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Unformatted text preview: Applied Models in Urban
and Regional Analysis NORBERT OPPENHEIM Polytechnic Institute of New York ‘ PRENTICE‘HALL, INC, Englewood Cliﬂs, New Jersey 07632 REFERENCES BENDAVID, A. Regional Economic Analysis for Practitioners. New York: Praeger
Publishers, Inc., 1974. CZAMANSKI, S. Methods of Regional Science. Lexington, Mass: D. C. Heath &
Company, 1975. HOOVER, E. M. An Introduction to Regional Economics. New York: Alfred A.
Knopf, Inc., 1971. LEONTIEF, W. Input—Output Analysis. New York: Oxford University Press, 1969. LEVEN, C., et al. An Analytical Framework for Regional Development Policy.
Cambridge, Mass: The MIT Press, 1970. RICHARSON, H. Input—Output of Regional Economics. London: George Weidenfeld
& Nicolson Ltd., 1972. TIC) 4 Models of Land Use
and Travel Demand 4.I Introduction In the preceding chapters, we ﬁrst learned how to project the evolution of the
composition and level of populations on the basis of the knowledge of their
internal characteristics (birth and death rates), and of the external factors of
change (as represented by the migration rates) We next learned how to
forecast the effect on the employment (and therefore indirectly on the migra
tion) of changes in the levels of economic activity These changes may them
selves, in turn, be related to changes in the population and its demands for
goods and services. The fact that changes in population level and/or changes in level of
economic activity at a given location may be inﬂuenced by concurrent
changes at other locations was taken into account by the development of the
multiregional versions of the population and economic analysis models.
Speciﬁcally, in the multiregional cohort survival model of population projec—
tion, the interregional migration rates in formula (2.37) which measurethe
proportion of people who have left a given area to go live in another thus
represent the relationship between the residential activity levels in two diﬁ‘er—
ent places or spaces. Thus, in this sense these coefﬁcients represent the level
of interaction between the various regions, in a spatial sense, with respect to
the activity of migrating, or choice of an area of residence. III 112 CH. 4 MODELS or LAND USE AND TRAVEL DEMAND Similarly, the interregional input/output coeﬂicients in formula (3.33)
reﬂect the level of spatial interaction in economic activity, since they rep
resent the proportions of goods, or resources, leaving a given physical space
to reach another for the purpose of economic production. Finally, in the
multiregional employment multiplier model, the coefﬁcients au‘ in formula
(3.45) measure the level of spatial interaction between areas i and j, with
respect to the choice of a place of residence (or of a place of work). The
coefﬁcients on, in the same model, in formula (3.49) represent the levels of
spatial interaction between spaces 1' and j with respect to the activity of
shopping. Thus far, we have assumed that these measures of spatial interaction were
given, or could be determined from observations, i.e., using empirical mea
surements obtained, for instance, through survey research. However, the
application of these models to forecast future conditions requires that we be
able to predict the values of these coefﬁcients at a future date. It is logical to
assume that they depend not only on the locational characteristics of the
activities (e.g., the relative levels of population, or production at various
locations), but also on the characteristics of the communication system con
necting these locations (principally, the transportation system). Consequently,
we cannot in general assume that these coefﬁcients are going to remain at
their present values, or even extrapolate them if these characteristics are
going to change. This might happen either because of natural evolution
(population, for example), or because of planned change (transportation
system), or even because of unforeseen changes (demands for production). In this chapter, we shall see several examples of simple models that
translate theories about the mechanics of spatial interaction which will
enable us to forecast such spatial interaction, given the levels of their deter
minants. In a sense, all spatial interaction ultimately is reﬂected in some
travel, or movement activity (or more generally communication activity), of
one sort or another. For instance, matrix A in (3.45), which represented the
choice of residence, is equivalent to the travel pattern of commuting to work
from home (or commuting home from work). Similarly, matrix C in (3.49),
which represented the choice of a shopping district is equivalent to a shopping
travel pattern. Finally, the matrix 61k, in formula (3.33) can be taken to rep—
resent the regional pattern of goods movements between regions k and 1. Consequently, the models we are going to see in this chapter can be
consideredeither models of land use or of travel demand or in some cases
both. This is consistent with the physical fact that land use determines travel
demand, which determines the characteristics of the transportation system,
which in turn inﬂuences the evolution of land use, etc. We shall therefore
begin with two general models of spatial interaction, the gravity model ﬁrst
and then the intervening opportunities model, and some of their variants. We
shall next survey brieﬂy some of the speciﬁc models of travel demand. 4.2 The Theory of the Gravity Model The ﬁrst theory of spatial interaction we shall examine is the gravity theory.
Its name comes from its historical origin as an application of Newton’s
fundamental law of gravity in physics. This law, which governs the move
ments of bodies in space, was in the ﬁeld of social science ﬁrst applied to the
description of the movements of people between areas. It has subsequently
been shown to replicate adequately various other kind of spatial phenomena.
Concurrently with its empirical validation, theoretical demonstrations from
a variety of points of View and assumptions have given it a logical justiﬁcation
which makes it one of the most widely used and important spatial models. 7 Let us now review brieﬂy the basic concept behind the law of gravity.
Newton’s law states that the force of attraction, or pull F, between two bodies
of respective masses M1 and M2, separated by a distance d, will be equal to F = 5.1%‘12. = gM,M,d~2 (4.1)
where g is a constant, or scaling factor which ensures that equation (4.1) is
balanced out, i.e., that it indeed represents an equality between the measure
ments of entities of a completely different nature. The verbal translation of the law of gravity is thus that the amount of interaction exerted by two physical bodies on each otheris proportional to
their respective masses, but also inversely proportional to the square of the
distance between them. In other words, as the mass of either of the bodies
grows, the interaction between them will grow linearly. However, as the
distance between them grows, the interaction will decrease parabolically.
