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Generalized power model for trip distribution
Transpn. Res:B Pnnad in Chat
Vol. 218. No. I. pp. 59-67. Bntaln.
GENERALIZED
0191-2615/87 0 1987 Pergamon
1987
POWER
MODEL
S3.00+ .@I Journals Ltd.
FOR TRIP DISTRIBUTION
B. ASHTAKALA Concordia
University,
Montreal,
(Received 23 July 1985; in revisedform
Canada 21 October 1985)
Abstract-In this paper a trip distribution model is proposed, based on the concepts of conventional trip distribution, linear regression analysis and power transformation on the independent variable. The proposed model takes into consideration the relationship between trips from an origin to a destination (dependent variable) and attraction of the destination (independent variable). Regression analysis is used to determine the parameters of the relationship which is optimized by varying the power parameter. Regression parameters and the optimum power parameter are used in formulating the trip distribution model. The functional form of the proposed model has a generalized application. Distinctive models can be derived from the generalized model by using appropriate independent variables to suit the given data. Therefore, this model is termed as Generalized Power model for trip distribution. In this paper two typical applications of the model are given as examples. Both models have been developed using cordon origindestination (O-D) survey data of Red Deer in Alberta. The paper includes discussion of results and conclusions.
INTRODUCTION
One of the key elements in transportation planning process is the estimation of the number of trips to be made between a pair of zones in a city or a wider area such as a province. The procedure for such an estimation is generally referred to as trip distribution because trips produced at an origin are considered as being distributed among various destinations. The objective of this study is to develop a trip distribution model based on the concepts of conventional trip distribution, linear regression analysis and power transformations on the independent variable. Such a model will assume a generalized functional form which would be the basis for generating specific models to suit the given data. These specific models will have to be validated by statistical measures and tests. In this paper a generalized trip distribution model is presented, which satisfies the requirements in the objective. Two typical applications of this model are developed using cordon origindestination survey of Red Deer in the province of Alberta, Canada. The paper includes discussion of results and conclusions.
FORMULATION
OF
MODEL
Let us suppose the study area is divided into several zones in which trips are interchanged. For trip distribution, one zone is identified as the origin and all zones including the origin are considered as destinations. Trips produced at the origin are distributed to various destinations, in proportion to the magnitude of attraction of each destination. The attraction of a destination is expressed in terms of one or more variables such as population, employment, etc. and its location with respect to the origin. The location of the destination is measured by travel time (distance, cost, etc.) and by its position in the sequential ascending order of travel time. This concept is similar to that of Intervening Opportunities, Heanue and Pyers (1966); however, it is used mainly to consider each destination uniquely following a recognizable order of occurrence. In order to consider destinations in sequential ascending order, the attractions of destinations are added cumulatively in the same ascending order. Similarly, trips to destinations are added cumulatively in the same ascending order. Cumulative summation as well as proportioning of quantities associated with destinations are important components in the proposed model. Let the total number of zones in the study area be n. Origin zones are identified by Oi, i = 1 to n and one origin zone at a time is considered for trip distribution. Destination zones are ordered and designated by J&, k = l,n in the ascending order of travel time from the given origin. Prefix i indicates that the sequential order is unique with respect to origin Oi. Also, prefix i is used for other terms which are associated with the sequential ascending order of destinations. 59
60
ASHTAKALA
B
above concept is illustrated with one origin, i = 1 in Fig. 1. In this figure, origin is 0, and the ordered destinations are ,D, to ,D,. ,D, is the destination under consideration. The travel time to ,D6 defines travel time contour. The destinations which are enclosed by the travel time contour (,D, to ,D5) are closer to origin than 1D6. The other terms in the figure will be referred to later in the text. For any destination iDk, the following terms are defined: The
iZk
=
number of attractions of destination k, to destination k.
fik = number of trips from origin i
Let IO, be the destination under consideration as shown in Fig. 1. The suffix j is used instead of to indicate the specific destination upto which cumulative addition of quantities is performed. However, it is to be noted that j (1 to n) indicates the same sequential ascending order as k (1 to n). The proportion of cumulative attractions of destinations upto j in the total attraction of all destinations is defined as k
izj
($,
izk)*
izk/$,
=
(1)
Similarly, the proportion of cumulative trips to destinations upto j in the total trips to all destinations is defined as
Tij =
($,
tik/$,
(2)
f.k).
,Z,_ , and T+ , are defined in a similar manner by excluding destination j from the cumulative totals. Trips from origin i to destination j is ti,, which is expressed as follows: 1-I
tij
=
i
ti,
-
By using (2) for T;j and a similar equation for tij =
2
c
t,k.
(3)
k=l
k=l
f,k
Tjj-,, (3) can be written as (Tij -
T;,-,).
