Design and construction of a subregional land use model

Socio-&on.Plm. Sci.Vol. 5, pp. 97-124(1971).Pergamon Press.Printi in GreatBritain

DESIGN AND CONSTRUCTION OF A SUBREGIONAL LAND USE MODEL MICHAEL BATTY

Urban Systems Research Unit, Department of Geography, Reading, England

University of Reading, Whiteknights,

(Received 30 July 1970) This paper describes the process of designing and constructing land use models for subregional planning. Several conceptual ideas concerned with the purpose and form of such models are first introduced, and these set the context for a description of a land use model designed for the Nottinghamshire-Derbyshire subregion, an area in central England. As this model has its origins in Lowry’s Model of Metropolis, Lowry’s model, and various developments of this model in North America and Britain are briefly discussed. A modification of this model is then proposed, and a method for calibrating such a model is outlined. The design of the zoning and information systems necessary to the model, and the various issues involved in defining the model’s variables show how the model has been adapted to the Nottinghamshire-Derbyshire subregion. The results of calibrating the model are then presented and evaluation of these results leads to proposals for redesign and further research into the model’s structure.

INTRODUCTION IN THE last

decade, approaches to urban and regional planning have been transformed by the development of systematic methodologies for understanding and forecasting change in spatial systems. One of the most significant of these developments has been the construction of symbolic models purporting to represent the dynamics of such systems. Following in the wake of developments in transportation planning in North America, several models have been designed to simulate the interactions between different land uses in spatial systems [19], and quite recently, this experience has been repeated in Britain [13]. Apart from the many problems posed by the integration of land use modelling in the planning process, there is a whole spectrum of problems associated with the detailed design and construction of such models. Although the basic structures on which different types of land use model have been built, are well documented, there seems to be very little explanation of the more detailed, and often more important decisions involved in model design. Therefore, it is the intention of this paper to describe and evaluate, both on a theoretical and practical level, various design considerations necessary to the development of a model of an urbanised subregion. In this paper the emphasis will first be upon the theoretical development of the model, and in particular on the elements of land use simulated by the model and on the simulation method itself. Important from both a theoretical and practical standpoint is the methodology used to calibrate and validate the model ; various calibration methods are reviewed, and in the light of their advantages and disadvantages, the method used to fit this model is outlined. 97

MICHAEL UAI~Y

98

The application of the model to the Nottinghamshire-Dcrbyshirc subrcginn, an urbanised area of central England, is then described. The zoning system used by the model, and the information requirements necessary to operate the model set the context for a description of decisions involved in constructing, calibrating and testing the performance of the model. This leads to an evaluation of the model’s performance, and to suggestions for redesign of the model and further research. THE

PROCESS

OF

MODEL

DESICiN

In general, the process of designing land use models is similar to the design processes involved in engineering and architecture. Sound principles of design rest upon a problemsolving process in which the design problem is first defined, and information relevant to the problem collected. Once the problem has been thoroughly understood, alternative solutions to the problem are hypothesised and then evaluated against the goals or criteria established in the early stages of the design. From this evaluation one solution is selected [2]. This design process which is referred to as a “problem-solving module” [28], may exist in various states of refinement for different design problems, and the process may be repeated many times before a final solution is reached. In land use modelling, this design process exists in a much suppressed form, although urban researchers have stressed the importance of defining the purpose for which the model is to be designed. Zwick [36] defines three purposes which involve the construction of land use models. The first is the design of a model for spatial forecasting, the second for research into spatial structure and the third for education. Any land use model will reflect the relative importance of each of these three purposes. For example, models for spatial forecasting will probably be closely oriented to available data and to the components of the spatial system which are critical to the projection. On the other hand, models which are required for research or educational programs may rely less upon available data and more upon the sensitivity of the model’s components to parameter change. Clearly, these three purposes are not mutually exclusive of each other and each may exist in different degrees in any model. The model considered in this paper has been designed primarily for spatial forecasting. A second facet of problem definition arises once the purpose of the model has been considered, and this concerns the form of the planning process within which the model is embedded. A distinction must first be made here between optimising and non-optimising models. An optimising model is a model which will search through a set of alternative spatial forecasts, and on the basis of some predetermined criteria will select an optimal solution by progressively eliminating sub-optimal solutions. The residential location model suggested in [20], based on a linear programming algorithm, is a good example of an optimising model. Partly because of the intrinsic difficulties in isolating the criteria necessary to define the optima in such models, and partly because of the difhculties in formulating such models in a mathematically tractable way, most of the effort in building land use models has been directed into non-optimising models which are designed to test the consequences of certain planning actions. The non-optimising model to be outlined here, has been designed to test alternative planning strategies in a subregion, and the model has been used in this way at several stages of the planning process to generate information on which such strategies can be evaluated [29]. The third facet of model design which is partly dependent upon the decisions to the two issues outlined above, involves the spatial system to be modelled. It is important to

Design and Construction of a Subregional Land Use Model

99

recognise that any land use model is the outcome of a scientific process in which observations and the search for order in those observations lead to a hypothesis about the organisation of the spatial system. The model is in fact a translation of the hypothesis into a form which can be manipulated and tested against the observations in the real world in the effort to establish a close correspondence between the hypothesis and the real world. The underlying process of land use model design and construction is focused upon this goal of hypothesis verification. In general terms, there is also a distinction between models which attempt to simulate the changing structure of land uses over a given period of time, referred to as dynamic models, and models which simulate the structure of land uses at only one point in time, referred to as static or equilibrium models. The choice between these two forms of model depends upon several factors and one of the main issues is related to available information. Time-series data is essential for the construction of a dynamic model. If only cross-sectional data is available, then it is likely that an equilibrium model is more suitable. There is however another important consideration, which involves the detailed structure of the model. Whether or not a dynamic or equilibrium model is chosen for spatial forecasting will depend upon the internal consistency of the model. In other words, the choice of a model should rest upon the model’s ability to produce a good simulation of the spatial system. It is not very often that several alternative models can be designed, calibrated, tested and the best of these chosen; therefore, the choice will probably be influenced by the success of past modelling efforts in different areas. One further issue may colour the choice between dynamic and equilibrium models-the need to use the model to forecast small changes in the spatial system or the need to use the model to test the impact of large changes in the structure of the system. If it is deemed necessary to forecast marginal changes in spatial structure, a dynamic model may be more suitable whereas equilibrium models are more suitable for conditional predictions and impact analysis [27]. Another important factor involves the types of land use to be simulated, and this will depend upon the definition of the planning problem to be solved. At one extreme, the model may be based upon only one land use, other land uses being provided exogenously to the model whereas at the other extreme, the model may attempt to simulate all the major land uses comprising the spatial system. For example, the San Francisco Housing Market model [35] simulates only the residential sector, whereas the Empiric model of the Greater Boston region simulates the industrial, commercial and residential sectors [21]. Furthermore, the degree of disaggregation of each land use or activity will depend upon the purpose to which the model will be put. The treatment of interactions between the land uses may be implicit or explicit, and will depend upon the mathematical techniques used to relate variables and upon the relative importance placed upon the study of spatial interaction. There are many issues involved here and the whole process of model design may be replicated several times before a consistent and useful model is designed. It is difficult to know how Well the model will perform before it is actually constructed and tested, and the major guidelines in model design may well depend upon the success of other modelling ventures. A basic constraint on model design has not so far been mentioned; this concerns the financial resources and time available. Such factors are certain to act as major constraints on design and will probably control the decision as to whether an existing land use model is adopted and developed, or whether an entirely different, and untried design is pursued.

