Delta operates a plane between ATL and DTW (Detroit). The plane has 150 seats for economy class with two price tiers: low fare at $300 and high fare at $450. There is an unlimited demand for low-fare seats among travelers. So seats offered at $300 will always be sold out so long as they are
offered the day before the flight. The high fare seats are aimed at business travelers because business travels plan their trips at the last minute and are willing to pay the higher price. The airline decides to reserve some seats for their sales to business travelers. Any seat that is reserved so that it could be
sold to a business traveler cannot be sold to other travelers. If a reserved seat doesn’t get bought by a business traveler, it will remain empty on the flight. Suppose that Delta determines that the number of business travelers that follow this route is Poisson with a mean of 25. Note that the PMF of a Poisson distribution with mean λ, P(X = x) can be calculated in Excel using the =POISSON.DIST(x,λ,FALSE) function and the CDF F(x) can be calculated using =POISSON.DIST(x,λ,TRUE).

(a) Suppose that Delta currently reserves 30 seats for business travelers. What is the expected number of seats sold to business travelers? What is the expected number of seats that sit empty? (You may leave Σ in the expression.)

(b) Derive an expected profit formula which is a function of the number of seats reserved for business
travelers, y.
• First, write this expression leaving E and D in the formula (where D is the distribution of
business travelers who follow this route).
• Then, simplify further but you may leave this formula with Σ in the expression.

(c) What is the optimal number of seats that the company should reserve for business travelers and what is the expected revenue associated with this policy? (The answers to these questions should be reported as numbers)
Hint: Here are two different ways you might approach this: 1) You might try to infer what the overstock and understock costs are in this setting. What does it mean to be over/understocked in this setting? 2) Alternatively, you may want to use your expected profit formula from the question above (still written in terms of E) to infer what the corresponding values of b,h, cv, and p are in this setting. Choose whichever approach makes the most sense to you.

(d) Now suppose that any seats that were reserved for business travelers that go unsold can now instead be sold on a discounted fare website the night before the flight to bargain-hunting cus- tomers. There will be unlimited demand for these tickets because they are sold for a low price of $80 per seat (these tickets can no longer be sold for $300 per seat because it is too late of notice for most customers). Does this change the optimal reservation policy? If so, what is the new policy and the new expected revenue? If not, why not and what is the new expected revenue?