## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the profit of a hot dog store by determining the optimal number of regular and premium hot dogs to produce daily.

### Decision Variables

- \(x_1\): The number of regular hot dogs produced per day.
- \(x_2\): The number of premium hot dogs produced per day.

### Objective Function

The profit from \(x_1\) regular hot dogs at $3 each and \(x_2\) premium hot dogs at $5 each needs to be maximized. Therefore, the objective function is:

\[ \text{Maximize:} \quad 3x_1 + 5x_2 \]

### Constraints

1. **Non-negativity constraints**: \(x_1 \geq 0\) and \(x_2 \geq 0\), as the number of hot dogs cannot be negative.
2. **Demand constraints**:
   - \(x_1 \leq 100\) (at most 100 regular hot dogs per day)
   - \(x_2 \leq 250\) (at most 250 premium hot dogs per day)
3. **Production capacity constraint**: \(x_1 + x_2 \leq 300\) (maximum of 300 hot dogs of either type per day)

## Gurobi Code

```python
import gurobi

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

    # Define the decision variables
    x1 = model.addVar(lb=0, ub=100, name="regular_hot_dogs")
    x2 = model.addVar(lb=0, ub=250, name="premium_hot_dogs")

    # Objective function: Maximize profit
    model.setObjective(3 * x1 + 5 * x2, gurobi.GRB.MAXIMIZE)

    # Production capacity constraint
    model.addConstraint(x1 + x2 <= 300, name="production_capacity")

    # Demand constraints are inherently handled by setting ub for x1 and x2

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution found. Produce {x1.x} regular hot dogs and {x2.x} premium hot dogs.")
        print(f"Maximum profit: ${3 * x1.x + 5 * x2.x}")
    else:
        print("The model is infeasible.")

solve_hot_dog_problem()
```

This code defines the optimization problem using Gurobi's Python interface, solves it, and prints out the optimal production levels for regular and premium hot dogs, along with the maximum achievable profit. If the problem is infeasible, it indicates that as well.