. In imitation of this gravity law of physics, the gravity concept of spatial
mteraction states in its simple form that the interaction between two areas,
numbered i and j (e. g., the number of people living in area j who work in area
i, or the amount of economic output of area j which is consumed in area i,
etc.) will be directly proportional to the “masses” of these areas (their size, or
population level, or level of expenditures, etc.) but will be inversely propor
tional to some function of the “distance” between them (travel time, or
cost of travel, etc). Thus, the amount of interaction between the two areas
will increase (all other things remaining equal), as the “importance” P, of the
ﬁrst area increases, and also as that P, of the second area increases, but will
decrease as the “separation” between them, d”, increases. In its general mathematical form, therefore, this theory can be stated as:
I”, the amount of interaction between areas, or spaces i and j, is equal to 1,, = k,lj0jD,F(d,,) (42) where k, is a scaling factor related to area i, 1, another scaling factor related
to area j, so that the product kJ, plays the role of the factor g in formula 113 114 CH. 4 MODELS or LAND USE AND TRAVEL DEMAND (4.1). 0, is a measure of the capacity for interaction producing of area j, and
0, is a measure of the capacity for interaction receiving of area 1. Finally, du
is a measure of the separation, or difficulty of interaction between space 1
and space j, and F (d, 1) is a mathematical function that can take on various
forms, to be speciﬁed, which represents the interaction impedance. Several points should be noted in connection with this general model.
First, it is important to note that the interaction between areas i and j is not
a symmetrical concept, i.e., that the action of area j on area i is not in general
equal to that of i on j. Thus, the order of the subscrlpts i and j in formula
(4.2) is important, and we shall refer to 1,] above as the interactionof area 1
on area i, or equivalently as the amount of interactlon from j to z (or by J
on i)‘. (This convention for the role of the order of the indices i. and j has been
adhered to previously in this text, in particular in the multiregional economic base model.) _
Also, the measures or scores 0, and D, can themselves be functions of (i.e., derived from) the values of selected characteristics of the zones i and j. For instance. D, = aan + azXn + ‘ . . + (1me (4.3)
01' D, = Xffoz' . . . XI; (4.4)
are two forms of simple functions (linear and power, respectively) which can
be used, where the Xu’s are characteristics of zone 1', such as their average
income, employment ratios, number of car registrations, etc. The same (or
other functions) of the same (or other) characteristics of zone j may (or may
not) be applied to the deﬁnition of 0,. By replacing D, and/or 0, by their
expressions (4.3) or (4.4), etc‘., in the model (4.2), the resulting spatial interac—
tion model is of a very general form. . The nature of the factors k, and I, should be distinguished from that of the
variables 0, and D,. The latter are given, i.e., are observabie quantitles,
which are usually obtained through empirical surveys. In modeling language,
they are exogeneously determined, and are used as inputs to the model. The
latter are endogeneously determined from the internal requirements of the
model, usually that the interactions “balance out” according to specrﬁed
conditions. [In fact, the two versions of the gravity model that we shall
examine later in this section represent the outcome of two such sets of con
ditions applied to the general form (4.2)]. ‘ Finally, the functional form of the interaction impedance function F (d,,) can be varied as well. Some functions commonly used are F(du) : I? (45)
F(d,,) = eMu (4.6)
F(d,,) = age—Mu (47) Each of these forms represents the translation of further hypotheses about
the law of interaction between space i and space j. Each of these hypotheses CH. 4 MODELS or LAND USE AND TRAVEL DEMAND 115 can be given a behavioristic basis. For instance, form (4.5), which represents a
generalization of the original gravity hypothesis, is the result of the assump
tion that (all other things being equal) residents in zone j will tend to select a
zone i for the location of the activity in geometrically decreasing proportion
to the amount of effort involved in reaching 1‘ from j, i.e., according to a form
of the principle of “least effort.” On the other hand, form (4.6) can be derived
from the assumption that the distribution of activities is the “most probable,”
consistent with a given total amount of travel effort between all zones. Finally,
form (4.7) can be interpreted to reﬂect both conﬂicting desires to choose an
activity zone that is away from the zone of residence j (the term e‘ﬁd"), but
at the same time not too far to involve a large travel effort (the term d}, . The practical determination of the speciﬁc form of the function to use, as
well as of the values of the parameters, is effected empirically, using observa
tional data pertaining to a speciﬁc situation. This process of ﬁtting, or
adjusting, the model is conducted through the application of the methods of
model calibration, which we shall survey in Chapter 6. In any case, this state
in the speciﬁcation of the model should not be confused with the determina
tion of the values of the scaling factors k, and/or 1,. These factors are not
parameters whose values can be changed to make the ﬁt between the output
of the model and the observed values as good as is possible, as are the afs
in (4.3), or at, [1, and y in (4.5), (4.6), or (4.7). On the contrary, they represent
structural factors, which can be computed once the logical requirements for
the consistency of the model are speciﬁed. 4. 3 First Application:
The Single—Constraint Form Since the single—constraint form discussed above surely must sound somewhat
abstract, we shall proceed with the application of these concepts to the
solution of a practical problem which is related to the residential and com
mercial location models (3.45) and (3.49) that were used in the‘ multiregional
employment multiplier model of Section 3.7. This will illustrate the single
constraint,” or “oneway,” form of the gravity model. It may be recalled that at that time, we distributed the various numbers of
workers E , working in n zones j = l to n according to a “work to home”
commuting pattern, represented by a matrix A, where the entry (1,, measures
the proportion of workers living in zone j who live in zone i. These entries
were assumed to be given, for the purpose of the model. However, they are
clearly related to a spatial interaction between zones j and zone i, in the sense
we have discussed above. The gravity model might then be the solution to the
problem mentioned above of ﬁnding a formal expression for matrix A that
may be used to forecast its values when, and if, the present conditions change. 116 CH. 4 MODELS OF LAND USE AND TRAVEL DEMAND Indeed, since the gravity model has been shown to adequately describe a large
variety of forms of spatial interaction under a wide range of conditions, we
might make the assumption that it also governs the commuting pattern
between places or work and places of residence in our case. Let us therefore assume that the spatial interaction represented by the
commuting of workers in the area from their places of work to their places of
residence follows a gravity—type law. Then, we would expect the number of workers WU who work in area j and live in area i to be of the form (4.2).
