## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'packs of paper plates' and 'cartons of milk', which we can denote as $x_1$ and $x_2$ respectively. The objective function to minimize is $1.68x_1 + 9.87x_2$. The constraints are:
- $16.51x_1 + 6.66x_2 \geq 35$
- $13.09x_1 + 14.76x_2 \geq 34$
- $-8x_1 + 3x_2 \geq 0$
- $16.51x_1 + 6.66x_2 \leq 102$
- $13.09x_1 + 14.76x_2 \leq 42$
- $x_1$ must be an integer
- $x_2$ must be an integer

## Step 2: Convert the problem into a symbolic representation
The symbolic representation of the variables is:
- $x_1$ represents 'packs of paper plates'
- $x_2$ represents 'cartons of milk'

The objective function in symbolic notation is: $1.68x_1 + 9.87x_2$

The constraints in symbolic notation are:
- $16.51x_1 + 6.66x_2 \geq 35$
- $13.09x_1 + 14.76x_2 \geq 34$
- $-8x_1 + 3x_2 \geq 0$
- $16.51x_1 + 6.66x_2 \leq 102$
- $13.09x_1 + 14.76x_2 \leq 42$

## 3: Express the problem in the required JSON format
```json
{
    'sym_variables': [('x1', 'packs of paper plates'), ('x2', 'cartons of milk')], 
    'objective_function': '1.68*x1 + 9.87*x2', 
    'constraints': [
        '16.51*x1 + 6.66*x2 >= 35', 
        '13.09*x1 + 14.76*x2 >= 34', 
        '-8*x1 + 3*x2 >= 0', 
        '16.51*x1 + 6.66*x2 <= 102', 
        '13.09*x1 + 14.76*x2 <= 42', 
        'x1 ∈ ℤ', 
        'x2 ∈ ℤ'
    ]
}
```

## 4: Implement the problem using Gurobi
We will now implement this problem using Gurobi in Python.

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="packs_of_paper_plates", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="cartons_of_milk", vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(1.68 * x1 + 9.87 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(16.51 * x1 + 6.66 * x2 >= 35, name="usefulness_rating")
    model.addConstr(13.09 * x1 + 14.76 * x2 >= 34, name="portability_rating")
    model.addConstr(-8 * x1 + 3 * x2 >= 0, name="milk_vs_plates")
    model.addConstr(16.51 * x1 + 6.66 * x2 <= 102, name="max_usefulness")
    model.addConstr(13.09 * x1 + 14.76 * x2 <= 42, name="max_portability")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("packs of paper plates: ", x1.varValue)
        print("cartons of milk: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```