## Step 1: Define the variables and their attributes
We have four variables: 'paper towel rolls', 'diapers', 'lightbulbs', and 'dish soap bottles'. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively. Their attributes are given as:
- $r_0$: usefulness rating with an upper bound of 172 and values $x_0 = 9$, $x_1 = 14$, $x_2 = 5$, $x_3 = 7$.
- $r_1$: sustainability score with an upper bound of 305 and values $x_0 = 13$, $x_1 = 11$, $x_2 = 8$, $x_3 = 3$.

## 2: Formulate the objective function
The objective function to minimize is: $1 \cdot x_0 + 9 \cdot x_1 + 4 \cdot x_2 + 6 \cdot x_3$.

## 3: Define the constraints
### Usefulness Rating Constraints
1. $14x_1 + 7x_3 \geq 39$
2. $9x_0 + 5x_2 \geq 19$
3. $9x_0 + 14x_1 + 5x_2 + 7x_3 \geq 19$
4. $9x_0 + 7x_3 \leq 161$
5. $9x_0 + 5x_2 \leq 155$
6. $9x_0 + 14x_1 + 5x_2 \leq 51$

### Sustainability Score Constraints
7. $13x_0 + 3x_3 \geq 55$
8. $13x_0 + 8x_2 \geq 44$
9. $11x_1 + 8x_2 \geq 61$
10. $11x_1 + 3x_3 \geq 40$
11. $11x_1 + 8x_2 + 3x_3 \geq 51$
12. $13x_0 + 11x_1 + 8x_2 \geq 51$
13. $11x_1 + 8x_2 + 3x_3 \geq 38$
14. $13x_0 + 11x_1 + 8x_2 \geq 38$
15. $13x_0 + 11x_1 + 8x_2 + 3x_3 \geq 38$
16. $13x_0 + 11x_1 + 8x_2 \leq 76$
17. $11x_1 + 8x_2 \leq 255$
18. $13x_0 + 11x_1 \leq 223$
19. $11x_1 + 3x_3 \leq 250$
20. $13x_0 + 8x_2 \leq 277$
21. $13x_0 + 3x_3 \leq 209$
22. $8x_2 + 3x_3 \leq 171$
23. $11x_1 + 8x_2 + 3x_3 \leq 174$

### Other Constraints
24. $5x_1 - 9x_2 \geq 0$
25. $x_0, x_1, x_2, x_3$ are integers.

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

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="paper_towel_rolls", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="diapers", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="lightbulbs", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="dish_soap_bottles", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(x0 + 9*x1 + 4*x2 + 6*x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(14*x1 + 7*x3 >= 39)
    model.addConstr(9*x0 + 5*x2 >= 19)
    model.addConstr(9*x0 + 14*x1 + 5*x2 + 7*x3 >= 19)
    model.addConstr(9*x0 + 7*x3 <= 161)
    model.addConstr(9*x0 + 5*x2 <= 155)
    model.addConstr(9*x0 + 14*x1 + 5*x2 <= 51)

    model.addConstr(13*x0 + 3*x3 >= 55)
    model.addConstr(13*x0 + 8*x2 >= 44)
    model.addConstr(11*x1 + 8*x2 >= 61)
    model.addConstr(11*x1 + 3*x3 >= 40)
    model.addConstr(11*x1 + 8*x2 + 3*x3 >= 51)
    model.addConstr(13*x0 + 11*x1 + 8*x2 >= 51)
    model.addConstr(11*x1 + 8*x2 + 3*x3 >= 38)
    model.addConstr(13*x0 + 11*x1 + 8*x2 >= 38)
    model.addConstr(13*x0 + 11*x1 + 8*x2 + 3*x3 >= 38)
    model.addConstr(13*x0 + 11*x1 + 8*x2 <= 76)
    model.addConstr(11*x1 + 8*x2 <= 255)
    model.addConstr(13*x0 + 11*x1 <= 223)
    model.addConstr(11*x1 + 3*x3 <= 250)
    model.addConstr(13*x0 + 8*x2 <= 277)
    model.addConstr(13*x0 + 3*x3 <= 209)
    model.addConstr(8*x2 + 3*x3 <= 171)
    model.addConstr(11*x1 + 8*x2 + 3*x3 <= 174)

    model.addConstr(5*x1 - 9*x2 >= 0)

    # Solve the problem
    model.optimize()

    # Output the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Paper Towel Rolls: ", x0.varValue)
        print("Diapers: ", x1.varValue)
        print("Lightbulbs: ", x2.varValue)
        print("Dish Soap Bottles: ", x3.varValue)
    else:
        print("The problem is infeasible")

optimize_problem()
```