# Complete Optimization Problem and Solution: election_representative

## 1. Problem Context and Goals

### Context  
A political party is strategically planning the allocation of its campaign resources to maximize the number of seats won in upcoming elections. The party's decision-making process involves determining the level of effort to allocate to each representative's campaign. This effort is represented as a continuous variable for each representative. The primary objective is to maximize the overall effectiveness of these efforts, which is directly linked to the number of seats each representative can potentially secure. The party operates under a fixed total budget for campaign resources, which serves as a constraint on the total effort that can be distributed. Each representative's campaign also has specific minimum and maximum effort requirements, ensuring that resources are allocated efficiently and effectively. The business configuration includes the total campaign resources available for allocation, which is a critical parameter in this optimization process.

### Goals  
The optimization goal is to maximize the total number of seats won by the party. This is achieved by optimizing the allocation of campaign efforts across representatives. The metric for success is the maximization of the sum of the product of seats associated with each representative and the effort allocated to them. This goal is articulated in natural language to emphasize the linear relationship between effort allocation and seat maximization, without resorting to mathematical notation.

## 2. Constraints    

The optimization process is subject to several constraints that ensure the feasibility and effectiveness of the resource allocation:

- The total effort allocated across all representatives must not exceed the total campaign resources available. This constraint ensures that the party operates within its budgetary limits.
- Each representative's campaign effort must fall within specified minimum and maximum bounds. This ensures that each campaign receives a sufficient level of support while preventing over-allocation of resources.

These constraints are described in business terms that naturally align with linear mathematical forms, avoiding any nonlinear relationships.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for missing optimization data and updating configuration logic for scalar parameters and formulas.

CREATE TABLE election (
  Seats INTEGER,
  Minimum_Effort FLOAT,
  Maximum_Effort FLOAT
);

CREATE TABLE representative_effort (
  Effort FLOAT
);
```

### Data Dictionary  
The data dictionary provides a comprehensive mapping of tables and columns to their business purposes and roles in the optimization process:

- **Election Table**: This table stores data related to each representative's potential election outcomes.
  - **Seats**: Represents the number of seats associated with each representative. This serves as the objective coefficient in the optimization process.
  - **Minimum Effort**: Indicates the minimum level of effort required for each representative's campaign. This acts as a lower bound constraint.
  - **Maximum Effort**: Specifies the maximum level of effort allowed for each representative's campaign. This serves as an upper bound constraint.

- **Representative Effort Table**: This table captures the level of campaign effort allocated to each representative.
  - **Effort**: Represents the decision variable in the optimization process, indicating the level of effort allocated to each representative's campaign.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical campaign resource allocations and historical performance metrics to ensure a realistic and solvable optimization problem.

-- Realistic data for election
INSERT INTO election (Seats, Minimum_Effort, Maximum_Effort) VALUES (1, 10.0, 50.0);
INSERT INTO election (Seats, Minimum_Effort, Maximum_Effort) VALUES (2, 15.0, 60.0);
INSERT INTO election (Seats, Minimum_Effort, Maximum_Effort) VALUES (3, 20.0, 70.0);

-- Realistic data for representative_effort
INSERT INTO representative_effort (Effort) VALUES (20.0);
INSERT INTO representative_effort (Effort) VALUES (30.0);
INSERT INTO representative_effort (Effort) VALUES (40.0);
```

## 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.