## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the profit from buying and selling two types of hoodies: zippered hoodies and pullover hoodies.

### Variables

- Let \(Z\) be the number of zippered hoodies.
- Let \(P\) be the number of pullover hoodies.

### Objective Function

The profit per zippered hoodie is $5, and the profit per pullover hoodie is $4. The objective is to maximize the total profit:

\[ \text{Maximize:} \quad 5Z + 4P \]

### Constraints

1. **Cost Constraint:** Zippered hoodies cost $20 each, and pullover hoodies cost $15 each. The total budget is $450.

\[ 20Z + 15P \leq 450 \]

2. **Sales Constraint:** The boy can sell at most 25 hoodies in total.

\[ Z + P \leq 25 \]

3. **Non-Negativity Constraint:** The number of hoodies cannot be negative.

\[ Z \geq 0, P \geq 0 \]
\[ Z \text{ and } P \text{ are integers} \]

However, since Gurobi can handle continuous variables, we will initially solve it as a continuous problem and then consider if integer constraints are needed.

## Gurobi Code

```python
import gurobi

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

    # Define variables
    Z = model.addVar(name="Zippered_Hoodies", lb=0)
    P = model.addVar(name="Pullover_Hoodies", lb=0)

    # Objective function: Maximize profit
    model.setObjective(5 * Z + 4 * P, gurobi.GRB.MAXIMIZE)

    # Cost constraint
    model.addConstr(20 * Z + 15 * P <= 450, name="Cost_Constraint")

    # Sales constraint
    model.addConstr(Z + P <= 25, name="Sales_Constraint")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal Solution: Zippered Hoodies = {Z.varValue:.2f}, Pullover Hoodies = {P.varValue:.2f}")
        print(f"Max Profit: ${model.objVal:.2f}")
    else:
        print("No optimal solution found.")

solve_hoodie_problem()
```

This code defines the problem in Gurobi, solving for the optimal number of zippered and pullover hoodies to maximize profit under the given constraints. Note that the solution values are printed with two decimal places; if integer solutions are required, further steps such as setting the variables as integer or using a different approach might be necessary. 

However, if an integer solution is strictly required, we should modify the code to reflect that:

```python
import gurobi

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

    # Define variables as integers
    Z = model.addVar(name="Zippered_Hoodies", lb=0, type=gurobi.GRB.INTEGER)
    P = model.addVar(name="Pullover_Hoodies", lb=0, type=gurobi.GRB.INTEGER)

    # Objective function: Maximize profit
    model.setObjective(5 * Z + 4 * P, gurobi.GRB.MAXIMIZE)

    # Cost constraint
    model.addConstr(20 * Z + 15 * P <= 450, name="Cost_Constraint")

    # Sales constraint
    model.addConstr(Z + P <= 25, name="Sales_Constraint")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal Solution: Zippered Hoodies = {Z.varValue}, Pullover Hoodies = {P.varValue}")
        print(f"Max Profit: ${model.objVal:.2f}")
    else:
        print("No optimal solution found.")

solve_hoodie_problem()
```