## 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 doors
- $x_2$ represents the number of bumpers

The objective is to maximize profit. Given that the profit per door is $200 and the profit per bumper is $150, the objective function can be written as:

Maximize: $200x_1 + 150x_2$

The constraints based on the given information are:
1. Machine time constraint: Each door takes 20 minutes and each bumper takes 10 minutes, with 3000 minutes available in a week.
   - $20x_1 + 10x_2 \leq 3000$

2. Door production limit: The plant can make at most 100 doors per week.
   - $x_1 \leq 100$

3. Bumper production limit: The plant can make at most 200 bumpers per week.
   - $x_2 \leq 200$

4. Non-negativity constraints: The number of doors and bumpers cannot be negative.
   - $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'doors'), ('x2', 'bumpers')],
    'objective_function': '200*x1 + 150*x2',
    'constraints': [
        '20*x1 + 10*x2 <= 3000',
        'x1 <= 100',
        'x2 <= 200',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=0, ub=100, name="doors")
    x2 = model.addVar(lb=0, ub=200, name="bumpers")

    # Define the objective function
    model.setObjective(200 * x1 + 150 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(20 * x1 + 10 * x2 <= 3000, name="machine_time")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Number of doors: {x1.varValue}")
        print(f"Number of bumpers: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The model is infeasible")

solve_optimization_problem()
```