## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize profit by determining the number of long-term cruise tickets and week-long cruise tickets to sell.

### Decision Variables

- $x$: Number of long-term cruise tickets sold
- $y$: Number of week-long cruise tickets sold

### Objective Function

The profit made on each long-term cruise ticket is $500, and on each week-long cruise ticket is $150. The objective is to maximize the total profit:

Maximize: $500x + 150y$

### Constraints

1. The cruise ship can take at most 1500 people:
   - $x + y \leq 1500$

2. There are a minimum of 35 long-term cruise tickets available:
   - $x \geq 35$

3. At least 4 times as many people prefer to buy week-long cruise tickets than long-term cruise tickets:
   - $y \geq 4x$

4. Non-negativity constraints:
   - $x \geq 0$
   - $y \geq 0$

Since $x$ and $y$ represent the number of tickets, they must be integers. However, for the purpose of this linear programming problem, we will first solve it without the integer constraint and then consider how to adjust for integer solutions if necessary.

## Gurobi Code

```python
import gurobi

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

    # Define the decision variables
    x = model.addVar(lb=35, name="long_term_tickets")  # Minimum of 35 long-term tickets
    y = model.addVar(name="week_long_tickets")

    # Objective function: Maximize profit
    model.setObjective(500 * x + 150 * y, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x + y <= 1500, name="capacity_constraint")
    model.addConstr(y >= 4 * x, name="preference_constraint")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal Solution: Long-term tickets = {x.varValue:.0f}, Week-long tickets = {y.varValue:.0f}")
        print(f"Maximum Profit: ${500 * x.varValue + 150 * y.varValue:.0f}")
    else:
        print("The model is infeasible.")

solve_cruise_ship_problem()
```

This code sets up the optimization problem with Gurobi, solves it, and prints out the optimal number of tickets to sell and the maximum profit achievable. Note that the `lb` parameter for `x` is set to 35 to reflect the minimum number of long-term cruise tickets available. The preference constraint $y \geq 4x$ and the capacity constraint $x + y \leq 1500$ are also included. 

The solution will automatically consider the non-negativity of $x$ and $y$ due to the nature of Gurobi's `addVar` method and the specified lower bound for $x$. 

If an integer solution is required, one could use Gurobi's MIP (Mixed-Integer Programming) capabilities by setting the `vtype` parameter of the variables to `gurobi.GRB.INTEGER` when adding them to the model. However, given that the objective coefficients and constraint coefficients are all integers, and the right-hand side values of the constraints are integers, the optimal solution to the LP relaxation will likely yield integer values for $x$ and $y$, or values very close to integers, which can then be rounded if necessary.