Statistical Problems

Question 1:
a) Briefly explain how the following features might be represented in the
classical 4-stage modelling process:
i. Decision making of travellers facing congestion
ii. Improvements to public transport links
10 marks
10 marks
b) Expansion factors used in trip generation modelling are typically
determined by changes in the characteristics of a zone, such as
population growth. These expansion factors do not typically take into
consideration policy changes and other causes of change; list and justify
FOUR such policies or other causes.
20 marks
c) A previous survey found the average number of trips per week made by
Households (HH), disaggregated by car ownership and household size,
as shown in Table 1 below. Calculate the average number of trips per
household.
Table 1. Household size vs car ownership survey results 2015
Cars owned by
HH
0 0 0 1 1 1 2+ 2+ 2+
HH Size 1 2 3+ 1 2 3+ 1 2 3+
% of HH 25% 12% 4% 25% 12% 8% 6% 4% 4%
Trips made by
HH
18 48 96 24 72 120 24 28 144
30 marks
d) However, a survey in 2020 showed the following changes compared to
those results registered in the 2015 survey:
– 50% of households with 1 car and with a HH size of 3 or more
acquired an additional car in the 2020 survey.
– 35% of households without a car and with a HH size of 2 or more
had acquired 1 car in the 2020 survey.
20 marks
Taking this new data into consideration, calculate the new average
number of trips per household.
e) Briefly discuss reasons for the differences between the results given in
answer to parts c) and d).
10 marks
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Question 2:
a) The gravity model is a well-known method used for trip distribution
when no reliable previous matrix exists but there are reliable 𝑂𝑖 and
𝐷𝑗
. The general form of the gravity model is given below:
𝑇𝑖𝑗 = 𝑘𝑂𝑖𝐷𝑗 𝑓(𝑐𝑖𝑗 )
i. Briefly describe in your own words what is the meaning and
purpose of the 𝑘 factor in the equation above.
15 marks
ii. Give an example of an equation of the 𝑓(𝑐𝑖𝑗) and explain
how its curve may affect the value of 𝑇𝑖𝑗
.
15 marks
b) A study area contains three zones X, Y and Z, and a trip matrix for
2015 as follows:
Table 2. Trip matrix 2015
O/D X Y Z
X 10 8 13
Y 5 14 7
Z 12 15 9
The forecast for 2035 yielded the following trip ends:
Forecast origin totals: Ox = 47, Oy = 35, Oz = 43
Forecast destination totals: Dx = 40, Dy = 49, Dz = 36
Using the Furness method, produce an estimate of the trip matrix
for 2035. Stop when row and column totals agree to ±5%.
30 marks
c) Given the cost matrix as shown in Table 3. below, produce a trip
matrix using a gravity model with
a single K 𝑖𝑗
factor and the deterrence function𝑓(𝑐𝑖𝑗) = 𝑐
−1
, and the
forecast origin and destination totals given in 2(c), above.
Table 3. Cost matrix
O/D X Y Z
X 3 5 3
Y 4 2 7
Z 3 6 5
30 marks
d) Briefly discuss reasons for the differences between the results given
in answer to parts b) and c).
10 marks
Question 3:
Consider the simple one-way network in Figure 1 where there are 1800 trips travelling
from O1 to D and 1200 from O2 to D. The link travel cost versus link flow relationships
are given in Table 4.
Figure 1: Network
Table 4: Link cost functions
Link Cost function
A 2.0 + 0.001 * flow
B 6.0 + 0.001 * flow
C 1.1 + 0.001 * flow
D 3.5 + 0.001 * flow
E 6.0 + 0.001 * flow
a) Carry out a Dial type assignment with β  , using empty link costs
and calculate the value of the disequilibrium indicator, calculate the
value of the disequilibrium indicator ().
