## Step 1: Define the symbolic representation of the variables
The variables are 'paper towel rolls', 'diapers', 'lightbulbs', and 'dish soap bottles'. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $1 \times x_0 + 9 \times x_1 + 4 \times x_2 + 6 \times x_3$.

## Step 3: List all the constraints in symbolic notation
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. $13x_0 + 3x_3 \geq 55$
5. $13x_0 + 8x_2 \geq 44$
6. $11x_1 + 8x_2 \geq 61$
7. $11x_1 + 3x_3 \geq 40$
8. $11x_1 + 8x_2 + 3x_3 \geq 51$
9. $13x_0 + 11x_1 + 8x_2 \geq 51$
10. $11x_1 + 8x_2 + 3x_3 \geq 38$
11. $13x_0 + 11x_1 + 8x_2 \geq 38$
12. $13x_0 + 11x_1 + 8x_2 + 3x_3 \geq 38$
13. $5x_1 - 9x_2 \geq 0$
14. $9x_0 + 7x_3 \leq 161$
15. $9x_0 + 5x_2 \leq 155$
16. $9x_0 + 14x_1 + 5x_2 \leq 51$
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. $11x_1 + 8x_2 + 3x_3 \leq 174$
24. $13x_0 + 11x_1 + 8x_2 \leq 76$

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'paper towel rolls'), ('x1', 'diapers'), ('x2', 'lightbulbs'), ('x3', 'dish soap bottles')],
    'objective_function': '1*x0 + 9*x1 + 4*x2 + 6*x3',
    'constraints': [
        '14*x1 + 7*x3 >= 39',
        '9*x0 + 5*x2 >= 19',
        '9*x0 + 14*x1 + 5*x2 + 7*x3 >= 19',
        '13*x0 + 3*x3 >= 55',
        '13*x0 + 8*x2 >= 44',
        '11*x1 + 8*x2 >= 61',
        '11*x1 + 3*x3 >= 40',
        '11*x1 + 8*x2 + 3*x3 >= 51',
        '13*x0 + 11*x1 + 8*x2 >= 51',
        '11*x1 + 8*x2 + 3*x3 >= 38',
        '13*x0 + 11*x1 + 8*x2 >= 38',
        '13*x0 + 11*x1 + 8*x2 + 3*x3 >= 38',
        '5*x1 - 9*x2 >= 0',
        '9*x0 + 7*x3 <= 161',
        '9*x0 + 5*x2 <= 155',
        '9*x0 + 14*x1 + 5*x2 <= 51',
        '11*x1 + 8*x2 <= 255',
        '13*x0 + 11*x1 <= 223',
        '11*x1 + 3*x3 <= 250',
        '13*x0 + 8*x2 <= 277',
        '13*x0 + 3*x3 <= 209',
        '8*x2 + 3*x3 <= 171',
        '11*x1 + 8*x2 + 3*x3 <= 174',
        '13*x0 + 11*x1 + 8*x2 <= 76'
    ]
}
```

## 5: Implement the optimization problem using 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(1*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(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(5*x1 - 9*x2 >= 0)
    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(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(13*x0 + 11*x1 + 8*x2 <= 76)

    # Solve the model
    model.optimize()

    # Print 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 model is infeasible")

optimize_problem()
```