## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Bobby', and 'hours worked by Ringo', which can be represented as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into symbolic notation
The objective function to minimize is $4 \times \text{hours worked by Hank} + 3 \times \text{hours worked by Bobby} + 1 \times \text{hours worked by Ringo}$, which translates to $4x_0 + 3x_1 + x_2$.

## 3: List the constraints in symbolic notation
1. $11x_0 \leq 263$
2. $20x_0 \leq 345$
3. $13x_0 \leq 215$
4. $20x_0 \leq 197$
5. $7x_1 \leq 263$
6. $18x_1 \leq 345$
7. $16x_1 \leq 215$
8. $5x_1 \leq 197$
9. $22x_2 \leq 263$
10. $5x_2 \leq 345$
11. $9x_2 \leq 215$
12. $x_2 \leq 197$
13. $11x_0 + 22x_2 \geq 38$
14. $11x_0 + 7x_1 \geq 67$
15. $11x_0 + 7x_1 + 22x_2 \geq 67$
16. $18x_1 + 5x_2 \geq 90$
17. $20x_0 + 18x_1 \geq 69$
18. $20x_0 + 18x_1 + 5x_2 \geq 69$
19. $13x_0 + 9x_2 \geq 58$
20. $13x_0 + 16x_1 \geq 25$
21. $13x_0 + 16x_1 + 9x_2 \geq 25$
22. $20x_0 + 5x_1 \geq 44$
23. $20x_0 + x_2 \geq 27$
24. $5x_1 + x_2 \geq 45$
25. $20x_0 + 5x_1 + x_2 \geq 45$
26. $10x_0 - 10x_2 \geq 0$
27. $11x_0 + 22x_2 \leq 242$
28. $7x_1 + 22x_2 \leq 96$
29. $20x_0 + 18x_1 + 5x_2 \leq 278$
30. $x_0 \in \mathbb{Z}$
31. $x_1 \in \mathbb{R}$
32. $x_2 \in \mathbb{R}$

## 4: Create the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'), 
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by Ringo')
    ], 
    'objective_function': '4*x0 + 3*x1 + x2', 
    'constraints': [
        '11*x0 <= 263', '20*x0 <= 345', '13*x0 <= 215', '20*x0 <= 197',
        '7*x1 <= 263', '18*x1 <= 345', '16*x1 <= 215', '5*x1 <= 197',
        '22*x2 <= 263', '5*x2 <= 345', '9*x2 <= 215', 'x2 <= 197',
        '11*x0 + 22*x2 >= 38', '11*x0 + 7*x1 >= 67', '11*x0 + 7*x1 + 22*x2 >= 67',
        '18*x1 + 5*x2 >= 90', '20*x0 + 18*x1 >= 69', '20*x0 + 18*x1 + 5*x2 >= 69',
        '13*x0 + 9*x2 >= 58', '13*x0 + 16*x1 >= 25', '13*x0 + 16*x1 + 9*x2 >= 25',
        '20*x0 + 5*x1 >= 44', '20*x0 + x2 >= 27', '5*x1 + x2 >= 45',
        '20*x0 + 5*x1 + x2 >= 45', '10*x0 - 10*x2 >= 0',
        '11*x0 + 22*x2 <= 242', '7*x1 + 22*x2 <= 96', '20*x0 + 18*x1 + 5*x2 <= 278',
        'x0 >= 0', 'x1 >= 0', 'x2 >= 0'
    ]
}
```

## 5: Write the Gurobi code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
    x1 = model.addVar(name="x1")  # hours worked by Bobby
    x2 = model.addVar(name="x2")  # hours worked by Ringo

    # Objective function
    model.setObjective(4 * x0 + 3 * x1 + x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(11 * x0 <= 263)
    model.addConstr(20 * x0 <= 345)
    model.addConstr(13 * x0 <= 215)
    model.addConstr(20 * x0 <= 197)

    model.addConstr(7 * x1 <= 263)
    model.addConstr(18 * x1 <= 345)
    model.addConstr(16 * x1 <= 215)
    model.addConstr(5 * x1 <= 197)

    model.addConstr(22 * x2 <= 263)
    model.addConstr(5 * x2 <= 345)
    model.addConstr(9 * x2 <= 215)
    model.addConstr(x2 <= 197)

    model.addConstr(11 * x0 + 22 * x2 >= 38)
    model.addConstr(11 * x0 + 7 * x1 >= 67)
    model.addConstr(11 * x0 + 7 * x1 + 22 * x2 >= 67)

    model.addConstr(18 * x1 + 5 * x2 >= 90)
    model.addConstr(20 * x0 + 18 * x1 >= 69)
    model.addConstr(20 * x0 + 18 * x1 + 5 * x2 >= 69)

    model.addConstr(13 * x0 + 9 * x2 >= 58)
    model.addConstr(13 * x0 + 16 * x1 >= 25)
    model.addConstr(13 * x0 + 16 * x1 + 9 * x2 >= 25)

    model.addConstr(20 * x0 + 5 * x1 >= 44)
    model.addConstr(20 * x0 + x2 >= 27)
    model.addConstr(5 * x1 + x2 >= 45)

    model.addConstr(20 * x0 + 5 * x1 + x2 >= 45)
    model.addConstr(10 * x0 - 10 * x2 >= 0)

    model.addConstr(11 * x0 + 22 * x2 <= 242)
    model.addConstr(7 * x1 + 22 * x2 <= 96)
    model.addConstr(20 * x0 + 18 * x1 + 5 * x2 <= 278)

    model.addConstr(x0 >= 0)
    model.addConstr(x1 >= 0)
    model.addConstr(x2 >= 0)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Hank: {x0.varValue}")
        print(f"Hours worked by Bobby: {x1.varValue}")
        print(f"Hours worked by Ringo: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```