100

MICHAEL BA’ITY

The model described in this paper, was designed under fairly stringent conditions. The lack of time-series data and the need to design a model suitable for testing the large scale impacts of changes in spatial structure lead to the adoption of an equilibrium model. Furthermore, the development strategies required for the subregion were to be conceived in terms of broad land uses-residential, commercial and industrial land uses tied together by the transport network. There was a further need to consider explicitly the interactions between these land uses and this meant that a model had to be designed in which trips between the various activities could be simulated. Because of the financial resource available, it was clearly impossible to evolve a completely new design for a land use model, and at the inception of the project, it seemed wise to adopt and modify a model which had already been designed and tested elsewhere. A model which eminently met the conditions posed by this project and which was eventually adopted in this study is the model pioneered by Lowry [26] for the Pittsburgh urban region. As Lowry’s model forms the basis of the model outlined here, it is proposed to treat his model in some detail describing the theory upon which the model relies and the solution method used to solve the model’s equations. THE

LOWRY

MODEL

The Lowry model organises the urban space-economy into activities on the one hand, and land uses on the other. The activities which the model defines are population, service employment and basic (manufacturing and primary) employment, and these activities correspond to the residential, service and industrial land uses. The model’s major operations are carried out at the level of activities and these activities are translated into appropriate land uses by means of land use/activity ratios. The division of employment into service and basic sectors is required because the model uses the traditional economic base method to generate service employment and population from basic employment [I]. Basic employment is defined as that employment which works in industries whose products are exported outside the region, whereas the products of service employment are consumed within the region. It is assumed that the location of basic industry is independent of the location of residential areas and service centres, and although this assumption appears to be weak, it is taken as a point of departure in the Lowry model. Besides deriving population and service employment, this model also allocates these activities to zones of the urban region. Population is allocated in proportion to the population potential of each zone and service employment in proportion to the employment or market potential of each zone. Constraints on the amount of land use accommodated in each zone are also built into the model. The model ensures that population located in any zone does not violate a maximum density constraint which is fixed on every zone. In the service sector, a minimum size constraint is placed on each category of service employment, and the model does not allow locations of service employment to build up which are below these thresholds. Service employment is disaggregated into three types: neighbourhood, local or district and metropolitan, each reflecting a different scale of activity in the urban region. Having located the various activities in accordance with the predetermined constraints, the model also tests the predicted distribution of population against the distribution used to compute potentials to find out whether the two distributions are consistent. Lowry argues that it is necessary to secure consistency between these distributions because the model uses distributions of population and employment to calculate the potentials which

Design and Construction

of a Subregional Land Use Model

101

indirectly affect the predicted location of these same variables. Consistency is secured by feeding back into the model predicted population and employment and reiterating the whole allocation procedure until the distributions input to the model are consistent with the outputs. To firm up the structure of this model and to emphasise the solution method adopted by Lowry, the model will be presented as a formal system of equations. This interpretation of the model adopts a different notation from that used in the original formulation. The particular notation and method of presentation used, is necessary so that the modification of the model proposed in a later section of this paper is consistent. The model divides up the spatial system into four sets of zones which differ in regard to the constraints imposed. Z, is the set of zones in which there are no locational constraints, Z, the set in which there are only residential constraints, Z, the set in which there are only service constraints and Z, the set in which there are both residential and service constraints. During the operation of the model, zones may be continually shifting between these four sets if the locational constraints are violated. The diagram below shows an abstracted division of the spatial system into these four sets.

The total set of zones is called Z and from the definitions above

z=J

z,

s=l

(1)

It is also obvious that the intersection of these four sets is equal to the empty set + 4 cp=

A

z,

(2)

S=l

Each variable will be defined as it appears in the text; the index m refers to the inner iterations of the model necessary to ensure that the locational constraints are satisfied whereas the index n refers to the outer iterations necessary to ensure a stable co-distribution of input and output variables. At the start of the model’s operations, m = 1 and n = 1, and it is important to note that Z = ZI, and that Z,, Z,, Z, E Z,. Also total employment in zone i, El(l), is equal to the basic employment Mr, and the total land for service uses, L{ (l), is equal to zero.

MICHAEL BATTY

102

Total population earlier.

is first calculated using the economic base relationship referred to

P(m) = a CM,(l -aC$)-l, I

ieZ

(3)

k

P is the total population in the region, a is the inverse activity rate and /I” is the ratio of service employment k to total population, called here the population-serving ratio. Total land use available for housing Ljb (n) is now calculated. Lib(n) = Lj-[LjY+Ljb+Lj’(n)],

jEZ

(4)

Lj is the total amount of land in each zone and the superscripts u and b on Lj denote

unusable land and land used for basic industry respectively. zones in proportion to the population potential.

Population is allocated to

Pj(mG)= P(m)[CEi(n)f’(cij)/CCEi(n)f’(cij)],i~Z,j~Z, 2’3 1

i

j

(5)

allocated to zone j, and f ’ (caj) is a function of generalised travel cost. At this point, population in each zone must be tested against the density constraint. If

Pj(m,n) is the population

Pj(m,n)28jLjb(n),

jEZ,,

Z,

(6)

then Pj(%n)

EZ2,

(7)

Z4

61 is a density coefficient which converts Ljb (n) into population. In the constrained sets 2, and Z4, population is set equal to the maximum population allowed. The index m is increased to mt 1. Pj(m+ 1, n) = sjLr(n),

jEZ,,

Z,

(8)

The population to be reallocated is found by taking the constrained population in Z, and Z, from total population. P(m+ 1) = P(m)-CPj(m+ j

l,n), jeZ,,

Z,

(9)

P(m+ 1) is substituted into equation (5) and equations (5)-(9) are reiterated until Pj(m+l,n)lGjLjb(tl),

jEZ

(10)

When equation (10) is satisfied. the allocation of population is in accord with the residential constraints. m is set equal to 1, and service employment in each class k (Sk) is calculated. Sk= fCPj(m,n), j

jeZ

(11)

Design and Construction

of a Subregional Land Use Model

103

Service employment is now allocated to each zone i (St’) in proportion to the employment or market potential S,k(m,n) = Sk[~~k~j(~~~~l)fZ(Cjj)+~k~~(~)/~~~kPj(~~~~l)fZ(C~j)+~kE~(~)]~i i j i

_iEz (12)

On the first iteration of the full model (n = I), P&n,n--1) is equal to the observed population Pj. gk and 4” are empirically determined coefficients showing the relative importance of population and employment in the index of market potential. f(crj) is a function of generalised travel cost. The quantity of service employment located in i must now be tested against the minimum size constraint min Sk. If
S,ktm,n>=

