Queation 1,2,3
IEOR 4601 Homework 2: Due Wednesday, February 13 (Due at the beginning of the class.)
1. Dynamic Booking Control: Consider a flight with a capacity of 50 seats and 4 fare classes. Suppose that we implement nested protection levels with protection levels (y1, y2, y3, y4) = (7, 20, 34, 50). The following table shows a series of booking requests.
a) Suppose that each request must be accepted on all-or-none basis, i.e., given a request of m units, we can only sell m units or none at all. Determine whether each request would be accepted, and update the booking limits and protection levels accordingly.
b) Redo part a) assuming that you can accept partial requests.
Booking Limits Protection Levels 1 2 3 4 1 2 3 4 Request Action
1 50 43 30 16 7 20 34 50 8 seats in Class 4 ? 2 10 seats in Class 4 ? 3 5 seats in Class 4 ? 4 10 seats in Class 3 ? 5 15 seats in Class 3 ? 6 4 seats in Class 2 ? 7 3 seats in Class 2 ? 8 6 seats in Class 2 ? 9 3 seats in Class 1 ? 10 5 seats in Class 1 ? 11
2. Optimal Policy, Heuristics and Bounds Suppose we have capacity of 220 seats and four fare classes. The fares and demand distribution for each fare class are given in the the following table. In all cases, except where noted, we will assume a low-to-high fare class arrival pattern.
Class Fare Demand Distribution 1 $500 Poisson(45) 2 $380 Poisson(55) 3 $215 Poisson(50) 4 $180 Poisson(100)
You can write a code in any computer language you want or you can modify the R code discussed in class and uploaded to Courseworks.
a) Determine the optimal protection levels using dynamic programming.
b) Determine the protection levels under the EMSR-a heuristic
c) Determine the protection levels under the EMSR-b heuristic
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d) Use simulation or the exact method to estimate the expected sales for each fare class and the total expected revenues for the policies determined in parts a), b), and c).
e) Find the expected revenue under a policy that does not protect inventory for higher fare classes assuming the arrival pattern is low-to-high.
f) Find the expected revenue of the policy in part e) if the fare class arrival pattern is high-to-low.
g) Solve the linear programming described in class to obtain an upper bound on the expected revenue of the optimal policy.
3. Callable Products
Suppose that you are the capacity provider for a popular event. The face value of the tickets is $100 per seat and the venue can hold 350 individuals. The $100 tickets go on sale a month before the event. Assume demand for $100 tickets is at least 350. You estimate that demand from people willing to pay $300 for a ticket the day of the event can be modeled as a negative binomial with parameters r = 36 and p = 1/4 (mean 144 and variance 432). More precisely, the probability mass function of demand for $300
tickets is P (D1 = k) = ( k−1 35
) (1/4)36(3/4)k−36 for integer values of k ≥ 36.
a) How many tickets should you reserve for sale at $300?
b) Evaluate the expected revenue of the strategy of part a) and determine the average number of unsold seats under the strategy of part a).
c) Suppose now that you sell the $100 tickets with a callable option that allows you to buy them back for $130 if needed (you can assume customers are willing to accept this deal). Suppose that you exercise the option of buying back $100 tickets at $130 when demand for $300 tickets exceeds the number you reserved for them in part a). Use simulation to evaluate the expected revenue of this strategy and determine the average number of unsold seats. You can continue to assume that demand for $100 tickets exceeds the capacity of the venue for the purpose of your calculations.
d) Consider now a refinement of the strategy in part c) where you can fine tune the number of tickets that you reserve for sale at $300. How many tickets would you reserve? Compute the expected profit under the new strategy and also the expected number of unsold seats.
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