DO STOCHASTIC TRAFFIC ASSIGNMENT MODELS CONSIDER
DIFFERENCES IN ROAD USERS' UTILITY FUNCTIONS ?
Otto Anker Nielsen
Institute of Planning (IFP), Technical University of Denmark (DTU)
1 INTRODUCTION
The early logit-based stochastic traffic assignment models (e.g. Dial, 1971) rest on
the assumption that different routes are independent. Thus, they lead to problems in
networks with overlapping routes (see Sheffi, 1985, pp.294-297). Daganzo & Sheffl
(1977) suggested the use of probit-based models to overcome this problem, and Sheffi
& Powell (1981) presented an operational solution algorithm in which the road users'
'perceived travel resistances' are simulated. A similar concept is used as a part of
the Stochastic User Equilibrium (SUE) suggested by Daganzo and Sheffi (1977) and
operationalized by Sheffi & Powell (1982).
In this paper it is discussed whether this modelling approach is sufficient to describe
the road users' behaviour. It is shown that the perceived travel resistances to a certain
point can make up for the road users lack of knowledge of the 'true' travel
resistances. However, it is also shown that it does not fully consider variations in the
road users' utility functions (e.g. the weighting of travel length versus time).
A heuristic modification of SUE is presented in which two types of stochastic
components occur - the first considers road users' perception of the traffic network
at link level (as in the traditional SUE) and the second considers differences in the
road users' utility functions. The traditional SUE is a special case of this model,
where the second stochastic component is zero. If the first stochastic component is
zero, each road user is assumed to have full knowledge of the road network (the
perceived travel resistances are equal to the 'true' resistances). However, the road
users might still use different routes as they might have different utility functions.
To illustrate the theoretically discussions in the paper, bundles of routes between two
zones in Copenhagen are presented according to the different principles and compared
with the results os a comprehensive traffic survey. In addition, the methods have been
tested on several full-scale traffic models.
2 DISCUSSION OF ROUTE CHOICE IN URBAN AREAS
A fundamental requirement for a traffic assignment model is that it reflects 1) Factors
affecting route choice, 2) How road users distinguish between different routes, and
3) That route choices consist of many 'stochastic' elements.
2.1 Factors affecting route choice
One of the fundamental problems in route choice models is to assign values to factors
affecting route choice (see figure 1). Usually, only the travel resistances (generalised
costs) are described by a linear function, where each cost is multiplied by a weight
factor (Ort6zar & Willumsen, 1990, p.262 and Thomas, 1991, pp.46-47).
Physical factors:
Traffic network
N
Land use
Socioeconomic factors:
) ~
Road users characteristlca
The societys characteristica
Personal characteristics:
Observations, knowledge,
habits, evaluations,
~
preferences and decisions
-
, .
[Route ChOiCes [
Normative factors:
Social values,norms,
expectationsand tradition
Figure 1 Basic factors influencing route choice.
2.2
H o w routes are c h o s e n
Bovy & Stem (1990, pp. 18-21) categorize route choices as simultaneous, sequential
or hierarchical. In most cases all three types occur. According to the simultaneous
route choice, the road users choose the entire route before the trip is started and do
not reconsider the choice during the trip. This occurs when there is no congestion and
road users do not change route choice due to habit or lack of knowledge of
alternatives. According to the sequential route choice, the road users reevaluate the
route choice at each decision point along the way, and the choices are independent of
previous choices. According to the hierarchical route choice, the traveller makes his
choices at the decision points, but they are dependent on previous choices. This is
probably the most realistic route choice: Normally, the pre-chosen route is followed,
but altered under certain conditions.
2.3
T y p e s o f 'stochastic'
The word stochastic may have associations with something random. However,
stochastic models encompass a number of factors not all due to randomness, such as:
1.
Road users do not have full knowledge about the traffic network, which means
they do not choose rationally according to their perceived preferences.
2.
Different routes are often chosen for the sake of variation.
3.
Travel times along different mutes may vary from day-to-day due to other road
users, traffic lights, queues or different degrees of congestion, the weather, etc.
4.
Different driving habits make different mutes optimal for different road users.
5.
The road users may have different preferences.
Often it is assumed that different route choices are due to variations in the way
drivers perceive the travel resistance, c~,~.,for each link, a (e.g. Sheffi, 1985, p.272).
c(,~ is assumed to consist of two components, a deterministic, c,, and a random, e,:
c(~,~ =
ca + e
,
(1)
In addition it is assumed, that E[e,] =0. Thus, the average perceived and deterministic
travel resistances equals (E[c~,~,]=c3.
