Spatial and Temporal Transferability of Trip Generation Demand
Models in Israel
ADRIAN V. COTRUS 1 JOSEPH
N. PRASHKER 2 YORAM
SHIFTAN 3,*
ABSTRACT
This research investigates the transferability of person-level
disaggregate trip generation models (TGMs) in time and space using
two model specifications: multinomial linear regression and Tobit.
The models are estimated for the Tel Aviv and Haifa metropolitan
areas based on data from the 1984 and 1996/97 Israeli National
Travel Habits Surveys. The paper emphasizes that Tobit models
perform better than regression or discrete choice models in
estimating nontravelers. Furthermore, the paper notes that variables
and file structures in household surveys need to be consistent.
Results of the study show that the estimated regression and Tobit
disaggregate person-level TGMs are statistically different in space
and in time. In spite of the transferred forecasts, the aggregate
forecasts were also similar.
KEYWORDS: Trip generation, transferability,
multinomial linear regression, Tobit model.
INTRODUCTION
Trip generation models (TGMs) are used as a first step in
classical four-step travel demand modeling and, therefore, any over-
or underprediction of trip generation rates can cause errors
throughout the entire transportation planning process. Inappropriate
decisionmaking due to these types of errors can account for
premature investments in infrastructure in the case of
overprediction and loss of labor hours, pollution, and low levels of
service in the case of underprediction.
TGMs are usually estimated based on periodic surveys of the
travel habits of individuals or households. They are expensive and
difficult to perform and are not conducted often. For our research,
we used the last Israeli National Travel Habits Surveys collected in
1984 and 1996/97. Transportation planners use models previously
estimated and sometimes in different contexts. The planners can
perform forecasts for the same areas and, if justifiable, transfer
the models to other areas. Hence, it is important to know whether
these models can be transferred in time and in space.
Recently, researchers and planning agencies began to implement
tour-based activity modeling systems rather than trip-based modeling
systems.1
The advantage in using an activity modeling approach is the ability
to model each individual's tours. However, in this paper, we were
only able to investigate the stability of individual predictions of
trips. This research presents the characteristics of trip generation
in Israel and tries to answer the question of whether linear
regression and Tobit TGMs can be transferred in time and space,
given the dynamic changes in metropolitan areas and socioeconomic
characteristics. The TGMs estimated and analyzed in this research
include only vehicular trips.
We estimated the models for the geographically diverse
metropolitan areas of Haifa and Tel Aviv in Israel and tested them
for transferability in space and in time. The topography of Haifa is
hilly, with the core of the city poorly connected to the rest of the
metropolitan area. Tel Aviv lies on level topography, with a well
connected road network. The metropolitan areas also differ
structurally: Tel Aviv is interconnected like a spider web,
including several minor cores with high-density population and
employment concentrations. On the other hand, Haifa's less connected
road network and rolling terrain give it a lower level of
accessibility. When comparing the areas by land use, the Tel Aviv
metropolitan area consists of neighborhoods that combine residential
shopping and personal business areas. Haifa, on the other hand,
contains highly separated areas with each area consisting of a
uniform land use.
Given the difference in accessibility and land use, the
calibrated models were restricted to demographic and socioeconomic
variables. The average number of daily trips per person was higher
in the Haifa metropolitan area (2.14 in 1984, and 2.03 in 1996/97)
than in the Tel Aviv metropolitan area (1.83 in 1984, and 1.91 in
1996/97). The difference in the average trip rates may be explained
based on the variation in land uses. The lack of mixed land uses in
Haifa may encourage the generation of more trips. Furthermore,
Haifa's hilly topography may encourage more vehicular trips than in
Tel Aviv, where shorter trips are probably done on foot. The
comparison between these metropolitan areas and the calibrated
models is possible due to the similarity of the distribution of most
demographic and socioeconomic variables. (See table
1 for a partial presentation of the comparison.) A more detailed
comparison of Haifa and Tel Aviv characteristics is presented in
Cotrus (2001).
LITERATURE REVIEW
Over the last few decades, several papers have discussed the
transferability of trip generation models. The debate among
researchers, in general, focused not only on the transferability of
models in space and time but also on the model specification and
level of aggregation. The aggregation levels are usually defined as
area (zonal), household, and person. Estimating the models (see,
e.g., Ortuzar and Willumsen 1994) at more disaggregate levels
improves the transferability of TGM.