(4)
k=l
The total number of trips from the origin to all destinations in the study area is equal to the trips produced, G,, at origin i. Therefore,
k=l
Trips
Fig. 1. Origin and ordered destinations.
trs
tripdistribution NOW, tij can be expressed by an identity in terms of Gi using (4) and (5): Generalized power model for
tij =
Gi(Tij - T,_,).
61
(6)
This relationship will be used later in developing the trip distribution model. Power transformations
A family of power transformations, Draper and Smith (1981) on a variable Z is stated as follows:
ZA’ =
ZA - 1 -
for A # 0
1nzX
for A = 0’
1
(7)
This family of transformations is continuous on a single parameter A, if it is specified as above. Transformations can be done on ZAinstead of (ZA - 1)/X. In such cases, only a scale difference and an origin shift are involved and the basic nature of subsequent analysis is unaffected. In this paper, transformation for the case of A # 0 always refers to ZA. The relationship between the dependent variable Y and the independent variable Z is of the form:
po+
Y =
pz?),
(8)
where POand B are parameters in the equation. Functional form The linear relationship in (8) and power transformations are used in the formulation of the
power model. The dependent variable Tij and the independent variable ,Zjare related as follows: Tij = (&)i + pi iq).
(9)
An equation similar to (9) can be obtained with T,_, and ,Z:?,. (PO),and pi are parameters in the regression equation. Suffix i is used to indicate that these parameters refer to variables Tij and ;Zj associated with origin i. The following equation is obtained from (9) and a similar one for T,,- ,: Tij -
Tij-1 = pi[iq”’ -
;q!l].
(10)
Substitute for (Tij - Tij- ,) in (6) using the relationship in (10) to get: tij = GiPi[iZJA’ -
,q:“-‘l].
(11)
The value of iZy’ at Tij = 0 is determined by setting Ti, = 0 and j = 0 in (9): i&l
=
-(PO)ilPi~
(12)
This value is to be used for $2, when j = 1 in (11). Equation (11) is the functional form of Generalized Power model. By performing transformations on JjA’ and ,q1’, , specific models can be derived. 1. Transformation: A # 0 Substitute for iZy’ and iZ:l,, in (11) using transformation for the case of A # 0: tij = GiPi( iZj’ - iZt_ I).
(13)
62
B.
ASHTAKALA
Different models can be obtained by giving specific values for X. 2. Transformation: h = 0 Substitute for @) and iZj?‘, in (11) using transformation for the case of A = 0: tij = G$i(ln
iZj -
(14)
In iZ,_,).
This is a special case of transformation which gives a logarithmic functional form for the model. In formulating the generalized model (1 I), travel time is used implicitly by considering destinations in an ascending order of travel time. However, this is not a necessary requirement. Travel time can be specified explicitly as shown later in one of the applications of this model. The independent variable iZ, in the generalized model (11) can be replaced by more than one variable. For example, several factors like population, employment and explicit specification of travel time or cost can be included in the list of independent variables. When more than one variable is used, transformations on individual variables can also be done. For this reason, the model is termed as Generalized Power model. It is beyond the scope of this paper to cover the entire range of applications of this model. However, two applications are presented in this paper as examples.
APPLICATIONS
Red Deer is a small city in Alberta, Canada, with a population of 3 1,580. Edmonton and Calgary are two major cities, each with a population of about half a million. These two cities are situated approximately 145 km from Red Deer in opposite directions. A major highway between Edmonton and Calgary passes through Red Deer. The rest of the cities and towns (240) account for less than a million population. Figure 2 shows the location of major cities and highway network in the province of Alberta. In this study, cities and towns which are within 240 minutes from Red Deer are considered. The data from a cordon origin-destination survey done at Red Deer is used for developing the two power models. The original survey data is large and it is therefore grouped to a smaller size. Power model 1
Red Deer is the origin and the cities, towns and rural zones are destinations in the cordon O-D survey. Passenger vehicle trips (all trip purposes) from Red Deer to external destinations (outbound direction) are taken into consideration. The location of each destination is measured by travel time from Red Deer. The destinations are considered in sequential ascending order of travel time from the origin. Since passenger vehicle trips of all purposes are used, population is taken as a measure of attraction at a destination. As only one origin (Red Deer) is taken into consideration in this model, suffix i is replaced by 1. Thus the origin is indicated by 0, and the ordered destinations are given by ,Dk, k = 1 to n. Let ,pk = population of destination k, = trips from origin 1 to destination tlk
k.
Figure 1 shows origin O,, ordered destinations ,D, to ,D6, and population ,p5 and trips t15of destination ,Ds. Let ,Dj be the destination under consideration. ,P, and ,P,- , are defined in terms of $?$ as follows:
(15) The above terms I/&, IPj and lP,_ 1correspond to irk, iZ, and iZ,_, for i = 1, respectively, have been defined earlier in the formulation of Generalized Power model. The above fractional proportions are converted into percentages: ,p, = ,P, x 100 and ,p,_,
= ,P,_,
x 100.
which
(16)
Generalized
power model for trip distribution
63
TRANSPORTATION
PRIMARY Scale:
HIGHWAYS
1 cm = 48 km
Fig. 2. Major cities and primary highways
Similarly,
T,j [eqn (2)] and T,j_ l are converted
in Alberta, Canada.
into percentages
for use in this model:
T, = T,j X 100 and TIj_, = Tt,_l X 100.
