# Math Help - airline ticketing problem - 3 variables.

1. ## airline ticketing problem - 3 variables.

Apologies if I have posted this in the wrong forum... open to suggestions if you think it should be moved elsewhere. Thanks.

So my problem is this:

I have to determine the best pricing structure for a 'simulated airline'.
There are 3 types of ticket;
Business Class seats @ $160 each, Premium Economy Class seats @$125 each, and Super Saver Class seats @ \$75 each.
A MINimum of 10% of the seats must be Business Class;
A MINimum of 50% must be Premium Economy Class tickets;
A MAXimum of 20% can be Super Saver Class tickets.
Of the Business Class seats it is predicted that only 52% will be sold, 65% of the Premium Economy Class tickets will be sold, and 76% of the Super Saver Class tickets will sell.

So the question is this... if I only have 108 seats on the aircraft what will be the best breakdown of tickets to get the maximum return given the restrictions on seat type allocations, expected sale %'s and different ticket costs?

So far I have only managed to work out an Excel based solution, using trial and error to work out the best results... but there must be a mathematical way of doing/illustrating this.

Thanks from New Zealand!

2. ## Re: airline ticketing problem - 3 variables.

Hey BuddhsNZ.

You should start by setting up a value function based on the random variables.

So think about a value function that is a function of your random variables [something like V = f(X,Y,Z)]

Now if you take the expectation of this, you will get E[V] = E[f(X,Y,Z)] and if you have coeffecients and other information that are independent of X, Y, Z (for example V = aX + bY + cZ has coeffecient variables a,b,c independent of X,Y,Z random variables), then what will happen is you will get a deterministic relationship between other variables and the expectations of the random variables.

You can then look at where this gets maximized with respect to your other information to figure out what f(X,Y,Z) should be and hence that becomes your value function.