## Step 1: Define the variables and their corresponding natural language objects
The variables are: 
- 'hours worked by Ringo' 
- 'hours worked by Hank' 
- 'hours worked by Bill' 
- 'hours worked by Jean' 
- 'hours worked by George' 
- 'hours worked by Peggy' 
- 'hours worked by John' 
- 'hours worked by Mary'

Let's denote them as $x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8$ respectively.

## Step 2: Formulate the objective function
The objective function to minimize is $7x_1 + 4x_2 + 9x_3 + 5x_4 + 4x_5 + 3x_6 + 8x_7 + 6x_8$.

## 3: List all the constraints
1. $8x_3 + 6x_4 \geq 28$
2. $14x_1 + 6x_4 \geq 24$
3. $14x_1 + 2x_8 \geq 11$
4. $8x_7 + 2x_8 \geq 21$
5. $18x_2 + 8x_3 \geq 17$
6. $6x_4 + 17x_5 \geq 12$
7. $14x_1 + 17x_5 \geq 31$
8. $17x_5 + 13x_6 \geq 17$
9. $8x_3 + 8x_7 \geq 13$
10. $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 \geq 13$
11. $-10x_4 + 3x_8 \geq 0$
12. $-2x_7 + 8x_8 \geq 0$
13. $-x_3 + x_5 \geq 0$
14. $6x_4 + 17x_5 \leq 122$
15. $6x_4 + 8x_7 \leq 55$
16. $8x_3 + 8x_7 \leq 72$
17. $14x_1 + 6x_4 \leq 191$
18. $14x_1 + 2x_8 \leq 103$
19. $18x_2 + 2x_8 \leq 153$
20. $8x_3 + 13x_6 \leq 84$
21. $8x_3 + 13x_6 + 8x_7 \leq 187$
22. $18x_2 + 6x_4 + 2x_8 \leq 95$
23. $6x_4 + 8x_7 + 2x_8 \leq 131$
24. $14x_1 + 6x_4 + 13x_6 \leq 201$
25. $14x_1 + 6x_4 + 8x_7 \leq 87$
26. $17x_5 + 13x_6 + 8x_7 \leq 88$
27. $18x_2 + 8x_7 + 2x_8 \leq 124$
28. $14x_1 + 18x_2 + 17x_5 \leq 108$

