Can someone please help me with this linear programming problem?

A vacation cruise liner has 4 classes of accommodations. The # of reservations, availability and cost of each of these classes is given below.


Class-- Reservation-- # available-- Cost





Super deluxe-- 100-- 160-- $3000


First class-- 145-- 150-- $2600


Tourist-- 170-- 180-- $2000


Economy-- 265 -- 230-- $1800


The number of reservations in each class does not match the availability. The problem is further complicated by a company policy that if a customer is holding a certain type reservation that cannot be filled, then the customer will receive the next better class of accommodation at no additional cost. For example, if a customer has a tourist reservation which must be satisfied with a first class accommodation, then the company will lose $600 in potential revenue by giving the customer a $2600 accommodation for only $2000.


Determine the number of accommodations in each class to be received by the customers that will minimize the loss of potential revenue for the company.

Can someone please help me with this linear programming problem?
Maybe someone else can see something in this problem that I can't, but this looks like a trivial arithmetic problem, not an LP. But we can still do an LP formulation.





We have 3 variables, the number of passengers elevated to the three higher classes of travel. So let's call the number of passengers elevated to Super Deluxe B1, and then B2 %26amp; B3 for the next two classes.





You should now be able to write the objective (sum of lost potential revenue * each variable).





As this is a minimisation, we need to put lower bounds on each variable to force the solver to accept all the reservations. So B3 %26gt;= 35 (the number of excess economy passengers), B2 %26gt;= B3 - 10, and B1 %26gt;= B2 - 5. You can rearrange these into the appropriate form.





Here's where it becomes trivial. B3 cannot be higher than 35, so B3 = 35. It's pretty obvious that you wouldn't elevate anyone unnecessarily, so all three constraint equations are really just equalities.





And just in case you end up using this stuff out in the real world, remind people like the one who formulated this problem (who is likely an accountant or a former manager of Soviet Cruise Lines, or both) that the game is to maximise profit, not minimise lost potential revenue. Otherwise you'd feed the 35 excess economy passengers to the sharks (OK, maybe just cancel their reservations).