2.4 Probit-based Assignment
Daganzo & Sheffi (1977) suggest the use of probit-based models to overcome the
problem with overlapping routes. Under the assumption that:
1.
Non-overlapping links (sections of roads) are perceived independently,
2.
Links with equal mean travel resistance have the same distribution of perceived
resistances,
3.
The perceived travel resistances, c,o,~, are normally' distributed with mean equal
to the travel resistance and with variance proportional to the resistance:
c(.)a e O(c~,err'ca)
or
c(p)~= c.-pa , p~ ~ O(1,err)
,
where ,I, symbolizes the normal distribution, c~ is the deterministic travel resistance and
e n ' o r t e n n . c(pl~and p~ are introduced to simplify the notation in the following,
(2)
err
is the
they show, that the probability of using a certain link or route can be described by
a multinominal normal distribution resulting in the probit model. Sheffi & Powell
(1981) present a practical solution algorithm:
1)
Initialization: Set the iteration number, n: = 1 and set the traffic flows T.(o,:= 0 for
all links, a. The ((9) in T~0~stands for iteration number zero (initialization).
2)
Update travel resistances: Sample c,.,~ ¢ ~(ca, err'c,) for all links, a, using a
Monte Carlo simulation.
3)
All-or-nothing assignment: Assign the trip matrix on the network with updated
c~,,.'s, resulting in new traffic flows, T ~ , for all links, a (Imp means a
temporary variable).
4)
Step Length is set: ~o, : = 1/n.
5)
Updating: T.(~ : = (1-(o~)" T.(... + ((o,. T,(~, for all links, a.
6)
Stop Criteria: Stop according to a set of stop criteria, otherwise set n: = n + I and
go to step 2.
2.5 Stochastic User Equilibrium (SUE)
The method abovedoes not consider flow-dependent travel resistances. In this case
an equilibrium should be reached where no travellers' perceived travel resistances can
be reduced by unilaterally changing routes. Sheffl (1985, pp.309-327) goes through
the formulation of SUE as a mathematical programme and presents a practical
solution algorithm:, which is the same as for the Probit Model, except for step 2 and
3:
2) Update Travel Resistances: Calculate c.(o~=f(c,o,.,Ta~,.,) V a, where f(.) is the
speed-flow curve.
3) Assignment: Assign the trip matrix on the network with a stochastic route choice
model based on the co~,'s. Hereby the T ~ , ' s are modelled for all links, a.
The solution algorithm consist of two loops: An outer of step 2-6 and the stochastic
assignment model in step 3. However, Sheffi (1985, pp.332-335) shows that the
number of calculations can be reduced by placing the majority of the iterations in the
outer loop. Actually a good result can be obtained with only one iteration in the inner
loop. This approach has been adopted in the following.
It is often discussed, when to use probit-based assignment, UE or both (SUE). The
probit-based assignment may be used in uncongested networks and UE may be a good
approximation for the SUE-solution in congested networks. The SUE approach should
then be used at intermediate congestion levels where neither approximation is accurate
(Sheffi, 1985, p.336-339). However, as all three circumstances occur in most cities,
and many routes pass both uncongested, intermediate and full congested links, it is
recommended to always use the SUE-approach.
3 MODIFICATIONS TO CONSIDER DIFFERENCES IN PREFERENCES
In the above formulation of SUE, the travel resistance, ca, is not discussed in detail.
Actually c, is treated as one homogeneous variable within the Monte Carlo simulation
performed for each link. Section 3.1 explains more fully the formulation of c~ within
this approach, while section 3.2 proposes an approach for handling differences in road
user preferences.
3.1 Link-based stochastic
The travel resistance, c,, consists of two types of components; a non-traffic dependent
(e.g. travel length) and a traffic dependent (typically travel time), e.g.:
(3)
where l~ is the length (often correlated to the out-of pocket cost) and t~ the travel time. k¢l~ and
k<o are weights.
Each of the two components can contain several variables with their own weights. If
the travel time follows the well-known BPR-formula, formula 3 can be rewritten as:
ca
=
k~o'l~
+ k¢o't(o)~"
/
1 + ~"
(4)
\ r(cap)a] I
where T. is the traffic at link a, Tt~.pl, is the practical capacity, tee), is the 'free' travel time (that
is at the level o f zero-traffic volume) and a and fl are parameters.