That is, WU = k,z,0,D,.F(d,,) (4.8) where 0, is some measure, to be speciﬁed, of the capacity of area of work j
to “produce” such workers, D, is some measure of the capacity of area 1' to
“receive” them, and d” is some measure of the difﬁculty to reach area i from
area j. Finally, we will also have to decide on a speciﬁc form for the function
F(d.;) It seems reasonable, in the absence of prior experiments or of any
particular insight, to adopt as a measure 0]., simply the number of workers in
the area, i.e, E, in the notation of Section 3.7. This is about the simplest
choice we can make for that measure, and it is logical to at least begin with it.
(Of course, later, if it turns out to prevent adequate validation of the repro—
duced or predicted commuting pattern against the realworld observed pattern,
this particular choice can always be modiﬁed.) Similarly, since we already
utilize the existing residential population levels P, in the various areas i, we
shall choose that characteristic as representing the measure D,. This means,
then, that we are assuming that the residential attractiveness of the zones i is
proportional to their population levels, or in other words, that the more
people already live in a zone, the more will‘be attracted to choose it as a place
of residence. Although there are undoubtedly other possible hypotheses
about the determinants of the capability of area i as a place of residence to
attract the workers, this is again one of the simplest we can make at this
point, and also one which offers the advantage that we already have the
values of the Pi’s. Next, we shall choose a measure of the difﬁculty (1,», of commuting between
zones j and i. A simple, logical choice here is to take the physical distance.
This measure should itself be closely related to other possible measures such
as travel time or cost of travel, and is also fairly simple to evaluate. Finally,
let us choose for the function F(du) the simplest expression we can, i.e., form
(4.5) with a parameter 7) = 1. In other words, we are making the assumption
that the difﬁculty of reaching area i from area j is inversely proportional to
the distance between them. The structure of our model for the number of commuters W“ from area
of work j to area of residence i is now completely speciﬁed as W” = k,.z,.E,1>,ar,,1 (4.9) CH. 4 MODELS OF LAND USE AND TRAVEL DEMAND 117 Since we have the values for all the variables E], P,, and d,,, the last step
before the model can be operational is the determination of the values of the
factors k, and I]. These, as was observed above, do not come from empirical
data but will be such that they ensure that the formulation (4.9) is consistent
with the speciﬁc constraints we have to observe. In this case, the only such
constraint is the fact that the number of workers who commute from a given
area of work j and all the areas of residence must be equal to the given number
of jobs in that area, i.e., E J. This set of constraints (one for each area of work
j) concerns the areas of work, i.e., production of interaction, and thus the
factors I, only. Since there are no comparable constraints for the attraction of interaction (i.e., conditions relating to the areas of residence i), the factors k,
can be dismissed, and the model (4.9) simpliﬁed as Wu = lejPida‘ (4.10) This onewaydistribution model is sometimes called the productioncon
strained version of the gravity model. Symmetrically, if the constraints con
cern the receiving areas i, but not the producing areas j (e.g., because of
restrictions on the numbers of housing units available in each area i) the
corresponding symmetrical model would be called attractionconstrained. (See Exercise 4.1.) In any case, for the ﬁrst area of work, the constraint can
be formulated as W11 W21 W31 Wu Wnl—i W11=E1
i=1
Similarly, for area of work 2,
W12 W22 W,2 ... W. " an = 1:21 Wu. z E2
and, in general for area of work j, Wu" W2} W3} Wu   ~ ‘“ n] z :21 Wt] 2" E1 (411) Replacing the Wil’s in the ﬁrst equation by their expression in (4.10), we get
llElPla’fl1 +11E1Pzdgf —i— .. . —l—11E1P,dfl1 + .. . —{11E1P,,a"1 — E1 "1— and by, factoring the constant terms 11E 1 in the sum, amends 1 Mg: 1 l Indy l l P.d;.‘)='l.E.iP.d,11=E1
i=1 All values are known in this equation but that of 1,. It can thus be used to
determine it by solving for the unknown. We thus get I l 1 n
E Pida‘
i=1 In the same fashion, for the general area of work j, we must have ’ 1,.E,.P,d;,1 +l,.E,.P2ci;j1 + . . . +1,E,P,d;j1 + . . . +1,E,P,,a’;jl = 11E, gPﬂg’ = E, 118 2 CH. 4 MODELS or LAND USE AND TRAVEL DEMAND and therefore, [I z (4.12) ___1___
2 Pida‘
i= 1 and of course for the last area, I. = 7—1——
2 P.d.;‘
1:! We have now evaluated the values for all the unknown factors I, through
the set of equations (4.12). (one for each value of j). The number of workers
who work in area j and live in area i can now be computed, sinCe there are no
remaining unknowns in the model (4.10). Indeed, by replacing I, in model
(4.10) by its general expression as given by (4.12), we have W, = "—1." E,P,d,j1 (4.13)
2 Pida‘
i=1
or, ﬁnally,
~1
W” = E}. TIM—21.. (4.14)
2 Pida‘
i=1 ”V To summarize the derivation above, if the spatial allocation of workers
from places of work to places of residence follows a gravity pattern, repre
sented by the model (4.10), and also if the requirements for consistency
represented by equations (4.11) are observed, the values of the scaling factors
I, must be equal to the expressions (4.12). It is worth noting that the foregoing
determination of the values of the scaling factors I j is equivalent to allocating
the workers to places of residence. Indeed, the lj’s, and consequently the
Wu’s as given by (4.14), are now completely speciﬁed in terms of the given
values oij, P,. and dij’s. The derivation of the expression for WU would have followed exactly
similar lines if the function F (d, 1) had assumed a form different from that we
have chosen above. In fact, the only difference in the ﬁnal model (4.14)
would be that dgl would be replaced by F(d,j). (See exercise 4.2.) Other
possible forms for the deﬁnition of the 01’s, for instance 0] = EESP?‘ =
A/EP; might have been used. Similarly, we might have adopted P} asha
measure of the attraction of zone 1', etc. The resulting model would again
have a form similar to (4.14). (See Exercise 4.2.) Although its derivation might seem involved, and its appearance for
bidding, formula (4.14) can be interpreted in a rather simple physrcal way.
