(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 1 and 2 with 100, 50, and 10 periods remaining. (d) Assume that there is a planned promotion that will be open between periods 80 to 100 (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. 3. (30 points) An airline offers two flights. Flight 1 from Antalya to Ankara and flight 2 from Ankara to Bursa. The flights have been scheduled so that passengers can connect from flight 1 to flight 2 in Ankara. The airline has assigned a 200-seat aircraft on flight 1 from Antalya to Ankara and a 100-seat aircraft on flight 2 from Ankara to Bursa. We assume that the airline offers two fare classes: discount fare and full-fare. The demand of each product is mutually independent and normally distributed. The fare of each product and the mean and standard deviation of its demand are given in the following table: (a) Since only flight 2 seems to be constrained (the other flight has enough seats to satisfy the whole demand), apply the greedy heuristic to find the optimal allocation of the seats of flight 2 among the products that use this flight. In the greedy heuristic use 5 the EMSR-a method to find the protection level. (b) Now assume that the forecast for the demand from Antalya to Bursa changes (while other forecasts remain unchanged): Both flights seem to be constrained under this forecast update. Formulate a linear program that maximizes the revenue by finding the planned allocations for each product. You can use the mean demand for each product as its point forecast. This is a small LP and can be solved with any solver (Excel, Python/Gurobi etc.)