## Step 1: Define the symbolic representation of the variables
The variables are defined as follows:
- $x_1$ represents 'packs of paper plates'
- $x_2$ represents 'lightbulbs'
- $x_3$ represents 'cookies'
- $x_4$ represents 'packs of napkins'
- $x_5$ represents 'cartons of milk'

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: $7x_1 + 8x_2 + 5x_3 + 1x_4 + 9x_5$

## Step 3: Define the constraints in symbolic notation
The constraints are:
1. $0.47x_1 + 0.26x_2 + 0.83x_5 \geq 8$
2. $8x_2 - 2x_3 + 10x_5 \geq 0$
3. $0.26x_2 + 0.78x_4 \leq 63$
4. $0.47x_1 + 0.29x_3 \leq 41$
5. $0.29x_3 + 0.83x_5 \leq 15$
6. $0.47x_1 + 0.78x_4 \leq 34$
7. $0.47x_1 + 0.83x_5 \leq 74$
8. $0.26x_2 + 0.83x_5 \leq 69$
9. $0.26x_2 + 0.29x_3 \leq 30$
10. $0.47x_1 + 0.26x_2 + 0.78x_4 \leq 70$
11. $0.29x_3 + 0.78x_4 + 0.83x_5 \leq 56$
12. $0.47x_1 + 0.26x_2 + 0.83x_5 \leq 41$
13. $0.47x_1 + 0.26x_2 + 0.29x_3 \leq 48$
14. $0.26x_2 + 0.78x_4 + 0.83x_5 \leq 70$
15. $0.26x_2 + 0.29x_3 + 0.83x_5 \leq 51$
16. $0.26x_2 + 0.29x_3 + 0.78x_4 \leq 61$
17. $0.47x_1 + 0.26x_2 + 0.29x_3 + 0.78x_4 + 0.83x_5 \leq 61$

## 4: Convert the problem into Gurobi code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name='packs_of_paper_plates', vtype=gurobi.GRB.INTEGER)  # packs of paper plates
x2 = m.addVar(name='lightbulbs', vtype=gurobi.GRB.INTEGER)  # lightbulbs
x3 = m.addVar(name='cookies', vtype=gurobi.GRB.INTEGER)  # cookies
x4 = m.addVar(name='packs_of_napkins', vtype=gurobi.GRB.INTEGER)  # packs of napkins
x5 = m.addVar(name='cartons_of_milk', vtype=gurobi.GRB.INTEGER)  # cartons of milk

# Define the objective function
m.setObjective(7 * x1 + 8 * x2 + 5 * x3 + x4 + 9 * x5, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(0.47 * x1 + 0.26 * x2 + 0.83 * x5 >= 8)
m.addConstr(8 * x2 - 2 * x3 + 10 * x5 >= 0)
m.addConstr(0.26 * x2 + 0.78 * x4 <= 63)
m.addConstr(0.47 * x1 + 0.29 * x3 <= 41)
m.addConstr(0.29 * x3 + 0.83 * x5 <= 15)
m.addConstr(0.47 * x1 + 0.78 * x4 <= 34)
m.addConstr(0.47 * x1 + 0.83 * x5 <= 74)
m.addConstr(0.26 * x2 + 0.83 * x5 <= 69)
m.addConstr(0.26 * x2 + 0.29 * x3 <= 30)
m.addConstr(0.47 * x1 + 0.26 * x2 + 0.78 * x4 <= 70)
m.addConstr(0.29 * x3 + 0.78 * x4 + 0.83 * x5 <= 56)
m.addConstr(0.47 * x1 + 0.26 * x2 + 0.83 * x5 <= 41)
m.addConstr(0.47 * x1 + 0.26 * x2 + 0.29 * x3 <= 48)
m.addConstr(0.26 * x2 + 0.78 * x4 + 0.83 * x5 <= 70)
m.addConstr(0.26 * x2 + 0.29 * x3 + 0.83 * x5 <= 51)
m.addConstr(0.26 * x2 + 0.29 * x3 + 0.78 * x4 <= 61)
m.addConstr(0.47 * x1 + 0.26 * x2 + 0.29 * x3 + 0.78 * x4 + 0.83 * x5 <= 61)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objVal)
    print('packs of paper plates: ', x1.varValue)
    print('lightbulbs: ', x2.varValue)
    print('cookies: ', x3.varValue)
    print('packs of napkins: ', x4.varValue)
    print('cartons of milk: ', x5.varValue)
else:
    print('The model is infeasible')
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'packs of paper plates'), 
        ('x2', 'lightbulbs'), 
        ('x3', 'cookies'), 
        ('x4', 'packs of napkins'), 
        ('x5', 'cartons of milk')
    ], 
    'objective_function': '7*x1 + 8*x2 + 5*x3 + 1*x4 + 9*x5', 
    'constraints': [
        '0.47*x1 + 0.26*x2 + 0.83*x5 >= 8',
        '8*x2 - 2*x3 + 10*x5 >= 0',
        '0.26*x2 + 0.78*x4 <= 63',
        '0.47*x1 + 0.29*x3 <= 41',
        '0.29*x3 + 0.83*x5 <= 15',
        '0.47*x1 + 0.78*x4 <= 34',
        '0.47*x1 + 0.83*x5 <= 74',
        '0.26*x2 + 0.83*x5 <= 69',
        '0.26*x2 + 0.29*x3 <= 30',
        '0.47*x1 + 0.26*x2 + 0.78*x4 <= 70',
        '0.29*x3 + 0.78*x4 + 0.83*x5 <= 56',
        '0.47*x1 + 0.26*x2 + 0.83*x5 <= 41',
        '0.47*x1 + 0.26*x2 + 0.29*x3 <= 48',
        '0.26*x2 + 0.78*x4 + 0.83*x5 <= 70',
        '0.26*x2 + 0.29*x3 + 0.83*x5 <= 51',
        '0.26*x2 + 0.29*x3 + 0.78*x4 <= 61',
        '0.47*x1 + 0.26*x2 + 0.29*x3 + 0.78*x4 + 0.83*x5 <= 61'
    ]
}
```