## Step 1: Define the variables and their attributes
We have four variables: 'hours worked by Jean', 'hours worked by Hank', 'hours worked by Bobby', and 'hours worked by Ringo'. Let's denote these as $x_0$, $x_1$, $x_2$, and $x_3$ respectively. Their attributes are given as:
- Productivity ratings: $x_0 = 23$, $x_1 = 32$, $x_2 = 17$, $x_3 = 12$
- Computer competence ratings: $x_0 = 27$, $x_1 = 15$, $x_2 = 10$, $x_3 = 10$

## 2: Formulate the objective function
The objective function to minimize is $2x_0 + 2x_1 + 9x_2 + 7x_3$.

## 3: Define the constraints
1. $32x_1 + 17x_2 \geq 205$
2. $23x_0 + 32x_1 + 17x_2 + 12x_3 \geq 205$
3. $10x_2 + 10x_3 \geq 64$
4. $15x_1 + 10x_3 \geq 70$
5. $27x_0 + 10x_3 \geq 67$
6. $27x_0 + 15x_1 + 10x_2 + 10x_3 \geq 67$
7. $-7x_0 + 4x_2 \geq 0$
8. $8x_2 - 6x_3 \geq 0$
9. $-10x_1 + 2x_2 + 4x_3 \geq 0$
10. $23x_0 + 32x_1 + 12x_3 \leq 632$
11. $15x_1 + 10x_3 \leq 331$
12. $10x_2 + 10x_3 \leq 170$
13. $27x_0 + 10x_3 \leq 228$
14. $27x_0 + 15x_1 \leq 130$
15. $27x_0 + 10x_2 \leq 118$
16. $27x_0 + 15x_1 + 10x_2 \leq 295$

## 4: Implement the problem in Gurobi
We will use Gurobi's Python API to model and solve this linear programming problem.

```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="hours_worked_by_Jean", lb=0)
    x1 = model.addVar(name="hours_worked_by_Hank", lb=0)
    x2 = model.addVar(name="hours_worked_by_Bobby", lb=0)
    x3 = model.addVar(name="hours_worked_by_Ringo", lb=0)

    # Define the objective function
    model.setObjective(2*x0 + 2*x1 + 9*x2 + 7*x3, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(32*x1 + 17*x2 >= 205)
    model.addConstr(23*x0 + 32*x1 + 17*x2 + 12*x3 >= 205)
    model.addConstr(10*x2 + 10*x3 >= 64)
    model.addConstr(15*x1 + 10*x3 >= 70)
    model.addConstr(27*x0 + 10*x3 >= 67)
    model.addConstr(27*x0 + 15*x1 + 10*x2 + 10*x3 >= 67)
    model.addConstr(-7*x0 + 4*x2 >= 0)
    model.addConstr(8*x2 - 6*x3 >= 0)
    model.addConstr(-10*x1 + 2*x2 + 4*x3 >= 0)
    model.addConstr(23*x0 + 32*x1 + 12*x3 <= 632)
    model.addConstr(15*x1 + 10*x3 <= 331)
    model.addConstr(10*x2 + 10*x3 <= 170)
    model.addConstr(27*x0 + 10*x3 <= 228)
    model.addConstr(27*x0 + 15*x1 <= 130)
    model.addConstr(27*x0 + 10*x2 <= 118)
    model.addConstr(27*x0 + 15*x1 + 10*x2 <= 295)

    # Optimize the model
    model.optimize()

    # Print the status of the optimization
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print("Hours worked by Jean: ", x0.varValue)
        print("Hours worked by Hank: ", x1.varValue)
        print("Hours worked by Bobby: ", x2.varValue)
        print("Hours worked by Ringo: ", x3.varValue)
        print("Objective value: ", model.objVal)
    elif model.status == gurobi.GRB.Status.INFEASIBLE:
        print("No feasible solution exists.")
    else:
        print("Optimization failed.")

solve_optimization_problem()
```