You're on the right track with your objective function, which is to minimize the foregone revenue due to bumped passengers. The objective function should be:
Min Z = $400Xs (super deluxe) + $600Xf (first class) + $200Xt (tourist) + $0Xe (economy)
Here, Xi represents the number of passengers bumped into class i. The coefficients represent the cost per room for each class.
Now let's move on to the constraints. Based on your observations, passengers will only be bumped up one class, and if the class they are to be bumped to is full, a passenger from the more expensive class will be bumped up to make room for them. With this in mind, we can set up the constraints as follows:
1. Super Deluxe: Xs + S ≤ 160, where S represents the original number of reservations made in the super deluxe class. This constraint ensures that the total number of reservations in the super deluxe class, including bumped passengers, does not exceed the total available reservations (160).
2. First Class: Xf + F - Xs ≤ 150, where F represents the original number of reservations made in the first class. This constraint accounts for passengers bumped into the first class from the super deluxe class and ensures that the total number of reservations in the first class, including bumped passengers, does not exceed the total available reservations (150).
3. Tourist: Xt + T - Xf ≤ 180, where T represents the original number of reservations made in the tourist class. This constraint accounts for passengers bumped into the tourist class from the first class and ensures that the total number of reservations in the tourist class, including bumped passengers, does not exceed the total available reservations (180).
4. Economy: Xe + E - Xt ≤ 230, where E represents the original number of reservations made in the economy class. This constraint accounts for passengers bumped into the economy class from the tourist class and ensures that the total number of reservations in the economy class, including bumped passengers, does not exceed the total available reservations (230).
5. Non-negativity: Xs ≥ 0, Xf ≥ 0, Xt ≥ 0, Xe ≥ 0. This constraint ensures that the number of bumped passengers in each class is non-negative.
By incorporating these constraints into your model, you will be able to determine the number of accommodations in each class that will minimize the loss of potential revenue.
Keep in mind that you need to substitute the actual numbers of reservations and available spaces when solving the model, as you mentioned that you will fill in the actual amounts later.