## Step 1: Define the symbolic representation of the variables
The variables are 'cartons of milk', 'packs of napkins', and 'packs of paper plates'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into symbolic notation
The objective function to minimize is $3x_0 + 3x_1 + 5x_2$.

## Step 3: Define the constraints in symbolic notation
1. $15x_0 + 8x_1 + 23x_2 \leq 304$ (total cost constraint)
2. $4x_0 + x_1 + 17x_2 \geq 340$ (total usefulness rating constraint)
3. $8x_1 + 23x_2 \geq 35$ (minimum spend on packs of napkins and packs of paper plates)
4. $15x_0 + 8x_1 \geq 96$ (minimum spend on cartons of milk and packs of napkins)
5. $15x_0 + 8x_1 + 23x_2 \geq 96$ (total spend constraint)
6. $4x_0 + 17x_2 \geq 84$ (usefulness rating from cartons of milk and packs of paper plates)
7. $4x_0 + x_1 \geq 58$ (minimum usefulness rating from cartons of milk and packs of napkins)
8. $4x_0 + x_1 + 17x_2 \geq 89$ (total usefulness rating constraint)
9. $-4x_0 + 7x_2 \geq 0$ (constraint on cartons of milk and packs of paper plates)
10. $15x_0 + 8x_1 + 23x_2 \leq 221$ (maximum spend constraint)

## 4: Specify the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'cartons of milk'), 
        ('x1', 'packs of napkins'), 
        ('x2', 'packs of paper plates')
    ], 
    'objective_function': '3*x0 + 3*x1 + 5*x2', 
    'constraints': [
        '15*x0 + 8*x1 + 23*x2 <= 304', 
        '4*x0 + x1 + 17*x2 >= 340', 
        '8*x1 + 23*x2 >= 35', 
        '15*x0 + 8*x1 >= 96', 
        '15*x0 + 8*x1 + 23*x2 >= 96', 
        '4*x0 + 17*x2 >= 84', 
        '4*x0 + x1 >= 58', 
        '4*x0 + x1 + 17*x2 >= 89', 
        '-4*x0 + 7*x2 >= 0', 
        '15*x0 + 8*x1 + 23*x2 <= 221'
    ]
}
```

## 5: Write the Gurobi code to solve the optimization problem
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="cartons_of_milk", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="packs_of_napkins", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="packs_of_paper_plates", vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(3*x0 + 3*x1 + 5*x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(15*x0 + 8*x1 + 23*x2 <= 304)
    model.addConstr(4*x0 + x1 + 17*x2 >= 340)
    model.addConstr(8*x1 + 23*x2 >= 35)
    model.addConstr(15*x0 + 8*x1 >= 96)
    model.addConstr(15*x0 + 8*x1 + 23*x2 >= 96)
    model.addConstr(4*x0 + 17*x2 >= 84)
    model.addConstr(4*x0 + x1 >= 58)
    model.addConstr(4*x0 + x1 + 17*x2 >= 89)
    model.addConstr(-4*x0 + 7*x2 >= 0)
    model.addConstr(15*x0 + 8*x1 + 23*x2 <= 221)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Cartons of milk: {x0.varValue}")
        print(f"Packs of napkins: {x1.varValue}")
        print(f"Packs of paper plates: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```