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ORIGINAL ARTICLE
Calibrating a trip distribution gravity modelstratiﬁed by the trip purposes for the cityof Alexandria
Mounir Mahmoud Moghazy Abdel-Aal
*
Transportation Engineering Department, Faculty of Engineering, Alexandria University, Egypt
Received 28 September 2013; accepted 5 April 2014Available online 23 June 2014
KEYWORDS
Transportation planning;Transportation modeling;Trip distribution;Gravity model;Trip length distribution;Weighted sample
Abstract
The trip distribution is the most important yet the most misunderstood model in theUrban Transportation Planning Process (UTPP). One overlooked aspect is the different sensitivitiesin choosing the destinations based on the trip purposes. This paper proposes a framework to cal-ibrate a doubly-constrained gravity model for the trip distribution of the city of Alexandria basedon a Household Travel Survey carried out in 2002. The trip ends are estimated from the availablecensus data. Important parameters for the trip attraction models were estimated and validated inthe course of this research. Since a small sample is used, a simple, effective weighing technique isapplied to mitigate the sample bias. The purpose-based dispersion parameters are estimated basedon the weighted sample. The model validation is also introduced in terms of trip length distribution,intrazonal trips and the distribution of the trip interchanges between city parts. The proposed modeldemonstrates the different patterns of trip distribution per purpose. It also shows a considerableshift toward non-compulsory trip purposes in the city of Alexandria.
ª
2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, AlexandriaUniversity.
1. Introduction
Urban Transportation Planning Process (UTPP) is a multi-stage process which includes transportation models such astrip generation, trip distribution, mode choice and trip assign-ment. A model is deﬁned as a simpliﬁed representation of partof the real world which concentrates on certain elements con-sidered important for the analysis from a particular point of view.Trip generation is a stage of a classical transportationmodels that aims at predicting the total number of trips gener-ated by (
O
i
) and attracted to (
D
j
) each zone of the study area(i.e., srcinated from and destined to each zone). This totalnumber of trips generated by households in a zone dependson the personal trip productions (e.g., car ownership, income,household structure and family size) and personal trip attrac-tions (e.g., number of employees and total areas of businesses).Trip distribution model is the second stage of transporta-tion models. This step matches trip maker srcins and destina-tions estimated by trip generation models to develop a ‘Trip
*Tel.: +20 1023348733.E-mail address: abdelaal61@yahoo.com.Peer review under responsibility of Faculty of Engineering, AlexandriaUniversity.
Production and hosting by Elsevier
Alexandria Engineering Journal (2014)
53
, 677
–
689
Alexandria University
Alexandria Engineering Journal
www.elsevier.com/locate/aejwww.sciencedirect.com1110-0168
ª
2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.http://dx.doi.org/10.1016/j.aej.2014.04.006
Tables’. A trip table is a matrix that displays the number of trips going from each srcin to each destination.The most well known models of trip distribution are gravitymodel, Logit model and Entropy Maximization model. Tripdistribution model depends on travel distance (time) betweeneach pair of zones.Trip distribution models of a different kind have beendeveloped to assist in forecasting future trip patterns whenimportant changes in network take place. Over the year, mod-elers have used several different formulations of trip distribu-tion. The ﬁrst was Frater or growth model. This structureextrapolated a base year trip table to the future based on agrowth factor(s), but took no account of changing spatialaccessibility due to increased supply or changes in travel pat-terns and congestion. The next well known model developedwas the gravity model, which was srcinally generated froman analog with Newton gravitation law. The ﬁrst use of thegravity occurred in 1950s.With the development of logit and other discrete choicetechniques as a derivative of the random utility model, new,demographically disaggregate approaches to travel demandmodeling were attempted. By including variables other thantravel time in determining the probability of making a trip, itis expected to have a better prediction of travel behavior.The logit model and the singly-constrained gravity model havebeen shown by Wilson [15] to be of essentially the same form.The application of these models differs in concept in that thegravity model uses impedance of travel time, perhaps stratiﬁedby socioeconomic variables, in determining the probability of trip making, while a discrete choice approach brings thosevariables inside the utility or impedance function. For exam-ple, the logit model expresses the destination choice as a func-tion of the utility of choosing one alternative over another. Ingeneral, discrete choice models require more information toestimate and more computational time.The notion of entropy maximization offers a theoreticalframework for spatial interaction models. The concept of entropy maximizing or minimum dispersion arose from theobservations that urban travel choices do not reﬂect the costminimizing behavior. Based on statistical mechanics, entropyis concerned with ﬁnding the degree of likelihood of the ﬁnalstate of a system. Data for urban systems are not usually abun-dantly available. Therefore, a method is needed for makingreasoned estimates of the likely state of an urban system usingthe available information. In this sense, the entropy is maxi-mized subject to constraints of known information. Theentropy maximization approach used in generating a widerange of models including the gravity model. In fact, theentropy-type constraint has been shown by Erlander [6] to beequivalent to a singly constrained gravity model and by Fisk[9] to be equivalent to logit choice function.The purpose of this research is to develop and calibrate adoubly-constrained trip distribution model for the city of Alex-andria, Egypt. A small sample of data has been collected forthe city of Alexandria. The proposed model is a frameworkof modeling trip distribution for the purpose of analyzingthe travel behavior of trip makers for different purposes. Asimple, optimizing and effective weighing technique is appliedto mitigate the sample bias associated with small sample usedin this paper. The dispersion parameters are estimated basedon the weighted sample for each purpose. The model valida-tion is also introduced in terms of – among other measures – the trip length distribution relative to the city size and the trippurpose.In order to calibrate trip distribution, an estimated trip gen-eration for each zone needed to be obtained. The average triprate for the entire city of 1.2 trips/inhabitant/day was esti-mated as a part of 1982 TranSyatem study [14]. A researchby the author in 2004 [1] indicated that this trip rate shouldbe increased by at least 10% to an updated rate of 1.32 trips/inhabitant/day. The estimates of trips produced from andattracted to each zone were essentially based on the latest aver-age trip rate.
