## Step 1: Define the optimization problem
The problem is to minimize the objective function: $7R + 4H + 9B + 5J + 4G + 3P + 8Jo + 6M$, where $R, H, B, J, G, P, Jo, M$ represent the hours worked by Ringo, Hank, Bill, Jean, George, Peggy, John, and Mary respectively.

## 2: List all the constraints
1. $8J + 8B \geq 28$
2. $14R + 6J \geq 24$
3. $14R + 2M \geq 11$
4. $8Jo + 2M \geq 21$
5. $18H + 8B \geq 17$
6. $6J + 17G \geq 12$
7. $14R + 17G \geq 31$
8. $17G + 13P \geq 17$
9. $8B + 8Jo \geq 13$
10. $R + H + B + J + G + P + Jo + M \geq 13$
11. $-10J + 3M \geq 0$
12. $-2Jo + 8M \geq 0$
13. $-B + G \geq 0$
14. $6J + 17G \leq 122$
15. $6J + 8Jo \leq 55$
16. $8B + 8Jo \leq 72$
17. $14R + 6J \leq 191$
18. $14R + 2M \leq 103$
19. $18H + 2M \leq 153$
20. $8B + 13P \leq 84$
21. $8B + 13P + 8Jo \leq 187$
22. $18H + 6J + 2M \leq 95$
23. $6J + 8Jo + 6M \leq 131$
24. $14R + 6J + 13P \leq 201$
25. $14R + 6J + 8Jo \leq 87$
26. $17G + 13P + 8Jo \leq 88$
27. $18H + 8Jo + 6M \leq 124$
28. $14R + 18H + 17G \leq 108$
29. $R, H, B, J, G, P, Jo, M$ are integers.

## 3: Implement the problem in Gurobi
We will use the Gurobi Python API to model and solve this problem.

```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
R = m.addVar(name="Ringo", vtype=gp.GRB.INTEGER)
H = m.addVar(name="Hank", vtype=gp.GRB.INTEGER)
B = m.addVar(name="Bill", vtype=gp.GRB.INTEGER)
J = m.addVar(name="Jean", vtype=gp.GRB.INTEGER)
G = m.addVar(name="George", vtype=gp.GRB.INTEGER)
P = m.addVar(name="Peggy", vtype=gp.GRB.INTEGER)
Jo = m.addVar(name="John", vtype=gp.GRB.INTEGER)
M = m.addVar(name="Mary", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(7*R + 4*H + 9*B + 5*J + 4*G + 3*P + 8*Jo + 6*M, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(8*J + 8*B >= 28)
m.addConstr(14*R + 6*J >= 24)
m.addConstr(14*R + 2*M >= 11)
m.addConstr(8*Jo + 2*M >= 21)
m.addConstr(18*H + 8*B >= 17)
m.addConstr(6*J + 17*G >= 12)
m.addConstr(14*R + 17*G >= 31)
m.addConstr(17*G + 13*P >= 17)
m.addConstr(8*B + 8*Jo >= 13)
m.addConstr(R + H + B + J + G + P + Jo + M >= 13)
m.addConstr(-10*J + 3*M >= 0)
m.addConstr(-2*Jo + 8*M >= 0)
m.addConstr(-B + G >= 0)
m.addConstr(6*J + 17*G <= 122)
m.addConstr(6*J + 8*Jo <= 55)
m.addConstr(8*B + 8*Jo <= 72)
m.addConstr(14*R + 6*J <= 191)
m.addConstr(14*R + 2*M <= 103)
m.addConstr(18*H + 2*M <= 153)
m.addConstr(8*B + 13*P <= 84)
m.addConstr(8*B + 13*P + 8*Jo <= 187)
m.addConstr(18*H + 6*J + 2*M <= 95)
m.addConstr(6*J + 8*Jo + 6*M <= 131)
m.addConstr(14*R + 6*J + 13*P <= 201)
m.addConstr(14*R + 6*J + 8*Jo <= 87)
m.addConstr(17*G + 13*P + 8*Jo <= 88)
m.addConstr(18*H + 8*Jo + 6*M <= 124)
m.addConstr(14*R + 18*H + 17*G <= 108)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("Ringo: ", R.varValue)
    print("Hank: ", H.varValue)
    print("Bill: ", B.varValue)
    print("Jean: ", J.varValue)
    print("George: ", G.varValue)
    print("Peggy: ", P.varValue)
    print("John: ", Jo.varValue)
    print("Mary: ", M.varValue)
else:
    print("The model is infeasible")
```