T (C C
min
)
ijr ijr ij
 
ijr
T C
min
ij ij
ij
30 marks
b) Use an incremental assignment technique with fractions 40%, 30%,
20% and 10% of the total demand to obtain an approximation to
equilibrium assignment. Use the empty link costs in the first increment,
and calculate the value of the disequilibrium indicator.
30 marks
c) Critically comment on:
i. The results from parts a) and b), including which specific
features of the problem favours one method over the other.
15 marks
ii. The suitability of alternative assignment methods for this
problem, and which one(s) you think would have performed
better than those in parts a) and b), and why.
25 marks
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SECTION B
This section of the exam contains two questions. Please check rubric to see how
many questions you need to answer.
Question 4:
a) There is a plan to introduce a fleet of automated vehicles to Leeds, which
can be reserved for individual trips in a similar way to a taxi service. An
aggregate four-stage modelling approach is considered for forecasting
the impacts of this new service. Identify THREE aspects of the impacts of
such a service that this approach would be able to model, and THREE
aspects that it would not. Briefly give reasons for your answers.
50 marks
b) A freight delivery service receives a request to deliver a product by 12.00.
The delivery service has the choice to reorganise other shipments so as
to leave the depot at either 08:00 (arriving at 11.20), or to leave later at
10:00 (arriving at 12:20). Taking into account the inconvenience of leaving
early (to the delivery company) or arriving late (to the customer), a
departure time choice model is estimated with utility function:
Ut = −0.14Tt − 3.9Pt− 0.13Et − 0.51Lt
where
Ut = utility associated with departure time t
Tt = travel time (minutes) at departure time t
Pt= late penalty indicator, = 1 for late arrival, = 0 otherwise
Et = amount of early arrival time (minutes) if departing at time t
Lt = amount of late arrival time (minutes) if departing at time t.
Assuming a binary logit choice model with scale parameter equal to 1,
calculate the choice probabilities of the two alternative departure times,
showing the key steps in your calculation.
50 marks
Page 6 of 6 End
Question 5:
A disaggregate binary logit model is estimated, in order to explain the choice between bus
and car for commuting trips to the centre of Leeds. In order to estimate the model, a stated
preference survey was conducted, with 659 responses used for calibration. The utility
functions assumed were of the following form:
𝑈𝐶𝐴𝑅 = 𝜃0 + 𝜃11𝑇𝐶𝐴𝑅 + 𝜃3𝐶𝐶𝐴𝑅
𝑈𝐵𝑈𝑆 = 𝜃12𝑇𝐵𝑈𝑆 + 𝜃2𝑊𝐵𝑈𝑆 + 𝜃3𝐶𝐵𝑈𝑆
where 𝑈𝐶𝐴𝑅 and 𝑈𝐵𝑈𝑆 are the utilities of the car and bus modes respectively, 𝑇𝐶𝐴𝑅 and 𝑇𝐵𝑈𝑆
are the in-vehicle travel times (minutes) by car and bus respectively; 𝑊𝐵𝑈𝑆 is the waiting
time (minutes) by bus; 𝐶𝐶𝐴𝑅 and 𝐶𝐵𝑈𝑆 are respectively the car parking charge and bus fare
(pence); and 𝜃0, 𝜃11, 𝜃12, 𝜃2 and 𝜃3 are coefficients of the model.
The results of the model estimation are given in Fout! Verwijzingsbron niet gevonden..
Table 5: Logit model estimation results for mode choice in Leeds
Coefficient Estimate t-statistic
𝜃0 0.82 6.81
𝜃11 –0.043 –8.24
𝜃12 –0.065 –4.93
𝜃2 –0.072 –3.24
𝜃3 –0.005 –12.42
𝜌
2 0.31
a) Comment on the results in Table 5, justifying any conclusions made. 40 marks
b) Estimate the value of time for in-vehicle time, for the car and bus modes. 20 marks
c) Estimate the mode share that would arise if all coefficients except 𝜃0 were
zero.
15 marks
d) Use your answer to c) to help explain the role of 𝜃0 and the real-life features
it could be capturing in this context