We can look at the product Pidg,I as the “residential potential” that area i,
with an attraction P,, and at a distance of d”. from area j, has for the workers CH. 4 MODELS or LAND USE AND TRAVEL DEMAND 119 in this latter area. In this sense, the factor of E , between brackets in formula
(4.14), which is speciﬁc to the couple (i, j), and which we can therefore call a”, 124.31
; Pida‘ is the potential of area i for area j, relative to the sum of the potentials of all
areas of residence i for area of work j. This quantity thus represents the
relative share of area of residence i of the market in area of work j. Another
equivalent interpretation of this factor is the (conditional) probability that a
worker in area of work j will choose area of residence i, given the presence
(i.e., potentials) of all the other areas. Both interpretations are consistent
with the fact that the number of workers which work in j and live in i is equal
to the total number of workers in areaj, E f, times the factor a”. This is easily
seen if we rewrite (4.14) using the deﬁnition in (4.15): WU : Eja” (4.16) These coefﬁcients a” are then seen to be precisely the values that charac
terize the work~to~home commuting pattern in model (3.45), and for which
we now have an expression. Indeed, we can now compute the total number of
workers W, who will live in residential area 1', coming from all areas of work,
as the sum of the numbers of workers Who come from area of work 1, plus
the number of workers who come from area of work 2, etc., a” — (4.15) Wi=Wil+Wi2+"'+"’+Wij+'+"'+Wln=j=21 W”
or, replacing the W,j’s by their expression in (4.16): W! = ailEl “i“ (1sz2 + . . . + aUEj + . . . + ainE" = 1:231 aqu (4.17) for all areas of residence i = l, 2, . . .n. (Note that we are now summing
with respect to the areas of work j, the “producing” areas.) The system of equalities (4.17) (one for each area of residence 1') can also be written in
matrix form: W1 all 012 013 al] “in E1
W2 1121 “22 “23 “21 ‘12» E2
W3 031 432 033 “31 “an E3
2 . (4.18)
W1 an 012 £113 a” (11,. E,
' ?
_Wn_ _an1 girl 11713 an] arm _En 120 CH. 4 MODELS or LAND USE AND TRAVEL DEMAND In this form, (4.18) is exactly similar to (3.45), but with the funda—
mental difference that we now have a formal expression for the a, ,’s. This is
a crucial improvement, since we are now in aposition to evaluate the changes
on the distribution of workers from the places of work to the places of
residence that would result from changes in the distribution of residential
population P,, and from changes in the “distances” d”, since these factors
are explicitly incorporated in the model, through formula (4.15) for the
au’s. As an illustrative example of this particular form of the gravity model,
let us assume that the development of a threezone area has been planned. The
existing numbers of residents (in tens of thousands, for instance) are P1 = 3,
P2 = 2, and P3 = 2, respectively. It is also projected that the numbers of
jobs in the three zones will increase by E1 =“ 2, E2 2 3, and E2 = 4, respec
tively (in hundreds), over the next 5 years. Finally, the distances in miles
between the three zones are as given below in Table 4.1. TABLE 4.1 Previous modeling experiments have shown that the workplace—home
spatial relationship could be adequately described for the area by a general gravity model of the form (4.10). Given the preceding information, in which . areas, and by how much, will the expected increase in employment result in
residential population increases? If we assume that the existing gravity pattern, which has so far been stable,
will also prevail at the date of the employment changes, then the forecasting
of the location of the development is the problem we have solved above of
spatially distributing the expected workers to areas of residence according to
the gravity pattern (4.10). The solution of this problem is thus, as we have
seen above, represented by the two sets of formulas (4.15) and (4.16). We
must therefore first compute the values of the au’s. To that effect, let us ﬁrst
compute the values of the terms Pida1 (the potentials, in the language of the
theoretical exposition). Beginning with the ﬁrst area of residence, the potential
of area of residence 1 for the workers of area 1 is equal to: Pldf} = Pl/a’11
= %. Similarly, the potential of area of residence 1 for workers in area of
work 2 is equal to P1 /d12 = %. Proceeding in the same fashion, we obtain the
set of values for the potentials in matrix form, as described in Table 4.2. CH. 4 MODELS or LAND USE AND TRAVEL DEMAND 121 TABLE 4.2 The next step is to compute the values of the relative potentials or shares
:1”. According to formula (4.15), these are easily derived from the absolute
potentials computed above by dividing them by their total with respect to all
the residential areas 1'. These totals are represented by the column totals in
Table 4.2. Thus, the values of the au’s are obtained by dividing each entry in
the matrix by the corresponding column total, as given in Table 4.3. TABLE 4.3 We can now compute the number of workers WU from area j who will
choose area i to reside. According to formula (4.16), this is equal to the
product of the coefﬁcient a”. computed above by the number E, of workers
in area j. Thus, the numbers Wu are easily obtained from the'entries in the
last matrix in Table 4.3 by multiplying each entry by the value E, corre
sponding to its column number, as given in Table 4.4. TABLE 4.4
2 3 Total = W,
3(3—9.) 4%) 3.37
393—3) 4%) 2.74
3(% 4(T46) 2.89 9.00 122 CH. 4 MODELS OF LAND USE AND TRAVEL DEMAND Finally, the numbers of new residents W, who should be found in a given
area of residence i will be given by summing the numbers of workers WU
coming from all areas j. This is obtained by computing the horizontal row
totals in Table 4.4. These are given in the last column of the table. Thus, area I should be chosen by 337 workers, i.e., receive 337/900 2
0.374, or 37.4 %, of the development of the region; area 2 should attract 274
workers, i.e, 274/900, or 30.4 %, of the total development; and area 3 should
get 32.2 %, the remainder. Indeed, as a (partial) check on the computations,
one will note that the total number of resident workers, i.e, 9.00 (9 hundreds)
for all three areas, is of course equal to the total number of workers in all areas. That is,
2' W, = 2]] E, (4.19) Thus, the gravity model shufﬂes the workers from places of work to places of
residence, with of course no loss of workers in the process. The numbers of total new residents (the workers and their dependents)
that will result from this distribution can from there on be estimated using
the approach of Section 3.4, i.e., converting the ﬁgures above into residential
population through the use of the workforce ratios 1/)», as in formula (3.46). The potential uses of the foregoing model for analytical purposes are
straightforward. Suppose, for instance, that the characteristics of the trans
portation system may be changed in the future, perhaps in anticipation of the
new commuting pattern. Is this going to change the residential distribution
from the projection above, and if so, how? If the characteristics of the trans
portation system change so that the travel times between zones are affected,
for instance because of increased capacity of service, resulting in lower travel time, or perhaps a new pattern of one way streets, etc., we can consider that the distances d” between zones will also be changed, and consequently the
residential potentials of the areas will be changed, through formula (4.15).
