## Step 1: Define the symbolic representation of the problem
The problem involves minimizing an objective function subject to various constraints. The variables are:
- $x_0$: hours worked by George
- $x_1$: hours worked by Paul
- $x_2$: hours worked by Ringo
- $x_3$: hours worked by Mary

The objective function to minimize is: $2x_0 + 5x_1 + 7x_2 + 7x_3$

## Step 2: List all constraints in symbolic notation
Constraints:
1. $7x_1 + x_2 \geq 17$
2. $7x_1 + 12x_3 \geq 23$
3. $x_2 + 12x_3 \geq 18$
4. $3x_0 + 7x_1 + x_2 + 12x_3 \geq 18$
5. $4x_0 + 9x_2 \geq 48$
6. $4x_0 + 12x_1 \geq 22$
7. $4x_0 + 7x_3 \geq 37$
8. $4x_0 + 12x_1 + 7x_3 \geq 46$
9. $12x_1 + 9x_2 + 7x_3 \geq 46$
10. $4x_0 + 9x_2 + 7x_3 \geq 46$
11. $4x_0 + 12x_1 + 9x_2 \geq 46$
12. $4x_0 + 12x_1 + 7x_3 \geq 50$
13. $12x_1 + 9x_2 + 7x_3 \geq 50$
14. $4x_0 + 9x_2 + 7x_3 \geq 50$
15. $4x_0 + 12x_1 + 9x_2 \geq 50$
16. $4x_0 + 12x_1 + 7x_3 \geq 53$
17. $12x_1 + 9x_2 + 7x_3 \geq 53$
18. $4x_0 + 9x_2 + 7x_3 \geq 53$
19. $4x_0 + 12x_1 + 9x_2 \geq 53$
20. $4x_0 + 12x_1 + 7x_3 \geq 51$
21. $12x_1 + 9x_2 + 7x_3 \geq 51$
22. $4x_0 + 9x_2 + 7x_3 \geq 51$
23. $4x_0 + 12x_1 + 9x_2 \geq 51$
24. $4x_0 + 12x_1 + 9x_2 + 7x_3 \geq 51$
25. $4x_0 + 4x_3 \geq 24$
26. $12x_1 + 5x_2 \geq 10$
27. $4x_0 + 12x_1 \geq 16$
28. $4x_0 + 12x_1 + 5x_2 + 4x_3 \geq 16$
29. $x_1 + 6x_2 \geq 29$
30. $9x_0 + 2x_3 \geq 24$
31. $x_1 + 2x_3 \geq 25$
32. $9x_0 + 6x_2 \geq 18$
33. $9x_0 + x_1 + 6x_2 + 2x_3 \geq 18$
34. $-x_1 + x_2 \geq 0$
35. $3x_0 + 7x_1 \leq 67$
36. $3x_0 + 7x_1 + 12x_3 \leq 39$
37. $3x_0 + x_2 + 12x_3 \leq 77$
38. $7x_1 + x_2 + 12x_3 \leq 87$
39. $12x_1 + 9x_2 + 7x_3 \leq 152$
40. $4x_0 + 5x_2 \leq 99$
41. $12x_1 + 4x_3 \leq 25$
42. $5x_2 + 4x_3 \leq 62$
43. $4x_0 + 12x_1 \leq 41$
44. $12x_1 + 5x_2 + 4x_3 \leq 59$
45. $4x_0 + 12x_1 + 4x_3 \leq 87$
46. $9x_0 + 2x_3 \leq 117$
47. $x_1 + 6x_2 + 2x_3 \leq 137$
48. $9x_0 + x_1 + 2x_3 \leq 140$

