## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Peggy', 'hours worked by Laura', 'hours worked by Hank', 'hours worked by Bobby']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $9.01x_0 + 4.3x_1 + 5.2x_2 + 7.19x_3$.

## Step 3: List all the constraints
1. $7x_0 \leq 61$
2. $x_0 \geq 0$ (Implicit, as hours cannot be negative)
3. $1x_0 \leq 86$
4. $2x_1 \leq 61$
5. $1x_1 \leq 86$
6. $7x_2 \leq 61$
7. $3x_2 \leq 86$
8. $8x_3 \leq 61$
9. $1x_3 \leq 86$
10. $2x_1 + 8x_3 \geq 13$
11. $7x_2 + 8x_3 \geq 13$
12. $7x_0 + 7x_2 \geq 6$
13. $7x_0 + 8x_3 \geq 11$
14. $2x_1 + 7x_2 + 8x_3 \geq 11$
15. $7x_0 + 2x_1 + 7x_2 \geq 11$
16. $2x_1 + 7x_2 + 8x_3 \geq 8$
17. $7x_0 + 2x_1 + 7x_2 \geq 8$
18. $7x_0 + 2x_1 + 7x_2 + 8x_3 \geq 8$
19. $x_0 + 3x_2 \geq 15$
20. $x_0 + x_1 \geq 14$
21. $x_0 + 3x_2 + x_3 \geq 21$
22. $x_0 + x_1 + 3x_2 \geq 21$
23. $x_0 + 3x_2 + x_3 \geq 10$
24. $x_0 + x_1 + 3x_2 \geq 10$
25. $x_0 + x_1 + 3x_2 + x_3 \geq 10$
26. $10x_1 - 2x_3 \geq 0$
27. $-6x_0 + 9x_1 \geq 0$
28. $2x_1 + 7x_2 \leq 24$
29. $7x_0 + 2x_1 \leq 52$
30. $2x_1 + 8x_3 \leq 34$
31. $x_0 + 3x_2 + x_3 \leq 68$
32. $x_0 \in \mathbb{Z}$ (Integer constraint for Peggy)
33. $x_2 \in \mathbb{Z}$ (Integer constraint for Hank)

## Step 4: Formulate the problem in Gurobi code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0, type=gurobi.GRB.INTEGER)  # hours worked by Peggy
x1 = m.addVar(name="x1", lb=0)  # hours worked by Laura
x2 = m.addVar(name="x2", lb=0, type=gurobi.GRB.INTEGER)  # hours worked by Hank
x3 = m.addVar(name="x3", lb=0)  # hours worked by Bobby

# Objective function
m.setObjective(9.01 * x0 + 4.3 * x1 + 5.2 * x2 + 7.19 * x3, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(7 * x0 <= 61)
m.addConstr(1 * x0 <= 86)
m.addConstr(2 * x1 <= 61)
m.addConstr(1 * x1 <= 86)
m.addConstr(7 * x2 <= 61)
m.addConstr(3 * x2 <= 86)
m.addConstr(8 * x3 <= 61)
m.addConstr(1 * x3 <= 86)
m.addConstr(2 * x1 + 8 * x3 >= 13)
m.addConstr(7 * x2 + 8 * x3 >= 13)
m.addConstr(7 * x0 + 7 * x2 >= 6)
m.addConstr(7 * x0 + 8 * x3 >= 11)
m.addConstr(2 * x1 + 7 * x2 + 8 * x3 >= 11)
m.addConstr(7 * x0 + 2 * x1 + 7 * x2 >= 11)
m.addConstr(2 * x1 + 7 * x2 + 8 * x3 >= 8)
m.addConstr(7 * x0 + 2 * x1 + 7 * x2 >= 8)
m.addConstr(7 * x0 + 2 * x1 + 7 * x2 + 8 * x3 >= 8)
m.addConstr(x0 + 3 * x2 >= 15)
m.addConstr(x0 + x1 >= 14)
m.addConstr(x0 + 3 * x2 + x3 >= 21)
m.addConstr(x0 + x1 + 3 * x2 >= 21)
m.addConstr(x0 + 3 * x2 + x3 >= 10)
m.addConstr(x0 + x1 + 3 * x2 >= 10)
m.addConstr(x0 + x1 + 3 * x2 + x3 >= 10)
m.addConstr(10 * x1 - 2 * x3 >= 0)
m.addConstr(-6 * x0 + 9 * x1 >= 0)
m.addConstr(2 * x1 + 7 * x2 <= 24)
m.addConstr(7 * x0 + 2 * x1 <= 52)
m.addConstr(2 * x1 + 8 * x3 <= 34)
m.addConstr(x0 + 3 * x2 + x3 <= 68)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
else:
    print("The model is infeasible")
```

## Step 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Peggy'),
        ('x1', 'hours worked by Laura'),
        ('x2', 'hours worked by Hank'),
        ('x3', 'hours worked by Bobby')
    ],
    'objective_function': '9.01*x0 + 4.3*x1 + 5.2*x2 + 7.19*x3',
    'constraints': [
        '7*x0 <= 61',
        'x0 >= 0',
        '1*x0 <= 86',
        '2*x1 <= 61',
        '1*x1 <= 86',
        '7*x2 <= 61',
        '3*x2 <= 86',
        '8*x3 <= 61',
        '1*x3 <= 86',
        '2*x1 + 8*x3 >= 13',
        '7*x2 + 8*x3 >= 13',
        '7*x0 + 7*x2 >= 6',
        '7*x0 + 8*x3 >= 11',
        '2*x1 + 7*x2 + 8*x3 >= 11',
        '7*x0 + 2*x1 + 7*x2 >= 11',
        '2*x1 + 7*x2 + 8*x3 >= 8',
        '7*x0 + 2*x1 + 7*x2 >= 8',
        '7*x0 + 2*x1 + 7*x2 + 8*x3 >= 8',
        'x0 + 3*x2 >= 15',
        'x0 + x1 >= 14',
        'x0 + 3*x2 + x3 >= 21',
        'x0 + x1 + 3*x2 >= 21',
        'x0 + 3*x2 + x3 >= 10',
        'x0 + x1 + 3*x2 >= 10',
        'x0 + x1 + 3*x2 + x3 >= 10',
        '10*x1 - 2*x3 >= 0',
        '-6*x0 + 9*x1 >= 0',
        '2*x1 + 7*x2 <= 24',
        '7*x0 + 2*x1 <= 52',
        '2*x1 + 8*x3 <= 34',
        'x0 + 3*x2 + x3 <= 68'
    ]
}
```