Atherton and Ben-Akiva (1976) emphasized that disaggregated
models tend to maintain the variance and behavioral context of the
response variable and, therefore, are expected to give better
estimates when transferred. Downes and Gyenes (1976) pointed out
that when the explanatory power of the model is of interest rather
than the aggregate forecasts, the disaggregate level should be
selected. Wilmot (1995) indicated that disaggregate models are
preferred because of their independence from zonal definitions. In
Supernak et al. (1983) and Supernak (1987), the person level was
preferred for TGM because of the identity of the response factor
(trip) and the generative (the person). One advantage of
disaggregate person-level models is the reduced amount of data
required for model estimation. (For more details, see Fleet and
Robertson 1968; and Ortuzar and Willumsen 1994.) Other types of
model specification techniques include cross-classification,
regression, logit-based models, artificial neural networks, fuzzy
logic, and simulations.
A number of studies found spatial transferability of models
satisfactory (Wilmot 1995; Atherton and Ben-Akiva 1976; Supernak
1982, 1984; Duah and Hall 1997; Walker and Olanipekun 1989; Rose and
Koppelman 1984; Caldwell and Demetsky 1980; and Kannel and
Heathington 1973). On the other hand, Smith and Cleveland (1976) and
Daor (1981) found spatial transferability unsatisfactory. We should
emphasize that Smith and Cleveland pointed out that although the
explanatory variables are distinctive, their effects vary in space.
A number of researchers found the transferability of models in time
(i.e., their temporal stability) satisfactory (Downes and Gyenes
1976; Yunker 1976; Walker and Peng 1991; Kannel and Heathington
1973; and Karasmaa and Pursula 1997). Unsatisfactory results,
however, were obtained in other studies (Doubleday 1977; Smith and
Cleveland 1976; and Copley and Lowe 1981).
While several international studies explored model
transferability in time and space, in Israel the transferability of
discrete mode choice models has been the main focus (Prashker 1982;
Silman 1981). This study deals with the investigation of trip
generation characteristics but also provides local estimates of TGMs
and their validation for transferability in time and space. The
study also explores the implementation of Tobit models in TGM.
We often approach trip generation from an economic viewpoint,
where trips are defined as the product and the person/household as
the customer. The strongest argument to model trips on a
disaggregate level is that any zonal outcome is based on the
aggregation of several customers, ignoring the heterogeneity among
them. The explanatory variables for the power of consumption of each
person/household can be found in several categories including
demographic, geographic, and economic.
As discussed above, several approaches exist to model trip
generation, including regression-based models such as multiple
linear regression (Wilmot 1995) and cross-classification (Walker and
Olanipekun 1989); discrete choice models such as probit, logit, and
ordered probit (Zhao 2000); simulations such as Smash, Amos, and the
Starchild System; fuzzy logic models; and artificial neural networks
(Huisken 2000). Clearly, the issue of trip generation can be
approached from several directions and tested for transferability in
time and space. Therefore, researchers will choose the modeling
approach based on the size of the database at hand, the nature and
structure of the variables, the aggregation level desired, as well
as other considerations. The main problem with using a regression
model is the treatment of trip rates as continuous rather than
discrete variables. Discrete choice models and spatially ordered
response models may better account for the behavioral process of
trip generation. However, due to practical reasons, in most models
to date, the dependent variable is treated as a continuous variable.
For this reason, we perform our analysis on such models.
METHODOLOGY
In this research, we first estimated regression models for each
metropolitan area for each year, taking into account the
inconsistency of the household surveys (table
2). We then estimated Tobit TGMs based on the same variables and
tested whether these models are suitable for trip generation
estimation and for transferability. The regression model form is
presented in equation 1.
y i = α + β
1 • x 1, i + β
2 • x 2, i … + β
k • x k, i + ξ
i
∀ i = 1, …, n
∀ k = 1, 2, …
k (1)
where
yi = trip rate generated by individual
i,
xk,i= explanatory variable k for
individual i,
n = the number of observations,
k = number of explanatory variables, and
ξi = error term of the ith
observation.