(13)

&Z

then S,k(m,n) G,

-G

(14)

and 0, kZ,,

“(m

Stk(m+ l,n),kZ,,

+ ” n, =

Z, is substituted

i

Z, S,k(m,n)Sk/CS”(m,n),

kZ,,

Z,

(15)

i

into equation (13) and equations (13) and (14) are

reiterated until S,“(m + 1,n) 2 min Sk, kZ

(16)

At this point in the model, all activities have been allocated. The index n is increased to 1, and service employment is converted to land use using the ratios ek.

nf

L,‘(n + 1) = vS,“(m,n),

kZ

(17)

ieZ

(18)

kZ

(19)

kZ

(20)

If L:(n+l)>L,-(LY+L,“), then L,‘(n+ 1) = Li-(LF+Lt),

Total employment is now calculated E,(n + 1) = Mi + ~S~(t?lgn),

The predicted distribution of population Pj(m,n) must be tested against the distribution which is used to compute the market potentials. The aim is to generate a consistent distribution of input and output variables, and if the predicted distribution is

Pj(m,n-1)

104

MICHAELBATTY

within a certain limit Z, of the input distribution, the two distributions are judged to be consistent. In other words, Pj(m,n) = Pj(m,n-l)+l,,

jEZ

(21)

If equation (20) is not satisfied, then L,‘(n+ 1) and E,(n+ 1) are substituted into equations (4) and (5) respectively, and equations (3)-(20) are reiterated until equation (21) is satisfied. At the beginning of each outer iteration of the model, the inner iteration m is set equal to 1, and Z1 = Z as before. A diagrammatic interpretation of this sequence of operations is presented in Fig. 1. The model outlined above is based on a widely accepted theory of spatial structure. The economic base mechanism which is used to derive service employment and population from basic employment, and the allocation of activities according to potentials reflect important determinants of spatial structure. Although it is obvious that the model is based on a highly simplified interpretation of spatial structure, the model is so organised as to permit further disaggregation of its variables. This flexibility means that the detailed structure of the model can be closely matched against available data, and this is frequently a critical factor in spatial forecasting. Moreover, a family of land use models has developed from this model; several of these have taken the development of this model further, and it is worthwhile reviewing some of these as they provide additional insights into the theory and application of the model.

A FAMILY

OF LAND

USE

MODELS

The first development of Lowry’s model was made by the CONSAD Research Corporation as part of the Pittsburgh Community Renewal Program [8]. This modelcalled the Time Oriented Metropolitan Model-adopted the basic structure of the model outlined above but also permitted a disaggregation of population into different socioeconomic groups. The authors felt that in disaggregating the model in this way, the explanatory power of the model would be increased. Furthermore, this model was restructured to explicitly account for the time element in forecasting. In using the Lowry model for forecasting, there is an assumption that all activities respond to changes in potential. This is obviously not the case because a certain proportion of activity will be stable during the forecast interval. The model was therefore revised to account for these stabilities. The model has also been used as an educational device in the METRO gaming simulation exercise at the University of Michigan [9]. The model is used to show participants in the game, the consequences of their decisions in terms of the spatial distribution of population and employment. Crecine [lo] has suggested a further revision of the Time Oriented Metropolitan Model. The structure of the revised model is basically the same as the original model although the allocation mechanisms have been made more realistic. Population and employment potentials are fairly crude measures of locational attraction, and Crecine has proposed that these be replaced by linear equations relating site rent, transport cost and other site amenities such as the availability of schools. Although this model has not been made operational as yet, it seems that the linear equations would first be fitted by regression analysis outside the model.

Design and Construction

of a Subregional

Land Use Model

BASIC EMPLOVMENI, IMVEL WE MAIRIX,ACIIVIIV S POPULA1lOll SERVING RAllOS, PARAHLIER VALUES, RESIOENIIAL L SLRVICE AIIIAClORS.tONSIRAINIS, INVtNlORV OF LAN0

SUBSlllUlE Pn~olc1fo POPULAIION L EMPtOVMfNl IN10 AUOCAIIOW lUllCllONS

KS

\

KS

FIO. 1. Generalised flow chart of the Lowry model.

105

106

MICHAELBATTY

Another major development of the Lowry model is due to Garin [17] who suggested that the economic base mechanism and the allocation rules used by the model could be integrated more closely. Garin replaced the potential models used by Lowry with simple gravity models and showed that a final distribution of population and employment could be derived analytically using a matrix formulation akin to input-output analysis. Following Garin’s development, the Projective Land Use Model designed by Goldner [18] for the Bay Area Transportation Study Commission, also uses simple gravity models for allocating activities, rather than potential models. Instead of disaggregating the population or service sector, Goldner disaggregates the parameters for each of nine counties in the Bay Area. Goldner also builds into his model zone specific activity rates and populationserving ratios to account for differences in population and employment structure. This necessitates the introduction of additional sets of scaling factors to adjust zonal populations and employment so that these activities sum to their respective regional totals. In Britain, two separate developments of the Lowry model have been undertaken. The first of these has been adapted to the town scale by Echenique [I41 who has integrated Garin’s version of the model with the constraints procedure used by Lowry. In this model, the repercussions of basic employment in deriving service employment and population are traced out explicitly, and at each iteration of the model, the predicted distributions of activity are tested against their constraints. The other development of Lowry’s model, including the model to be discussed here, is based upon Garin’s version. The models built for the Bedfordshire subregion [l 11 and the Central Lancashire subregion [3] are straightforward applications of Garin’s model, although considerable emphasis has been placed upon the form of the gravity models used to allocate activities. One problem in using Garin’s model arises because constraints on population and service employment are not built into the system. As Garin’s model makes use of simple gravity models to allocate activities, it is logical to suppose that constraints on the location of activity can be handled in a way consistent with these models. Wilson [33] has devised a procedure to handle constraints and this method has been built into the model to be outlined here. Another innovation has been made in the model at present being developed for the Cheshire County Planning Authority [6]. The authors of this model argue that at the subregional scale, the spatial structure of activities can be divided into mutually disjoint areas which correspond to a particular level in the central place hierarchy. If interactions between these areas are weak, then there are computational advantages in organising the model to simulate the fine pattern of interaction within such areas, but only the coarse pattern of interaction between the areas. There is an added advantage in that the activity and population-serving ratios can be specific to each area without affecting the total activity in the subregion. Apart from the slight differences in the structure of each of these developments, there are two distinct differences between the applications in North America and in Britain. In Britain, the emphasis has been on developing land use models for systems of interdependent cities whereas in North America, many of the models have been developed for metropolitan areas. This difference in scale means that the average zone size used in these models is considerably larger in Britain. Furthermore, the British versions have used fewer zones; this is mainly due to the size of the computational facilities available for the development of these models. These differences in structure and in the scale of application can be better appreciated in the outline of the Nottinghamshire-Derbyshire model which follows.