The BPR-formula'has several advantages (Nielsen, 1994, 2, pp. 3.31-3.42):
1.
It equals the 'free speed' when traffic is zero.
2.
It can be rewritten as a function of speed independent of the length. This makes
it possible to estimate the parameters globally for all links of a certain type.
3.
It contains two independent parameters, a and t ; the first defining the level of
decrease of the speed, and the second the shape of the speed-flow curve.
4.
It is not restricted by link capacity. According to Sheffi (1985, p.359), 'this
means that the equilibrium algorithm does not have to be adjusted to maintain
flows feasibility on all links, a process that can slow the algorithm considerably,
with no appreciable change in accuracy' (see also Boyce et.al., 1981).
In addition, formula 4 has shown useful in practice. In Denmark it has been validated
by driving tests (Matth~ii & Stanton, 1993) as well as part of traffic assignment
models (Nielsen, 1994, 2) 3. Many assignment models use global values for c~ and ft.
As the speed-flow curves are different for different road types, the method should
reflect this by using link or type specific parameters instead: cq and fla. As an
example one slow car can cause delays on a small secondary road at low traffic levels
(low r ) while this is not the case on a motorway (large fl). Finally, one might want
to assign very low speeds for traffic volumes above the practical capacity as the BPRcurve then tends to overestimate the speeds. A simple heuristic approach is to use
fixed queue speeds beyond the practical capacity. Clearly, better approaches can and
have been formulated.
Figure 2 Link-based SUE: Routes between zone-pair in Copenhagen. e,=0.25. Left:
kt=0.4, k,=0.6. Right: kt=O.O, k~=l.0.
Figure 2 shows at the left side an example of the link-based SUE, where the bundle
of routes between a zone in the northern Copenhagen ('Farum') and the International
Airport has been extracted 4. This case is very suitable for tests, since the results can
be compared with a comprehensive survey of route choices (Vejdirektoratet, 1990).
In the example most routes go through the central part of Copenhagen, which reflects
the given weight of travel length versus travel time. If the error term is increased, the
traffic is distributed to more sub-alternatives to the route through Copenhagen.
However, very few trips are assigned to the alternative much longer but slightly faster
ring-motorway. This was not reflected in the survey.
3.2 Preference-based stochastic (whole route)
In the case above, changing the weights of time versus length (k, and k~) can give the
result shown at the right side of figure 2. Here, most trips are assigned to the ringmotorway with few sub-alternatives, and too few trips are assigned through central
Copenhagen. No values of k, and k, could be found that render a reasonable split
between the two main routes (the ring-motorway versus the several alternative routes
through the city).
If a route is considered for one road user, the perceived resistance, c,o,ob, can be
described as:
C(p)ab
Pa " Cab = Pa "( k(l)b'la + k(t)b'ta ) '
(5)
where c~b = kc3)b'l~ + kctlb.t~ is road user b's travel resistance function and p. symbolizes the
stochastic simulation of the travel resistance for road a.
If k,~b and k~t~bvary for different road users, this variation itself could be modelled by
a stochastic simulation of the weights (represented by p~ and p,,). For a specific route
for one road user, the weights can be assumed to be the same for all road segments.
Thus, it is assumed that his utility function does not change during the specific trip.
This gives the following perceived travel resistances for all road users:
cto}a
= P~ ' ( P(0 "k(0"la + P('}"k0)"ta ) '
(6)
A heuristic modification of SUE can then be formulated by adjusting step 2 and 3:
2)
Update Travel Resistances: Sample Pa for all links, a, and sample P{t) and 9,,
once. Calculate c~,~c.,according to formula 6, where ta follows a speed-flow curve
(n is the iteration number).
3)
All-or-nothing Assignment: Assign the trip matrix on the network with updated
c~p~(.~'s, resulting in new traffic flows, T . ~ , for all links, a.
If the error terms for Pat and 90~ are
zero, the model is equal to the traditional SUE. Here, the road users will
follow routes according to the fixed
weights (as in the two examples mentioned above). If, at the same time, the
error term for p, is relatively big, there
may be many sub-alternatives to these
routes.
On the other hand, if the error terms
for Pc~ and 9(,~ are relatively big, the
road users will also choose between
different main alternatives (in the
example the route through Copenhagen
or the ring motorway around the City).
