## Problem Description and Formulation

The problem is an optimization problem with multiple variables and constraints. The goal is to maximize an objective function subject to various constraints on resources and attributes.

### Variables
- $x_0$: apple pies
- $x_1$: bowls of pasta
- $x_2$: bowls of cereal
- $x_3$: ham sandwiches

### Objective Function
Maximize: $5.64x_0 + 5.52x_1 + 2.47x_2 + 8.43x_3$

### Constraints
1. $28x_0 + 24x_2 \geq 116$
2. $24x_2 + 14x_3 \leq 387$
3. $28x_0 + 24x_2 \leq 192$
4. $28x_0 + 24x_1 \leq 179$
5. $24x_1 + 24x_2 \leq 480$
6. $28x_0 + 24x_1 + 24x_2 + 14x_3 \leq 480$
7. $23x_1 + 4x_2 \leq 68$
8. $27x_0 + 23x_1 \leq 132$
9. $27x_0 + 2x_3 \leq 133$
10. $23x_1 + 2x_3 \leq 80$
11. $23x_1 + 4x_2 + 2x_3 \leq 139$
12. $27x_0 + 23x_1 + 4x_2 + 2x_3 \leq 139$

### Variable Bounds
- $x_3$ is an integer (Implicitly handled in Gurobi by setting `x3.VarType = GRB.INTEGER`)
- Other variables are continuous

## Gurobi Code

```python
import gurobi as grb

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

# Define variables
x0 = m.addVar(name="apple_pies", lb=0)  # No lower bound specified, assuming 0
x1 = m.addVar(name="bowls_of_pasta", lb=0)
x2 = m.addVar(name="bowls_of_cereal", lb=0)
x3 = m.addVar(name="ham_sandwiches", lb=0, vtype=grb.GRB.INTEGER)

# Objective function
m.setObjective(5.64 * x0 + 5.52 * x1 + 2.47 * x2 + 8.43 * x3, grb.GRB.MAXIMIZE)

# Constraints
m.addConstr(28 * x0 + 24 * x2 >= 116, name="tastiness_apple_pies_cereal_min")
m.addConstr(24 * x2 + 14 * x3 <= 387, name="tastiness_cereal_ham_sandwiches_max")
m.addConstr(28 * x0 + 24 * x2 <= 192, name="tastiness_apple_pies_cereal_max")
m.addConstr(28 * x0 + 24 * x1 <= 179, name="tastiness_apple_pies_pasta_max")
m.addConstr(24 * x1 + 24 * x2 <= 480, name="tastiness_pasta_cereal_max")
m.addConstr(28 * x0 + 24 * x1 + 24 * x2 + 14 * x3 <= 480, name="tastiness_total_max")
m.addConstr(23 * x1 + 4 * x2 <= 68, name="cost_pasta_cereal_max")
m.addConstr(27 * x0 + 23 * x1 <= 132, name="cost_apple_pies_pasta_max")
m.addConstr(27 * x0 + 2 * x3 <= 133, name="cost_apple_pies_ham_sandwiches_max")
m.addConstr(23 * x1 + 2 * x3 <= 80, name="cost_pasta_ham_sandwiches_max")
m.addConstr(23 * x1 + 4 * x2 + 2 * x3 <= 139, name="cost_pasta_cereal_ham_sandwiches_max")
m.addConstr(27 * x0 + 23 * x1 + 4 * x2 + 2 * x3 <= 139, name="cost_total_max")

# Optimize
m.optimize()

# Print solution
if m.status == grb.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Apple pies: ", x0.varValue)
    print("Bowls of pasta: ", x1.varValue)
    print("Bowls of cereal: ", x2.varValue)
    print("Ham sandwiches: ", x3.varValue)
else:
    print("No optimal solution found")
```