We can then evaluate these changes by computing the new values of the aij’s,
and from them the new values of the Wij’s and of the W,’s, in the same man
ner as above. (See Exercise 4.3.) The gravity model can, of course, be applied to the spatial distribution of
activities other than residential. In particular, the shopping activity of resi—
dents of various areas in a given region can be allocated to these areas in the
same fashion as above. In this case, the interaction capability of the producing
areas (residential) could be represented by the total amount of expenditures
coming out of these areas, by the average incomes, by the propensities to
consume, etc. The measure of attraction of the receiving areas (commercial)
could be represented by their total volumes of sales, or their accessibility, or
by their ﬂoor space, range of services, etc. The derivation of the “hometo . shop” matrix C in the multiregional employment multiplier [formula (3.49)] CH. 4 MODELS or LAND USE AND TRAVEL DEMAND 123 can then be effected in the same fashion as that of the matrix seen above,
thus supplying the other input into the model. A caveat should be injected in this connection. If the output of a given
iteration of the model, i.e., the P,’s and the E,’s or other quantities directly
connected, are themselves used as inputs into the gravity model to derive the
matrices A and S as above, these matrices will not remain constant, as was
assumed in the model, but change at every iteration of the distributive pro
cedure represented in Figure 3.3. This is because the coefﬁcients in these
matrices (the aij’s and the c,,’s) are assumed to depend, through expressions
of the type (4.15) on the existing levels of population, or of service jobs, and
that these will then change at every iteration. Therefore, the compact expres
sions for the ﬁnal employment and residential distributions (3.51) through
(3.53) are not valid anymore. This slight diﬁiculty is, of course, removed if
one uses measures of production and attraction that depend on ﬁxed factors,
such as the area size or a set value derived from a planning goal. (See
Exercise 4.10.) The gravity model, being a model for the allocation of interactions
between areas, can then be used for distributing several urban and regional
activities to various spaces. Thus, it can be used also as a model for predicting
landuse patterns. By combining the preceding analysis for several activities
and taking into account the interrelationships, such as conﬂicts or competi
tion for the same spaces, a general spatial development model can be built. For instance, a new residential distribution might, in turn, provoke
changes in the transportation system characteristics, in the commercial dis
tribution, in the availability of housing, and therefore the rent structure, etc.
These and similar effects will change the relative desirabilities, or residential
potentials of the respective areas. If these changes are reﬂected in the model,
for instance through submodels of rent structure as a function of the housing
demand, i.e., of the residential desirability values, then we have an interactive
model that can represent these feedback effects. This approach, extended to
all the principal components of urban and regional systems, such as popula
tion, employment, land, housing, services, transportation, etc., is the basis
for the comprehensive or general models of urban and regional systems. 4.4 Second Application: The DoubleConstraint Form In some applications of the general gravity model to modeling situations of
the type above, it is necessary to consider constraints on both sets of regions
at once. For instance, a standard and very important such case comes from
the need to forecast transportation requirements on the basis of separate CH. 4 MODELS or LAND USE AND TRAVEL DEMAND 148 can be tried, where the determinination of the values of the parameters can be
effected through a linear regression. (See Chapter 7.) A typical form might be, for instance,
MM) = 1.5105N:°>5e15”‘ (4'54) Otherwise, the calibration of the model and its application follow the same
principles. 4.8’ Travel Generation Models In the remainder of this chapter, we shall examine models speciﬁc to travel
demand analysis. As we already noted, those are usually the corollary of
spatial analysis problems. In fact, although the models we have seen above
can be characterized as general activity distribution models, as a particular
case they can be applied to distribute the travel corresponding to the location
of a speciﬁc activity. For instance, a model of residential distribution from
areas of work is also a model of travel distribution from home to work.
Similarly, a model of the distribution of shopping activity from residential
zones to commercial zones can be used as a model of travel distribution from
home to shopping areas. These models can therefore also be used for trans—
portation planning purposes. However, these distribution models will require the prior estimation of
how much travel there is to distribute, i.e., the estimation of the input values
of the 01’s and of the Di’s in the models described above. Furthermore,
more information than just the distribution of travel will be needed. The
transportation planner might, for instance, be interested in which modes are
used to travel from zone j to zone i, or in what itineraries are followed, etc.