If, at the same time, the error term for
0. is small, the road users will only use
few or no sub-alternatives to these main Figure 3 Modified SUE: k~=0.2, k,=0.8,
routes. If the error term for 13, is zero, p,=0.2, pl,~=l.O and p,,l=/.O.
only optimal routes according to different weight-profiles are used. In the example of Copenhagen, only five such routes are
possible for any combinations of weights~. Figure 3 shows the route-split with a
reasonable combination of p~, p(,, and p,,, that corresponds very well to the traffic
survey.
When the speed-flow curve is included in formula 6, the travel resistance function can
be formulated as:
c(p)~
po. o(o.k(o.l *
/
k (car(a)/
[[
(7)
] ]
P<a>and p~p~provide the possibility for a stochastic simulation of the speed-flow curve.
Because the speed-flow curve is part of the travel time, p~.) and p(p>do not have the
same character as p., p(,) and p(,). However, if the speed-flow curve is uncertain (due
to estimation problem, day-to-day variations, etc.) it might be reasonable to assume
that different road users perceive or have experienced different speed-flow relationships. Whether this is reasonable or not has not been investigated in detail. However, the simulation of the speed-flow curve may seriously affect the UE-part of SUE
and thus affect the convergence of the solution algorithm.
3.3 Convergence problems
The convergence of UE and SUE has been discussed several places in the literature
(e.g. by Powell & Sheffi, 1982, Sheffi, 1985, pp. 327-331 and Van Vuren, 1995).
Two main type of convergence or stop-criteria have been formulated:
1.
To compare results (flows or travel times) from the last two iterations.
2.
To compare results with traffic counts or measured travel times.
The first criteria tests convergence to some extent (although this is not necessarily the
case), while the second secures a reasonable fit at the road level. However, it is the
author's experience when testing the different measures in the literature as well as
other measures, that these convergence criteria do not necessarily secure a 'good'
model. As a test, SUE was run twice on several full-scale models (with different
Monte Carlo simulations). Then the differences between the two runs were calculated.
Although the stop-criteria was fulfilled for each run, quite large marginal differences
between the two runs were discovered (see figure 4 and section 3.4). This can cause
problems, when the model is used to evaluate impacts of proposed infrastructure
projects. Often, it will be impossible to differ between marginal changes and
stochastic differences between two model runs. Of cause, fixed seed-values could be
used for the random-number generator, although this might hide potential convergence
problems for the user. In addition, the term 'systematic random numbers' may not be
theoretically sound.
In the solution of User Equilibrium (UE) by the convex combinations method (Frank
& Wolfe, 1956), (<ojis solved as a optimization problem to speed up convergence (see
Sheffi, 1985, p. 130 for references). It would be tempting to solve SUE the same way
in stead of using the fixed 1/n. Chen & Alfa (1991) have tried to do this, where step
3 was the Fisk (1980) logit model. However, this introduces the problems of the logit
model (see e.g. Sheffi, 1985, pp.294-297). Optimizing (~ in SUE based on the probit
model, is problematic due to the stochastic simulation from iteration to iteration (see
Sheffi, 1985, p. 323-324).
The Copenhagen case:
The N~estved case:
I~00
1000
8:]0
500
%%~%%
400
~
~-
2eo
~0
100
250
Num~r ol Itlration~
5~
i~
0
Figure 4 Marginal differences between two run of the model with- or without the
modification of SUE at the Ncestved model as well as the Copenhagen model.
3.4 Modification to speed up convergence
To speed up convergence, an inner loop was introduced in the modified SUE:
1)
Initialisation of Outer Loop: Set the iteration number n: = 1, and Taco~:= 0 k/a.
2)
Initialisation of Inner Loop: Take the first zone, i.
3)
Update Travel Resistances: Sample p, for all links, a, and sample Pc~ and Pc0
once. Calculate c ( ~ according to formula 7.
4)
All-or-nothing Assignment: Assign the vector of trips from zone i to the network
with updated c<p~c,)'s, resulting in new traffic flows, T.(~'s.
5)
Inner Loop control: If there are more zones, take the next zone, i, and go to step
2. Otherwise continue to step 6.
6)
Step Length is set: ~'c.~: = 1/n.
7)
Updating: T~.) : = (1-~c,~)"T,<~,, + ~.~'T,c~,
8)
Stop Criteria: Stop according to a set of stop criteria, otherwise set n: = n + 1 and
go to step 2.
