## Step 1: Define the variables and their corresponding natural language objects
Let's denote the hours worked by Ringo, Paul, Peggy, and Mary as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Formulate the objective function
The objective function to minimize is $7x_0 + 1x_1 + 8x_2 + 6x_3$.

## Step 3: List all the constraints
1. $2x_0 \leq 95$
2. $8x_0 \leq 163$
3. $14x_1 \leq 95$
4. $7x_1 \leq 163$
5. $12x_2 \leq 95$
6. $1x_2 \leq 163$
7. $4x_3 \leq 95$
8. $2x_3 \leq 163$
9. $14x_1 + 4x_3 \geq 7$
10. $2x_0 + 12x_2 \geq 7$
11. $14x_1 + 12x_2 \geq 14$
12. $2x_0 + 14x_1 \geq 9$
13. $2x_0 + 14x_1 + 12x_2 \geq 21$
14. $2x_0 + 14x_1 + 4x_3 \geq 21$
15. $2x_0 + 14x_1 + 12x_2 \geq 19$
16. $2x_0 + 14x_1 + 4x_3 \geq 19$
17. $2x_0 + 14x_1 + 12x_2 + 4x_3 \geq 19$
18. $7x_1 + 1x_2 \geq 32$
19. $1x_2 + 2x_3 \geq 28$
20. $7x_1 + 2x_3 \geq 35$
21. $8x_0 + 7x_1 + 2x_3 \geq 34$
22. $8x_0 + 7x_1 + 1x_2 \geq 34$
23. $7x_1 + 1x_2 + 2x_3 \geq 34$
24. $8x_0 + 7x_1 + 2x_3 \geq 33$
25. $8x_0 + 7x_1 + 1x_2 \geq 33$
26. $7x_1 + 1x_2 + 2x_3 \geq 33$
27. $8x_0 + 7x_1 + 2x_3 \geq 35$
28. $8x_0 + 7x_1 + 1x_2 \geq 35$
29. $7x_1 + 1x_2 + 2x_3 \geq 35$
30. $8x_0 + 7x_1 + 1x_2 + 2x_3 \geq 35$
31. $8x_0 - 3x_2 \geq 0$
32. $14x_1 + 4x_3 \leq 81$
33. $14x_1 + 12x_2 \leq 86$
34. $14x_1 + 12x_2 + 4x_3 \leq 56$
35. $2x_0 + 14x_1 + 4x_3 \leq 62$
36. $7x_1 + 1x_2 \leq 95$
37. $1x_2 + 2x_3 \leq 152$
38. $8x_0 + 1x_2 + 2x_3 \leq 113$

## 4: Convert the problem into a symbolic representation
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'),
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Peggy'),
        ('x3', 'hours worked by Mary')
    ],
    'objective_function': '7*x0 + 1*x1 + 8*x2 + 6*x3',
    'constraints': [
        '2*x0 <= 95',
        '8*x0 <= 163',
        '14*x1 <= 95',
        '7*x1 <= 163',
        '12*x2 <= 95',
        '1*x2 <= 163',
        '4*x3 <= 95',
        '2*x3 <= 163',
        '14*x1 + 4*x3 >= 7',
        '2*x0 + 12*x2 >= 7',
        '14*x1 + 12*x2 >= 14',
        '2*x0 + 14*x1 >= 9',
        '2*x0 + 14*x1 + 12*x2 >= 21',
        '2*x0 + 14*x1 + 4*x3 >= 21',
        '2*x0 + 14*x1 + 12*x2 >= 19',
        '2*x0 + 14*x1 + 4*x3 >= 19',
        '2*x0 + 14*x1 + 12*x2 + 4*x3 >= 19',
        '7*x1 + 1*x2 >= 32',
        '1*x2 + 2*x3 >= 28',
        '7*x1 + 2*x3 >= 35',
        '8*x0 + 7*x1 + 2*x3 >= 34',
        '8*x0 + 7*x1 + 1*x2 >= 34',
        '7*x1 + 1*x2 + 2*x3 >= 34',
        '8*x0 + 7*x1 + 2*x3 >= 33',
        '8*x0 + 7*x1 + 1*x2 >= 33',
        '7*x1 + 1*x2 + 2*x3 >= 33',
        '8*x0 + 7*x1 + 2*x3 >= 35',
        '8*x0 + 7*x1 + 1*x2 >= 35',
