2. (40 points) In this exercise, we develop a dynamic programming based solution (using a spreadsheet or Python) similar to the one in Exercise 1. Let us consider a 3-class dynamic capacity allocation problem for a flight that has 90 seats under the following assumptions: at most one booking request takes place in a period and the probability that this arrival is of class i is given in the below table along with respective class fares (the same assumptions from the lectures): (a) Write the optimality equation vt(x) with t periods to go and x seats remaining and find the expected optimal revenue from this flight starting with 200 periods to go and all 90 seats available. (b) Find the expected marginal value of a seat with 20 seats remaining with 50,40,30, 20 and 10 periods remaining. (c) Report the protection levels for classes 2 and 3 with 100,50 , and 10 periods remaining. (d) Assume that there is a planned promotion that will be open between periods 80 to 20 (remaining until the deadline). The arrival probabilities during this period will change as: These arrival probabilities only apply to the promotion period. At other times, the previous probability distribution is valid. Note that you have to modify the recursion during the promotion period. Find the the expected optimal revenue starting with 200 periods to go and all 90 seats available when the promotion is taken into account. Find the expected marginal value of a seat with 20 seats remaining with 50,40,30, 20 and 10 periods remaining