## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Peggy', and 'hours worked by Paul'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function is to minimize $3 \times$ ('hours worked by Hank') $+ 6 \times$ ('hours worked by Peggy') $+ 3 \times$ ('hours worked by Paul'). In symbolic terms, this is $3x_1 + 6x_2 + 3x_3$.

## 3: Define the constraints in symbolic representation
The constraints are:
- $1x_1 \leq 109$ (Hank's organization score constraint, but it's more about the resource usage)
- $1x_2 \leq 109$ (Peggy's organization score constraint)
- $8x_3 \leq 109$ (Paul's organization score constraint)
- $1x_1 + 1x_2 \geq 22$ (combined organization score from Hank and Peggy)
- $1x_1 + 1x_2 + 8x_3 \geq 19$ (combined organization score from all)
- $1x_1 + 1x_2 + 8x_3 \geq 19$ is the same as the previous one, so we keep only one
- $1x_1 + 8x_3 \leq 72$ (combined organization score from Hank and Paul)

## 4: Correct and Simplify Constraints
Given that $x_1$, $x_2$, and $x_3$ represent hours worked and not directly organization scores, but rather contribute to an organization score at rates of 1, 1, and 8 respectively, we interpret the constraints as:
- The organization score contribution from Hank is $1x_1$
- The organization score contribution from Peggy is $1x_2$
- The organization score contribution from Paul is $8x_3$
So, the constraints based on organization scores are:
- $x_1 \geq 0$ (Implicit, as hours cannot be negative)
- $x_2 \geq 0$ (Implicit)
- $x_3 \geq 0$ (Implicit)
- $x_1 + x_2 \geq 22$
- $x_1 + x_2 + 8x_3 \geq 19$
- $x_1 + 8x_3 \leq 72$

## 5: Create the symbolic representation of the problem
The symbolic representation is:
```json
{
'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Peggy'), ('x3', 'hours worked by Paul')],
'objective_function': '3*x1 + 6*x2 + 3*x3',
'constraints': [
    'x1 + x2 >= 22',
    'x1 + x2 + 8*x3 >= 19',
    'x1 + 8*x3 <= 72',
    'x1 >= 0',
    'x2 >= 0',
    'x3 >= 0'
]
}
```

## 6: Write the Gurobi code to solve the optimization problem
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name='x1', lb=0)  # hours worked by Hank
    x2 = model.addVar(name='x2', lb=0)  # hours worked by Peggy
    x3 = model.addVar(name='x3', lb=0)  # hours worked by Paul

    # Define the objective function
    model.setObjective(3*x1 + 6*x2 + 3*x3, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(x1 + x2 >= 22)
    model.addConstr(x1 + x2 + 8*x3 >= 19)
    model.addConstr(x1 + 8*x3 <= 72)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        print(f'Hours worked by Hank: {x1.varValue}')
        print(f'Hours worked by Peggy: {x2.varValue}')
        print(f'Hours worked by Paul: {x3.varValue}')
        print(f'Objective function value: {model.objVal}')
    else:
        print('No optimal solution found.')

solve_optimization_problem()
```