02q1sol

# 02q1sol - MASSACHUSETTS INSTITUTE F T ECHNOLOGY O...

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MASSACHUSETTS OF TECHNOLOGY INSTITUTE 1.203J/6.281J/13.665J/15.073J/16.76J/ESD.216J Logistical and Transportation P lanning Methods Fall 2002 Quiz 1 Solutions (a)(i) Without loss of generality we can pin down X1 at any fixed point. X2 is still uniformly distributed over the square. Assuming that the police car will always follow a shortest route to the emergency incident, the max possible distance between X1 and X2 is 2 km. The travel distance is thus uniformly distributed between 0 and 2 km. (a)(ii) Following similar logic, the max possible distance is now 4 km. The travel distance is thus uniformly distributed between 0 and 4 km. (b) Let's number the links as shown in Figure 1. There is a chance that the emergency incident will be on any one of the 12 links. Thus if we can determine the conditional pdf for the travel distance from X1 (conditioned to be uniformly distributed on link 7) to X2 for each possible link for X2, we are done. All we do then is add the resulting conditional pdfs, multiplying each by A, the probability of occurrence. Careful inspection of the problem reveals that with regard to computing the conditional travel distance pdf between X1 and X2 there are three sets of links Figure 1: Link Numbering Set 1: 1,3,4,5, 6,8,9, 10, 11

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