Designand Construction of a Subregional Land Use Model A MODIFICATION

OF THE LOWRY

107

MODEL

This model is largely based upon Garin’s theoretical development mentioned above. The full model will be outlined using a presentation similar to that used earlier. An index m denotes the value of a variable on the inner iteration of the model. This inner iteration is used to derive the increments of service employment and population from basic employment. This is different from the Lowry model in which the sum of these increments was derived analytically using the economic base relation in equation (3). The outer iteration, denoted by the index n is used to make the model satisfy the locational constraints. The previous notation will be used, and further notation will be defined when necessary. At the start of the model’s operations, m = 1, n = 1 and Et(l,n) is equal to Mr. The weight, I&(n)j~2, on residential attraction is set equal to 1 and the weight on service centre attraction K&)&Z is also equal to 1. All zones belong to ZI at this stage. First, the basic employees &(l,n) are distributed to their zones of residence T,j(m,n) = Ai(n)B,~n)Ei(m,n)f’(Fj,cij),i,j~Z

4 =

l/CBj(n)f'(Fj,C,j),i,jEZ

i

(22) (23)

Tif is the number of workers employed at i and living at j. f’(F,,cu) is a function relating the attraction of area j,Fj, to the generalised cost of travel, C~Jbetween i and j. The population living at j is found by summing equation (22) over i and applying the inverse activity rate a Pj(m,ll)

=

CtCTij(m,n),i,jEZ L

(24)

The population at j demands to be serviced and the number of service employees demanded, &(m,n) is found by applying the population-serving ratio B.

In this application of the model, the service sector was not disaggregated; in terms of the original Lowry model, /I = x9. The service employees demanded at j now have to be distributed to their places of kork

&j(m,n) = R,@)D,Xm9n)Ki(nl.f 2(Fi,c&i~Z R,(n) = 1/~KXnlf2(Fj,cjj),i,j~Z

(26) (27)

I

,SsJ is the number of service employees working at i demanded by the population at j, andf(Fa,cu) is a function relating the attraction of service centres to the generalised cost of travel. Service employment in i can now be calculated by summing equation (26) over j E,(m+ 1,n) = zS,km,n),i,jeZ i

(28)

108

MICHAEL BATTY

At this point, the first increments of population and service employment have been generated. It is now necessary to allocate service employees to zones of residence, and Et(m+l,n) is substituted for &(m,n) in equation (22). Equations (22)-(28) are reiterated until

CEi(m+ l,n)I 1

and l,,jeZ

CPj(m,n)<

j

(30)

I, and IP are limits below which further increments of service employment and population are small enough to ignore. Total population and employment predicted by the model will be approximately equal to their respective regional totals; these are calculated as follows: Pj(n)

=

CPj(m,tl),jEZ

In

E,(n) = CEi(m,n),kZ m

By summing equations (22) and (26) over m, matrices giving the interactions workplace and residential areas and between residential areas and service centres be calculated. At this point, tests must be made on the allocation of activities to find out the density constraints on population or the minimum size constraints on service ment have been violated. Firstly, if Pj(n)>djLjh,jeZ

(31) (32)

between can also whether employ-

(33)

then Pj(n)

(34)

Ez2,z4

Secondly if [Ei(n) - Mi] < min S,kZ

(35)

then [Ei(n)-Mi]

EZ,,Z,

(36)

If equations (33) and (35) do not hold for any zone, then the constraints are satisfied and the simulation terminates. However, if equations (33) and (35) do hold then Bj(n)GjLjh/Pj(n),j~Z,,Z,

Bj(?l + 1) = i

mz, 2,

(37)

and (38)

Design and Construction

of a Subregional Land Use Model

109

Bj(n+ 1) and K&+1) are now substituted into equations (22) and (26) respectively and the whole system of equations from (22) to (38) is reiterated until Pj(n)lGjLjh,jEZ

(39)

and [El(n) - Mi] 2 min S&Z, ,Z,

(40)

The sequence of operations in this model is presented in Fig. 2. Examination of the model reveals that the total population generated approximates the population predicted by the economic base relationship given in equation (3). The residential location model, is subject to constraints on the numbers of workers leaving any zone, and also constraints on the maximum population density allowed in any zone. Formally, (41) crCCT,j(m,n)IGjLjh,i,j~Z mi

(42)

In the service centre location model, constraints on the numbers of service employees demanded by the population in each zone, and the minimum size constraint are observed. CCSij(flI,n) = CDj(Wl,~~),i,jd mi

(43)

In

~pij(nz,f,){ =o?z+ 2min S,tEZI,ZZ mj

There are three important differences between this model and the Lowry model. Firstly, this model uses interaction models of a gravity type to allocate activity, in contrast to the potential functions used by Lowry. This means that trips between home and work and between home and service centre are calculated. Besides the obvious advantages of generating this information, it means that calibration procedures used in trip distribution studies can be used in fitting the model. Secondly, the method of securing consistency between the input and output distributions used by Lowry, is not- used here. Assuming that population and employment were used as proxies for locational attraction in the model, it would be a simple matter to feed back these variables in the outer iteration of the model. However, this would considerably complicate the calibration procedure and in this case, the difference between the inputs and outputs was not sufficient to warrant using this option. Thirdly, there is more emphasis on activities than on land use in the model. Although land use enters the model through the maximum density constraint on population, there is no priority given to the location of service employment as in the Lowry model. This is largely due to the scale of the zoning system used although the maximum density constraint is affected by the existing amount of service land use.

MICHAELBATN

110

BASIC EHPLtlVHElll,lRAVEL HE HAIRIX,AtllVlIV A POPULAIION SERVING RATIOS PARAMEIER VAtlIES. lESlOENIlAL L SEKVICf

+ AUOCAIE THE INCREMENI IX EMPLOYMEN 10 IHE RLSIDLNCE ZONES

CALCULAIE RLSIOENIIAL

y

1 lNtRlHLll1 OF POPULATION

1 CALCULAIE IHE INCREMENI OF SIRVICE EHPLOVHENl DEMANDlO BV RESIDENIIAL POPULATION I ALLOCATE INCREM(EN1 SfRVlCE EMPLOVHENl

CAWJUIE LOVHlNl

OF

1 SERVICE IMPIN CENVRES

IS MAXIMUM OENSIIV CONSlRAlNlON IHE ~PULAIION SAIlSFlED “::-‘=I ?

YES

STOP dJ

OUIPUl OAIA : IOIAL POPULATION AN0 IUIAL EHPLOVHENl WORK IRIP AND StRVlCL IRIP NAVRICES IRIP tENGIll OlSlRlBUIlON CURVES 4

v FIG.2.

Generalised flow chart of the modified Lowry model.

Certain details of the model structure still have to be defined. In the NottinghamshireDerbyshire model the minimum size constraint on service employment was not necessary and only two sets of zones, 2, and 2, were defined. In effect

The form of the attraction and deterrence functions in the gravity models follows the theoretical derivation proposed by Wilson [32]. Using a probability-maximising methodology, Wilson has shown that the function of travel cost in the gravity model should be

Design and Construction of a SubregionalLand Use Model

111

negative exponential. It has also been demonstrated that population provides a sensible proxy for locational attraction in the residential model; similarly, service employment can be used to measure attraction in the service centre model [7]. In the residential location model f'(Fj,Cij) =

Pj

exp(-I2f.&),i,j&!

