This post may be a little lengthy, so please bare with me if you can help. The problem is: A cruise liner has 4 classes of accommodations. Following are the # of reservations made, # of reservations available, and cost per room for each class:
Super Deluxe, 100, 160, $3000
First Class, 145, 150, $2600
Tourist, 170, 180, $2000
Economy, 265, 230, $1800
The company has a policy that if a customer is holding a certain type reservation that cant be filled, then the customer will be bumped up to the next class at no additional cost.
I need to determine the number of accommodations in each class to be received by customers that will MINIMIZE THE LOSS OF POTENTIAL REVENUE.
My observations: To minimize foregone revenue due to bumped passengers, a passenger will only be bumped up one class. 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. Therefore, if we let Xi=the # passengers bumped into class i, I think the objective function should be "Min Z=$400Xs(super deluxe)+ $600Xf(first class)+ $200Xt(tourist)+ $0Xe(economy)". The only constraints I see are regarding the total # reservations for each class, so I came up with: (all < or > are intended as less/greater than or equal to)
Min Z=$400Xs + $600Xf + $200Xt + $0Xe
s.t. Xs<160, Xf<150, Xt<180, Xe=0, Xs,f,t,e>0
Does this model look right, or am I way off? I came up with these constraints by figuring that the maximum amount of money you could possibly lose in each category would be in the event that the more expensive class had 0 reservations and you completely filled it with bumps from the next lower class. Remember, I am trying to MINIMIZE FOREGONE REVENUE, not necessarily to maximize profit. I have to enter the data into the Management Scientist software we are using and print out the solution it gives. I am not worried about the actual # of reservations given in the problem, as the model needs to work for any given # of reservations. I will fill in the actual amounts later. If anyone can give me some feedback as to whether I am on the right track or not, I would really appreciate it. I only have a few days to complete this, so any ideas you can offer will help greatly. Thanks for your time.
The objective function looks right, the constraints are not right. First, you want the constraints at <= You say tha Xs is the number bumped to super deluxe. Let S be the original number in superdeluxe. So the constraint is S+Xs <= 160. Let F be the original number in first class. The constraint here is F-Xs+Xf<=150. And so on.
Ah. I had thought about F+Xf<=150 and T+Xt<=180 before but forgot to subtract out the people bumped out of it. Thanks for the help!
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.