Hald (1949) first presented the model that, in its final form, is
called the Tobit model (1958). Tobit models differentiate from
regression models by the incorporation of truncated or censored
dependent variables. Tobit analysis assumes that the dependent
variable has a number of its values clustered at a limiting value,
usually zero. The Tobit model can be presented as a
discrete/continuous model that first makes a discrete choice of
passing the threshold and second, if passed, a continuous choice
regarding the value above the threshold. This approach is
appropriate for trip generation, as an individual must decide
whether to make any trips and, if so, how many trips to make.
Tobit analysis uses all observations when estimating the
regression line, including those at the limit (no trips) and those
above the limit (those who chose to travel). As shown by McDonald
and Moffitt (1980), Tobit analysis can be used to determine the
changes in the value of the dependent variable if it is above the
limit, as well as changes in the probability of being above the
limit. Since the surveys include observations at the limit (i.e.,
persons that are not traveling), it was also interesting to find out
how well the Tobit model can predict persons doing no travel at all.
The Tobit model form is presented in equation 2:
y i = X i •
β + ξ
i if X
i • β + ξ i > 0
y i =
0 if X
i • β + ξ i ≤ 0
∀ i = 1, 2, 3, …, (N - 1),
N (2)
where
N = number of observations,
yi= trip rate generated by
observation i,
Xi = vector of independent
variables,
β = vector of coefficients, and
ξ = independently distributed error term ~ (0,
σ2).
Because Tobit models have not been used previously in the context
of trip generation, this research investigates their suitability for
that purpose. We also compared the predictions obtained using
regression models with those produced using the analogous Tobit
model. The best specification of the regression model is not
necessarily the best specification of the Tobit model. However, in
order to allow for basic comparisons of the model parameters, we
estimated the regression models first; then, after the determination
of the final variables in the model, we estimated Tobit models with
the same variables.
All models were estimated at the disaggregate person level. At
the person level of modeling, we maintained the heterogeneity among
observations and kept a good identity between the consumer of the
product (the person) and the outcome (number of daily trips taken by
the person). As discussed above, disaggregate models tend to show
better transfer results than aggregate models and also incorporate
the power to understand and control the production of trips. The
models were estimated for a 24-hour period2
and tested for transferability in space and in time. Figure
1 presents the sequence of the analysis.
Statistical tests were conducted to determine the spatial and
temporal stability of the estimated models by assessing the
transferability of the coefficients from one area to another, and
for each metropolitan area between the two survey years.
Transferability was also tested by comparing the overall aggregate
prediction obtained by the transferred model with the local model.
Furthermore, we analyzed the ability of Tobit models to represent
and evaluate nontravelers, that is, people who do not generate trips
based on the given survey data.
Data Sources and Descriptions
The Israeli Central Bureau of Statistics (CBS) conducted some
limited scope3
Traveling Habits Surveys in the 1960s. Comprehensive National
Traveling Habits Surveys have been conducted by CBS every 12 years
since 1972. Because the 1972 survey is not available on magnetic
media, it was not possible to do a computer-based statistical
analysis. Therefore, we based this research on the 1984 and 1996/97
household surveys.
The main problems we encountered in doing this research were
related to the inconsistency in the investigated variables, the
structure of the surveys, the definition of variables, the period of
investigation, the geographic deployment, and the database structure
(see table 2 again). The 1984 and 1996/97 household surveys differ
in several ways: the geographic deployment (number and size of
jurisdictions in the survey), the size of the survey (number of
households), the definitions of the investigation period, and the
variables that were excluded from the surveys. For example, income
is included in the 1984 survey but is omitted from the 1996/97
survey.
Despite definition and database differences in the two surveys
(1984 was an activity survey and 1996/97 was a trip survey), we were
able to bring the variables in the models to a common basis. In
particular, the 1984 survey included bicycle and walking trips among
the means to accomplish the activities, while the later survey
excluded them. To resolve this difference, we excluded walking and
biking trips from the 1984 database; only motorized trips were
considered for each person.
The 1984 survey files included data for 5,420 persons in the Tel
Aviv metropolitan area and 4,056 persons in the Haifa metropolitan
area. The final files used for model calibration after sieving
incomplete and anomalous observations totaled 4,385 and 3,258
persons, respectively. The 1996/97 files included data for 20,436
persons in the Tel Aviv metropolitan area and 6,417 persons in the
Haifa metropolitan area. The final files used for model calibration
totaled 15,729 and 5,041 persons, respectively.
