## Step 1: Define the variables and their attributes
We have four variables: 'hours worked by Laura', 'hours worked by John', 'hours worked by George', and 'hours worked by Hank'. Let's denote them as $L$, $J$, $G$, and $H$ respectively.

## 2: Formulate the objective function
The objective function to minimize is $3.4L + 3.75J + 9.58G + 9.08H$.

## 3: List all constraints
1. $28L + 8J + 27G + 6H$ is not a constraint but attribute.
2. $4L + 2J + 6G + 11H$ is not a constraint but attribute.
3. $8J + 6H \geq 45$
4. $28L + 8J \geq 55$
5. $28L + 8J + 27G + 6H \geq 55$
6. $4L + 2J \geq 30$
7. $4L + 11H \geq 44$
8. $6G + 11H \geq 31$
9. $4L + 6G \geq 30$
10. $4L + 6G + 11H \geq 32$
11. $4L + 2J + 11H \geq 32$
12. $4L + 6G + 11H \geq 45$
13. $4L + 2J + 11H \geq 45$
14. $4L + 2J + 6G + 11H \geq 45$
15. $-10L + 3H \geq 0$
16. $28L + 6H \leq 261$
17. $27G + 6H \leq 273$
18. $28L + 8J \leq 203$
19. $2J + 6G \leq 93$
20. $4L + 11H \leq 103$

## 4: Define variable bounds and types
- $L$ is continuous.
- $J$ is integer.
- $G$ is continuous.
- $H$ is continuous.

## 5: Implement the problem in Gurobi
```python
import gurobi as gp

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

# Define variables
L = m.addVar(lb=0, name="hours_worked_by_Laura", vtype=gp.GRB.CONTINUOUS)
J = m.addVar(lb=0, name="hours_worked_by_John", vtype=gp.GRB.INTEGER)
G = m.addVar(lb=0, name="hours_worked_by_George", vtype=gp.GRB.CONTINUOUS)
H = m.addVar(lb=0, name="hours_worked_by_Hank", vtype=gp.GRB.CONTINUOUS)

# Objective function
m.setObjective(3.4*L + 3.75*J + 9.58*G + 9.08*H, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(8*J + 6*H >= 45, name="constraint_1")
m.addConstr(28*L + 8*J >= 55, name="constraint_2")
m.addConstr(28*L + 8*J + 27*G + 6*H >= 55, name="constraint_3")
m.addConstr(4*L + 2*J >= 30, name="constraint_4")
m.addConstr(4*L + 11*H >= 44, name="constraint_5")
m.addConstr(6*G + 11*H >= 31, name="constraint_6")
m.addConstr(4*L + 6*G >= 30, name="constraint_7")
m.addConstr(4*L + 6*G + 11*H >= 32, name="constraint_8")
m.addConstr(4*L + 2*J + 11*H >= 32, name="constraint_9")
m.addConstr(4*L + 6*G + 11*H >= 45, name="constraint_10")
m.addConstr(4*L + 2*J + 11*H >= 45, name="constraint_11")
m.addConstr(4*L + 2*J + 6*G + 11*H >= 45, name="constraint_12")
m.addConstr(-10*L + 3*H >= 0, name="constraint_13")
m.addConstr(28*L + 6*H <= 261, name="constraint_14")
m.addConstr(27*G + 6*H <= 273, name="constraint_15")
m.addConstr(28*L + 8*J <= 203, name="constraint_16")
m.addConstr(2*J + 6*G <= 93, name="constraint_17")
m.addConstr(4*L + 11*H <= 103, name="constraint_18")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Laura: {L.varValue}")
    print(f"Hours worked by John: {J.varValue}")
    print(f"Hours worked by George: {G.varValue}")
    print(f"Hours worked by Hank: {H.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```