## Symbolic Representation

To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation.

Let's define the symbolic variables as follows:

- $x_1$ represents the number of carry-on suitcases produced per day.
- $x_2$ represents the number of large suitcases produced per day.

The objective is to maximize profit. The profit from $x_1$ carry-on suitcases is $100x_1$ and from $x_2$ large suitcases is $150x_2$. Therefore, the objective function to maximize is:

\[ \text{Maximize:} \quad 100x_1 + 150x_2 \]

The constraints based on the production rates and the sewing machine capacity are:

1. The carry-on suitcase team can produce at most 15 suitcases per day: $x_1 \leq 15$
2. The large suitcase team can produce at most 20 suitcases per day: $x_2 \leq 20$
3. The sewing machine can process at most 25 suitcases per day: $x_1 + x_2 \leq 25$
4. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'carry-on suitcases'), ('x2', 'large suitcases')],
    'objective_function': '100*x1 + 150*x2',
    'constraints': [
        'x1 <= 15',
        'x2 <= 20',
        'x1 + x2 <= 25',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=0, ub=15, name="carry-on_suitcases")
    x2 = model.addVar(lb=0, ub=20, name="large_suitcases")

    # Define the objective function
    model.setObjective(100 * x1 + 150 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(x1 + x2 <= 25, name="sewing_machine_capacity")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Carry-on suitcases: {x1.varValue}")
        print(f"Large suitcases: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_luggage_problem()
```