RESULTS
The selected trip generation models included six categorical
variables: age, car availability, possession of a driver's license,
employment, education, and status in the household. Data fell into
five age categories: 8–13, 14–17, 18–29, 30–64, and 65 and over.
Ortuzar and Willumsen (1994) found that life cycle variables were an
important factor for explaining trip generation. Different trip
rates can be expected for households and people at various stages of
life. Furthermore, age should correlate with employment, having a
driver's license, and marital status. Car availability included
three categories: 0, 1, and ≥ 2 cars in the household. Clearly,
households with more cars available will generate more trips. The
driver's license category has only one variable: whether the person
has a license (including motorcycle) or not.
The employment variable indicates whether the person was employed
or not. Employed persons were expected to generate more trips,
because they usually make at least two trips: to and from work.
Household status refers to whether the person defines himself or
herself as the head of the household. This variable indicates the
responsibility and availability of household resources as an
incentive for consumption of trips.
Finally, four education categories were defined based on the
number of years of study (0, 1–8, 9–12, and 13 or more). The
literature shows good correlation between education and income. In
the absence of a pure economic indicator, education is used also as
a proxy for income. Respondents with higher education (hence higher
income) were expected to generate more trips. All variables were
found significant and the coefficients corresponded with our
expectations.
Table
3 shows the estimation results for the regression models for
1984 for both metropolitan areas. As can be seen from the table, all
coefficients were found to be significant at the 95% level. The
number of observations remaining in the estimation process resulted
from the limited scope of this survey and the elimination of
incomplete observations in the original database.
Estimation results for these models show that all variables
affected trip generation as expected. The education coefficients
show that people with higher education generated more trips. This
can be explained not only by the assumption of the relationship
between education and income but also by assuming that a person with
higher education is more likely to pursue culture and perhaps
leisure activities. Also, as expected, persons with driver's
licenses and employed persons tended to generate more trips than the
equivalent nonworking and/or nondriving persons. Heads of household
tended to generate more trips, as assumed, because of the
responsibility and availability of resources. The coefficients of
the age categories indicate that persons aged 14 to 17 travel more
than people with similar characteristics of other age groups,
probably because they are young and active and have less household
or work responsibilities.
The overall R2 of the 1984
models was 0.33 for the Tel Aviv model and 0.34 for the equivalent
Haifa model. These R2 values are
modest but not anomalous for trip generation modeling. They indicate
that a substantial portion of trip generation can be explained by
nonhousehold factors, such as relative location of residence,
employment, and other parameters. Statistical Z tests
(assuming known variances, normal distributions, and independence of
populations) for the transferability of the coefficients (without
updating) show that, at a 95% level of confidence, the coefficients
differ, except for the coefficient defining the head of household.
To verify the results we also conducted Chow tests for the
transferability of the models. The calculated statistic was 7.86 in
comparison with the critical 1.72 F-value (at a 95% level of
confidence), and yield the same conclusion that the 1984 models are
not transferable in space.
Table
4 shows the results of transferring the models in space by
showing the predictions from the estimated models and the 1984
database, each applied for both metropolitan areas. The table shows
that the Haifa model overpredicts the actual trip rate when applied
to the Tel Aviv database, in comparison with the Tel Aviv model
applied to the Haifa database. This was expected, as the Haifa trip
rate is higher than that for Tel Aviv.
Table
5 presents the estimation results for the 1984 Tobit models
using the 1984 household survey data. When we transferred the
estimated Tobit models in space and used them to predict the average
trip rate in the other city, we found that the Haifa Tobit model
overestimated the trips in Tel Aviv by 21.9% (table
6). When we used the estimated Tel Aviv Tobit model to predict
the average trip rate for Haifa, we found that it underestimated the
total trips by 27.5%. However, transferability t-tests at a
95% level of confidence showed that most of the coefficients are not
significantly different in space, except for the license variable
and the 8–13 and 30–64 age category variables. On the other hand,
χ2 tests at the 95% level of
confidence strengthen the alternative hypothesis, that the models
vary in space (x 213, 0.95 = 22.36 <
91.198) .
An important issue was to find out whether the Tobit model could
explain and capture the nontravelers in the population. As can be
seen in table
7, the results are not consistent for the two models. The Haifa
1984 model correctly estimated only 13.5% of the observed
nontravelers in the Haifa data and 18.5% in the Tel Aviv data. The
Tel Aviv model obtained better results estimating correctly 41.9% of
the observed nontravelers in the Tel Aviv data and 34.7% in the
Haifa data. These results encourage further research.