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Ringo'), 
        ('x2', 'hours worked by Hank'), 
        ('x3', 'hours worked by Bill'), 
        ('x4', 'hours worked by Jean'), 
        ('x5', 'hours worked by George'), 
        ('x6', 'hours worked by Peggy'), 
        ('x7', 'hours worked by John'), 
        ('x8', 'hours worked by Mary')
    ], 
    'objective_function': '7*x1 + 4*x2 + 9*x3 + 5*x4 + 4*x5 + 3*x6 + 8*x7 + 6*x8', 
    'constraints': [
        '8*x3 + 6*x4 >= 28',
        '14*x1 + 6*x4 >= 24',
        '14*x1 + 2*x8 >= 11',
        '8*x7 + 2*x8 >= 21',
        '18*x2 + 8*x3 >= 17',
        '6*x4 + 17*x5 >= 12',
        '14*x1 + 17*x5 >= 31',
        '17*x5 + 13*x6 >= 17',
        '8*x3 + 8*x7 >= 13',
        'x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 >= 13',
        '-10*x4 + 3*x8 >= 0',
        '-2*x7 + 8*x8 >= 0',
        '-x3 + x5 >= 0',
        '6*x4 + 17*x5 <= 122',
        '6*x4 + 8*x7 <= 55',
        '8*x3 + 8*x7 <= 72',
        '14*x1 + 6*x4 <= 191',
        '14*x1 + 2*x8 <= 103',
        '18*x2 + 2*x8 <= 153',
        '8*x3 + 13*x6 <= 84',
        '8*x3 + 13*x6 + 8*x7 <= 187',
        '18*x2 + 6*x4 + 2*x8 <= 95',
        '6*x4 + 8*x7 + 2*x8 <= 131',
        '14*x1 + 6*x4 + 13*x6 <= 201',
        '14*x1 + 6*x4 + 8*x7 <= 87',
        '17*x5 + 13*x6 + 8*x7 <= 88',
        '18*x2 + 8*x7 + 2*x8 <= 124',
        '14*x1 + 18*x2 + 17*x5 <= 108',
        'x1 % 1 == 0',
        'x2 % 1 == 0',
        'x3 % 1 == 0',
        'x4 % 1 == 0',
        'x5 % 1 == 0',
        'x6 % 1 == 0',
        'x7 % 1 == 0',
        'x8 % 1 == 0'
    ]
}
```

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

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

    # Define the variables
    x1 = model.addVar(name='x1', vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo
    x2 = model.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
    x3 = model.addVar(name='x3', vtype=gurobi.GRB.INTEGER)  # hours worked by Bill
    x4 = model.addVar(name='x4', vtype=gurobi.GRB.INTEGER)  # hours worked by Jean
    x5 = model.addVar(name='x5', vtype=gurobi.GRB.INTEGER)  # hours worked by George
    x6 = model.addVar(name='x6', vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
    x7 = model.addVar(name='x7', vtype=gurobi.GRB.INTEGER)  # hours worked by John
    x8 = model.addVar(name='x8', vtype=gurobi.GRB.INTEGER)  # hours worked by Mary

    # Define the objective function
    model.setObjective(7 * x1 + 4 * x2 + 9 * x3 + 5 * x4 + 4 * x5 + 3 * x6 + 8 * x7 + 6 * x8, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(8 * x3 + 6 * x4 >= 28)
    model.addConstr(14 * x1 + 6 * x4 >= 24)
    model.addConstr(14 * x1 + 2 * x8 >= 11)
    model.addConstr(8 * x7 + 2 * x8 >= 21)
    model.addConstr(18 * x2 + 8 * x3 >= 17)
    model.addConstr(6 * x4 + 17 * x5 >= 12)
    model.addConstr(14 * x1 + 17 * x5 >= 31)
    model.addConstr(17 * x5 + 13 * x6 >= 17)
    model.addConstr(8 * x3 + 8 * x7 >= 13)
    model.addConstr(x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 >= 13)
    model.addConstr(-10 * x4 + 3 * x8 >= 0)
    model.addConstr(-2 * x7 + 8 * x8 >= 0)
    model.addConstr(-x3 + x5 >= 0)
    model.addConstr(6 * x4 + 17 * x5 <= 122)
    model.addConstr(6 * x4 + 8 * x7 <= 55)
    model.addConstr(8 * x3 + 8 * x7 <= 72)
    model.addConstr(14 * x1 + 6 * x4 <= 191)
    model.addConstr(14 * x1 + 2 * x8 <= 103)
    model.addConstr(18 * x2 + 2 * x8 <= 153)
    model.addConstr(8 * x3 + 13 * x6 <= 84)
    model.addConstr(8 * x3 + 13 * x6 + 8 * x7 <= 187)
    model.addConstr(18 * x2 + 6 * x4 + 2 * x8 <= 95)
    model.addConstr(6 * x4 + 8 * x7 + 2 * x8 <= 131)
    model.addConstr(14 * x1 + 6 * x4 + 13 * x6 <= 201)
    model.addConstr(14 * x1 + 6 * x4 + 8 * x7 <= 87)
    model.addConstr(17 * x5 + 13 * x6 + 8 * x7 <= 88)
    model.addConstr(18 * x2 + 8 * x7 + 2 * x8 <= 124)
    model.addConstr(14 * x1 + 18 * x2 + 17 * x5 <= 108)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('x1: ', x1.varValue)
        print('x2: ', x2.varValue)
        print('x3: ', x3.varValue)
        print('x4: ', x4.varValue)
        print('x5: ', x5.varValue)
        print('x6: ', x6.varValue)
        print('x7: ', x7.varValue)
        print('x8: ', x8.varValue)
    else:
        print('No solution found')

solve_optimization_problem()
```