## 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 packages of LED lightbulbs.
- $x_2$ represents the number of packages of Halogen lightbulbs.

The objective function to maximize profit is:
\[ \text{Maximize:} \quad 30x_1 + 50x_2 \]

The constraints based on the machine availability are:
\[ 6x_1 + 9x_2 \leq 500 \] (plastics machine constraint)
\[ 12x_1 + 10x_2 \leq 500 \] (wiring machine constraint)
\[ x_1 \geq 0 \] and $x_2 \geq 0$ (non-negativity constraints, as the number of packages cannot be negative)

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'packages of LED lightbulbs'), ('x2', 'packages of Halogen lightbulbs')],
    'objective_function': '30*x1 + 50*x2',
    'constraints': [
        '6*x1 + 9*x2 <= 500',
        '12*x1 + 10*x2 <= 500',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

# Create a new model
model = gp.Model("Lighting_Company_Optimization")

# Define the variables
x1 = model.addVar(name="LED_packages", lb=0)  # packages of LED lightbulbs
x2 = model.addVar(name="Halogen_packages", lb=0)  # packages of Halogen lightbulbs

# Define the objective function
model.setObjective(30*x1 + 50*x2, gp.GRB.MAXIMIZE)

# Add constraints
model.addConstr(6*x1 + 9*x2 <= 500, name="plastics_machine_constraint")
model.addConstr(12*x1 + 10*x2 <= 500, name="wiring_machine_constraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution: LED_packages = {x1.varValue}, Halogen_packages = {x2.varValue}")
    print(f"Maximum profit: ${model.objVal:.2f}")
else:
    print("The model is infeasible.")
```