Table
8 presents the estimation results of the 1996/97 regression
models. As can be seen, the 1996/97 coefficients differ
substantially from those of 1984, and most of the coefficients are
significant at the 95% confidence level. The best model
specification for 1984 was found to be also the best specification
for the 1996/97 model, indicating that the most important variables
affecting trip generation are similar in both models. The main
problem raised during the basic comparison was the difference in the
geographic scope (definition of the metropolitan survey area) for
the two.
The overall R2 of the 1996/97
models, 0.21 for the Tel Aviv model and 0.23 for the equivalent
Haifa model, are even smaller than the values achieved for the 1984
models. But they are still not anomalous in the field of trip
generation modeling. Statistical Z-tests conducted at the 95%
level of confidence show that none of the coefficients are the same
for the two metropolitan areas; that is, the coefficients differ in
space. To verify the results, we conducted Chow tests for the
transferability of the models. The calculated statistic was 6.91 in
comparison with the 1.72 tabular F-value (at a 95% level of
confidence), thus yielding the same conclusion, that the 1996/97
models are not transferable in space.
Table
9 presents the predicted average daily trips using the 1996/97
data for each model and each metropolitan area. As in 1984, the
1996/97 Haifa model overpredicts trips compared with the Tel Aviv
model, however, the differences are smaller. One should remember
that in regression models, the regression line always passes through
the average (center of gravity of the observations). Since the
observed average number of trips (in Tel Aviv and Haifa) was equal
in the 1996/97 metropolitan files, the estimation difference was
expected to be smaller.
However, the similar predicted aggregate trip rates indicate
overrepresentation of particular sections of the population. For
example, the calculated average car availability per household in
metropolitan Tel Aviv was higher in the 1996/97 surveys than in
metropolitan Haifa (0.60 > 0.55), but in 1984 the average car
availability per household was almost the same (0.489 ≈ 0.484).
Statistically, different definitions of the sampling areas could
affect the transferability of the estimated models
Table
10 shows the Tobit model results for 1996/97. A comparison with
table 8 shows the resemblance in the effect of the explanatory
variables and the difference in the magnitude of the coefficients
between the Tobit and the regression models. Transferring the models
in space and evaluating the estimated average trip rate from the
models, for 1996/97, we found that the Haifa Tobit model
overestimated the trips in the Tel Aviv file by 7.2% and the Tel
Aviv Tobit model underestimated the trips in the Haifa file by 7.1%
(table 6). These values are quite similar to the over- and
underprediction of the equivalent regression models shown in table
9. Spatial transferability t-tests held at the 95% level of
confidence show that most of the coefficients are not significantly
different between the metropolitan areas, except age categories and
the education "non-educated" category. χ2 tests for the spatial transferability of
the models at the same level of confidence reach the same conclusion
(x 213, 0.95 = 22.36 << 82.01).
In table
11, the 1996/97 Tobit model prediction of nontravelers is even
worse than in the analogous 1984 models. When trying to represent
nontravelers, the Haifa 1996/97 Tobit model captured only 6.8% of
the observed nontravelers in the Tel Aviv file and 11.8% in the
Haifa file. The Tel Aviv 1996/97 Tobit model captured only 3.8% of
the nontravelers in the Haifa file and 9.8% in the Tel Aviv file. A
point worth mentioning is the resemblance in the proportion of
nontravelers in the two surveys (about 35% of the persons
represented in the sample files did not generate trips). Finally,
about 70% to 75% of the estimated nontravelers are observed
nontravelers.
Table
12 shows the estimation results for temporal transferability of
the regression and Tobit models. When we tested for temporal
transferability using the 1984 models to predict 1996/97 trip rates,
we observed that the 1984 Tel Aviv regression model underestimated
the observed total number of trips in Tel Aviv in 1996/97 by 7%. The
Haifa 1984 regression model overestimated the observed total number
of trips in 1996/97 Haifa data by only 2.8%. Taking into account
that the average number of daily trips in the Haifa 1984 survey was
2.17 and in the 1996/97 survey it was 2.07, the difference is not
surprising. However, it may also be affected by the different
definition of the geographic scope of the two household surveys.