(46)

and in the service centre location model f'(Fi,Cij)= Si eXp(-/&j),i,

j6Z

(47)

I and p are parameters of the functions, and daj, the travel time between i and j, is used as a proxy for the generalised travel cost cl+ Having defined the model, the procedures for calibration and testing will be outlined. There are several issues to resolve in the choice of an appropriate calibration procedure, and these will be described in the following section. PROCEDURES FOR CALIBRATION Although there has been considerable research into different land use model designs, there has been relatively little emphasis on the possible procedures for calibrating such models. In the case of models based upon linear equation systems, econometric techniques can be used for parameter estimation [23]. However, in land use models of the type discussed here which involve sets of non-linear equations, techniques to estimate the parameters and statistics to measure the goodness of fit are fairly crude. In this section, guidelines for calibrating this type of model will be established based on the results of some research into the calibration process. In several modelling efforts, measurements of the goodness of fit between observed and predicted distributions, have been based upon conventional statistics such as the coefficient of determination (R*) and chi-square statistic (x2). Such statistics when used to measure the fit of population and employment in this model can be extremely insensitive to variations in the parameter values [4]. In models based on spatial interaction, it seems logical to measure the goodness of the fit of the interactions, rather than the activities which are summations of interaction. Hyman [22] has produced theoretical evidence that the best statistic to use in measuring the goodness of fit in spatial interaction models is the mean trip length. Wilson [32] has also shown that the mean trip length is related to the parameter of the trip distribution function. In a model which is built from different submodels, it is impossible to find one statistic which sununarises the goodness of fit for all the distributions predicted by the model. In this application, the Ra statistic measuring the fit of the work trip distribution, and the mean trip lengths of both the work and service trip distributions were used. For the work trip distribution, the mean trip length is defined as C. C

=

CCT,jd,jlCCT,j,i,jd i

j

i

j

(48)

The mean service trip length (s) is found using an equation similar to (48) with Stf substituted for Tg+ To provide an assessment of the fit between predicted and observed population and employment, R* statistics have been calculated but these were not used in calibrating the model.

MICHAELBATTY

112

The method of calibration is extremely important to good model design. As this model has two parameters which are theoretically interdependent, a method must be devised which takes account of this interdependence. The traditional method of calibrating this type of model is to fit the parameters of each function outside the model [25]. The main techniques which have been used include graphical approximations to the trip frequency curves, and numerical methods of iteratively approximating the parameter values. However, these methods ignore the interdependence between the two parameter values, and if they are used it is by no means certain that the best fit will be found. No analytic means for deriving the parameters of this model are known to the author and to take account of the relationships between the parameters, a numerical method can be used. First, a range of values for each parameter is fixed, and the model is run under all combinations of the two parameter values at different intervals within each range. Second, the intervals within which the best fit occurs are selected, and tbe model is rerun under different combinations of parameters in these intervals. The process continues until the values of the parameters are judged to be within the prescribed limits of accuracy. This search procedure is based on the assumption that there is a unique optima whose coordinates are located by the two parameter values. This procedure is fairly lengthy, and experience with this model at the subregional scale suggests that the two parameter values are relatively independent of each other [4]. A procedure which is a compromise between the independent and interdependent methods has been devised, thus reducing the computational time involved without destroying the accuracy of the calibration. Briefly, the method first finds the values of each parameter independently using iteration. These values are then used as first approximations and a gradient search procedure is used to test the uniqueness of these values. If the fit improves, then the gradient search is initiated once again, and the process stops when the fit does not improve. The method can be described in different stages with a residential parameter R and a service centre parameter p. 1, As a first approximation, the best values 1” and pL”are found independently by running the model through a range of values for each parameter. 2. Two narrow ranges of values, PfAl and pX&A,? are fixed. 3. The model is then run through all combinations of values at intervals in the ranges P&AA, and $kAp. 4. If the fit does not improve, ;i” and pLxare the best parameter values. If the fit does improve, then new parameter values ;IX+yand pX”” are selected. 5. Pz”+Yand ,uX’Yare substituted for II” and ,uXrespectively in stage 2 above. 6. Stages 2-5 are repeated until there is no further improvement in the model’s fit. The best parameter values ;IX+y+Zand P’+~+’ are then selected. In the case of a tie between two or more best values at any stage, the above procedure is applied to those two or more values. This method was used to calibrate the Nottinghamshire-Derbyshire model, but before this is described the design of the zoning and information systems necessary to the model will be outlined. ZONING

AND

INFORMATION

SYSTEMS

Although the factors affecting the number and size of zones, and the data describing activities within zones are closely intertwined, Broadbent [5] has suggested a fundamental

Design and Construction of a SubregionalLand Use Model

113

principle affecting zone size. When designing spatial interaction models, Broadbent argues that it is important to have a “ sufficient ” number of interactions between zones in order that the model describes trip behaviour in an accurate way. In other words, the smaller the ratio of interzonal to intrazonal interaction, the less need there is for a model describing spatial interaction. As a general rule, Broadbent suggests that the average radius of a zone should be less than the mean trip length. Although this principle is basic to zone definition, it must be tempered against other factors. If possible, zone geometry should be fairly regular for this can ensure that as many zones as possible are packed into the system. The smaller and more regular the zone, the more accurate is the location of the zone centroid, although zone centroids should be placed on or near the main transport network used in measuring the travel times between zones. The spatial distribution of activities within zones should be as homogeneous as possible, especially in the types of model discussed here in which the variables are not disaggregated. The zoning system should also follow topographical barriers as far as possible. The zoning system used in Nottingbamshire-Derbyshire is shown in Fig. 3, and in Table 1 the major characteristics of this zoning system are presented. The decisions affecting the zoning system have not only been made with regard to the factors mentioned above; the area1 units for which data is available was another critical factor in these decisions. As mentioned previously, the development of this model was undertaken under a strict limit on the supply of data. No special surveys could be commissioned to collect data. This is reflected in the fact that the model is designed to operate with only data available from published sources. The Census of Population (1966) provided the main data source for the model. Population was available for enumeration districts from the Ward and Parish Library, and this provided a fine zoning system which was later aggregated to form larger zones. Total employment data for Local Authorities (aggregations of enumeration districts) was used as a control total, and a detailed classification of the location and size of firms employing over 5 persons was available from the Employment Exchange Record. The classification of these firms was by Minimum List Headings of the Standard Industrial Classification, and this was used as a basis for the division of employment into basic and service sectors. The maximum density constraints on each zone were calculated using the inventory of land uses compiled by the local Planning Authorities. Data on the spatial distribution of work trips was available from the Census of Population. A special tabulation of work trips had already been carried out by the Census for this area and, furthermore, a cordon survey around the main towns in the subregion was available. No data was available for the trips made between home and service centre, but from other similar studies, mean service trip lengths were available. It was therefore decided to calibrate the model against a range of mean service trip lengths before one particular service trip length was adopted. The constructed variables such as activity and population-serving ratios were calculated from the data. A particularly good data bank had been assembled by the Nottinghamshire-Derbyshire Subregional Planning Unit and all data was stored in this bank. The zoning system was designed several times, taking account of the principles outlined above, and the final design shown in Fig. 3 is the result of this lengthy process. Before the results of applying the model are presented, it is necessary to look in more detail at some of the issues involved in sectoring employment and in choosing the variables measuring locational attraction.