(17)
In Table 1, Tlj and ,p, are given in columns 1 and 2. A plot of Tv and ,p, is shown in Fig. 3. The shape of the curve indicates that the relationship is nonlinear. The relationship between T,j and ,pj can be expressed as in (9) by substituting , ?f, for ,Z,: Tlj
=
(PO)1 +
Pl(lpj)'.
(18)
The parameters (Do), and PI in the above equation are determined by assuming a value for A and performing regression analysis on the above variables. In the same manner, regression analysis is carried out assuming different values for A in the range of - 1.O and 1 .O. It is to be noted that at A = 0, (,~j) becomes In (,pj). The coefficient of determination (R’), correlation coefficient
64
B. ASHTAKALA Table
1. Red Deer survey data for Power model 1
Cumulative trips (a) T,,
Cumulative population (a) IPI
11.69 23.22 49.32 56.05 59.61 60.47 67.06 73.63 80.69 83.68 90.52 90.88 98.05 98.35 98.54 98.90 99.24 99.47 99.81 100.00
0.08 0.12 0.58 0.79 0.90 0.92 1.34 1.86 3.34 6.05 42.92 45.14 86.93 88.13 89.34 91.72 92.68 93.90 98.81 100.00
,p,p= 1.874 1.699 1.146 1.061 1.027 1.021 0.929 0.856 0.740 0.638 0.391 0.386 0.327 0.326 0.325 0.323 0.322 0.321 0.317 0.316
Note: Total outbound trips (G,) = 5550 per day; Total population data is grouped to 20 observations for regression analysis.
of destinations
= 1,228,307;
Survey
(r) and standard deviation (S.D) as obtained from various runs of regression analysis are shown in Table 2. It can be seen from this table that R2 has a maximum value of 0.99 at h = - 0.25. A plot of A and R2 values is shown in Fig. 4. The standard deviation at X = -0.25 is 2.52 which is the minimum value. Thus the optimum value for A is - 0.25. The values of (PO>, and 6, from regression analysis at optimum X are as follows: (&,), = 116.53 Regression
equation
and
PI = -55.42.
(19)
is stated as follows: Tv = 116.53
-
55.42(,p,)-“.2s.
(20) IPj and ,Pj_ , for ,Z, and
The functional form of Power model 1 is obtained by substituting ,Z,_ , in (11) and using a transformation h # 0:
GPILP: -
ll, =
I
1
t
20
40
60
Cumulative Population Fig. 3. Cumulative
(21)
,P:_,l.
,
60
(%)
plot of Red Deer O-D survey data.
100
Generalized
power model for
tripdistribution
65
Table 2. A and R* values for Power model 1 A -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00
R2
S.D
r
0.73 0.82 0.94 0.99 0.92 0.80 0.71 0.66 0.62
14.10 11.37 6.59 2.52 7.70 12.09 14.57 15.93 16.61
-0.85 -0.90 - 0.97 - 0.99 0.96 0.89 0.84 0.81 0.79
The above equation is expressed in terms of ,Fj and , i”/_,, using ( 16):
s
t.. 'I =
[,F$ - &I.
(22)
Substituting the value for p, from (19) and the value for A = - 0.25 in the above equation: flj = -0.55G,[(,Fj)-0.*’
- (,FJ_,)-0.*5].
(23)
The initial value for (,pj)-o.25 is determined by setting T,j = 0 andj = 0, in (20): (,p,>-“.” = 2.1.
(24)
The value for (,~o)-o~25is to be used in (23) whenj = 1. Thus the Power model for Red Deer is given by (23). The above model estimates the trips from Red Deer to external destinations. Trips in the opposite direction can be determined by assuming one inbound trip for every outbound trip. Therefore the inbound trips will be equal to the outbound trips t,, between each O-D pair. The total trip generation at the origin is 2G,. Power model 2
This model takes into consideration population and travel time for defining the attraction of destination. An additional power parameter 1 is used for performing transformations on the travel time variable. Since one origin is taken into consideration in this model, suffix i is replaced by 1 in all the terms. Destinations are considered in sequential ascending order of travel time
I
-1 0
I
-0.75
1
-050
I
-0.25
I
0
0.25
I
I
0.75
050
Power X Fig. 4. A versus R2 plot for Power model
I.
J 1.0
66
B. ASHTAKALA
from the origin and they are denoted by ,Dr, k = 1 to n. The concept and the notations are shown in Fig. 1. Let ,pk = population of destination k, d,t = travel time from origin 1 to destination k. Attraction of destination is defined by ,uk in terms of ,pa, dlk and 5; as ,uk
=
,pkd(;'.