        '7*x1 + 1*x2 + 2*x3 >= 35',
        '8*x0 + 7*x1 + 1*x2 + 2*x3 >= 35',
        '8*x0 - 3*x2 >= 0',
        '14*x1 + 4*x3 <= 81',
        '14*x1 + 12*x2 <= 86',
        '14*x1 + 12*x2 + 4*x3 <= 56',
        '2*x0 + 14*x1 + 4*x3 <= 62',
        '7*x1 + 1*x2 <= 95',
        '1*x2 + 2*x3 <= 152',
        '8*x0 + 1*x2 + 2*x3 <= 113'
    ]
}
```

## 5: 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 Ringo
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Paul
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Peggy
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Mary

    # Objective function
    model.setObjective(7 * x0 + 1 * x1 + 8 * x2 + 6 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(2 * x0 <= 95)
    model.addConstr(8 * x0 <= 163)
    model.addConstr(14 * x1 <= 95)
    model.addConstr(7 * x1 <= 163)
    model.addConstr(12 * x2 <= 95)
    model.addConstr(1 * x2 <= 163)
    model.addConstr(4 * x3 <= 95)
    model.addConstr(2 * x3 <= 163)
    model.addConstr(14 * x1 + 4 * x3 >= 7)
    model.addConstr(2 * x0 + 12 * x2 >= 7)
    model.addConstr(14 * x1 + 12 * x2 >= 14)
    model.addConstr(2 * x0 + 14 * x1 >= 9)
    model.addConstr(2 * x0 + 14 * x1 + 12 * x2 >= 21)
    model.addConstr(2 * x0 + 14 * x1 + 4 * x3 >= 21)
    model.addConstr(2 * x0 + 14 * x1 + 12 * x2 >= 19)
    model.addConstr(2 * x0 + 14 * x1 + 4 * x3 >= 19)
    model.addConstr(2 * x0 + 14 * x1 + 12 * x2 + 4 * x3 >= 19)
    model.addConstr(7 * x1 + 1 * x2 >= 32)
    model.addConstr(1 * x2 + 2 * x3 >= 28)
    model.addConstr(7 * x1 + 2 * x3 >= 35)
    model.addConstr(8 * x0 + 7 * x1 + 2 * x3 >= 34)
    model.addConstr(8 * x0 + 7 * x1 + 1 * x2 >= 34)
    model.addConstr(7 * x1 + 1 * x2 + 2 * x3 >= 34)
    model.addConstr(8 * x0 + 7 * x1 + 2 * x3 >= 33)
    model.addConstr(8 * x0 + 7 * x1 + 1 * x2 >= 33)
    model.addConstr(7 * x1 + 1 * x2 + 2 * x3 >= 33)
    model.addConstr(8 * x0 + 7 * x1 + 2 * x3 >= 35)
    model.addConstr(8 * x0 + 7 * x1 + 1 * x2 >= 35)
    model.addConstr(7 * x1 + 1 * x2 + 2 * x3 >= 35)
    model.addConstr(8 * x0 + 7 * x1 + 1 * x2 + 2 * x3 >= 35)
    model.addConstr(8 * x0 - 3 * x2 >= 0)
    model.addConstr(14 * x1 + 4 * x3 <= 81)
    model.addConstr(14 * x1 + 12 * x2 <= 86)
    model.addConstr(14 * x1 + 12 * x2 + 4 * x3 <= 56)
    model.addConstr(2 * x0 + 14 * x1 + 4 * x3 <= 62)
    model.addConstr(7 * x1 + 1 * x2 <= 95)
    model.addConstr(1 * x2 + 2 * x3 <= 152)
    model.addConstr(8 * x0 + 1 * x2 + 2 * x3 <= 113)

    # Solve the model
    model.optimize()

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

optimize_problem()
```