## Step 1: Define the optimization problem and its constraints
The problem requires minimizing an objective function subject to various constraints. The objective function to minimize is: $8B^2 + 6BP + 6P^2 + 3PH + B + 9P + H$, where $B$, $P$, and $H$ represent the hours worked by Bobby, Paul, and Hank, respectively.

## 2: List all constraints
1. $B_{org} = 5$
2. $B_{prod} = 18$
3. $B_{quit} = 11$
4. $B_{cost} = 13$
5. $P_{org} = 16$
6. $P_{prod} = 15$
7. $P_{quit} = 13$
8. $P_{cost} = 6$
9. $H_{org} = 15$
10. $H_{prod} = 12$
11. $H_{quit} = 5$
12. $H_{cost} = 16$
13. $B^2 + P^2 \geq 55$
14. $B + P + H \geq 55$
15. $B^2 + H^2 \geq 50$
16. $B + P \geq 22$
17. $B + P + H \geq 22$
18. $P_{quit}P + H_{quit}H \geq 25$
19. $B_{quit}B + P_{quit}P \geq 27$
20. $B^2_{quit}B^2 + P^2_{quit}P^2 + H^2_{quit}H^2 \geq 27$
21. $B_{quit}B + P_{quit}P + H_{quit}H \geq 27$
22. $B_{cost}B + H_{cost}H \geq 48$
23. $P_{cost}P + H_{cost}H \geq 57$
24. $B^2_{cost}B^2 + P^2_{cost}P^2 + H^2_{cost}H^2 \geq 88$
25. $B_{cost}B + P_{cost}P + H_{cost}H \geq 88$
26. $P - 4H \geq 0$
27. $B_{prod}B + H_{prod}H \leq 107$
28. $B_{prod}B + P_{prod}P + H_{prod}H \leq 102$
29. $P^2_{quit}P^2 + H^2_{quit}H^2 \leq 47$
30. $B_{quit}B + H_{quit}H \leq 59$
31. $B_{quit}B + P_{quit}P \leq 34$
32. $B_{quit}B + P_{quit}P + H_{quit}H \leq 59$
33. $B$ must be an integer.

## 3: Formulate the problem in Gurobi
We will use Gurobi's Python API to formulate and solve this problem.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
B = m.addVar(name="Bobby", vtype=gp.GRB.INTEGER)  # Integer hours for Bobby
P = m.addVar(name="Paul")  # Continuous hours for Paul
H = m.addVar(name="Hank")  # Continuous hours for Hank

# Objective function
m.setObjective(8*B**2 + 6*B*P + 6*P**2 + 3*P*H + B + 9*P + H, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(B == 5, name="Bobby_org")
m.addConstr(18*B == 18, name="Bobby_prod") # This constraint is always true and can be removed
m.addConstr(11*B == 11, name="Bobby_quit")
m.addConstr(13*B == 13, name="Bobby_cost")

m.addConstr(P == 16, name="Paul_org")
m.addConstr(15*P == 15, name="Paul_prod") # This constraint is always true and can be removed
m.addConstr(13*P == 13, name="Paul_quit")
m.addConstr(6*P == 6, name="Paul_cost")

m.addConstr(H == 15, name="Hank_org")
m.addConstr(12*H == 12, name="Hank_prod") # This constraint is always true and can be removed
m.addConstr(5*H == 5, name="Hank_quit")
m.addConstr(16*H == 16, name="Hank_cost")

m.addConstr(B**2 + P**2 >= 55, name="org_score")
m.addConstr(B + P + H >= 55, name="total_org_score")

m.addConstr(B**2 + H**2 >= 50, name="prod_rating_BH")
m.addConstr(B + P >= 22, name="prod_rating_BP")
m.addConstr(B + P + H >= 22, name="prod_rating_BPH")

m.addConstr(13*P + 5*H >= 25, name="quit_index_PH")
m.addConstr(11*B + 13*P >= 27, name="quit_index_BP")
m.addConstr(11**2*B**2 + 13**2*P**2 + 5**2*H**2 >= 27, name="quit_index_BPH_squared")
m.addConstr(11*B + 13*P + 5*H >= 27, name="quit_index_BPH")

m.addConstr(13*B + 16*H >= 48, name="cost_BH")
m.addConstr(6*P + 16*H >= 57, name="cost_PH")
m.addConstr(13**2*B**2 + 6**2*P**2 + 16**2*H**2 >= 88, name="cost_BPH_squared")
m.addConstr(13*B + 6*P + 16*H >= 88, name="cost_BPH")

m.addConstr(P - 4*H >= 0, name="Paul_Hank")

m.addConstr(18*B + 12*H <= 107, name="prod_rating_BH_max")
m.addConstr(18*B + 15*P + 12*H <= 102, name="prod_rating_BPH_max")

m.addConstr(13**2*P**2 + 5**2*H**2 <= 47, name="quit_index_PH_max")
m.addConstr(11*B + 5*H <= 59, name="quit_index_BH_max")
m.addConstr(11*B + 13*P <= 34, name="quit_index_BP_max")
m.addConstr(11*B + 13*P + 5*H <= 59, name="quit_index_BPH_max")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Bobby hours: ", B.varValue)
    print("Paul hours: ", P.varValue)
    print("Hank hours: ", H.varValue)
else:
    print("The model is infeasible")
```