## Step 1: Define the optimization problem and the goal
The goal is to maximize the objective function $2.18x_0 + 6.25x_1 + 8.27x_2$, where $x_0$, $x_1$, and $x_2$ represent the hours worked by John, Peggy, and Mary, respectively.

## 2: List all the constraints
1. $6x_0 \geq 6$ (John's computer competence rating is 6, but this seems to be an attribute, not a constraint in the traditional sense. We will consider it as given: $x0 = 6$),
2. $1x_0 \geq 1$ (John's productivity rating is 1, similarly, $x0 = 1$),
3. $19x_0 \geq 19$ (John's dollar cost per hour is 19, $x0 = 19$),
4. $4x_1 \geq 4$ (Peggy's computer competence rating is 4, $x1 = 4$),
5. $11x_1 \geq 11$ (Peggy's productivity rating is 11, $x1 = 11$),
6. $13x_1 \geq 13$ (Peggy's dollar cost per hour is 13, $x1 = 13$),
7. $18x_2 \geq 18$ (Mary's computer competence rating is 18, $x2 = 18$),
8. $6x_2 \geq 6$ (Mary's productivity rating is 6, $x2 = 6$),
9. $7x_2 \geq 7$ (Mary's dollar cost per hour is 7, $x2 = 7$),
10. $6x_0 + 4x_1 + 18x_2 \geq 36$,
11. $19x_0 + 13x_1 \geq 27$,
12. $6x_0 + 18x_2 \leq 75$,
13. $6x_0 + 4x_1 + 18x_2 \leq 102$,
14. $11x_1 + 6x_2 \leq 48$,
15. $1x_0 + 6x_2 \leq 74$,
16. $1x_0 + 11x_1 + 6x_2 \leq 98$,
17. $19x_0 + 13x_1 \leq 46$,
18. $19x_0 + 7x_2 \leq 131$,
19. $19x_0 + 13x_1 + 7x_2 \leq 134$,
20. $x_0$ is an integer,
21. $x_1$ is an integer,
22. $x_2$ is an integer.

However, upon closer inspection, constraints 1 through 9 seem to describe attributes rather than constraints. The actual constraints from the problem statement that we need to consider in the optimization are:

## 3: Correctly formulate the constraints and objective function for Gurobi
The objective function to maximize is: $2.18x_0 + 6.25x_1 + 8.27x_2$

Constraints:
- $6x_0 + 4x_1 + 18x_2 \geq 36$
- $19x_0 + 13x_1 \geq 27$
- $6x_0 + 18x_2 \leq 75$
- $6x_0 + 4x_1 + 18x_2 \leq 102$
- $11x_1 + 6x_2 \leq 48$
- $x_0 + 6x_2 \leq 74$
- $x_0 + 11x_1 + 6x_2 \leq 98$
- $19x_0 + 13x_1 \leq 46$
- $19x_0 + 7x_2 \leq 131$
- $19x_0 + 13x_1 + 7x_2 \leq 134$
- $x_0, x_1, x_2 \geq 0$ (Non-negativity constraints, although not explicitly stated, they are implied for hours worked)
- $x_0$ is an integer,
- $x_1$ is an integer,
- $x_2$ is an integer.

## 4: Implement the optimization problem in Gurobi
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="hours_worked_by_John", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="hours_worked_by_Peggy", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="hours_worked_by_Mary", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(2.18*x0 + 6.25*x1 + 8.27*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(6*x0 + 4*x1 + 18*x2 >= 36)
    model.addConstr(19*x0 + 13*x1 >= 27)
    model.addConstr(6*x0 + 18*x2 <= 75)
    model.addConstr(6*x0 + 4*x1 + 18*x2 <= 102)
    model.addConstr(11*x1 + 6*x2 <= 48)
    model.addConstr(x0 + 6*x2 <= 74)
    model.addConstr(x0 + 11*x1 + 6*x2 <= 98)
    model.addConstr(19*x0 + 13*x1 <= 46)
    model.addConstr(19*x0 + 7*x2 <= 131)
    model.addConstr(19*x0 + 13*x1 + 7*x2 <= 134)

    # Non-negativity constraints
    model.addConstr(x0 >= 0)
    model.addConstr(x1 >= 0)
    model.addConstr(x2 >= 0)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Hours worked by John: {x0.varValue}")
        print(f"Hours worked by Peggy: {x1.varValue}")
        print(f"Hours worked by Mary: {x2.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```