~/a.
Compared with SUE, this heuristic approach includes as many more Monte Carlo
simulations of travel resistances per outer iteration as there are zones'. This results
in a slight increase in calculation time for the all-or-nothing algorithm, but with the
potential of reducing the number of outer loops. This turned out to be the case for the
case-studies: Figure 4 shows as an example the average absolute deviations at link
level between two calculations with the same preconditions. Note that the curves
should be compared 'horizontally', e.g that SUE needed about 200 iterations to get
an average absolute deviation of 100 for the N~estved case7, while the modified SUE
only needed about 40 iterations to get the same result. In addition, the results were
compared with traffic counts and the two approaches with each other. The result was,
that the two approaches gave comparable results (e.g that 500 outer iteration with
SUE gave the same flow-pattern as 100 with the modified approach for the
Copenhagen case~). The introduction of the inner-loop increased the calculation time
for each outer iteration with a factor two in both cases8.
4 I N T E R P R E T A T I O N AND IMPLICATION O F RESULTS
The main interpretation of the modified SUE is that it describes two types of
'stochastic behaviour'. The first, Pa, considers road users perception of the traffic
network at link-level, and the second, 9~1~and lac,~,considers differences in the road
users' preferences.
The link-basedperception corresponds to some extent to the sequential route choice
outlined in section 2.2. In addition, it can handle overlapping routes. This covers
most of points 1, 2 and 3 mentioned in section 2.3. The preference-basedperception
corresponds to some extent to the simultaneous route choice and makes it possible to
model differences in road users utility functions and differences in driving habits
(point 4 and 5 in section 2.3). The link- andpreference-basedperception may then
be said to reflect the hierarchical route choice. This principle covers the factors
outlined in figure 1 better than the two other principles.
In the ease-study from Copenhagen, the introduction of p<~ and Pc,~made it possible
to model the main route choice of the ring motorway versus the bundle of routes
through the city. In addition, the model could account for that sub-routes were chosen
in a geographically systematic way, e.g. that road users choosing different bridges
between Sealand and Amager had different utility functions, which affected the degree
and types of sub-routes in the different catchment areas of the bridges. If the model
does not take the preference-based perception into account, it might give systematic
biased results even at a strategic level. In the case-study, it might lead to a wrong
conclusion on the impacts of a proposed high-class road in a new tunnel under the
Harbour.
5 P R A C T I C A L EXPERIENCES
The modified SUE has been tested in several real-size cases, all implemented in a
Geographical Information System (GIS) 7. In all cases the model performed well. Trees
and bundles of routes were examined, all showing a more reasonable route-split
compared with the traditional SUE. In addition, the modified approach presented in
section 3.4 speeded up the convergence.
Some experiences need to be referred regarding model estimation. Normally it is
desired that the deviations between modelled and counted traffic are minimized
according to some measure of deviation. However, even though the deviation is low,
the flow pattern might be biased (as many OD-patterns can result in the same traffic
at link-level). This might first be reflected in the forecast, but with no data-set to
compare with. Thus, beside comparing with traffic counts additional knowledge about
route choice behaviour is needed in order to estimate the model. Figure 5 illustrates
the problem. Note that optimal parameters might be found in rather large intervals.
For the Copenhagen case, p,, might be found in the interval of 0.1 - 0.4 and p,~ in
the interval of 0.5 1.0. However, by common sense, Po, should not be larger than
0.3, and P~ not larger than 1.0, as the behaviour otherwise would be very stochastic.
The N~stved case: Average relative deviations
between modelled and counted traffic
eo,,~"'o,~
The Copenhagen ease: Average relative deviations
between modelled and counted traffic
,0 , o ~ "
'~ae
,0~
Figure 5 Surfaces of deviations at different combinations of p. and 9,~ (Pc,~was set
equal to PoO in the Ncestved and Copenhagen cases.
An additional problem is that the method by which the trip matrix is estimated should
reflect the route choice. Once a matrix estimation method based on one assignment
model has been used, the structure of the matrix will affect this (see Nielsen, 1994,
1), and thus cause problems when it is reassigned with another assignment model.
