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Unformatted text preview: 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 Albertos 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 . 6 = 0 . 36; the second has probability 0 . 4 . 3 = . 12. So that the total probability is 0 . 36 + 0 . 12 = 0 . 48. In other words, P 2 A,A = 0 . 48, where P = . 6 0 . 4 . 3 0 . 7 . In particular, P 2 = . 48 0 . 52 . 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 ) = . 6 . 6 . 6 . 6 + 0 . 4 . 3 = . 36 . 48 = 3 4 . - (c) What is the long-run proportion of trips starting in zone A ? - We want to compute = P . We get = ( A , B ) = (3 / 7 , 4 / 7) = (0 . 429 , . 571) - (c) Albertos expected profit for a trip entirely in zone A is $6; Albertos expected profit for a trip entirely in zone B is $8; and Albertos expected profit for a trip that involves both zones A and B is $12. What is Albertos long-run average profit per trip? - He makes an A A trip with probability A P A,A = (3 / 7) . 6 = 18 / 70. He makes an A B trip with probability A P A,B = (3 / 7) . 4 = 12 / 70. He makes an B A trip with probability B P B,A = (4 / 7) . 3 = 12 / 70....
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This note was uploaded on 04/08/2008 for the course IEOR 4106 taught by Professor Whitt during the Spring '08 term at Columbia.
- Spring '08
- Operations Research