B

114

MICHAEL BAI-IY

CHESlERFlELO

URBAN NUMBERS

NATIONAL INOlCAlE

AREAS:

MAJOR

MOTORWAY ZONES

POPULATION

CENTRES

SHOWN Ok====!

Ml

: SEE

FIGURE

5 AN0 IABLE

Total No. of zones Total land area (mile*) Average land area per zone Total population Average population zone

per

N

3

FIG. 3. Zoning system for the Nottinghamshire-Derbyshire TABLE 1. MAJOR

IaLES

subregion.

CHAF~ACI-ERIS-IXX OF THE STUDY AREA

62 Population density 1456 Basic employment

1168.303 438,830

23.484 Total employment 1,701,050 Ratio of basic to total employment Ratio of interzonal to 27,436 intrazonal work trips

751,260 0.584 0.907

De&

and Construction of a Subregional Land Use Model

DETAILED

DECISIONS

IN MODEL

115

DESIGN

It was stated earlier that the division of employment into basic and service reflected a weak assumption in this model. The weakness of this assumption is largely due to the practical difficulties involved in dividing employment into these two sectors. Although methods such as the minimum-requirements approach [30] have been suggested to overcome some of these difficulties, such methods break down when applied to the local scale. As data was available on the location of each firm and its classification by minimum List Heading, the division into basic and service employment was carried out by examining each firm from these files and deciding whether or not the fu-m belonged to the basic or service sectors. As a general rule, the primary and manufacturing sectors constituted basic employment. But there were exceptions to this. For example certain publishing establishments classified under manufacturing were shifted to the service sector; some military and government establishments were transferred from the service to the basic sector on account of their functions and site requirements. An attempt was also made to restrict the amount of employment in the basic sector, for as the ratio of basic to total employment approaches 1, all employment is exogenous to the model, and the service centre location model is redundant. If this ratio is large, then the performance of the model may be biased towards a good fit, especially if fit is measured in terms of the total distribution of population and employment. In this case, the ratio was 0.58 and this was judged to be low enough to ensure that the model’s fit would not be biased. The measurement of locational attraction is one of the most important problems in model design, and in the types of model discussed here this problem has never been resolved satisfactorily. In this model, population is used to measure the attraction of residential areas and service employment the attraction of service centres [see equations (46) and (47)]. There have been some cogent criticisms of the use of population and service employment in this way [15] ; these arguments refer to the fact that models which use one variable to predict the same variable are tautological. Against this can be set two comments supporting these measures of locational attraction. Firstly, as these models are equilibrium models summarising the whole history of spatial structure by simple equations at one point in time, it is logical to use variables which describe the history of locational attraction. Population and service employment are variables which achieve this description. Secondly, in the case of residential location, other variables which summarise attraction such as the number of houses or households tend to be highly correlated with population. The same high correlations are found between service employment, and variables such as floorspace and sales .in the service centre location model [12]. These problems seem to be a feature of equilibrium models, and may only be resolved in a dynamic context when such variables as site rent, and amenities (accessibility to schools, recreational resources, etc.) may provide more realistic measures of locational attraction. There are, however, two mechanisms in the model which change the measures of locational attraction. The first is the constraints procedure which determines weights on residential and service centre attraction, and the second is the procedure used by Lowry to establish consistency between the input and output variables in the model. Although these procedures have only an operational, not theoretical meaning, they act as a brake on the model’s simulation process and ensure that the model’s predictions fall within reasonable limits. The measurements of general&d travel cost within and between zones were based on the travel times along the shortest routes in the subregional road network. The travel

116

MICHAELBATTY

times were computed from average driving speeds by car which varied over the network according to the density of population. The higher the density of population in areas adjacent to the route, the lower the average driving speed. The travel times were also weighted by parking times at each end of the trip. dij

=

ti+

tij+

tj,i,jEZ

(49)

dij is the total average travel time between and j, ts and tj are at i and j, and tif is the average travel time on the shortest route between i and j. The shortest routes in the network were calculated using a minimum path algorithm based on the cascade method, developed by Foot [16]. Although the use of travel time as a proxy for generalised travel cost was a fairly crude measure, in further work it is hoped to construct a more valid function based upon the function developed by Wingo [31]. The model is programmed to run on the ATLAS I installation at the University of Manchester. The program is especially simple in structure, and the shortest route algorithm referred to above is built in as a subroutine to the main program. No magnetic tape is needed for data storage, as the computer is able to accommodate both the program and data in core for a problem of less than 70 zones. Approximate running (computing) time is 65 set for the set of iterations comprising the inner loop of the NottinghamshireDerbyshire model. Input time is about 50 set and output time varies between 30 and 60 set dependent upon the outputs required. For problems larger than 70 zones, the program structure would have to be redesigned to account for the storage of data on magnetic tapes, although the structure of this model is well-suited for division into several subroutines.

APPLICATION

OF THE

MODEL

The model was first run and calibrated without constraints on residential or service centre location, and the best fit was found using the gradient search procedure described above. The model was run in this way so that any systematic bias in the model’s predictions could be isolated and studied without the distortions caused by locational constraints. As mentioned above, the measures of locational attraction are crude for they do not take into account the influence of land availability on attraction. Without the use of maximum density constraints, it was thought that the model’s predictions would overestimate the amount of population locating in zones which had reached their holding capacities and underestimate the amount of population locating in neighbouring zones. If this hypothesis proved to be correct, then the use of capacity constraints would be validated. If however the bias in the predictions was of a different nature, then the model could be redesigned to account for this. The model’s predictions were examined and a simple test was designed in the hope that any systematic bias could be revealed. The test was based on a comparison of two variables, the first describing the model’s predictions of population, the second describing the proportion of land not yet developed in each zone. The ratio of predicted to observed population was used for the first variable. 5 = CPj(m,l)/Pj,jCZ m

(50)

Design and Construction of a Subregional Land Use Model

117

The proportion of land not yet developed in each zone was defined as

(51)

Xj = [Lj-(L~+L~+L:)]iL,,j~Z

The coefficient of determination, R',between Yj and Xf was 0.7459, revealing a highly significant relationship. A regression of Yf against Xf yielded the following equation Yi = 1.8062-2.0659

Xj,jEZ

(52)

