## Step 1: Define the optimization problem
The problem is to maximize the objective function: $9 \times \text{milkshakes} + 4 \times \text{oreos} + 9 \times \text{bowls of pasta} + 5 \times \text{apple pies}$.

## Step 2: List all the constraints
The constraints are as follows:
- $\text{milkshakes} \times 23.23 + \text{oreos} \times 13.58 + \text{bowls of pasta} \times 20.74 + \text{apple pies} \times 14.49 \leq 446$ (calcium upper bound)
- $\text{milkshakes} \times 20.53 + \text{oreos} \times 11.23 + \text{bowls of pasta} \times 10.41 + \text{apple pies} \times 23.66 \leq 184$ (sourness index upper bound)
- $\text{bowls of pasta} \times 20.74 + \text{apple pies} \times 14.49 \geq 87$ (calcium from pasta and apple pies)
- $\text{oreos} \times 13.58 + \text{apple pies} \times 14.49 \geq 68$ (calcium from oreos and apple pies)
- $\text{milkshakes} \times 23.23 + \text{bowls of pasta} \times 20.74 \geq 104$ (calcium from milkshakes and pasta)
- $\text{milkshakes} \times 23.23 + \text{oreos} \times 13.58 + \text{apple pies} \times 14.49 \geq 69$ (calcium from milkshakes, oreos, and apple pies)
- $\text{bowls of pasta} \times 10.41 + \text{apple pies} \times 23.66 \geq 16$ (sourness from pasta and apple pies)
- $\text{oreos} \times 13.58 + \text{bowls of pasta} \times 20.74 \leq 277$ (calcium from oreos and pasta)
- $\text{milkshakes} \times 23.23 + \text{oreos} \times 13.58 \leq 387$ (calcium from milkshakes and oreos)
- $\text{milkshakes} \times 23.23 + \text{bowls of pasta} \times 20.74 \leq 295$ (calcium from milkshakes and pasta)
- $\text{bowls of pasta} \times 20.74 + \text{apple pies} \times 14.49 \leq 137$ (calcium from pasta and apple pies)
- $\text{milkshakes} \times 23.23 + \text{oreos} \times 13.58 + \text{apple pies} \times 14.49 \leq 331$ (calcium from milkshakes, oreos, and apple pies)
- $\text{milkshakes} \times 23.23 + \text{oreos} \times 13.58 + \text{bowls of pasta} \times 20.74 \leq 256$ (calcium from milkshakes, oreos, and pasta)
- $\text{milkshakes} \times 23.23 + \text{oreos} \times 13.58 + \text{bowls of pasta} \times 20.74 + \text{apple pies} \times 14.49 \leq 256$ (total calcium)
- $\text{milkshakes} \times 20.53 + \text{bowls of pasta} \times 10.41 \leq 68$ (sourness from milkshakes and pasta)
- $\text{oreos} \times 11.23 + \text{apple pies} \times 23.66 \leq 107$ (sourness from oreos and apple pies)
- $\text{bowls of pasta} \times 10.41 + \text{apple pies} \times 23.66 \leq 50$ (sourness from pasta and apple pies)
- $\text{milkshakes} \times 20.53 + \text{apple pies} \times 23.66 \leq 133$ (sourness from milkshakes and apple pies)
- $\text{oreos} \times 11.23 + \text{bowls of pasta} \times 10.41 + \text{apple pies} \times 23.66 \leq 61$ (sourness from oreos, pasta, and apple pies)
- $\text{milkshakes} \times 20.53 + \text{oreos} \times 11.23 + \text{bowls of pasta} \times 10.41 \leq 143$ (sourness from milkshakes, oreos, and pasta)
- $\text{milkshakes} \times 20.53 + \text{oreos} \times 11.23 + \text{apple pies} \times 23.66 \leq 157$ (sourness from milkshakes, oreos, and apple pies)
- $\text{milkshakes} \times 20.53 + \text{oreos} \times 11.23 + \text{bowls of pasta} \times 10.41 + \text{apple pies} \times 23.66 \leq 157$ (total sourness)

## Step 3: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
milkshakes = m.addVar(name="milkshakes", lb=0)
oreos = m.addVar(name="oreos", lb=0)
bowls_of_pasta = m.addVar(name="bowls_of_pasta", lb=0)
apple_pies = m.addVar(name="apple_pies", lb=0)

# Define the objective function
m.setObjective(9 * milkshakes + 4 * oreos + 9 * bowls_of_pasta + 5 * apple_pies, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(23.23 * milkshakes + 13.58 * oreos + 20.74 * bowls_of_pasta + 14.49 * apple_pies <= 446)
m.addConstr(20.53 * milkshakes + 11.23 * oreos + 10.41 * bowls_of_pasta + 23.66 * apple_pies <= 184)

m.addConstr(20.74 * bowls_of_pasta + 14.49 * apple_pies >= 87)
m.addConstr(13.58 * oreos + 14.49 * apple_pies >= 68)
m.addConstr(23.23 * milkshakes + 20.74 * bowls_of_pasta >= 104)
m.addConstr(23.23 * milkshakes + 13.58 * oreos + 14.49 * apple_pies >= 69)

m.addConstr(10.41 * bowls_of_pasta + 23.66 * apple_pies >= 16)
m.addConstr(13.58 * oreos + 20.74 * bowls_of_pasta <= 277)
m.addConstr(23.23 * milkshakes + 13.58 * oreos <= 387)
m.addConstr(23.23 * milkshakes + 20.74 * bowls_of_pasta <= 295)
m.addConstr(20.74 * bowls_of_pasta + 14.49 * apple_pies <= 137)
m.addConstr(23.23 * milkshakes + 13.58 * oreos + 14.49 * apple_pies <= 331)
m.addConstr(23.23 * milkshakes + 13.58 * oreos + 20.74 * bowls_of_pasta <= 256)
m.addConstr(23.23 * milkshakes + 13.58 * oreos + 20.74 * bowls_of_pasta + 14.49 * apple_pies <= 256)

m.addConstr(20.53 * milkshakes + 10.41 * bowls_of_pasta <= 68)
m.addConstr(11.23 * oreos + 23.66 * apple_pies <= 107)
m.addConstr(10.41 * bowls_of_pasta + 23.66 * apple_pies <= 50)
m.addConstr(20.53 * milkshakes + 23.66 * apple_pies <= 133)
m.addConstr(11.23 * oreos + 10.41 * bowls_of_pasta + 23.66 * apple_pies <= 61)
m.addConstr(20.53 * milkshakes + 11.23 * oreos + 10.41 * bowls_of_pasta <= 143)
m.addConstr(20.53 * milkshakes + 11.23 * oreos + 23.66 * apple_pies <= 157)
m.addConstr(20.53 * milkshakes + 11.23 * oreos + 10.41 * bowls_of_pasta + 23.66 * apple_pies <= 157)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Milkshakes: ", milkshakes.varValue)
    print("Oreos: ", oreos.varValue)
    print("Bowls of pasta: ", bowls_of_pasta.varValue)
    print("Apple pies: ", apple_pies.varValue)
else:
    print("The model is infeasible")
```