(25)
Let ,D, be under consideration as shown in Fig. 1. Cumulative proportion ,V, is defined as
(26) Cumulative proportion of trips T, is defined as (2). ,V, and Tlk are expressed in the form of percentages rEj and Tik as in Power model 1 for convenience. Red Deer cordon O-D survey data is used again to develop the model. The values of rEj are computed from this data assuming a specific value for x in the range of 4.0 and -4.0. Regression analysis is done using these ,Uj values and the corresponding TX values. In this manner several runs of regression analysis are done using different values of A. The values of RZ and S.D obtained from each run are tabulated as shown in Table 3. It can be seen from this table, that highest value of R2 (0.98) and least value of S.D (3.68) are obtained at i = -3. The values of regression parameters ( &Jr and p, , at i = - 3, are - 22.18and 1.22, respectively. The functional form of Power model 2 is obtained by substituting ,V, and ,V,_ , for ,Z, and ,Z,_ ,, respectively, in (11) and using a transformation X = 1, tlj
Substitute for Itij and ,e,_,
in teInE
=
%
[,vj
-
,Uj_,].
(27)
of pk and dlk consider transformation x f 0:
-- GP,
[l, -
(28)
I()()
The difference of terms in the numerator in (28) can be replaced by (,p, d:,) since it is the amount of attraction of destination ,D,, as indicated in Fig. 1. The denominators in (28) can be written with a suffix j instead of k. Thus equation (28) can be simplified as
(2%
Table 3. h and R* values for Power model 2 i -4.0 -3.0 - 2.0 - 1.0 0.0 1.0 2.0 3.0 4.0
R2
0.84 0.98 0.79 0.66 0.61 0.59 0.55 0.49 0.40
S.D
r
10.69 3.68 12.36 15.68 16.90 17.38 18.13 19.32 20.96
0.91 0.99 0.89 0.81 0.78 0.11 0.74 0.70 0.63
Generalized
power model for trip distribution
Power model 2 with Red Deer data is obtained by substituting for A in (29):
tlj =
1.23
61
the value of 1.23 for p, and - 3
cl{tPjdij’/$ tPjd6’).
(30)
The functional form of Power model 2 is similar to a Production constrained gravity model if p, = 1. dG3 is the impedence function associated with the above type of gravity model (Wilson, 1970). Thus Power model 2 has the characteristics of a Production constrained gravity model. Thus two applications of Generalized Power model are developed using Red Deer cordon O-D data. The models are significantly different from each other. They have a high degree of predictability and are equally acceptable.
DISCUSSION
AND
CONCLUSION
The parameters in the Generalized Power model are determined by regression analysis and the accuracy of the model’s performance is verified by statistical measures associated with regression analysis. For the sake of simplicity, R2 is used as a statistical measure for determining the optimum value of X; however, other techniques such as maximization of likelihood function can be used. The type of power transformations used in the Generalized model has been utilized in the past by Gaudry and Wills (1978) for developing travel demand models. The transformations are continuous on power A and so the model that best suits the given data can be obtained by optimizing h. Power models 1 and 2 confirm this fact. The advantage of this approach is that it eliminates the need for assuming a pre-determined power function for a model. A trip table which includes trip interchanges from every town as origin to all the towns as destinations and vice versa, can be developed for an entire study area using individual trip distribution models like Power model 1. A methodology similar to the one given in Ashtakala (1983) can be adopted for developing such a trip table. In conclusion, a model which satisfies the requirements stated in the objective, is proposed. It has a generalized functional form for application to trip distribution problems. Distinctive models can be derived from the generalized model by using appropriate independent variables to suit the given data. Regression analysis for determination of parameters, statistical measures to evaluate the model’s performance and application of power transformations have strengthened the model for trip distribution. It is therefore appropriately termed as Generalized Power model for trip distribution. Acknowledgments-The author acknowledges the Grant in aid of Research awarded by the Natural Sciences and Engineering Research Council of Canada. A. S. Narasimha Murthy, doctoral student in the Dept. of Civil Engineering, Concordia University Montreal, has assisted in regression analysis of the data. The cordon O-D survey data used in this paper is taken from unpublished reports of Alberta Transportation Department, Edmonton, Alberta, Canada.
REFERENCES Ashtakala B. (1983) An Empirical model for provincial scale trip distribution. RTAC Conference, Edmonton, Alberta, Canada. Draper N. R. and Smith H. (1981) Applied Regression Ano/.vsis, 2nd ed. John Wiley & Sons, Inc., New York. Gaudry 1. I. and Wills M. J. (1978) Estimating the functional form of travel demand models. Tramp. Res. 12, 257289. Heanue K. E. and Pyers C. E. (1966) A comparative evaluation of trip distribution procedures, H.R. Record 114. HRB, Washington, D.C. Wilson A. G. (1970) Advances and problems in distribution modelling. Tramp. Res. 4, l-18.
