Here's our approach to formulating and solving this linear programming problem with Gurobi in Python:

**Decision Variables:**

* `x`: Number of heated seats sold
* `y`: Number of regular seats sold

**Objective Function:**

Maximize profit: `30x + 20y`

**Constraints:**

* **Capacity:** `x + y <= 300`  (The arena holds at most 300 people)
* **Minimum Heated Seats:** `x >= 50` (At least 50 heated seats)
* **Regular Seat Preference:** `y >= 3x` (At least 3 times as many regular seats as heated)
* **Non-negativity:** `x >= 0`, `y >= 0` (Cannot sell negative seats)


```python
import gurobipy as gp

# Create a new model
model = gp.Model("HockeyArenaOptimization")

# Create decision variables
x = model.addVar(lb=0, vtype=gp.GRB.INTEGER, name="heated_seats") # Integer number of heated seats
y = model.addVar(lb=0, vtype=gp.GRB.INTEGER, name="regular_seats") # Integer number of regular seats

# Set the objective function
model.setObjective(30*x + 20*y, gp.GRB.MAXIMIZE)

# Add constraints
model.addConstr(x + y <= 300, "capacity")
model.addConstr(x >= 50, "min_heated")
model.addConstr(y >= 3*x, "regular_preference")


# Optimize the model
model.optimize()

# Check for infeasibility
if model.status == gp.GRB.INFEASIBLE:
    print("The model is infeasible.")
else:
    # Print the optimal solution
    print(f"Optimal Solution:")
    print(f"Number of heated seats (x): {x.x}")
    print(f"Number of regular seats (y): {y.x}")
    print(f"Maximum profit: ${model.objVal}")

```