6 PERSPECTIVE FOR FURTHER RESEARCH
Many subjects need to be addressed further:
The normal distribution is not totally fit to describe the distribution of weights, as
they should never be negative. This is a bigger problem than for the perceived costs,
since the error term according to the author's experience needs to be bigger in order
to get a reasonable split on routes. Instead of truncating the normal distribution it
might be better to use a non-negative distribution.
The trip matrix estimation should reflect the route choice model. A SUE-based
matrix-estimation method which takes inconsistent traffic counts into account has been
implemented by the author, but not yet published (an earlier heuristic approach is
documented in Nielsen, 1994, 1).
All the tests in Copenhagen emphasized a need for including delays at intersections
and other nodes. These and not the link-capacity were found to be the bottlenecks in
many cases. Thus, an intersection delay model needs to be implemented as a part of
step 2 and 3 in the algorithm. In addition, dependencies between links need in some
cases to be included in the algorithm, e.g. for long queues and green waves. Some
guidelines for this are given in Sheffi (1985).
Finally, the approach may be adopted for public transportation assignment. Refer to
Sheffi (1985) and Cantarella (1996) for discussion of this assignment problem as well
as multi-mode assignment.
7 CONCLUSION
In the paper a heuristic modification on SUE is presented in which two types of stochastic components occur - the first considers road users' perception of the traffic
network at link level (as in the traditional SUE) and the second considers differences
in the road users' utility functions. This turned out to give a more realistic flow
pattern in cases, where both travel length and time are a significant part of the utility
function (or less ambitious the travel resistance function). In addition, the paper
presents a modification of the solution algorithm to speed up convergence. This has
been tested with success on several full-scale cases.
For car traffic assignment the modified SUE may come in handy in countries, where
the travel length (or cost) is a significant part of the utility function due to fuel prices,
tax policies, or the economic level of the drivers. Thus, it has been used with success
in Danish cases (high taxes) and in Indonesia (low income level). For transit
assignment the method might be useful as the utility function usually is more complex
with several weights for waiting time, travel time, transfer time, e.g. Future research
will address this issue.
Acknowledgements. The author wishes to thank R.D.Frederiksen & N.Simonsen for implementing the
proposed algorithm and for help in some of the case-studies, E.R.Nielsen & T.lsraelsen f o r tests
conducted in the Bandung case study, and Steen Leleur, Bill Stanton and Susan Galsae for fruitful
comments and for proofreading. The Danish Transport Council is thanked for economic support.
NOTES
The Normal distribution is not totally fit to describe the distribution of perceived costs, as they should
never be negative. However, with a reasonable error term, err, this is seldom a problem. The few
negative values shall be truncated (see Sheffi, 1985, p.298 for a discussion).
2 SUE was originally suggested by Daganzo & Sheffi (1977), and the solution algorithm by Sheffi &
Powell (1982).
3 Values ofct and fl at 0.8 and 1.5 gave good results. The much lower ct (0.15) often used in USA was
found to underestimate the effect of congestion, while the bigger fl (4) was found to underestimate the
point, when congestion will appear.
4 A SUE-programme implemented for the author by Frederiksen & Simonsen (1996) was used. One
of the features is the ability to save information on traffic leaving one specific zone, arriving at one
specific zone, passing a certain road segment or a combination of these three criteria.
s The ring motorway was chosen with, k~lI e [0.00; 0,07] (klu= 1-kill), and four different routes through
the city in intervals of [0.08; 0.22; 0,90; 0,96; 1,00] all using the same bridge, "Langebro'.
The inner loop is somewhat like the principle in Burrells assignment (1968), although he sampled for
each path as well. However, doing this would lead to a prohibitive increase in the calculationcomplexity of the all-or-nothing assignment.
7 The methods were tested at the following cases: A model for car traffic to and from the airport ( 121
links in the main study area, 25 zones), a model for the city of N~estved with surrounding area
(versions of 134 links, 33 zones as well as 314 links and 106 zones, 120,000 inhabitants), a model for
Copenhagen (versions of 2,422 links, 97 zones as well as 2,765 links, 269 zones, 1,7 Million
inhabitants) and finally a model for the City of Bandung Indonesia (980 links, 180 zones, 5 Million
inhabitants). See Nielsen (1994, 1) for references to the Danish studies and Nielsen & Israelsen (1996)
to the Bandung Study.
s The calculation times for each outer loop in the N~estved case were 0.42 seconds for SUE and 0.9
seconds for the new approach. The similar times for the Copenhagen case were 7.7 and 18.2 seconds.
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