To solve this optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote the number of cars produced as `x1` and the number of bikes produced as `x2`.

The objective is to maximize profit. Given that each car nets $5000 in profit and each bike nets $1000 in profit, the objective function can be represented algebraically as:
\[ 5000x_1 + 1000x_2 \]

The constraints are based on the available resources:
1. **Steel Constraint**: Both cars and bikes require 30 kg of steel. With 1000 kg of steel available each week, this constraint can be written as:
\[ 30x_1 + 30x_2 \leq 1000 \]
Simplifying it gives:
\[ x_1 + x_2 \leq \frac{1000}{30} \]
\[ x_1 + x_2 \leq \frac{100}{3} \]

2. **Engineering Time Constraint**: A car requires 3 hours of engineering time, and a bike requires 1 hour of engineering time, with a total of 400 hours available:
\[ 3x_1 + x_2 \leq 400 \]

Thus, the symbolic representation of our problem is:
```json
{
  'sym_variables': [('x1', 'number of cars'), ('x2', 'number of bikes')],
  'objective_function': '5000*x1 + 1000*x2',
  'constraints': ['x1 + x2 <= 100/3', '3*x1 + x2 <= 400', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll use the following code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, name="cars")
x2 = m.addVar(lb=0, name="bikes")

# Set objective function
m.setObjective(5000*x1 + 1000*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 100/3, "steel_constraint")
m.addConstr(3*x1 + x2 <= 400, "engineering_time_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Cars: {x1.x}")
    print(f"Bikes: {x2.x}")
    print(f"Maximum Profit: ${m.objVal}")
else:
    print("No optimal solution found")
```