06q1sol

# 06q1sol - Logistical and Transportation Planning QUIZ 1...

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Logistical and Transportation Planning± QUIZ 1± Solutions± Problem 1. Patrolling Police Car. A patrolling police car is assigned to the rectangular sector shown in the figure. The sector is bounded on all four sides by a roadway that requires 50% of the police car’s patrolling time. The other 50% of the time, the car patrols the inner rectangular part of the sector. Thus, at a random time when the police car is available for dispatch, the police car’s location is equally likely to be drawn from a uniform distribution over the bounding roadway or by a uniform distribution over the rectangular part of the sector inside the bounding roadway. Travel is right-angle or the Manhattan metric, with directions of travel parallel to the sides of the rectangle. 911 calls for service are also distributed randomly over the rectangular sector, in the same way as the police car and independently of the location of the police car. That is, 50% of the 911 calls are uniformly distributed over the bounding roadway and 50% uniformly distributed over the inner rectangular part of the sector. Given a random call for service at a give location, the police car will follow a minimum distance right-angle path from its current location to the location of the call. Thus, we assume that the police car can exit the bounding 2 km bounding roadway 1 km roadway at any point, and – as is usual with the right angle metric, we ignore the complication of ‘city blocks’, assuming instead an infinitely divisible right-angle travel space within the region. (a) Find the mean distance traveled by the police car in response to a random call for service. Key methods:² - “Divide & Conquer”± - Expected distance between two random points±

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Logistical and Transportation Planning± The police car can either be on the highway or in the inner part of the sector, and it is the same for the random call for service. Therefore, we have four different cases. The probabilities of those events are summarized in the table: Police Car Highway Inner Part 911 Call Highway Case 1: 1/4 Case 2: 1/4 Inner Part Case 3: 1/4 Case 4: 1/4 For each case, we will determine the expected distance. •± Case 1: We have to use the method “Divide and Conquer” again. If the call and the car are on two opposite portions of the highway, then the car has to cross the inner rectangular region, whereas if they are on the same portion of the highway, the car does not have to travel across that region. Thus, the expected distance depends on which part of the highway the call and cars are. Note: the car can go off the highway, therefore, the problem cannot be reduced to a 1-D problem. Same portions rectilinear portion Different Different 1 1 E [ D ] = P = 2 3 Both on small rectilinear portion E [ D ] = 1 + 2 = 7 2 2 1 P = 1 3 3 P = = 2 6 6 9 One the same one 1 1 2 E [ D ] = 2 = P = 2 3 3 Both on long 4 4 = E [ D ] = 1 5 2 + 1 = P = 3 3 6 6 9 1 P = 2 One on small and one on± long portion± 2 4 P = 2 = 6 6 9 1 1 3 E [ D ] = 2 + 1 = 2 2 2
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## This note was uploaded on 11/08/2011 for the course AERO 16.72 taught by Professor Hansman during the Fall '06 term at MIT.

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06q1sol - Logistical and Transportation Planning QUIZ 1...

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