## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be the effort allocated to the \( i \)-th representative's campaign. This is a continuous decision variable representing the level of effort.

#### Objective Function
Maximize the total number of seats won by the party:
\[ \text{Maximize } Z = \sum_{i=1}^{n} \text{Seats}_i \times x_i \]

Where:
- \( \text{Seats}_i \) is the number of seats associated with the \( i \)-th representative, obtained from the `election.Seats` column.

#### Constraints
1. Total effort constraint:
   \[ \sum_{i=1}^{n} x_i \leq \text{Total Campaign Resources} \]

2. Minimum and maximum effort constraints for each representative:
   \[ \text{Minimum Effort}_i \leq x_i \leq \text{Maximum Effort}_i \quad \forall i \]

Where:
- \( \text{Minimum Effort}_i \) and \( \text{Maximum Effort}_i \) are the minimum and maximum effort levels for the \( i \)-th representative, obtained from the `election.Minimum_Effort` and `election.Maximum_Effort` columns, respectively.
- \( \text{Total Campaign Resources} \) is a parameter representing the total budget available for campaign efforts.

Data Source Verification:
- Objective coefficients (\( \text{Seats}_i \)) are from `election.Seats`.
- Minimum and maximum bounds (\( \text{Minimum Effort}_i \) and \( \text{Maximum Effort}_i \)) are from `election.Minimum_Effort` and `election.Maximum_Effort`.
- Total campaign resources are a business configuration parameter not explicitly listed in the tables but assumed to be known for the optimization problem.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def election_representative_optimization():
    """Optimize campaign resource allocation to maximize seats won."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("election_representative")
    
    # Data from the database (hardcoded for this example)
    seats = [1, 2, 3]
    min_effort = [10.0, 15.0, 20.0]
    max_effort = [50.0, 60.0, 70.0]
    total_campaign_resources = 100.0  # Example total budget
    
    n_representatives = len(seats)
    
    # CRITICAL: Validate array lengths before loops
    assert len(seats) == len(min_effort) == len(max_effort) == n_representatives, "Array length mismatch"
    
    # 2. VARIABLES
    # Variable dictionaries for efforts
    effort = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"effort_{i}", lb=min_effort[i], ub=max_effort[i]) 
              for i in range(n_representatives)}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of seats won
    model.setObjective(gp.quicksum(seats[i] * effort[i] for i in range(n_representatives)), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total effort constraint
    model.addConstr(gp.quicksum(effort[i] for i in range(n_representatives)) <= total_campaign_resources, name="total_effort_limit")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_representatives):
            if effort[i].x > 1e-6:
                print(f"Effort for representative {i}: {effort[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_representative_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 260.0
**Execution Time**: 0.22 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_representative_optimization(total_campaign_resources):
    """Optimize the allocation of campaign resources to maximize seats won."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="election_representative")
    
    # Data from the database
    seats = [1, 2, 3]
    min_effort = [10.0, 15.0, 20.0]
    max_effort = [50.0, 60.0, 70.0]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(seats) == len(min_effort) == len(max_effort), "Array length mismatch"
    safe_range = range(min(len(seats), len(min_effort), len(max_effort)))  # Safe indexing
    
    # 2. VARIABLES
    # Variable dictionaries (most common)
    effort = {i: mdl.continuous_var(name=f"effort_{i}", lb=min_effort[i], ub=max_effort[i]) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of seats won
    objective = mdl.sum(seats[i] * effort[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total effort constraint
    total_effort = mdl.sum(effort[i] for i in safe_range)
    mdl.add_constraint(total_effort <= total_campaign_resources, ctname="total_effort_limit")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(effort[i])
            if value > 1e-6:
                print(f"effort[{i}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Example usage
total_campaign_resources = 100.0  # Example total campaign resources
election_representative_optimization(total_campaign_resources)
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 260.0
**Execution Time**: 1.05 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_representative_optimization():
    """Optimize campaign resource allocation to maximize seats won."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Data from the database
    seats = [1, 2, 3]
    min_effort = [10.0, 15.0, 20.0]
    max_effort = [50.0, 60.0, 70.0]
    total_campaign_resources = 100.0  # Example total budget
    
    # CRITICAL: Validate array lengths before indexing
    n_representatives = len(seats)
    assert len(min_effort) == len(max_effort) == n_representatives, "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.RangeSet(1, n_representatives)  # 1-based indexing
    
    # 4. PARAMETERS
    model.seats = pyo.Param(model.I, initialize={i+1: seats[i] for i in range(n_representatives)})
    model.min_effort = pyo.Param(model.I, initialize={i+1: min_effort[i] for i in range(n_representatives)})
    model.max_effort = pyo.Param(model.I, initialize={i+1: max_effort[i] for i in range(n_representatives)})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.seats[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Total effort constraint
    def total_effort_rule(model):
        return sum(model.x[i] for i in model.I) <= total_campaign_resources
    model.total_effort_constraint = pyo.Constraint(rule=total_effort_rule)
    
    # Minimum and maximum effort constraints
    def effort_bounds_rule(model, i):
        return (model.min_effort[i], model.x[i], model.max_effort[i])
    model.effort_bounds_constraint = pyo.Constraint(model.I, rule=effort_bounds_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("\nVariable values:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"x[{i}] = {x_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_representative_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 260.0
**Execution Time**: 0.80 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a moderate execution time, consistent with the other solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 260.00 | 0.22s | N/A | N/A |
| Docplex | OPTIMAL | 260.00 | 1.05s | N/A | N/A |
| Pyomo | OPTIMAL | 260.00 | 0.80s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 260.0

### Final Recommendation
**Recommended Optimal Value**: 260.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its fastest execution time and high reliability, making it suitable for scenarios requiring quick decision-making.

### Business Interpretation
**Overall Strategy**: The optimal allocation of resources will maximize the number of seats won, achieving the best possible outcome given the constraints.
**Objective Value Meaning**: Achieving an optimal objective value of 260.0 indicates the maximum number of seats that can be won with the given resources.
**Resource Allocation Summary**: Resources should be allocated to maximize seat gains while adhering to individual effort constraints.
**Implementation Recommendations**: Implement the solution by allocating resources as per the optimal decision variables, ensuring compliance with constraints and monitoring outcomes.