): 25, (1, 'Vice President'): 18, (1, 'Secretary'): 22, (1, 'Treasurer'): 15, (1, 'Class President'): 30, (1, 'Class Senator'): 20,
        (2, 'President'): 30, (2, 'Vice President'): 25, (2, 'Secretary'): 20, (2, 'Treasurer'): 18, (2, 'Class President'): 28, (2, 'Class Senator'): 22,
        (3, 'President'): 20, (3, 'Vice President'): 30, (3, 'Secretary'): 25, (3, 'Treasurer'): 22, (3, 'Class President'): 18, (3, 'Class Senator'): 25
    }
    
    # 3. SETS
    model.I = pyo.RangeSet(1, n_students)  # Students
    model.J = pyo.Set(initialize=positions)  # Positions
    
    # 4. VARIABLES
    model.x = pyo.Var(model.I, model.J, within=pyo.Binary)
    
    # 5. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(votes[i, j] * model.x[i, j] for i in model.I for j in model.J)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 6. CONSTRAINTS
    
    # Each position must be filled by exactly one student
    def position_filled_rule(model, j):
        return sum(model.x[i, j] for i in model.I) == 1
    model.position_filled_constraint = pyo.Constraint(model.J, rule=position_filled_rule)
    
    # Each student can hold at most one position
    def student_limit_rule(model, i):
        return sum(model.x[i, j] for j in model.J) <= 1
    model.student_limit_constraint = pyo.Constraint(model.I, rule=student_limit_rule)
    
    # 7. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    
    # Solve the model
    results = solver.solve(model, tee=True)
    
    # 8. 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("\nElection results:")
        for i in model.I:
            for j in model.J:
                if pyo.value(model.x[i, j]) > 0.5:  # Binary variable, check if elected
                    print(f"Student {i} is elected as {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

# Run the optimization
voter_2_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo confirms the infeasibility, reinforcing the conclusion that the problem setup may be flawed.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.18s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 2.30s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 1.14s | 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**: HIGH
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently report infeasibility, suggesting the issue lies within the problem formulation rather than solver performance.

### Business Interpretation
**Overall Strategy**: The current problem setup does not allow for a feasible solution, indicating a need to revisit the constraints or data inputs.
**Objective Value Meaning**: The objective value represents the total number of votes received by the elected representatives, which cannot be maximized due to infeasibility.
**Resource Allocation Summary**: Currently, no resources can be allocated as no feasible assignment of students to positions exists.
**Implementation Recommendations**: Review and potentially relax constraints, verify data accuracy, and ensure the problem formulation aligns with business requirements.