ps2sol

# ps2sol - 1.203J 6.281J 13.665J 15.073J 16.76J ESD.216J...

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1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J Logistical and Transportation Planning Methods Problem Set #2 Issued: September 25, 2006 Due: October 4, 2006 Problem 1 (i). In this problem, we should consider random incidence. There are three different interval lengths: 4, 5 or 6 minutes. i i Let A i be the event that he arrives in interval of length i , then P ( A i ) = = . 4 + 5 + 6 15 Let T be the number of minutes he waits. 0 for i = 4; We have: P ( T ∈[ 4,5 ] | A i ) = 5 4 1 = for i = 5 or 6. i i Therefore: 6 P ( T ∈[ 4,5 ]) = P ( T ∈[ 4,5 ] | A i )⋅ P ( A i ) i = 4 1 1 5 1 6 P ( T ∈[ 4,5 ]) = 0* + * + * 4 5 15 6 15± P ( T ∈[ 4,5 ]) = 15 ii). If the intervals between trains were exactly 5 minutes, the probability for Mendel to arrive in an interval 1 of length 5 would be 1, but his probability to wait between 4 and 5 minutes would remain the same, i.e. . 5 1 (using the same notations as in the previous question). The probability increases. 5 Thus, P ( T ∈[ 4,5 ]) = From the previous question, we have the result: P ( T ∈[ 4,5 ] | A 5 ) f P ( T ∈[ 4,5 ] | A 6 ) f P ( T ∈[ 4,5 ] | A 4 ) . Therefore, if the probability of arriving in an interval of length 6 or 4 decreases in favor of the probability of arriving in an interval of 5, the chances of waiting between 4 and 5 minutes increases. However, the probability of waiting more than 5 minutes is now zero. Problem Set #2 Fall 2006 1/6

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1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J Logistical and Transportation Planning Methods Problem 2 a) x Directions of travel 1 y 1 0 A Fig. 1 : Urban Area± The area of the triangular region is . Since the demand is uniformly distributed of the region, the joint 2 pdf is f X , Y ( x , y ) = 2. Let D be the travel distance from a point in the region to the facility at A, and D x and D y the distance from A to demand along x axis and y axis: E [ D ] = E [ D x ]+ E [ D y ]= 2 + 1 = 1 3 3 b) There are two different areas. If the demand is in area I, it travels to A. If the demand is in area II, it uses
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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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ps2sol - 1.203J 6.281J 13.665J 15.073J 16.76J ESD.216J...

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