Equation (51) shows that as the proportion of available land in any zone decreases, the ratio of predicted to observed population rises. This relationship supports the use of capacity constraints on the location of population, and a decision was made to calibrate the model described in equations (22x40) above. The model was not completely recalibrated, for the values of the parameters 1 and ~1 fixed by the preliminary calibration of the model were used as first approximations. Other work carried out with a traffic distribution model having a similar form to the residential location model used here has revealed that the parameters of such a model are relatively insensitive to locational constraints [34]. Therefore the parameter values were not altered independently of each other; the values of Iz and p were then substituted in the model, and the model was run through all combinations of values in a small range around R.and CL. In other words, the model was run through stages 2-6 of the calibration method outlined above. The process of appIying constraints to the mode1 involves a lengthy set of outer iterations in which the original predictions are gradually distorted to satisfy the constraints. At each iteration of the model, population was tested against the maximum density constraint in each zone. If the constraint was violated, the zone was included within the constrained set 2%. During this process, zones in which the ratio Yf was greater than 1, were constrained first. The surplus population then tended to locate in neighbouring zones and these eventually had to be constrained. In effect, surplus activity was redistributed outward from the central urban cores in the subregion until all constraints were satisfied. Convergence towards an equilibrium in which all constraints were satisfied, was extremely slow. Twenty iterations were required before all the predictions affected by constraints were within 1 per cent of the maximum density constraint. This procedure is very expensive; it could probably be speeded up by raising the ratio s,Ljh/Pj(n) to some power greater than 1, or by building in the consistency procedure used by Lowry. In future applications of the model, these options may be tried. As in Lowry’s model, there is no guarantee that convergence will be achieved. Theoretically, the model could oscillate around certain predicted values, and zones could be continually shifting between the two sets Z, and Z,. In practice, however, the zones belonging to the set Z, did not change after about three iterations. Convergence in the first five iterations was fairly rapid; in Fig. 4, cross sections of population across the region show the changes caused by constraints in the first three and final iterations. In Table 2, the parameters and statistics measuring the goodness of fit of the model, with and without constraints, are presented. The critical statistics concern the trip distributions. It is interesting to note that there is little change between these statistics for the unconstrained and constrained predictions, thus revealing a certain insensitivity between

118

MICHA~LBA~

SECIION

ACROSS

SUBREGION A-A

SECIION 27

29 :

32 j

33 :

:

:

:

: : : :

: : : :

:

1 : :

: : :

B-B

60

:

:: IllI i 9; ; !

SUBREGION

52

I

1:.

ACROSS

Lo

i

30

:

\.

@=IIRSl

IlElAllDH

@-

@-1NlRO

IlERAllOW

@

ZUlliS

WNERE

NAXINUN

-

SICOHO

IIERAIION

FIR11

IILRAIION

OEWSllV

CONS1RAIkl

-70 IS OPElAlIVE

Fro. 4. Cross-section profiles of the predicted population in the subregion.

Design

and Construction of a Subregional Land Use Model TABLE 2.

119

GOODNESS OF FIT OF n-m MODEL

Statistics or parameter Residential parameter, 1 Service centre parameter, p Ratio of observed to predicted mean work trip lengths Ratio of assumed to predicted mean service trip lengths P between log transformed population distribution R* behwm log transformed employment distribution

Calibration without constraints

Calibration with constraints

Et& value

Best value

0.2300 0.1600

0.2400 0.1600

1.0775

1.0711

0.9389

0.9648

0.9364

0.9402

0.9764

0.9802

these two types of prediction. This, however, would vary between applications and the greater the difference between unconstrained and constrained predictions, the greater the difference between the fits. Rs statistics were used to measure the fit of population and employment. As distributions such as population and employment tend to be lognormal the R’ statistics were calculated using the logs of these distributions. The fits of these distributions are good, although for reasons stated earlier, and because of the strict density constraints on population, these statistics were not used to calibrate the model. In Table 3, the actual predictions of population and service employment for a subset of typical zones in the subregion are displayed. These statistics reflect the overall performance of the model at the macro-scale. At the micro-scale, the performance is not as good. This can be demonstrated in relation to the work trip distribution, and in Fig. 5 the observed and predicted trip frequency distributions are compared for 4 typical zones. The distributions shown were constructed by interpolation from histograms recording the percentage of trips made from employment centre to zone of residence in intervals of 5 min. In terms of this sample of zones, Fig. 5 shows that the model is unable to predict the detailed pattern of interaction as accurately as the general pattern of interaction given in the statistics above. This is to be expected for statistical models such as this one, only attempt to simulate the general not the detailed organisation of a spatial system. This analysis does however, highlight the need for thorough evaluation of the model’s performance both at the macro- and micro-scale. EVALUATION

AND REDESIGN

Already some comments have been made on the model’s performance, although performance as measured by the goodness of fit is too narrow a guide to evaluation. A comprehensive evaluation of the model can only be made if the variables and parameters are subjected to sensitivity testing. The calibration procedure revealed that the measures of spatial interaction were most sensitive to parameter variation. Small changes in the intrazonal travel times lead to large differences between the predictions, and this suggests that the ratio of inter to intra-zonal interaction is too low. Only by reducing the zone size and by packing more zones into the subregion can this problem be overcome. If a variable such as intra-zonal distance is too sensitive, then this means that the use of the model in

120

MICHAEL BA-I-IY

prediction is suspect if large changes in intrazonal distances occur. In any redesign of the model, this would be one of the most important problems to tackle. TABLE 3.

PREDI~D

WPULATION

AND SERVICE EMPLOYMENT FOR A SAMPLE OF ZONES

Observed values (1966) Zone No. 1 3 5 9 11 16 17 19 22 26 29 31 33 35 37 39 41 45 47 52 57 59 62

Population

Name

Chesterfield Eckington Bolsover Worksop Grinalev Woodhouse Mansfield Sutton Alfreton Heanor Nottingham West Bridgeford Long Baton Derby Spondon Mickleover Breaston Arnold Wirksworth Broughton Blidworth Ollerton East Leake

79,080 27,510 11,070 33,010 11,450 22,640 55,610 40,840 32,050 24,130 305,050 65,130 31,090 125,900 30,430 22,010 21,900 29,840 8160 6620 21,880 23,830 8540

Service employment 22,080 1850 430 8740 1220 2410 18,050 4200 3410 3410 81,410 10,609 5090 49,340 200 3620 2510 4130 1060 930 2160 1730 2170

Calibration with constraints (1966)

Calibration without constraints (1966) Population

Service employment

111,298 18,879 8703 36,591 6786 30,249 77,194 47,445 21,297 21,607 439,397 43,283 23,068 202,644 24,190 13,475 12,579 25,735 5064 1858 13,828 22,519 2698

27,956 1736 360 8540 656 3350 24,778 5201 2715 3242 102,293 7600 3704 50,940 163 1840 1612 3697 475 225 1733 1343 406

* Shows zones in which the maximum density constraint on population

Population 79,175* 28,103 12,103 33,047* 7059 22,712* 55,799* 40,985 37,035 24,297 307,930* 65,731’ 31,339* 126,782* 32,642 23,158 35,463 30,113* 6225 5083 35,363 23,013 5854

Service employment 26,334 2221 455 8797 703 3349 24,313 5227 3583 3771 92,904 8470 4485 48,403 191 2077 2246 3463 579 298 2773 1415 517

is necessary.

