## Step 1: Define the optimization problem
The problem is to minimize the objective function: $8.56M + 5.65L + 7.58R + 7.27J$, where $M$, $L$, $R$, and $J$ represent the hours worked by Mary, Laura, Ringo, and John, respectively.

## 2: List all the constraints
The constraints are:
1. $6.26M + 1.6L + 5.57R + 3.99J$ (productivity ratings are given but not directly constrained)
2. $0.76M + 5.92L + 4.04R + 5.11J$ (paperwork competence ratings are given but not directly constrained)
3. $1.6L + 3.99J \geq 11$
4. $6.26M + 1.6L \geq 31$
5. $1.6L + 5.57R \geq 26$
6. $5.57R + 3.99J \geq 30$
7. $6.26M + 1.6L + 5.57R + 3.99J \geq 30$
8. $0.76M + 4.04R \geq 51$
9. $4.04R + 5.11J \geq 50$
10. $0.76M + 5.92L \geq 26$
11. $5.92L + 4.04R \geq 36$
12. $0.76M + 5.92L + 4.04R + 5.11J \geq 36$
13. $-5M + 5J \geq 0$
14. $1.6L + 3.99J \leq 50$
15. $6.26M + 5.57R \leq 95$
16. $1.6L + 5.57R \leq 79$
17. $6.26M + 5.57R + 3.99J \leq 99$
18. $5.92L + 4.04R \leq 75$
19. $5.92L + 5.11J \leq 174$
20. $0.76M + 4.04R \leq 227$
21. $0.76M + 5.92L \leq 188$
22. $4.04R + 5.11J \leq 67$

## 3: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define the variables
    M = model.addVar(lb=0, name="hours_worked_by_Mary")
    L = model.addVar(lb=0, name="hours_worked_by_Laura")
    R = model.addVar(lb=0, name="hours_worked_by_Ringo")
    J = model.addVar(lb=0, name="hours_worked_by_John")

    # Define the objective function
    model.setObjective(8.56 * M + 5.65 * L + 7.58 * R + 7.27 * J, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(1.6 * L + 3.99 * J >= 11)
    model.addConstr(6.26 * M + 1.6 * L >= 31)
    model.addConstr(1.6 * L + 5.57 * R >= 26)
    model.addConstr(5.57 * R + 3.99 * J >= 30)
    model.addConstr(6.26 * M + 1.6 * L + 5.57 * R + 3.99 * J >= 30)
    model.addConstr(0.76 * M + 4.04 * R >= 51)
    model.addConstr(4.04 * R + 5.11 * J >= 50)
    model.addConstr(0.76 * M + 5.92 * L >= 26)
    model.addConstr(5.92 * L + 4.04 * R >= 36)
    model.addConstr(0.76 * M + 5.92 * L + 4.04 * R + 5.11 * J >= 36)
    model.addConstr(-5 * M + 5 * J >= 0)
    model.addConstr(1.6 * L + 3.99 * J <= 50)
    model.addConstr(6.26 * M + 5.57 * R <= 95)
    model.addConstr(1.6 * L + 5.57 * R <= 79)
    model.addConstr(6.26 * M + 5.57 * R + 3.99 * J <= 99)
    model.addConstr(5.92 * L + 4.04 * R <= 75)
    model.addConstr(5.92 * L + 5.11 * J <= 174)
    model.addConstr(0.76 * M + 4.04 * R <= 227)
    model.addConstr(0.76 * M + 5.92 * L <= 188)
    model.addConstr(4.04 * R + 5.11 * J <= 67)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Mary: {M.varValue}")
        print(f"Hours worked by Laura: {L.varValue}")
        print(f"Hours worked by Ringo: {R.varValue}")
        print(f"Hours worked by John: {J.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```