Vol. 218. No. I. pp. 59-67. Bntaln.
GENERALIZED
0191-2615/87 0 1987 Pergamon
1987
POWER
MODEL
S3.00+ .@I Journals Ltd.
FOR TRIP DISTRIBUTION
B. ASHTAKALA Concordia
University,
Montreal,
(Received 23 July 1985; in revisedform
Canada 21 October 1985)
Abstract-In this paper a trip distribution model is proposed, based on the concepts of conventional trip distribution, linear regression analysis and power transformation on the independent variable. The proposed model takes into consideration the relationship between trips from an origin to a destination (dependent variable) and attraction of the destination (independent variable). Regression analysis is used to determine the parameters of the relationship which is optimized by varying the power parameter. Regression parameters and the optimum power parameter are used in formulating the trip distribution model. The functional form of the proposed model has a generalized application. Distinctive models can be derived from the generalized model by using appropriate independent variables to suit the given data. Therefore, this model is termed as Generalized Power model for trip distribution. In this paper two typical applications of the model are given as examples. Both models have been developed using cordon origindestination (O-D) survey data of Red Deer in Alberta. The paper includes discussion of results and conclusions.
INTRODUCTION
One of the key elements in transportation planning process is the estimation of the number of trips to be made between a pair of zones in a city or a wider area such as a province. The procedure for such an estimation is generally referred to as trip distribution because trips produced at an origin are considered as being distributed among various destinations. The objective of this study is to develop a trip distribution model based on the concepts of conventional trip distribution, linear regression analysis and power transformations on the independent variable. Such a model will assume a generalized functional form which would be the basis for generating specific models to suit the given data. These specific models will have to be validated by statistical measures and tests. In this paper a generalized trip distribution model is presented, which satisfies the requirements in the objective. Two typical applications of this model are developed using cordon origindestination survey of Red Deer in the province of Alberta, Canada. The paper includes discussion of results and conclusions.
FORMULATION
OF
MODEL
Let us suppose the study area is divided into several zones in which trips are interchanged. For trip distribution, one zone is identified as the origin and all zones including the origin are considered as destinations. Trips produced at the origin are distributed to various destinations, in proportion to the magnitude of attraction of each destination. The attraction of a destination is expressed in terms of one or more variables such as population, employment, etc. and its location with respect to the origin. The location of the destination is measured by travel time (distance, cost, etc.) and by its position in the sequential ascending order of travel time. This concept is similar to that of Intervening Opportunities, Heanue and Pyers (1966); however, it is used mainly to consider each destination uniquely following a recognizable order of occurrence. In order to consider destinations in sequential ascending order, the attractions of destinations are added cumulatively in the same ascending order. Similarly, trips to destinations are added cumulatively in the same ascending order. Cumulative summation as well as proportioning of quantities associated with destinations are important components in the proposed model. Let the total number of zones in the study area be n. Origin zones are identified by Oi, i = 1 to n and one origin zone at a time is considered for trip distribution. Destination zones are ordered and designated by J&, k = l,n in the ascending order of travel time from the given origin. Prefix i indicates that the sequential order is unique with respect to origin Oi. Also, prefix i is used for other terms which are associated with the sequential ascending order of destinations. 59
60
ASHTAKALA
B
above concept is illustrated with one origin, i = 1 in Fig. 1. In this figure, origin is 0, and the ordered destinations are ,D, to ,D,. ,D, is the destination under consideration. The travel time to ,D6 defines travel time contour. The destinations which are enclosed by the travel time contour (,D, to ,D5) are closer to origin than 1D6. The other terms in the figure will be referred to later in the text. For any destination iDk, the following terms are defined: The
iZk
=
number of attractions of destination k, to destination k.
fik = number of trips from origin i
Let IO, be the destination under consideration as shown in Fig. 1. The suffix j is used instead of to indicate the specific destination upto which cumulative addition of quantities is performed. However, it is to be noted that j (1 to n) indicates the same sequential ascending order as k (1 to n). The proportion of cumulative attractions of destinations upto j in the total attraction of all destinations is defined as k
izj
($,
izk)*
izk/$,
=
(1)
Similarly, the proportion of cumulative trips to destinations upto j in the total trips to all destinations is defined as
Tij =
($,
tik/$,
(2)
f.k).
,Z,_ , and T+ , are defined in a similar manner by excluding destination j from the cumulative totals. Trips from origin i to destination j is ti,, which is expressed as follows: 1-I
tij
=
i
ti,
-
By using (2) for T;j and a similar equation for tij =
2
c
t,k.
(3)
k=l
k=l
f,k
Tjj-,, (3) can be written as (Tij -
T;,-,).
(4)
k=l
The total number of trips from the origin to all destinations in the study area is equal to the trips produced, G,, at origin i. Therefore,
k=l
Trips
Fig. 1. Origin and ordered destinations.
trs
tripdistribution NOW, tij can be expressed by an identity in terms of Gi using (4) and (5): Generalized power model for
tij =
Gi(Tij - T,_,).