Chow tests of the temporal stability of the 1984 models compared
with the 1996/97 show that the statistic for the 1984 Tel Aviv model
was 4.53, bigger than the 1.72 tabular F-statistic at the 95%
level of confidence, meaning that the 1984 coefficients differ from
the 1996/97 coefficients. The statistic for the temporal stability
of the 1984 Haifa model was 8.39 compared with the 1.72 tabular
F, reaching the same conclusion. Transferability χ2 tests of the Tobit models in time show
that, at the 95% level of confidence, we can reject the null
hypothesis; that is, the models for the two time points are
different (x 213, 0.95 = 22.36 <
128.72).
CONCLUSIONS
In our research, statistical tests indicated that the regression
and Tobit models estimated for two metropolitan areas and two time
periods differ statistically in time and in space. One exception was
the Tobit transferability in space, where the coefficients from the
two models for the same year were not significantly different. The
distinction cannot be well explained, but it might be due partially
to geographic, demographic, socioeconomic, and spatial structure
differences between the two metropolitan areas. The smaller sample
size and scope of the 1984 household survey compared with the
1996/97 household survey (as shown in table 2) did not allow us to
represent the ethnicity of the survey participants, a variable
believed to be related to trip generation. Also, the incorporation
in the models of a pure economic variable such as income was not
possible, because it was not included in the 1996/97 survey.
We ascribed the temporal instability of the estimated models to
changes in the structure and development of the metropolitan areas
of Tel Aviv and Haifa, changes in lifestyle and socioeconomic
variables that are not all accounted for in the model, as well as
the inconsistency of the two surveys. A partial explanation may be
that 1984 was an economically unstable year, featuring high
inflation rates and uncertainty, while 1996/97 was considered to be
economically stable.
An important conclusion based on our results is that in order for
trip generation models to be transferable they need to account for
variables not included in the current models: income, land use and
spatial structure, the economy, the transportation system and
accessibility, and more detailed socioeconomic and life style
variables. If we could estimate a perfect disaggregate model
accounting for all factors that affect trip generation and with
appropriate segmentation, it would likely be transferable. With this
data lacking, models are not transferable, because unobserved
variables affect coefficients of observed variables with which they
are correlated.
Another conclusion is that household surveys conducted on a
regular basis will be more useful if the design stays constant.
Differences in the structure, variables, range, investigation
period, definition of the variables, and database structure affect
the transferability of the estimated models.
We also would emphasize the need for further research on the
implementation of Tobit models in the context of trip generation.
Tobit models tend to represent the mechanism of trip generation more
realistically, capturing and estimating (partially) nontravelers. As
a combination of regression and discrete choice models, the Tobit
model may be more suitable for implementation in TGM than discrete
choice or regression models, particularly because Tobit is better
formulated to differentiate nontravelers from travelers. The
underestimation of nontravelers may be partly due to the fact that
we did not necessarily estimate the best Tobit model.
For the linear regression models, almost all variables were
significant at the 95% confidence level, but the coefficients were
shown to vary in time and space. For the Tobit model, while almost
all variables were significant at the 95% confidence level, the
coefficients of the models of the two metropolitan areas were
statistically similar but they differed in time for each city.
The nature of the local household surveys raises a need to
validate the results of this study in future research. In
particular, further research can identify what makes two study areas
"similar enough" to justify transferring a model from one to the
other. We also suggest further research incorporating Tobit models
in TGM and for investigating the characteristics of
nontravelers.
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END NOTES
1. Tours refer to a sequence of
trips usually starting and ending at home. Trips refer to
just the movement between an origin and destination.
2. The 1984 household survey contained
1.5 days of data for each person; the 1996/97 survey contained 3 to
4 days of data for each person.
3. These surveys were restricted to
work-related activities only.
ADDRESSES FOR CORRESPONDENCE
1 A. Cotrus, Department
of Civil Engineering, Technion, Israel Institute of Technology,
Haifa 32000, Israel. E-mail: cotrus@walla.co.il
2 J. Prashker,
Transportation Research Institute,Technion, Israel Institute of
Technology, Haifa 32000, Israel. E-mail: prashker@netvision.net.il
3 Corresponding Author:
Y. Shiftan, Transportation Research Institute,Technion, Israel
Institute of Technology, Haifa 32000, Israel. E-mail: shiftan@tx.technion.ac.il
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