## Step 3: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [
        ['x0', 'hours worked by George'],
        ['x1', 'hours worked by Paul'],
        ['x2', 'hours worked by Ringo'],
        ['x3', 'hours worked by Mary']
    ],
    'objective_function': '2*x0 + 5*x1 + 7*x2 + 7*x3',
    'constraints': [
        '7*x1 + x2 >= 17',
        '7*x1 + 12*x3 >= 23',
        'x2 + 12*x3 >= 18',
        '3*x0 + 7*x1 + x2 + 12*x3 >= 18',
        '4*x0 + 9*x2 >= 48',
        '4*x0 + 12*x1 >= 22',
        '4*x0 + 7*x3 >= 37',
        '4*x0 + 12*x1 + 7*x3 >= 46',
        '12*x1 + 9*x2 + 7*x3 >= 46',
        '4*x0 + 9*x2 + 7*x3 >= 46',
        '4*x0 + 12*x1 + 9*x2 >= 46',
        '4*x0 + 12*x1 + 7*x3 >= 50',
        '12*x1 + 9*x2 + 7*x3 >= 50',
        '4*x0 + 9*x2 + 7*x3 >= 50',
        '4*x0 + 12*x1 + 9*x2 >= 50',
        '4*x0 + 12*x1 + 7*x3 >= 53',
        '12*x1 + 9*x2 + 7*x3 >= 53',
        '4*x0 + 9*x2 + 7*x3 >= 53',
        '4*x0 + 12*x1 + 9*x2 >= 53',
        '4*x0 + 12*x1 + 7*x3 >= 51',
        '12*x1 + 9*x2 + 7*x3 >= 51',
        '4*x0 + 9*x2 + 7*x3 >= 51',
        '4*x0 + 12*x1 + 9*x2 >= 51',
        '4*x0 + 12*x1 + 9*x2 + 7*x3 >= 51',
        '4*x0 + 4*x3 >= 24',
        '12*x1 + 5*x2 >= 10',
        '4*x0 + 12*x1 >= 16',
        '4*x0 + 12*x1 + 5*x2 + 4*x3 >= 16',
        'x1 + 6*x2 >= 29',
        '9*x0 + 2*x3 >= 24',
        'x1 + 2*x3 >= 25',
        '9*x0 + 6*x2 >= 18',
        '9*x0 + x1 + 6*x2 + 2*x3 >= 18',
        '-x1 + x2 >= 0',
        '3*x0 + 7*x1 <= 67',
        '3*x0 + 7*x1 + 12*x3 <= 39',
        '3*x0 + x2 + 12*x3 <= 77',
        '7*x1 + x2 + 12*x3 <= 87',
        '12*x1 + 9*x2 + 7*x3 <= 152',
        '4*x0 + 5*x2 <= 99',
        '12*x1 + 4*x3 <= 25',
        '5*x2 + 4*x3 <= 62',
        '4*x0 + 12*x1 <= 41',
        '12*x1 + 5*x2 + 4*x3 <= 59',
        '4*x0 + 12*x1 + 4*x3 <= 87',
        '9*x0 + 2*x3 <= 117',
        'x1 + 6*x2 + 2*x3 <= 137',
        '9*x0 + x1 + 2*x3 <= 140'
    ]
}
```

## Step 4: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by George
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Paul
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Ringo
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Mary

    # Objective function
    model.setObjective(2 * x0 + 5 * x1 + 7 * x2 + 7 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(7 * x1 + x2 >= 17)
    model.addConstr(7 * x1 + 12 * x3 >= 23)
    model.addConstr(x2 + 12 * x3 >= 18)
    model.addConstr(3 * x0 + 7 * x1 + x2 + 12 * x3 >= 18)
    model.addConstr(4 * x0 + 9 * x2 >= 48)
    model.addConstr(4 * x0 + 12 * x1 >= 22)
    model.addConstr(4 * x0 + 7 * x3 >= 37)
    model.addConstr(4 * x0 + 12 * x1 + 7 * x3 >= 46)
    model.addConstr(12 * x1 + 9 * x2 + 7 * x3 >= 46)
    model.addConstr(4 * x0 + 9 * x2 + 7 * x3 >= 46)
    model.addConstr(4 * x0 + 12 * x1 + 9 * x2 >= 46)
    model.addConstr(4 * x0 + 12 * x1 + 7 * x3 >= 50)
    model.addConstr(12 * x1 + 9 * x2 + 7 * x3 >= 50)
    model.addConstr(4 * x0 + 9 * x2 + 7 * x3 >= 50)
    model.addConstr(4 * x0 + 12 * x1 + 9 * x2 >= 50)
    model.addConstr(4 * x0 + 12 * x1 + 7 * x3 >= 53)
    model.addConstr(12 * x1 + 9 * x2 + 7 * x3 >= 53)
    model.addConstr(4 * x0 + 9 * x2 + 7 * x3 >= 53)
    model.addConstr(4 * x0 + 12 * x1 + 9 * x2 >= 53)
    model.addConstr(4 * x0 + 12 * x1 + 7 * x3 >= 51)
    model.addConstr(12 * x1 + 9 * x2 + 7 * x3 >= 51)
    model.addConstr(4 * x0 + 9 * x2 + 7 * x3 >= 51)
    model.addConstr(4 * x0 + 12 * x1 + 9 * x2 >= 51)
    model.addConstr(4 * x0 + 12 * x1 + 9 * x2 + 7 * x3 >= 51)
    model.addConstr(4 * x0 + 4 * x3 >= 24)
    model.addConstr(12 * x1 + 5 * x2 >= 10)
    model.addConstr(4 * x0 + 12 * x1 >= 16)
    model.addConstr(4 * x0 + 12 * x1 + 5 * x2 + 4 * x3 >= 16)
    model.addConstr(x1 + 6 * x2 >= 29)
    model.addConstr(9 * x0 + 2 * x3 >= 24)
    model.addConstr(x1 + 2 * x3 >= 25)
    model.addConstr(9 * x0 + 6 * x2 >= 18)
    model.addConstr(9 * x0 + x1 + 6 * x2 + 2 * x3 >= 18)
    model.addConstr(-x1 + x2 >= 0)
    model.addConstr(3 * x0 + 7 * x1 <= 67)
    model.addConstr(3 * x0 + 7 * x1 + 12 * x3 <= 39)
    model.addConstr(3 * x0 + x2 + 12 * x3 <= 77)
    model.addConstr(7 * x1 + x2 + 12 * x3 <= 87)
    model.addConstr(12 * x1 + 9 * x2 + 7 * x3 <= 152)
    model.addConstr(4 * x0 + 5 * x2 <= 99)
    model.addConstr(12 * x1 + 4 * x3 <= 25)
    model.addConstr(5 * x2 + 4 * x3 <= 62)
    model.addConstr(4 * x0 + 12 * x1 <= 41)
    model.addConstr(12 * x1 + 5 * x2 + 4 * x3 <= 59)
    model.addConstr(4 * x0 + 12 * x1 + 4 * x3 <= 87)
    model.addConstr(9 * x0 + 2 * x3 <= 117)
    model.addConstr(x1 + 6 * x2 + 2 * x3 <= 137)
    model.addConstr(9 * x0 + x1 + 2 * x3 <= 140)

    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by George: {x0.varValue}")
        print(f"Hours worked by Paul: {x1.varValue}")
        print(f"Hours worked by Ringo: {x2.varValue}")
        print(f"Hours worked by Mary: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```