61
(6)
This relationship will be used later in developing the trip distribution model. Power transformations
A family of power transformations, Draper and Smith (1981) on a variable Z is stated as follows:
ZA’ =
ZA - 1 -
for A # 0
1nzX
for A = 0’
1
(7)
This family of transformations is continuous on a single parameter A, if it is specified as above. Transformations can be done on ZAinstead of (ZA - 1)/X. In such cases, only a scale difference and an origin shift are involved and the basic nature of subsequent analysis is unaffected. In this paper, transformation for the case of A # 0 always refers to ZA. The relationship between the dependent variable Y and the independent variable Z is of the form:
po+
Y =
pz?),
(8)
where POand B are parameters in the equation. Functional form The linear relationship in (8) and power transformations are used in the formulation of the
power model. The dependent variable Tij and the independent variable ,Zjare related as follows: Tij = (&)i + pi iq).
(9)
An equation similar to (9) can be obtained with T,_, and ,Z:?,. (PO),and pi are parameters in the regression equation. Suffix i is used to indicate that these parameters refer to variables Tij and ;Zj associated with origin i. The following equation is obtained from (9) and a similar one for T,,- ,: Tij -
Tij-1 = pi[iq”’ -
;q!l].
(10)
Substitute for (Tij - Tij- ,) in (6) using the relationship in (10) to get: tij = GiPi[iZJA’ -
,q:“-‘l].
(11)
The value of iZy’ at Tij = 0 is determined by setting Ti, = 0 and j = 0 in (9): i&l
=
-(PO)ilPi~
(12)
This value is to be used for $2, when j = 1 in (11). Equation (11) is the functional form of Generalized Power model. By performing transformations on JjA’ and ,q1’, , specific models can be derived. 1. Transformation: A # 0 Substitute for iZy’ and iZ:l,, in (11) using transformation for the case of A # 0: tij = GiPi( iZj’ - iZt_ I).
(13)
62
B.
ASHTAKALA
Different models can be obtained by giving specific values for X. 2. Transformation: h = 0 Substitute for @) and iZj?‘, in (11) using transformation for the case of A = 0: tij = G$i(ln
iZj -
(14)
In iZ,_,).
This is a special case of transformation which gives a logarithmic functional form for the model. In formulating the generalized model (1 I), travel time is used implicitly by considering destinations in an ascending order of travel time. However, this is not a necessary requirement. Travel time can be specified explicitly as shown later in one of the applications of this model. The independent variable iZ, in the generalized model (11) can be replaced by more than one variable. For example, several factors like population, employment and explicit specification of travel time or cost can be included in the list of independent variables. When more than one variable is used, transformations on individual variables can also be done. For this reason, the model is termed as Generalized Power model. It is beyond the scope of this paper to cover the entire range of applications of this model. However, two applications are presented in this paper as examples.
APPLICATIONS
Red Deer is a small city in Alberta, Canada, with a population of 3 1,580. Edmonton and Calgary are two major cities, each with a population of about half a million. These two cities are situated approximately 145 km from Red Deer in opposite directions. A major highway between Edmonton and Calgary passes through Red Deer. The rest of the cities and towns (240) account for less than a million population. Figure 2 shows the location of major cities and highway network in the province of Alberta. In this study, cities and towns which are within 240 minutes from Red Deer are considered. The data from a cordon origin-destination survey done at Red Deer is used for developing the two power models. The original survey data is large and it is therefore grouped to a smaller size. Power model 1
Red Deer is the origin and the cities, towns and rural zones are destinations in the cordon O-D survey. Passenger vehicle trips (all trip purposes) from Red Deer to external destinations (outbound direction) are taken into consideration. The location of each destination is measured by travel time from Red Deer. The destinations are considered in sequential ascending order of travel time from the origin. Since passenger vehicle trips of all purposes are used, population is taken as a measure of attraction at a destination. As only one origin (Red Deer) is taken into consideration in this model, suffix i is replaced by 1. Thus the origin is indicated by 0, and the ordered destinations are given by ,Dk, k = 1 to n. Let ,pk = population of destination k, = trips from origin 1 to destination tlk
k.
Figure 1 shows origin O,, ordered destinations ,D, to ,D6, and population ,p5 and trips t15of destination ,Ds. Let ,Dj be the destination under consideration. ,P, and ,P,- , are defined in terms of $?$ as follows:
(15) The above terms I/&, IPj and lP,_ 1correspond to irk, iZ, and iZ,_, for i = 1, respectively, have been defined earlier in the formulation of Generalized Power model. The above fractional proportions are converted into percentages: ,p, = ,P, x 100 and ,p,_,
= ,P,_,
x 100.
which
(16)
Generalized
power model for trip distribution
63
TRANSPORTATION
PRIMARY Scale:
HIGHWAYS
1 cm = 48 km
Fig. 2. Major cities and primary highways
Similarly,
T,j [eqn (2)] and T,j_ l are converted
in Alberta, Canada.
into percentages
for use in this model:
T, = T,j X 100 and TIj_, = Tt,_l X 100.
