## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The vehicle company produces two products: bikes and cars. Each product requires a certain amount of time on two machines: an assembly machine and a painting machine. The goal is to maximize profit given the constraints on machine availability.

Let's define the decision variables:
- \(x\): the number of bikes produced
- \(y\): the number of cars produced

The objective function to maximize profit is:
\[ \text{Maximize:} \quad 1000x + 3000y \]

The constraints based on machine availability are:
- Assembly machine: \( x + 3y \leq 10 \)
- Painting machine: \( 0.5x + y \leq 8 \)
- Non-negativity: \( x \geq 0, y \geq 0 \)

## Gurobi Code

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

```python
import gurobi

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

    # Define the decision variables
    x = model.addVar(name="bikes", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    y = model.addVar(name="cars", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Objective function: Maximize profit
    model.setObjective(1000*x + 3000*y, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x + 3*y <= 10, name="assembly_machine")
    model.addConstr(0.5*x + y <= 8, name="painting_machine")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: Produce {x.varValue} bikes and {y.varValue} cars.")
        print(f"Maximal profit: ${1000*x.varValue + 3000*y.varValue}")
    else:
        print("The model is infeasible.")

if __name__ == "__main__":
    solve_vehicle_production_problem()
```

This code defines the problem in Gurobi, solves it, and prints out the optimal production levels for bikes and cars, along with the maximal profit achievable. If the problem is infeasible (i.e., there are no production levels that satisfy all constraints), it indicates that as well.