## Problem Description and Formulation

The problem is to maximize profit from selling VIP and general seat tickets for a concert at Nolan Center. The key constraints and information are:

- The total seating capacity is 200 people.
- The profit per VIP seat ticket is $30.
- The profit per general seat ticket is $14.
- At least 20 seats must be reserved as VIP seats.
- At least 4 times as many people prefer sitting in general seats than in VIP seats.

## Symbolic Representation

Let's denote:
- \(x\) as the number of VIP seat tickets sold.
- \(y\) as the number of general seat tickets sold.

The objective is to maximize profit \(P = 30x + 14y\).

Subject to the constraints:
1. \(x + y \leq 200\) (Total seats constraint)
2. \(x \geq 20\) (Minimum VIP seats constraint)
3. \(y \geq 4x\) (General seats vs. VIP seats preference constraint)
4. \(x \geq 0\) and \(y \geq 0\) (Non-negativity constraint, as tickets cannot be negative)

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x = model.addVar(lb=0, name="VIP_seats")  # Number of VIP seats
    y = model.addVar(lb=0, name="general_seats")  # Number of general seats

    # Objective function: Maximize profit
    model.setObjective(30*x + 14*y, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x + y <= 200, name="total_seats")
    model.addConstr(x >= 20, name="min_VIP_seats")
    model.addConstr(y >= 4*x, name="general_vs_VIP")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: VIP seats = {x.varValue}, General seats = {y.varValue}")
        print(f"Maximum profit: ${30*x.varValue + 14*y.varValue}")
    else:
        print("No optimal solution found")

maximize_profit()
```