## Problem Description and Symbolic Representation

The problem requires minimizing the cost of producing backpacks and handbags under certain constraints. Let's denote:
- $x_1$ as the number of backpacks
- $x_2$ as the number of handbags

The objective is to minimize the total cost, which is $5x_1 + 8x_2$.

The constraints are:
1. The machine must be operated for at least 3000 minutes per week: $20x_1 + 15x_2 \geq 3000$
2. The factory must make a minimum of 180 items total: $x_1 + x_2 \geq 180$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation

```json
{
    'sym_variables': [('x1', 'backpacks'), ('x2', 'handbags')],
    'objective_function': '5*x1 + 8*x2',
    'constraints': [
        '20*x1 + 15*x2 >= 3000',
        'x1 + x2 >= 180',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="backpacks", lb=0, vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="handbags", lb=0, vtype=gurobi.GRB.INTEGER)

    # Objective function: minimize 5*x1 + 8*x2
    model.setObjective(5*x1 + 8*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(20*x1 + 15*x2 >= 3000, name="machine_time")
    model.addConstr(x1 + x2 >= 180, name="total_items")

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of backpacks: {x1.varValue}")
        print(f"Number of handbags: {x2.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_problem()
```