Although an adequate coverage of the range of methods speciﬁc to trans
portation systems planning and management would require much more space
than we can allocate to them in this chapter, we shall now survey brieﬂy
several methods used to answer such questions. Let us then begin with a description of the overall transportation planning
process. This is basically composed of four phases, as represented in Figure
4.3. The ﬁrst phase, travel generation, consists of estimating the level of
travel demand generated and attracted by various zones. This will in particular
provide the input 0] and D, to the spatial distribution models. This evalua
tion can be performed either on a dissagregated (or micro) basis, i.e., from
characteristics of the travelers or of the activity sites, or on an aggregated
(or macro) basis, from the characteristics of the zones. In the former case,
the method, sometimes called category analysis or crossclassiﬁcation analy
sis, essentially consists of identifying in the population homogeneous groups,
or “types” of travelers, for whom the travel needs (frequency, purpose, mode, CH. 4 MODELS or LAND USE AND TRAVEL DEMAND Population and
economic activity
. Spatial activity
structure 149 Travel generation
Travel distribution
Transportation I
system ‘1 Split
Travel assignment Figure 4.3 Anal tical s '
process. y equence of the transportation planning :iiguagievgtjble (i.e.ghvary little within groups) and well differentiated (i e
cover e range of travel needs) Char ' ‘ . n
. . . acteristics such as '
:tvznelisinpijlevel of income, number of employed members of the househdlili1
110185 wilt/h fwtsed, refulting in such groups as “small upperincome house”
cars, 1V1ng in highly residential “ ' "
families with no cars liVin ' ' ’ areas, 01‘ large IOW'lncomE’
. , g in highdenSIty areas ” etc If the '
. . , . rates of tr .
Elizfsiilirgegfaglaeefrlious group; in a typology of this kind can be accurately2
' s1 y proyecte , and if the future com 't'
tion in terms of the numbers of h p08} 1011 Of the P0131113“
ouseholds belongin to h
assessed, this approach will rovid g 63C type can be
e a ' '
future levals of travel demandp procedure for the estimation of the
T . . . . .
he resulting information (in a very Simpliﬁed case of 20 categories) might then be of the form re '
' . ‘ presented in Table 4.21. T
tion in a given zone will then be equal to he tOtal tFaVel PrOdUC' 20
T = k; nkrk (4.55) where nk is the number of hou '
seho ’ ‘
trip rate. ld in category k and rk IS the corresponding
em In‘prlactical' terms, the identiﬁcation of a “good” classiﬁcation is an
roplirica ,.heuristic problem of ﬁnding descriptors that will lead to as few
gb ps is I; necessary to account, for instance, for 95 % of the travel demand
serve . tatistical techniques such as factor analysis and cluster analysis CH. 4 MODELS or LAND USE AND TRAVEL DEMAND
150 TABLE 4.21 Trip Rates Per Household Per Day Numbers of Cars Owned 0 1 2 3 or more
1 1.1 2.5 4.2 _
2 1.7 4.8 6.6 —
”maﬁa” 3 2.5 6.2 8.4 11.1
Size 4 2.7 7.4 12.1 14.2
5+ 5.2 9.3 14.4 17.6 and other taxonomic procedures may be used to that effect. Also, dissagregate
methods such as adaptations of cohort survival projection methods may be
used to forecast the future makeup of the population in terms of the chosen
groups. Conversely, the levels of travel attraction can also be estimated on a
dissagregated basis, this time in relation to a class1ﬁcation of the attracting
activities in the recipient zones. For instance, these act1V1ties might be groupe
in classes of industrial, educational, commercial, governmental, serVice
oriented, and residential, each corresponding to a certain unit rat’e'of travel
production. The method again is to ﬁnd a typology of .such actiVities whiclh
is at once precise (i.e., corresponds to travel attractions rates With litt e
variance within the class) but also exhaustive (i.e., Wthh covers all causes
for travel attraction), and which is compatible with the analytical activ1ty
forecasting models, such as input/output, shift and share, and employment
multiplier, which might be used to estimate the future distribution of aCthltlesi
At the aggregate level, travel demand estimation methods usuallyconSisd
of applying the multivariate linear regression techniques to the detection an
modeling of a relationship between such zonal characteristics as averafgulej car
ownership, average income, average family Size, distance from centre;1 1::—
ness district (C.B.D), industrial mix, residential character, etc., to eit er e
rate of trip making from a given zone (generation) or to a given zone (attraﬁ
tion). (Regression methods will be described in more detail in Chaptefr .2
the travel generation is measured in terms of absolute levels and not 0 ra :15,
such variables as population levels, car registrations, etc., may be used for tf 6
production of travel, and level of employment, level of retail sales, etc., or the attraction of travel. .
It may be important in this case to include descriptors of the transporta— tion system. Indeed, it is reasonable to assume that, just as in the diisiagregati
approach where the level of car ownership might be a determinant o ravet,em
the aggregate approach the level of service of the transportation sys 111
(i.e., density of public transportation stops, capaCity of streets, etc.) w affect travel demand. . . . .
The typical model for travel generation estimation usmg this approach CH. 4 MODELS or LAND USE AND TRAVEL DEMAND 151 would then be Y1: (101' + #21 ”HA/i] (4.56) where X,j is the ith zonal characteristic for zone j, and the au’s are empir1
ically determined coefﬁcients which are produced by the regression analysis.
For instance, a typical model might be, in a speciﬁc situation, Y: 4.3 + 3.9X1 — 0.005X2 —— 0.13X3 — 0.012X4 (4.57) where Y is the total rate of trip generation per household in the area, X1 the
level of car ownership per household, X2 the number of residential dwelling
units per acre, X 3 the distance from the CED. in miles, X 4 the family income
in thousands of dollars. It is important to note that these models are in general not replicable
from geographical area to geographical area. Also, the deﬁnition of the
spatial zones will affect the value of the resulting coefﬁcients, just as the
deﬁnition of the categories in the cross—classiﬁcation analysis would'aﬁ‘ect
the resulting rates of travel generated. The results of this type of analysis are
highly local—speciﬁc. The foregoing travel generation analysis may be dissagregated by trip
purpose, such of travel to work, to shop, etc, to improve the level of precision.