(17)
In Table 1, Tlj and ,p, are given in columns 1 and 2. A plot of Tv and ,p, is shown in Fig. 3. The shape of the curve indicates that the relationship is nonlinear. The relationship between T,j and ,pj can be expressed as in (9) by substituting , ?f, for ,Z,: Tlj
=
(PO)1 +
Pl(lpj)'.
(18)
The parameters (Do), and PI in the above equation are determined by assuming a value for A and performing regression analysis on the above variables. In the same manner, regression analysis is carried out assuming different values for A in the range of - 1.O and 1 .O. It is to be noted that at A = 0, (,~j) becomes In (,pj). The coefficient of determination (R’), correlation coefficient
64
B. ASHTAKALA Table
1. Red Deer survey data for Power model 1
Cumulative trips (a) T,,
Cumulative population (a) IPI
11.69 23.22 49.32 56.05 59.61 60.47 67.06 73.63 80.69 83.68 90.52 90.88 98.05 98.35 98.54 98.90 99.24 99.47 99.81 100.00
0.08 0.12 0.58 0.79 0.90 0.92 1.34 1.86 3.34 6.05 42.92 45.14 86.93 88.13 89.34 91.72 92.68 93.90 98.81 100.00
,p,p= 1.874 1.699 1.146 1.061 1.027 1.021 0.929 0.856 0.740 0.638 0.391 0.386 0.327 0.326 0.325 0.323 0.322 0.321 0.317 0.316
Note: Total outbound trips (G,) = 5550 per day; Total population data is grouped to 20 observations for regression analysis.
of destinations
= 1,228,307;
Survey
(r) and standard deviation (S.D) as obtained from various runs of regression analysis are shown in Table 2. It can be seen from this table that R2 has a maximum value of 0.99 at h = - 0.25. A plot of A and R2 values is shown in Fig. 4. The standard deviation at X = -0.25 is 2.52 which is the minimum value. Thus the optimum value for A is - 0.25. The values of (PO>, and 6, from regression analysis at optimum X are as follows: (&,), = 116.53 Regression
equation
and
PI = -55.42.
(19)
is stated as follows: Tv = 116.53
-
55.42(,p,)-“.2s.
(20) IPj and ,Pj_ , for ,Z, and
The functional form of Power model 1 is obtained by substituting ,Z,_ , in (11) and using a transformation h # 0:
GPILP: -
ll, =
I
1
t
20
40
60
Cumulative Population Fig. 3. Cumulative
(21)
,P:_,l.
,
60
(%)
plot of Red Deer O-D survey data.
100
Generalized
power model for
tripdistribution
65
Table 2. A and R* values for Power model 1 A -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00
R2
S.D
r
0.73 0.82 0.94 0.99 0.92 0.80 0.71 0.66 0.62
14.10 11.37 6.59 2.52 7.70 12.09 14.57 15.93 16.61
-0.85 -0.90 - 0.97 - 0.99 0.96 0.89 0.84 0.81 0.79
The above equation is expressed in terms of ,Fj and , i”/_,, using ( 16):
s
t.. 'I =
[,F$ - &I.
(22)
Substituting the value for p, from (19) and the value for A = - 0.25 in the above equation: flj = -0.55G,[(,Fj)-0.*’
- (,FJ_,)-0.*5].
(23)
The initial value for (,pj)-o.25 is determined by setting T,j = 0 andj = 0, in (20): (,p,>-“.” = 2.1.
(24)
The value for (,~o)-o~25is to be used in (23) whenj = 1. Thus the Power model for Red Deer is given by (23). The above model estimates the trips from Red Deer to external destinations. Trips in the opposite direction can be determined by assuming one inbound trip for every outbound trip. Therefore the inbound trips will be equal to the outbound trips t,, between each O-D pair. The total trip generation at the origin is 2G,. Power model 2
This model takes into consideration population and travel time for defining the attraction of destination. An additional power parameter 1 is used for performing transformations on the travel time variable. Since one origin is taken into consideration in this model, suffix i is replaced by 1 in all the terms. Destinations are considered in sequential ascending order of travel time
I
-1 0
I
-0.75
1
-050
I
-0.25
I
0
0.25
I
I
0.75
050
Power X Fig. 4. A versus R2 plot for Power model
I.
J 1.0
66
B. ASHTAKALA
from the origin and they are denoted by ,Dr, k = 1 to n. The concept and the notations are shown in Fig. 1. Let ,pk = population of destination k, d,t = travel time from origin 1 to destination k. Attraction of destination is defined by ,uk in terms of ,pa, dlk and 5; as ,uk
=
,pkd(;'.
