## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_i \) represent the amount of campaign resources allocated to the \( i \)-th representative, where \( i = 1, 2, 3 \).

#### Objective Function
Maximize the total number of seats won by the party:
\[
\text{Maximize } Z = 1 \cdot x_1 + 2 \cdot x_2 + 3 \cdot x_3
\]
**Data Source Verification**: The coefficients \( 1, 2, 3 \) are from the `election.Seats` table.

#### Constraints
1. **Total Budget Constraint**: The total resources allocated to all representatives must not exceed one million units.
\[
x_1 + x_2 + x_3 \leq 1000000
\]
**Data Source Verification**: The constant \( 1000000 \) is from the business configuration (total budget).

2. **Minimum Resource Allocation Constraint**: Each representative must receive at least one thousand units of resources.
\[
x_1 \geq 1000, \quad x_2 \geq 1000, \quad x_3 \geq 1000
\]
**Data Source Verification**: The constant \( 1000 \) is from the business configuration (minimum resource allocation).

3. **State Resource Limit Constraint**: The total resources allocated to representatives within any single state must not exceed the state-specific resource cap.
\[
x_1 \leq 50000, \quad x_2 \leq 60000, \quad x_3 \leq 70000
\]
**Data Source Verification**: The constants \( 50000, 60000, 70000 \) are from the `state_resource_limits.Maximum_Resource_Allocation` table.

#### Summary of the Complete Linear Programming Model
\[
\text{Maximize } Z = 1 \cdot x_1 + 2 \cdot x_2 + 3 \cdot x_3
\]
Subject to:
\[
x_1 + x_2 + x_3 \leq 1000000
\]
\[
x_1 \geq 1000, \quad x_2 \geq 1000, \quad x_3 \geq 1000
\]
\[
x_1 \leq 50000, \quad x_2 \leq 60000, \quad x_3 \leq 70000
\]

This is a complete, immediately solvable linear programming model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy 12.0.2 Implementation for Election Representative Optimization
"""

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
    seats = [1, 2, 3]  # From election.Seats
    resource_allocation = [1000.0, 2000.0, 3000.0]  # From resource_allocation.Resource_Amount
    state_limits = [50000, 60000, 70000]  # From state_resource_limits.Maximum_Resource_Allocation
    
    # Business configuration
    total_budget = 1000000
    min_resource = 1000
    
    # CRITICAL: Validate array lengths before loops
    n_representatives = len(seats)
    assert len(resource_allocation) == n_representatives, "Resource allocation array length mismatch"
    assert len(state_limits) == n_representatives, "State limits array length mismatch"
    
    # 2. VARIABLES
    x = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"x_{i}", lb=min_resource) 
         for i in range(n_representatives)}
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(seats[i] * x[i] for i in range(n_representatives)), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Total Budget Constraint
    model.addConstr(gp.quicksum(x[i] for i in range(n_representatives)) <= total_budget, name="total_budget")
    
    # State Resource Limit Constraints
    for i in range(n_representatives):
        model.addConstr(x[i] <= state_limits[i], name=f"state_limit_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_representatives):
            print(f"x[{i}] = {x[i].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
if __name__ == "__main__":
    election_representative_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 380000.0
**Execution Time**: 0.21 seconds
**Reliability**: high
**Analysis**: Gurobipy found the optimal solution quickly with the lowest execution time.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCplex implementation for election_representative optimization problem
"""

from docplex.mp.model import Model

def election_representative_optimization():
    """Optimize campaign resource allocation to maximize seats won"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="election_representative")
    
    # Data from the problem description
    seats = [1, 2, 3]  # From election.Seats
    min_resource = 1000  # Minimum resource allocation per representative
    total_budget = 1000000  # Total budget constraint
    state_limits = [50000, 60000, 70000]  # From state_resource_limits.Maximum_Resource_Allocation
    
    # CRITICAL: Validate array lengths to prevent IndexError
    n_reps = len(seats)
    assert len(state_limits) == n_reps, "Array length mismatch"
    safe_range = range(min(len(seats), len(state_limits)))  # Safe indexing
    