Also, further dissagregations might be necessary, depending on the speciﬁc,
local situation, such as, for instance, a distinction between home~based trips
and workbased trips, downtown and suburbs, interurban and intraurban,
etc. The same general approach is applicable. Finally, account should also be
taken of special travel generators such as airports, governmental facilities,
major shopping centers, manufacturing plants, etc. In each case, typical
rates of trip production and attraction can be obtained. The estimation of passenger travel demand may also be supplemented by
the estimation of the freight movements between zones. In each case appro—
priate modiﬁcations of the approach may be required, such as using an
input/output model in connection with a regional goods movements estimate,
etc. In summary, the travel generation models require as inputs the outputs
of the models of urban and regional activity analysis that we have examined
in the preceding chapters. In that sense, they constitute the link between the
general urban system models and the subset of transportation systems models
we are presently considering. The next stage in the transportation planning process is the spatial dis
tribution of travel, given the levels generated and attracted as determined
above. The spatial distribution models that we have seen in Sections 4.2 to
4.7 can be utilized to that effect, and we shall therefore not elaborate further. 4.9 Models of Mode Choice The next and third phase in the transportation planning analytical process
is the determination of the modal split, i.e., the estimation of the fractions of
the travel distributed between the zones which will utilize the respective
transportation modes available. Although the theory can be extended to the case of several modes, we
shall present it in the context of a choice between two competing modes,
mainly the private car and public transportation. This case is in some respects
the most important, since one of the critical problems in urban transporta
tion planning is the diversion of travel from the private car to mass transit. The ﬁrst model of mode choice is the probit model. This model equates the ' probability that a given traveler chooses travel mode 1 instead of mode 2 to
the cumulative probability that a standard normal random variable (see
Section 1.4) takes on a value less than a given value, called the utility of mode
1. This quantity is evaluated as a function of the characteristics of the traveler
and of the relative advantages of mode 1 over mode 2. Formally, the prob
ability of the choice of mode 1 is given by were @(x) is the value of the cumulative standard normal probability dis
tribution function for the value x, and Gk is the value of the “utility” of mode 1
for travelers in population group k. The probability P2 of the choice of the
second mode is, of course, the complement of P1 ; that is, P2 = l — P1 = 1 " ‘1)(Gk) (4'59) The rationale for such a formulation derives from the application of principles
of mathematical biology to the responses of travelers to the utilities oﬂ'ered
by both modes. In spite of its unusual appearance, the model of mode choice (4.58) is
very simple to use, once the expression for the utility Gk has been obtained,
since the probability of choice of mode 1 is then computed as the probability
that a standard normal random variable takes a value at most equal to Gk.
This probability is easily read off Table A2 in Appendix A, as we shall
learn in Section 6.3. The expression of the utility is usually taken to be a linear function of the
relative attributes of the two transportation modes (their levels of service,
costs, travel times, comfort, etc.) and of the socioeconomic characteristics
of the traveler. It is then speciﬁed as Gk = a + Z bs(Xl — X?) + 2 CtYl‘ (4.60) 1‘2 CH. 4 MODELS or LAND USE AND TRAVEL DEMAND I53 17
glint: the X S s are. the values of the characteristics or attributes of mode 1
, e corresponding characteristics of mode 2 and the Y"’ ’
) t characteristics of travelers in group k. Other formulations
on the ratios X ,1 /X 3 instead of the differences (X §
competition between the two modes. The plot of the values of the probability of choice of mode 1 a ainst th
values of the utility can be represented as in Figure 4.4. When the vaglue of the
utility lS highly negative, mode 2 will be chosen with high probabilit and
conversely, when it is highly positive, mode 1 will be chosen. The breaklpoint
3e, when mode 1 has the same probability (50 %) of being chosen as mode by'a traveler in group k, is for a relative utility of zero. Figure 4 4 can b
consrdered to be equivalent to an empirical diversion curve which . lot the
observed fraction of the total volume of trafﬁc between given are}; S 6
mode 1, as a function of the relative characteristics of the two modes S, s are personal
might be based
—— X ,2), as measures of the using P1 1.0 0.5 0 Figure 4.4 Probit model. in practical terms, probit models of mode choice are calibrated on th
baSis of empirical observations of the variables above using the techni ue :‘
maximum likelihood estimation of the coefﬁcients do b and c in 3460C;
(This technique will be described in detail in Section ’6.7“when we consider methods of model calibration.) W :3 at; example of the application of a probit model, let us assume that
e ave orecasted that the level of travel between areas j and i is 1500 trips p:1:e:la3}; lit the present time, the only mode available is the private car. It
to t 1 , inu es (on the average) and costs about $2.25 (at $0.15 per mile) rave by car between the two zones. A public transportation system is
planned, which would enable residents to travel between the two zones f0; 154 CH. 4 MODELS or LAND USE AND TRAVEL DEMAND $0.85 and would take 55 minutes. Let us assume further that on the basis of
research in areas with similar situations in terms of transportation alterna
tives and population, it has been determined that the probability P, of a traveler choosing the private car can be expressed by the probit model where
the utility G is represented by G = 0.8 + 0.11(T2 — T1) + 0.1(C2 ~ C1) + 0011+ 0.00_5A where T2 and T1 are the travel times in hours on the given route with
transit and the car, respectively, and C2 and C1 the corresponding costs in
dollars. I is the income of the traveler, in thousands of dollars and A is his or
her age, in years. Given the above, what percentage of the segment of the
traveling population of age 35 with an income of $25,000 would be diverted to
transit use? Let us compute the value of the probit corresponding to the
foregoing comparison of transportation service characteristics (where mode 1
is the car) and to the given traveler. From the above formula, the probit is
equal to G = 0.80 + 0.11 (55 6‘0 35) + O.1(O.85 — 2.25) + 001(25) + 0.005(35)
= 1.12 L The probability of choosing the car is equal to the entry in Table A2
corresponding to a value of 1.12. This is equal to 08686, or about 87 %. At
this level of service and fare, public transportation would then attract only
13 % of the ridership in that particular segment of the population. The model can be used for the usual analytical purposes by varying the
level of the variables. For instance, one could ask how cheap transit would
have to become (or faster, or both) before it becomes competitive with the
car in the sense of having a higher probability of being used by that particular
type of traveler. Similar questions can also be asked. (See Exercise 4.7.) The other model of mode choice that we shall describe is the logit model.
