## Step 1: Define the symbolic representation of the variables
The variables are 'hot dogs', 'blueberry pies', 'milkshakes', 'bowls of cereal' which can be represented as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $8x_0 + 7x_1 + 8x_2 + 3x_3$.

## Step 3: List all the constraints in symbolic notation
1. $22x_0 + 17x_1 + 23x_2 + 11x_3 \leq 435$ (total protein constraint)
2. $4x_0 + 19x_1 + 14x_2 + 13x_3 \leq 315$ (total calcium constraint)
3. $17x_1 + 23x_2 \geq 73$ (minimum protein from blueberry pies and milkshakes)
4. $22x_0 + 23x_2 \geq 107$ (minimum protein from hot dogs and milkshakes)
5. $22x_0 + 17x_1 + 23x_2 + 11x_3 \geq 107$ (minimum total protein)
6. $4x_0 + 14x_2 \geq 45$ (minimum calcium from hot dogs and milkshakes)
7. $14x_2 + 13x_3 \geq 28$ (minimum calcium from milkshakes and bowls of cereal)
8. $19x_1 + 14x_2 \geq 27$ (minimum calcium from blueberry pies and milkshakes)
9. $4x_0 + 19x_1 \geq 67$ (minimum calcium from hot dogs and blueberry pies)
10. $4x_0 + 13x_3 \geq 43$ (minimum calcium from hot dogs and bowls of cereal)
11. $4x_0 + 19x_1 + 14x_2 + 13x_3 \geq 43$ (minimum total calcium)
12. $-8x_0 + 8x_3 \geq 0$ (relationship between hot dogs and bowls of cereal)
13. $x_1 - 10x_2 \geq 0$ (relationship between blueberry pies and milkshakes)
14. $22x_0 + 11x_3 \leq 143$ (maximum protein from hot dogs and bowls of cereal)
15. $22x_0 + 23x_2 \leq 247$ (maximum protein from hot dogs and milkshakes)
16. $22x_0 + 17x_1 + 11x_3 \leq 222$ (maximum protein from hot dogs, blueberry pies, and bowls of cereal)
17. $22x_0 + 23x_2 + 11x_3 \leq 138$ (maximum protein from hot dogs, milkshakes, and bowls of cereal)
18. $22x_0 + 17x_1 + 23x_2 \leq 166$ (maximum protein from hot dogs, blueberry pies, and milkshakes)
19. $14x_2 + 13x_3 \leq 187$ (maximum calcium from milkshakes and bowls of cereal)

## Step 4: Define the symbolic representation of the problem
```json
{
'sym_variables': [
    ('x0', 'hot dogs'), 
    ('x1', 'blueberry pies'), 
    ('x2', 'milkshakes'), 
    ('x3', 'bowls of cereal')
],
'objective_function': '8*x0 + 7*x1 + 8*x2 + 3*x3',
'constraints': [
    '22*x0 + 17*x1 + 23*x2 + 11*x3 <= 435',
    '4*x0 + 19*x1 + 14*x2 + 13*x3 <= 315',
    '17*x1 + 23*x2 >= 73',
    '22*x0 + 23*x2 >= 107',
    '22*x0 + 17*x1 + 23*x2 + 11*x3 >= 107',
    '4*x0 + 14*x2 >= 45',
    '14*x2 + 13*x3 >= 28',
    '19*x1 + 14*x2 >= 27',
    '4*x0 + 19*x1 >= 67',
    '4*x0 + 13*x3 >= 43',
    '4*x0 + 19*x1 + 14*x2 + 13*x3 >= 43',
    '-8*x0 + 8*x3 >= 0',
    'x1 - 10*x2 >= 0',
    '22*x0 + 11*x3 <= 143',
    '22*x0 + 23*x2 <= 247',
    '22*x0 + 17*x1 + 11*x3 <= 222',
    '22*x0 + 23*x2 + 11*x3 <= 138',
    '22*x0 + 17*x1 + 23*x2 <= 166',
    '14*x2 + 13*x3 <= 187'
]
}
```

## Step 5: Write the Gurobi code for the optimization problem
```python
import gurobi

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

# Define the variables
x0 = model.addVar(name='hot_dogs', vtype='I')  # Integer variable for hot dogs
x1 = model.addVar(name='blueberry_pies', vtype='C')  # Continuous variable for blueberry pies
x2 = model.addVar(name='milkshakes', vtype='C')  # Continuous variable for milkshakes
x3 = model.addVar(name='bowls_of_cereal', vtype='I')  # Integer variable for bowls of cereal

# Define the objective function
model.setObjective(8*x0 + 7*x1 + 8*x2 + 3*x3, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(22*x0 + 17*x1 + 23*x2 + 11*x3 <= 435)  # Total protein constraint
model.addConstr(4*x0 + 19*x1 + 14*x2 + 13*x3 <= 315)  # Total calcium constraint
model.addConstr(17*x1 + 23*x2 >= 73)  # Minimum protein from blueberry pies and milkshakes
model.addConstr(22*x0 + 23*x2 >= 107)  # Minimum protein from hot dogs and milkshakes
model.addConstr(22*x0 + 17*x1 + 23*x2 + 11*x3 >= 107)  # Minimum total protein
model.addConstr(4*x0 + 14*x2 >= 45)  # Minimum calcium from hot dogs and milkshakes
model.addConstr(14*x2 + 13*x3 >= 28)  # Minimum calcium from milkshakes and bowls of cereal
model.addConstr(19*x1 + 14*x2 >= 27)  # Minimum calcium from blueberry pies and milkshakes
model.addConstr(4*x0 + 19*x1 >= 67)  # Minimum calcium from hot dogs and blueberry pies
model.addConstr(4*x0 + 13*x3 >= 43)  # Minimum calcium from hot dogs and bowls of cereal
model.addConstr(4*x0 + 19*x1 + 14*x2 + 13*x3 >= 43)  # Minimum total calcium
model.addConstr(-8*x0 + 8*x3 >= 0)  # Relationship between hot dogs and bowls of cereal
model.addConstr(x1 - 10*x2 >= 0)  # Relationship between blueberry pies and milkshakes
model.addConstr(22*x0 + 11*x3 <= 143)  # Maximum protein from hot dogs and bowls of cereal
model.addConstr(22*x0 + 23*x2 <= 247)  # Maximum protein from hot dogs and milkshakes
model.addConstr(22*x0 + 17*x1 + 11*x3 <= 222)  # Maximum protein from hot dogs, blueberry pies, and bowls of cereal
model.addConstr(22*x0 + 23*x2 + 11*x3 <= 138)  # Maximum protein from hot dogs, milkshakes, and bowls of cereal
model.addConstr(22*x0 + 17*x1 + 23*x2 <= 166)  # Maximum protein from hot dogs, blueberry pies, and milkshakes
model.addConstr(14*x2 + 13*x3 <= 187)  # Maximum calcium from milkshakes and bowls of cereal

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', model.objval)
    print('Hot dogs: ', x0.varValue)
    print('Blueberry pies: ', x1.varValue)
    print('Milkshakes: ', x2.varValue)
    print('Bowls of cereal: ', x3.varValue)
else:
    print('No optimal solution found')
```