    # 2. VARIABLES
    # Decision variables: amount of resources allocated to each representative
    x = {i: mdl.continuous_var(name=f"x_{i}", lb=min_resource) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total seats won: sum(seats[i] * x[i])
    objective = mdl.sum(seats[i] * x[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Total budget constraint: sum(x[i]) <= total_budget
    total_usage = mdl.sum(x[i] for i in safe_range)
    mdl.add_constraint(total_usage <= total_budget, ctname="total_budget")
    
    # State resource limits: x[i] <= state_limits[i]
    for i in safe_range:
        mdl.add_constraint(x[i] <= state_limits[i], ctname=f"state_limit_{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])
            print(f"x[{i}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
if __name__ == "__main__":
    election_representative_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 380000.0
**Execution Time**: 1.24 seconds
**Reliability**: high
**Analysis**: DOCplex found the optimal solution but 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():
    """Optimization model for maximizing seats won by allocating campaign resources."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Data from the database schema and realistic data
    seats = [1, 2, 3]  # From election.Seats
    min_resource = 1000  # Minimum resource allocation
    total_budget = 1000000  # Total budget
    state_limits = [50000, 60000, 70000]  # From state_resource_limits.Maximum_Resource_Allocation
    
    # CRITICAL: Validate array lengths before indexing
    n_reps = len(seats)
    assert len(state_limits) == n_reps, "Array length mismatch"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.RangeSet(1, n_reps)  # 1-based indexing for representatives
    
    # 4. PARAMETERS (data containers)
    model.seats = pyo.Param(model.I, initialize={i+1: seats[i] for i in range(n_reps)})
    model.state_limits = pyo.Param(model.I, initialize={i+1: state_limits[i] for i in range(n_reps)})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeReals)  # Resource allocation to each representative
    
    # 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 Budget Constraint
    def budget_rule(model):
        return sum(model.x[i] for i in model.I) <= total_budget
    model.budget_constraint = pyo.Constraint(rule=budget_rule)
    
    # Minimum Resource Allocation Constraint
    def min_resource_rule(model, i):
        return model.x[i] >= min_resource
    model.min_resource_constraint = pyo.Constraint(model.I, rule=min_resource_rule)
    
    # State Resource Limit Constraint
    def state_limit_rule(model, i):
        return model.x[i] <= model.state_limits[i]
    model.state_limit_constraint = pyo.Constraint(model.I, rule=state_limit_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(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nResource allocations:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            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

# Execute the optimization
if __name__ == "__main__":
    election_representative_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 380000.0
**Execution Time**: 0.92 seconds
**Reliability**: high
**Analysis**: Pyomo found the optimal solution with an execution time between Gurobipy and DOCplex.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 380000.00 | 0.21s | N/A | N/A |
| Docplex | OPTIMAL | 380000.00 | 1.24s | N/A | N/A |
| Pyomo | OPTIMAL | 380000.00 | 0.92s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 380000.0

### Final Recommendation
**Recommended Optimal Value**: 380000.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its faster execution time while still providing the same optimal solution as the other solvers.

### Optimal Decision Variables
- **x_1** = 50000.000
  - *Business Meaning*: Allocation of resources to the first representative, capped at 50,000 units.
- **x_2** = 60000.000
  - *Business Meaning*: Allocation of resources to the second representative, capped at 60,000 units.
- **x_3** = 70000.000
  - *Business Meaning*: Allocation of resources to the third representative, capped at 70,000 units.

### Business Interpretation
**Overall Strategy**: The optimal allocation of resources maximizes the number of seats won by the party, ensuring that each representative receives the maximum allowed resources within their state limits.
**Objective Value Meaning**: The optimal objective value of 380,000 represents the maximum number of seats the party can win with the given resource allocation.
**Resource Allocation Summary**: Resources should be allocated as follows: 50,000 units to the first representative, 60,000 units to the second representative, and 70,000 units to the third representative.
**Implementation Recommendations**: Ensure that the resource allocation adheres to the state-specific caps and the total budget constraint. Monitor the execution to confirm that the allocated resources are effectively utilized to maximize seat wins.