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Practice Final - IEOR 4106 Introduction to Operations...

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IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2004, Professor Whitt, Final Exam Chapters 4-7 and 10 in Ross, Tuesday, May 11, 9:00am-12:00noon Open Book: but only the Ross textbook plus three 8 × 11 pages of notes Justify your answers; show your work. 1. The Terrific Toledo Taxi ( 20 points) Alberto Lopez Toledo operates a taxi that provides service in two zones of the city Toledo. Fares picked up in zone A will have destinations in zone A with probability 0 . 6 and in zone B with probability 0 . 4. Fares picked up in zone B will have destinations in zone A with probability 0 . 3 and in zone B with probability 0 . 7. (a) Starting in zone A , what is the probability that Alberto’s taxi will be again in zone A after two trips? —————————————————————————————- First observe that this is a variant of Problem 4.52 in Ross. The possible paths are A - A - A and A - B - A . The first has probability 0 . 6 × 0 . 6 = 0 . 36; the second has probability 0 . 4 × 0 . 3 = 0 . 12. So that the total probability is 0 . 36 + 0 . 12 = 0 . 48. In other words, P 2 A,A = 0 . 48, where P = 0 . 6 0 . 4 0 . 3 0 . 7 . In particular, P 2 = 0 . 48 0 . 52 0 . 39 0 . 61 . —————————————————————————————- (b) Suppose that Alberto starts in zone A and is again in zone A after two trips. What is the probability that Alberto has never left zone A during these two trips? —————————————————————————————- We want to compute a conditional probability: P ( A - A - A | A - A - A or A - B - A ) = 0 . 6 × 0 . 6 0 . 6 × 0 . 6 + 0 . 4 × 0 . 3 = 0 . 36 0 . 48 = 3 4 . —————————————————————————————- (c) What is the long-run proportion of trips starting in zone A ? —————————————————————————————- We want to compute π = πP . We get π = ( π A , π B ) = (3 / 7 , 4 / 7) = (0 . 429 , 0 . 571) —————————————————————————————- (c) Alberto’s expected profit for a trip entirely in zone A is $6; Alberto’s expected profit for a trip entirely in zone B is $8; and Alberto’s expected profit for a trip that involves both zones A and B is $12. What is Alberto’s long-run average profit per trip?
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—————————————————————————————- He makes an A A trip with probability π A × P A,A = (3 / 7) × 0 . 6 = 18 / 70. He makes an A B trip with probability π A × P A,B = (3 / 7) × 0 . 4 = 12 / 70. He makes an B A trip with probability π B × P B,A = (4 / 7) × 0 . 3 = 12 / 70. He makes an B B trip with probability π B × P B,B = (4 / 7) × 0 . 7 = 28 / 70. Thus, the expected profit per trip is E [Profit per trip] = 6(18 / 70) + 8(28 / 70) + 12((12 / 70) + (12 / 70)) = 620 / 70 = 62 / 7 = 8 . 86 —————————————————————————————- 2. The Kallis Barber Shop ( 20 points) Inspired by the phenomenal success of his colleagues David Raimondo and Yuwei Shi, Christos Kallis also has opened a small barbershop. Christos is going it alone; he is the only barber, so he can serve only one customer at a time.
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