MIT1_204S10_assn8

MIT1_204S10_assn8 - Computer Algorithms in Systems...

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Computer Algorithms in Systems Engineering Spring 2010 Problem Set 8: Nonlinear optimization: Bus system design Due: 12 noon, Wednesday, May 12, 2010 Problem statement The analytical equations in lecture 23 are approximate solutions of a system of nonlinear equations. In this homework we will obtain numerical solutions of these equations. We use Cartesian coordinates. We have a set of parallel bus routes operating in a region of uniform density. All trips are bound to or originate from a point beyond the region (typically a downtown area) that the bus routes serve. The bus routes are spaced a distance g apart, operate at a headway h and charge a fare f. Users are uniformly distributed in the area of dimension X times Y, and walk in a perpendicular direction to the nearest bus route. We ignore bus stop spacing along the routes, the dimensions of the street grid, etc. All of those issues can be handled but make the model more complex. X Y (To downtown) g Bus route To maximize profit, we maximize the difference of revenues minus costs. Revenue= TpXYf(a 0 -a 2 (kh +g/(4j))-a 4 f) Cost= 2XTcY/(ghv) Figure by MIT OpenCourseWare.
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Variable name Value Definition Units p 3.59 Trip density Trips/mi 2 /day j 0.05 Walk speed Miles/minute k 0.4 Wait/headway ratio c 50 Bus operating cost Cents/minute T 1050 Length of day Minutes V 0.167 Bus speed Miles/minute a 0 0.41 Bus market share if service equal to auto a 2 0.0081 Bus wait time coefficient a 4 0.0014 Bus fare coefficient X 4.0 Width of analysis area Miles Y 6.0 Length of analysis area Miles The total number of trips by all modes of transport (bus, auto, etc.) is TpXY, or the trip density times the area and the time period. The number of bus trips is the bus market share times the total trips; the bus market share is a 0 -a 2 (kh +g/(4j))-a 4 f). This is a linear approximation to a logit demand model that estimates the market share as a function of the headway h, the route spacing g (which determines the average walking distance to the bus route), and the fare f. The bus
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MIT1_204S10_assn8 - Computer Algorithms in Systems...

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