CE 3025: Traffic Engineering
Indian Institute of Technology Madras
CE 3025: Homework on Transporation Planning
Due Date: April 29, 2023 11:59 PM
Instructions
• The assignment should be typed or neatly handwritten. If you decide to type your solution, you can
use any word processor or LaTeX (a template is uploaded on Gihub or use Overleaf). The final version
should be submitted as a PDF document using the URL https://shorturl.at/bcpHZ.
• The PDF document should contain the solutions and figures/graphs (if any). Submit excel or computer
codes as a supplementary file (if any).
• The front page of the homework file should have the name, student ID, and IITM email address of the
student submitting the document.
• You are encouraged to collaborate with other students but you must submit your solutions individually
in your own words. Copying homework is academic dishonesty and will be penalized.
• There are two types of problems, namely, Concepts check and Reflection and critical thinking. Concepts
check problems are similar to the ones solved in class and Reflection and critical thinking requires a
little more effort than the class problems.
• If you have any questions related to problems in this homework, feel free to email me (Pramesh Kumar
(prameshk@iitm.ac.in)) or come to my office (Room 340).
Problem 1 Travel diary (8 pt) (Reflection and critical thinking)
Keep a travel diary for the next two weekdays. For each trip, record the following information:
A. the start and end time (rounded to the nearest minute)
B. start and end location of each trip by street/intersection name or landmark,
C. primary mode you took (drive alone, car driver with passenger, car passenger, campus bus, kiss-andride, walk, cycle, motorcycle, taxi, Ola/Uber, other). (use only these codes)
D. purpose (to work/class, return home, eating, work-related business, shopping, family/personal business,
school, worship, medical/dental, vacation, visit friends or relatives, other social recreational, other) (use
the codes provided)
E. whether you traveled with anyone else (1-Yes, 0-No)
Use the Travel Diary Sample (provided to you with this HW) to help you. Analyze your personal travel
diary by answering the following questions:
(a) Create a frequency histogram for the following:
– end locations with the frequency of end location on y-axis and end locations on the x-axis
– mode used with the frequency of mode usage on y-axis and modes on the x-axis.
(b) Similarly, create histogram plots for activity purpose and time of the day . Use a class range of 3 hours
for creating a histogram of time of the day.
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CE 3025: Traffic Engineering
Indian Institute of Technology Madras
(c) Which trip has the highest travel time? Give its purpose and travel time.
Upload the answers and histograms within the same PDF and spreadsheet (separately) to the URL
https://shorturl.at/bcpHZ .
Problem 2 Trip Generation (8 pt) (Concepts check)
The trips originated from zone i (i.e. Tio ) and destined to zone j (i.e. Tjd ) in an area depend on four factors
– number of households(n), number of jobs (N ), median income (in thousands) (I) of households in that
zone, and access to transit (a) which can be expressed as;
Tio = 80 + 0.2n + 0.2N + 0.2I + 50a
(1)
Tjd = 60 + 0.1n + 0.5N + 0.1I + 50a
(2)
Following table shows the number of households(n), number of jobs (N ), median income (in thousands) (I)
of households, and access to transit (a takes the value 1 if transit is accessible within walking distance, 0
otherwise) for different zones.
Zone
Number of
households (n)
Number of
jobs (N )
Median Income
(in thousands) (I)
access to
transit (a)
1
2
3
100
200
100
50
20
40
40
75
50
1
1
0
For income, use the numbers without thousands (for e.g. in Zone 1, put I = 40). Using information given
above, calculate the following:
(a) Number of trips originated and destined for different zones.
(b) Check if the sum of trips generated is equal to sum of trips destined. If they are not equal, normalize
it using the method discussed in class.
Problem 3 Trip Distribution (8 pt)
Assume that the number of trips starting and ending at zone MP and SP are given in Table 1. The travel
impedence function is f = t13 , where tij is the travel time between zone i and zone j which is given in Table
ij
2. Answer the following questions:
Zone
Trips starting
Trips Ending
MP
SP
1100
1200
800
1500
Table 1: Trips origins and destinations
Zone j
Zone i
MP
SP
MP
SP
12
15
14
12
Table 2: Travel cost between zone i and j
(a) (Concepts check) Use gravity model to find the distribution of trips i.e. TM P,M P , TM P,SP , TSP,M P ,
TSP,SP in Table 3.
