An airline company operates a plane between atl and dc this...

An airline company operates a plane between ATL and DC. This plane has 120 seats for economy class with two price tiers: low fare at $250 and high fare at $420. The low fare seats target for travelers and there is an unlimited demand from travelers. So seats offered at $250 will always be sold out. The high fare seats target for business travelers and the company believes that the number of business travelers for this route follows Poisson with mean 20. As business travelers plan for their travels in the last minute, any seats assigned to business travelers must be remained and cannot be offered to travelers. 1). How many seats should be assigned to the business travelers to maximize the revenue? 2). Calculate the expected revenue when the company offers the optimal number of seats to business travelers.

Transcript

00:01 All right, so in this question, we have a company that operates a plane with 120 seats, and there are price tiers.

00:09 One is 250, and if we make a seat in this price tier, it's 100 % to be filled.

00:15 And the next one is 420, and if we make it in this price tier, it's not 100 % to be filled.

00:22 We're told that the number of business travelers which would fill the 420 seats, $420 seats, are poisson distributed with mean 20.

00:33 So you remind yourself that the poisson distribution is defined if we say, let's say b is the number of business travelers, the probability that b equals some number k is the normal poisson distribution, this.

01:01 And so, we can just do this kind of explicitly.

01:05 We can say, okay, let's suppose we choose n seats for business.

01:10 Then our total profit is going to basically be, we take 120 minus n, and we make 250 off of it.

01:24 And then we have to kind of go through all these probabilities.

01:27 We have this sum where we take the, let's say from k equals zero to infinity of the probability that the business travelers equal to k.

01:43 And if they're k business travelers, we make basically the, we make 420 times k up to n, right? so sorry, this should be the min of k and n.

02:10 Right, let's make sure this makes sense.

02:16 Right.

02:16 So for example, if it turns out that there are only one, only one business traveler, then we would only make 420 times one, right? because k would be one here.

02:38 But if there are like 2n business travelers, then we would make only, this thing would only go up, we would just, this would eventually become just n, because we would only be able to fill those seats and we make 420 times n.

02:57 Okay.

02:57 So one thing we can do, the easiest way to think about this is that this is our kind of profit and we say, okay, so we want to choose n to make this great.

03:09 And one thing you can think to do this is like, should we increase n, right? intuitively, if we keep increasing n, eventually like the marginal value of that seat will go down.

03:17 So if we take n to n plus one, what happens to this, right? what happens is that this term results in a minus $250, right? because we increase n plus one, so we're subtracting a bigger number here.

03:30 Um, this sum is maybe a little less obvious, right? we can think about it at the, the margin, right? somewhere here, there's a term probability billy b equals n times 420 n plus probability of b equals n plus one times 420 n plus one, um, and so on and so forth.

03:58 Right.

03:59 And when we increased n plus, sorry, this is n, that's what it is to start right now, right? this is just n.

04:04 And when we increased to n plus one, what we're really doing is we're adding on value where, where, um, this one is unchanged, but all of these guys, which have n will now be n plus one, right? so we basically are, are the sum from k equals n plus one to infinity, probably b equals k of 420.

04:30 Right now, this is, this tail is just n and it's now changing to n plus one.

04:35 So it's, you know, it's going up by actually just this much...

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