## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'packs of napkins' and 'dish soap bottles'. Let's denote 'packs of napkins' as $x_1$ and 'dish soap bottles' as $x_2$. The objective function to minimize is $4x_1 + x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
1. $3x_1 + 3x_2 \geq 16$ (minimum spend of $16),
2. $x_1 + 8x_2 \geq 8$ (minimum usefulness rating of 8),
3. $x_1 + x_2 \geq 13$ (minimum sustainability score of 13),
4. $-x_1 + 2x_2 \geq 0$ (relationship between packs of napkins and dish soap bottles),
5. $3x_1 + 3x_2 \leq 32$ (maximum spend of $32),
6. $x_1 + 8x_2 \leq 26$ (usefulness rating not exceeding 26),
7. $x_1 + x_2 \leq 32$ (sustainability score not exceeding 32, corrected from 40 to match given upper bound),
8. $x_1$ must be an integer (packs of napkins are whole numbers),
9. $x_2$ must be an integer (dish soap bottles are whole numbers).

## 3: Correct and Clarify Constraints
Upon review, there seems to be a typo in the sustainability score upper bound, which should be 40 according to the initial problem statement, not 32. Thus, the correct constraint should be $x_1 + x_2 \leq 40$.

## 4: Symbolic Representation
The symbolic representation of the problem is:
- Variables: $x_1$ for 'packs of napkins', $x_2$ for 'dish soap bottles'.
- Objective function: Minimize $4x_1 + x_2$.
- Constraints:
  1. $3x_1 + 3x_2 \geq 16$,
  2. $x_1 + 8x_2 \geq 8$,
  3. $x_1 + x_2 \geq 13$,
  4. $-x_1 + 2x_2 \geq 0$,
  5. $3x_1 + 3x_2 \leq 32$,
  6. $x_1 + 8x_2 \leq 26$,
  7. $x_1 + x_2 \leq 40$.

## 5: Gurobi Code
```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="packs_of_napkins", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="dish_soap_bottles", vtype=gurobi.GRB.INTEGER)

    # Objective function: Minimize 4*x1 + x2
    model.setObjective(4 * x1 + x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(3 * x1 + 3 * x2 >= 16, name="min_spend")
    model.addConstr(x1 + 8 * x2 >= 8, name="min_usefulness")
    model.addConstr(x1 + x2 >= 13, name="min_sustainability")
    model.addConstr(-x1 + 2 * x2 >= 0, name="relationship")
    model.addConstr(3 * x1 + 3 * x2 <= 32, name="max_spend")
    model.addConstr(x1 + 8 * x2 <= 26, name="max_usefulness")
    model.addConstr(x1 + x2 <= 40, name="max_sustainability")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("packs_of_napkins: ", x1.varValue)
        print("dish_soap_bottles: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```

## 6: Final Output
```json
{
    'sym_variables': [('x1', 'packs of napkins'), ('x2', 'dish soap bottles')],
    'objective_function': '4*x1 + x2',
    'constraints': [
        '3*x1 + 3*x2 >= 16',
        'x1 + 8*x2 >= 8',
        'x1 + x2 >= 13',
        '-x1 + 2*x2 >= 0',
        '3*x1 + 3*x2 <= 32',
        'x1 + 8*x2 <= 26',
        'x1 + x2 <= 40'
    ]
}
```