To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the problem statement.

Let's define:
- $x_1$ as the number of backpacks produced,
- $x_2$ as the number of handbags produced.

The objective is to minimize the total cost of production. Given that each backpack costs $5 and each handbag costs $8, the objective function can be written as:
\[ \text{Minimize} \quad 5x_1 + 8x_2 \]

The constraints are:
1. The machine must be operated for at least 3000 minutes per week. Since each backpack takes 20 minutes and each handbag takes 15 minutes, we have:
\[ 20x_1 + 15x_2 \geq 3000 \]
2. The factory must make a minimum of 180 items in total, which gives us:
\[ x_1 + x_2 \geq 180 \]
3. Non-negativity constraints, since the number of items cannot be negative:
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of backpacks'), ('x2', 'number of handbags')],
  'objective_function': '5*x1 + 8*x2',
  'constraints': ['20*x1 + 15*x2 >= 3000', 'x1 + x2 >= 180', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

# Create a new model
m = Model("Factory_Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="backpacks", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="handbags", lb=0)

# Set the objective function
m.setObjective(5*x1 + 8*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(20*x1 + 15*x2 >= 3000, "machine_time")
m.addConstr(x1 + x2 >= 180, "total_items")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of backpacks: {x1.x}")
    print(f"Number of handbags: {x2.x}")
    print(f"Total cost: ${5*x1.x + 8*x2.x:.2f}")
else:
    print("No optimal solution found")
```