(25)
Let ,D, be under consideration as shown in Fig. 1. Cumulative proportion ,V, is defined as
(26) Cumulative proportion of trips T, is defined as (2). ,V, and Tlk are expressed in the form of percentages rEj and Tik as in Power model 1 for convenience. Red Deer cordon O-D survey data is used again to develop the model. The values of rEj are computed from this data assuming a specific value for x in the range of 4.0 and -4.0. Regression analysis is done using these ,Uj values and the corresponding TX values. In this manner several runs of regression analysis are done using different values of A. The values of RZ and S.D obtained from each run are tabulated as shown in Table 3. It can be seen from this table, that highest value of R2 (0.98) and least value of S.D (3.68) are obtained at i = -3. The values of regression parameters ( &Jr and p, , at i = - 3, are - 22.18and 1.22, respectively. The functional form of Power model 2 is obtained by substituting ,V, and ,V,_ , for ,Z, and ,Z,_ ,, respectively, in (11) and using a transformation X = 1, tlj
Substitute for Itij and ,e,_,
in teInE
=
%
[,vj
-
,Uj_,].
(27)
of pk and dlk consider transformation x f 0:
-- GP,
[l, -
(28)
I()()
The difference of terms in the numerator in (28) can be replaced by (,p, d:,) since it is the amount of attraction of destination ,D,, as indicated in Fig. 1. The denominators in (28) can be written with a suffix j instead of k. Thus equation (28) can be simplified as
(2%
Table 3. h and R* values for Power model 2 i -4.0 -3.0 - 2.0 - 1.0 0.0 1.0 2.0 3.0 4.0
R2
0.84 0.98 0.79 0.66 0.61 0.59 0.55 0.49 0.40
S.D
r
10.69 3.68 12.36 15.68 16.90 17.38 18.13 19.32 20.96
0.91 0.99 0.89 0.81 0.78 0.11 0.74 0.70 0.63
Generalized
power model for trip distribution
Power model 2 with Red Deer data is obtained by substituting for A in (29):
tlj =
1.23
61
the value of 1.23 for p, and - 3
cl{tPjdij’/$ tPjd6’).
(30)
The functional form of Power model 2 is similar to a Production constrained gravity model if p, = 1. dG3 is the impedence function associated with the above type of gravity model (Wilson, 1970). Thus Power model 2 has the characteristics of a Production constrained gravity model. Thus two applications of Generalized Power model are developed using Red Deer cordon O-D data. The models are significantly different from each other. They have a high degree of predictability and are equally acceptable.
DISCUSSION
AND
CONCLUSION
The parameters in the Generalized Power model are determined by regression analysis and the accuracy of the model’s performance is verified by statistical measures associated with regression analysis. For the sake of simplicity, R2 is used as a statistical measure for determining the optimum value of X; however, other techniques such as maximization of likelihood function can be used. The type of power transformations used in the Generalized model has been utilized in the past by Gaudry and Wills (1978) for developing travel demand models. The transformations are continuous on power A and so the model that best suits the given data can be obtained by optimizing h. Power models 1 and 2 confirm this fact. The advantage of this approach is that it eliminates the need for assuming a pre-determined power function for a model. A trip table which includes trip interchanges from every town as origin to all the towns as destinations and vice versa, can be developed for an entire study area using individual trip distribution models like Power model 1. A methodology similar to the one given in Ashtakala (1983) can be adopted for developing such a trip table. In conclusion, a model which satisfies the requirements stated in the objective, is proposed. It has a generalized functional form for application to trip distribution problems. Distinctive models can be derived from the generalized model by using appropriate independent variables to suit the given data. Regression analysis for determination of parameters, statistical measures to evaluate the model’s performance and application of power transformations have strengthened the model for trip distribution. It is therefore appropriately termed as Generalized Power model for trip distribution. Acknowledgments-The author acknowledges the Grant in aid of Research awarded by the Natural Sciences and Engineering Research Council of Canada. A. S. Narasimha Murthy, doctoral student in the Dept. of Civil Engineering, Concordia University Montreal, has assisted in regression analysis of the data. The cordon O-D survey data used in this paper is taken from unpublished reports of Alberta Transportation Department, Edmonton, Alberta, Canada.
REFERENCES Ashtakala B. (1983) An Empirical model for provincial scale trip distribution. RTAC Conference, Edmonton, Alberta, Canada. Draper N. R. and Smith H. (1981) Applied Regression Ano/.vsis, 2nd ed. John Wiley & Sons, Inc., New York. Gaudry 1. I. and Wills M. J. (1978) Estimating the functional form of travel demand models. Tramp. Res. 12, 257289. Heanue K. E. and Pyers C. E. (1966) A comparative evaluation of trip distribution procedures, H.R. Record 114. HRB, Washington, D.C. Wilson A. G. (1970) Advances and problems in distribution modelling. Tramp. Res. 4, l-18.