2. Model structure
2.1. Model formulation
The gravity model is much like Newton’s theory of gravity.The gravity model assumes that the trips produced at an srcinand attracted to a destination are directly proportional to thetotal trip productions at the srcin and the total trip attrac-tions at the destinations and inversely proportional to the dis-tance (separation) between the srcin and destination. Theseparation between the srcin and destination zones is betterbe formulated as a decreasing function which is known asdeterrence function. For a study area divided into
Z
zones,the model can be represented in the following functional form[12]:
T
ij
¼
a
O
i
D
j
f
ð
c
ij
Þ 8
i
;
j
2
Z
ð
1
Þ
where
T
ij
is the trips produced in an srcin zone
i
and destina-tion zone
j
,
O
i
,
D
j
the total trip ends produced at
i
andattracted at
j
,
f
(c
ij
) the generalized function of travel costsbetween any pair of zones
i
and
j
,
a
is the a proportionality fac-tor.One version of this travel cost (deterrence) function thatwill be used throughout this paper is given as follows:
f
ð
c
ij
Þ ¼
e
ð
b
d
ij
Þ
8
i
;
j
2
Z
ð
2
Þ
where
b
is the dispersion parameter,
d
ij
the distance betweenzones
i
and
j.
The sum of the trips produced between any srcinzone
i
and all destination zones
j
e
Z
should be equal to thetotal trip ends produced at the srcin zone. Similar statementcan be made for any destination zone. These are known asthe ﬂow conservation constraints and are given as follows:
X
j
T
ij
¼
O
i
8
i
2
Z
ð
3
Þ
X
i
T
ij
¼
D
j
8
j
2
Z
ð
4
Þ
To ensure the ﬂow conservation constraints given in Eqs. (3)and (4), the single proportionality factor
a
should be replacedby two sets of balancing factors
A
i
and
B
j
. Introducing thesebalancing factors in Eq. (1) results in the classical version of the doubly constrained gravity model which is given as follows[12]:
T
ij
¼
A
i
O
i
B
j
D
j
f
ð
c
ij
Þ 8
i
;
j
2
Z
ð
5
Þ
where
A
i
¼
1
X
j
B
j
D
j
f
ð
c
ij
Þ8
i
2
Z
ð
6
Þ
678 M.M.M. Abdel-Aal
B
j
¼
1
P
i
A
i
O
i
f
ð
c
ij
Þ 8
j
2
Z
ð
7
Þ
The balancing factors are, clearly, interdependent whichsuggests that the calculation of one set of the balancing factorsrequires the values of the other set. This indicates an iterativeprocess.
2.2. Trip purposes
The formulation of the trip distribution discussed above isbased on the assumption that distance (or time) spent travelingis perceived negatively; the more distant the destination is, themore burdensome the trip becomes. Most rips produced in agiven zone will be attracted to the surrounding or nearbyzones; of course some will attracted to moderately distantzones and very few will be attracted to very distant zones. Thisis exactly what the structure of the deterrence function isindicating.Intuitively, the trip purpose has an effect on the tripdistribution in a sense that the effect of the travel distance(or time) in discouraging trips is more profound for non-work trips than for work trips which fall off less sharplywith distance. In other words, the compulsory (or manda-tory) trips such as going to work or school are a lot lesssensitive to how distant their destinations are than thediscretionary (or optional) trips such as shopping, social orrecreational trips.To capture the effect on the destination choice the trip pur-pose should be included as a variable in the trip distributionmodel. The trip distribution model can be split into severalsub-models each of which models the destination choice of the trips of a certain purpose.Hence, for a set of purposes,
P
, Eq. (5) can be rewritten as follows:
Figure 1
Study area (city of Alexandria).
Calibrating a trip distribution gravity model stratiﬁed 679
T
pij
¼
A
pi
O
pi
B
p j
D
p j
f
p
ð
c
ij
Þ 8
i
;
j
2
Z
;
8
p
2
P
ð
8
Þ
where
f
p
ð
c
ij
Þ ¼
e
ð
b
p
d
ij
Þ
8
i
;
j
2
Z
;
8
p
2
P
ð
9
Þ
Eqs. (6), (7) can also be rewritten as follows:
A
pi
¼
1
X
j
B
p j
D
p j
f
p
ð
c
ij
Þ8
i
2
Z
;
8
p
2
P
ð
10
Þ
B
p j
¼
1
X
i
A
pi
O
pi
f
p
ð
c
ij
Þ8
j
2
Z
; ;
8
p
2
P
ð
11
Þ
Finally, the ﬂow conservation constraints can be given asfollows:
X
j
X
p
T
pij
¼
X
p
O
pi
¼
O
i
8
i
2
Z
ð
12
Þ
X
i
X
p
T
pij
¼
X
p
D
p j
¼
D
j
8
j
2
Z
ð
13
Þ
The purposes considered for the purpose of this paper areas follows: – HBW: home-based work trips. – HBEdu: home-based educational trips (it is rather dividedinto two sub-purposes.
HBSch: home-based school trips.
HBUniv: home-based university trips. – HBShop: Home-based shopping trips. – HBO: home-based other trips. – NHBW: non-home-based work trips. – NHBO: non-home-based other trips.
Figure 2
Zone zystem (sections) of the study area (city of Alexandria).
680 M.M.M. Abdel-Aal

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