(b) (Concepts Check) Balance this trip matrix using the method discussed in class. Please conduct only
as maximum as 5 iterations.
(c) (Concepts Check) If the growth rate is 1.05 for next year, find the distribution of trips of different
origin-destination pairs for next year. You can use the following formula to update.
next year
previous year
TM
= growth factor*TM
P,M P
P,M P
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CE 3025: Traffic Engineering
Indian Institute of Technology Madras
Zone j
Zone i
MP
SP
MP
SP
TM P,M P
TSP,M P
TM P,SP
TSP,SP
Table 3: Trip Distribution
(d) (Reflection and critical thinking) As your trip matrix is doubly constrained, we can write the following
conservation equations. Is it possible to solve the following conservation equations to find the unknowns
in Table 3? If yes, how? If no, why?
TM P,M P + TM P,SP = 1100,
TSP,M P + TSP,SP = 1200,
TM P,M P + TSP,M P = 800,
TM P,SP + TSP,SP = 1500
Fun fact: Estimating Origin-Destination for different modes of transportation is a classic problem for
which various methods are proposed in transportation literature. It remains an unsolved problem to solve it
exactly.
Problem 4 Mode Choice (8 pt)
(a) (Reflection and critical thinking) Suppose you have two alternative modes available (transit and auto)
to go to BSB. Write any three factors that derives utility of each mode for you.
(b) (Concepts Check) Let us assume that the disutility for auto and transit can be expressed as following:
Auto : VA = −0.46 − 0.35T T − 0.08W T − 0.005C
(3)
Transit : VT = −0.07 − 0.35T T − 0.08W T − 0.005C
(4)
where,
T T = total travel time in minutes, W T = waiting time in minutes, and C = cost. The travel characteistics between two zones are given below:
TT
WT
C
Auto
Transit
20
8
320
30
6
100
Calculate the probability of taking each mode using binary logit choice model.
(c) (Concepts Check) If 300 students go to BSB everyday between these two zones, use the results calculated in (b) to find the number of students commuting by each mode.
(d) (Reflection and critical thinking) Suppose rising fuel prices lead to an increase of Rs. 10 for each mode.
Will utilities and mode shares be affected? A yes or no answer is required with an explanation.
Fun Fact: Discrete choice theory is used in various fields to describe, explain, and predict choices between
two or more discrete alternatives. Daniel Little McFadden, an American econometrician and a University
of Minnesota graduate won Nobel Memorial Prize in Economic Sciences for his development of theory and
methods for analyzing discrete choice. Check out his picture on the Internet!
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CE 3025: Traffic Engineering
Indian Institute of Technology Madras
Problem 5 Traffic Assignment (6 pt)
There are three routes (highway, local, and transit route) available to go from r to s, travel time functions
for which are given in Figure 1.
Local
tl = 40 + xl
Transit
tt = 50
r
s
Highway
th = 20 + 0.005x2h
Figure 1: Transportation network for Problem 2
(a) (Concepts Check) Given total demand = 100, calculate the flows on each route by using the definition
of UE. Also calculate the total system travel time.
(b) (Reflection and critical thinking) Write the System Optimal (SO) mathematical program model for
above network. You do not need to solve the mathematical program to calculate the flows. The solution
should be in the following form (you only need to input the travel time functions and demand values
below):
minimize xl tl (xl ) + xt tt (t ) + xh th (xh )
xl ,xh ,xt
subject to xl + xt + xh = d
(5)
xl , xt , xh ≥ 0
where d is the demand.
(c) (Reflection and Critical Thinking) Write one limitation of BPR function.
Fun Fact: A more geneal concept of UE is known as Nash Equilibrium (named after late mathematician
John Forbes Nash Jr.) in Game theory, which is a proposed solution of a non-cooperative game involving
two or more players in which each player is assumed to know the equilibrium strategies of the other players,
and no player has anything to gain by changing only their own strategy (shortest path in our case). A good
example is Prisoner’s dilemma in the movie Batman : The Dark Knight (ferry scene). Can you find out
how the solution found in the movie different from Nash Equilibrium? Check out John Nash’s picture on the
Internet (not the actor Russell Crowe of the movie ”A Beautiful Mind” made on the biography of John Nash).
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