This model, which can be derived from the application of principles of mathematical psychology, expresses the probability of the choice of mode 1 as . Pk 3G; 1
1 — eGi‘ + eG§ _ l —— 66'5"Gi ' (4.61) where as in the probit model above, G’f is a function of the characteristics of
mode 1', as evaluated by a traveler of type k. The expression of this function
can take on various forms, linear or otherwise. It is interesting to note that
model (461) closely resembles the logistic model of population projection
(2.13). Thus, the plot of the probability P, as a function of the values of the
variable (0’; — G’f) has the shape represented in Figure 4.5. Although the
general shape of this plot is similar to that of the probit model in Figure
4.4, the speciﬁc values of the probabilities are different. CH. 4 MODELS or LAND USE AND TRAVEL DEMAND 155 P1 (0’; ~ G’f) Figure 4.5 Logit model. Also, the same transformation that enabled us to calibrate the logistic model can be used to evaluate the parameters of the function G’f. Indeed
from (4.61) we have ’ Gk Gk
1—P’f=1—— 6“ e“ G’c Gk Gk G):
e 1 + e 2 e 1 + e 2
and therefore : _ er—G’z‘ so that Pk
10g (1‘3) = 10g (171) =1°g 66“”? = (G’f — G5) (4.62) 2
. Usually, the function G!‘, the utility function of mode 1' for traveler k,
Will be taken to be linear, i.e., of the form G11: : a1 .__ Z blsXs + 2 GUY? and (4.63) G’; = a2 —— Z b2,X, + 2 chf where as in (4.60) the X,’s are the attributes, or levels of service common to
both the ﬁrst mode and to the second mode, and the Yf’s the characteristics of
the traveler 1n group k. In this (linear) case, (4.62) can be written Pk
log (“P729 : €11 — 02 ‘i‘ ; (b1; ‘— [723)XS —— ; (Ci: — 523}? (464) The value of the dilTerences between the coefﬁcients (a1 —— a2), (b1, — b2,), and (c1, — 02:): which are the parameters in model (4.61), can then be esti mated empirically from the results of a multivariate linear regression of 6 CH. 4 MODELS or LAND USE AND TRAVEL DEMAND
IS the quantity log (P’f/P’z‘) (as observed from the mode choices of travelers of
type k) against the values of the X’s and the Y’s. . . As an illustration, let us assume that the calibration of a logit model of
binary choice between the private car (mode 1) and commuter train (mode 2)
has resulted in a utility diﬁerence between the two modes given by G2 — G, = —0.7 + 0.3(c1 — c,) + 0.2(t1 — t,) (4.65) where c, is the cost of travel by mode i in dollars and .t, is the travel time for
that mode in hours. [Note that this is a simpliﬁed ver5ion of (4.64), where the
characteristics Y, of the traveler do not intervene. Consequently, this can be
considered to be applicable to all traveler types k, With therefore no index k.]
Given this model and the fact that the difference in travel time on a given
route and the car is equal to t2 —— t1 = 0.4 hour, and that the cost of travel
by car is equal to $1.50, what should be the level of the train fare so that it
could attract onethird of the commuters? This question can be rephrased by;
asking what should the value of C2 be so that the probability P2 of ch01ce 0 mode 2 is equal to a}, or P1 = % From (4.61), we have 1
P1 = W
and replacing (G2 —— G) from (4.65):
1 P1 — 1 + e—[0.7+0.3(ca—c1)+0.2(tz—h)] (Note the importance of the order of the indices.) With the given values, 1 ._ 2 P1 "‘ 1 + e[0.7+0.3(ci—l.5)+0.2(0.4)] 3 01‘
1+ e~(o.33+o.3c2) = g = 1.5
Thus,
e—(o.33+o.3c2) z 05
Ol'
"(0‘3“0'3“) =  0.33 + 0.3c2) : log 0.5 = —0.693
log [e ] ( so that 0.3c2 = 0.693 — 0.330 = 0.363 and c2 = 1.21 or $1.21 in our units. Thus the disadvantage of the train with respect to travel time could in
principle be compensated by a lower fare. Similar questions can be answered in the. same fashion. (See Exercise 4.8.) 4.10 Travel Assignment Methods Let us now go on to the fourth and last phase in the transportation planning
analytical sequence of Figure 4.3. We now have the levels of the travel ﬂows
T” which are expected between each origin j and each destination 1‘, by mode,
possibly by trip purpose, type of traveler, etc. We must now in this ﬁnal stage
assign these travel demands to the various links in the transportation network.
Indeed, there may in general be several possible itineraries, or routes to
reach area i from area j, each with its own characteristics (capacity, distance
or travel time, etc). These individual characteristics of the links between zones
j and i may already have intervened in the determination of the separation,
or friction factor F (d, j), in the travel distribution phase (possibly through an
averaging of their values over all links connecting areas i from area j). How
ever, the problem of travel assignment is to determine which fraction of the
total demand Tu each of these links will be made to carry. There are several possible approaches to that problem. However, they
all generally rely on the basic principle that the transportation network: _ loading should be such that it minimizes travel eﬁ‘ort (i.e,, a composite of time, cost, inconvenience, etc.) between zones, both for an individual traveler
choosing between routes for a given trip and for the total effort of the
entire population of travelers. Thus, the ﬁrst step in the methods of travel
assignment is the determination of the leasteffort route between any given
zones, given the efforts associated with each link. This can be accomplished
through the application of the shortestpath method, as explained below. Let
us ﬁrst, however, deﬁne a few basic terms in network theory. A network will, in general, be a set of nodes (or points) in space, con
nected by links (or arcs). A network can be directed, as in the case of oneway
streets, or they can be undirected. Similarly, a network can be capacitated,
meaning that there will be limits on the amounts each link can carry, or it
can be uncapacitated, or mixed. There are a number of ways to represent a network. The graph of a network will simply be its diagrammatic representation, such’as in Figure 4.6,
which represents a directed, capacitated network. Arrows along the links Figure 4.6 The graph of a network. ...
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