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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View Full DocumentVariable 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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 Spring '10
 GeorgeKocur
 Systems Engineering, bus service, bus market share

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