Another feature of the model revealed by the calibration involves the use of systemwide parameters in residential and service centre location. The model tends to overallocate population and service employment in urban areas and under-allocate these activities in rural areas. This suggests that the locational parameters should be higher in rural than in urban areas to account for the greater resistance to travel. The constraints procedure partly obscures this problem for the maximum density constraint ensures that urban areas do not attract too much activity. However, those areas which are not rural and which are not constrained tend to attract too much activity from rural areas. One way of overcoming this is to use Goldner’s [18] method of disaggregating the parameters spatially. Disaggregation of employment and population would probably help to alleviate this problem. The problem of measuring locational attraction has already been discussed. Considerable research is required into this problem so that suitable measurement techniques and variables can be defined [24]. As mentioned previously, the measurement of attraction may be resolved more simply within the structure of dynamic models, but this does not remove the problem of attraction in equilibrium models. Any advance in this area will involve the role of constraint procedures in such models, and will involve an evaluation of the importance of gravity or potential models within the general model structure. Another difficulty arises from the use of the model in forecasting. At each forecast interval, the

Design and Construction

of a Subregional Land Use Model

121

model will reallocate all the existing activity along with any additional activity. To mitigate against the severe effects of total reallocation, only the increment (or decrement) jn activity has been reallocated with this model. In some ways, this is as unrealistic as the original method of reallocating all activity, and some compromise between the methods is required. This suggests that the model should be redesigned to explicitly account for the pool of activity which is to be reallocated, and if the problem is deemed to be critical, a dynamic model along the lines of the Time Oriented Metropolitan Model [8] may be suitable. ZONE 29:

I 11

ZONE 17: HANSFIELO 60 -

POPULAIION 55610 EMPLOYMENI 29680

= i;

0

10

IRAVR

60

IlNt

20

FROM

30

ZONE

CtNlROJD

LO

0

50

ZONE 37 : SPONOON

I

POPULAIION ~MPLOYHfNI

59

IO

20

30

LO

so

I LO

1 39

IN MINIJI~S

2ONE 59: OLLEAlOlj

I I’

60 -

30130 1360

POPULAIION tHPLOYMEN1 ___--

23830 12930

50 -

I I /r

I 30-I

’

y----PhtDlcItn ’

20 -

i

10 -

0 0

I 10

I 20

FIG. 5. Observed

I 30

I LO

I 50

0

and predicted trip distributions

IO

20

30

for four typical zones.

Redesign of the model to resolve the problems mentioned above would conflict in several instances with the arguments presented earlier in this paper. For example, the need for smaller zones to lower the inter to intrazonal interaction ratio may not be possible from available information sources. This example serves to underline the fact that redesign of the model, like the initial design, involves a series of compromises. Without

122

MICHAEL BATTY

additional data, it is difficult to see how the model could be made dynamic or its variables disaggregated. Certain improvements such as the introduction of different parameters for each zone could be made but against this must be set the need to keep the calibration process manageable. The number of parameters in such models should be kept to a minimum so that the time and cost of calibrating the model is at a minimum. It is easy to make suggestions which would probably lead to more realistic models, but when disaggregation of the models variables and parameters is possible, this must be weighed against the time involved in design and construction, and the ability of the model user to digest the model’s output. CONCLUSIONS

This paper has attempted to outline several theoretical and practical considerations affecting the development of a subregional land use model. The design of a model under fairly limited resources of time, money and information, has been traced through, and some of the problems in adapting and modifying an existing model have been demonstrated. Methods for calibrating and testing the performance of the model have revealed major problem areas which may be overcome by redesigning the model in particular ways. There is one essential point which has emerged from this project, and this concerns the simplicity of the model. The structure of this model is sufficiently simple to enable a clear analysis of its deficiencies to be made. In more complicated models, this may not be possible as different variables may be so interdependent that their individual effects upon the model’s predictions cannot be defined. It is important to disentangle these effects for the performance of the model and its validity in forecasting are intimately related to the effect of each variable on the allocation of activities. It is exceedingly difficult to generalise about the design of land use models but simplicity and clarity of structure seem to be good design principles to follow. As urban phenomena has an alarming number of facets, the structure of the model must be orientated around the particular phenomena to be studied or manipulated. If the purposes for which the model is designed, are well defined, and if the constraints on what is intrinsically impossible are observed, then relevant models can be designed and applied intelligently to problems of spatial planning.

REFERENCES 1. R. ARTLE, On some methods and problems in the study of metropolitan economics, Pap. Proc. Reg. Sci. Ass. 8, 71-87 (1961). 2. M. A~IMOV,Introduction to Design. Prentice Hall (1962). 3. M. BATTY, Models and projections of the space-economy, Tn Plunn. Rev. 41, 121-147 (1970). 4. M. BATT”Y, Some problems of calibrating the Lowry model, Envir. Plunn. 2,95-l 14 (1970). 5. T. A. BROADBENT,Zone Size and Spatial Interaction in Operational Models, CES-WN106, Centre for Environmental Studies, London (1969). 6. M. CORDEY-HAYES,T. A. BROADBENTand D. B. MASSEY,Towards Operational Urban Development Models, CES-WP-60, Centre for Environmental Studies, London (1970). 7. M. GXDEY-HAYES and A. G. W~SON, Spatial Interaction, CES-WP-57, Centre for Environmental Studies, London (1970) 8. J. P. CRECINE,TOMM: Time Oriented Metropolitan Model, CRP Technical Bulletin No. 6, CONSAD Research Corporation, Pittsburgh (1964). 9. J. P. Cnnowr, Computer Simulation in Urban Research, P-3734, RAND Corporation, Santa Monica (1967).

Design and Construction of a Subregional Land Use Model 10. J. P. Cw, A Dynamic Model of Urban Structure, P-3803, RAND Corporation, Santa Monica (1968). 11. E. L. Crrrpps and D. H. S. FOOT,A land use model for subregional planning, Reg. Stud. 3, 243-268 (1969). 12. R. L. DAVIES,Variable relationships in central place and retail potential models, Reg. stud. 4,49-61 (1970). 13. R. Daawarr, (editor), Urban and regional models in British planning research, Reg. Stud. 3, 233-358 (1969). 14. M. ECHENIQU~, D. Caowrnaa, W. LINDSAYand R. Srraas, Model of a Town: Reading, Working Paper No. 12, Centre for Land Use and Built Form Studies, Cambridge (1969). 15. M. ECHENIQUE, D. CROWTHWand W. LINDSAY,Development of a Model of a Town, Working Paper No. 26, Centre for Land Use and Built Form Studies, Cambridge (1969). 16. D. H. S. FOOT, The Shortest Route Problem: Algol Programmes and a Discussion of Computational Problems in Large Network Applications, Discussion Paper No. 10, Department of Economics, University of Bristol, Bristol (1965). 17. R. A. GARIN,A matrix formulation of the Lowry model for intra-metropolitan activity location, J. Am. Inst. Ph. 32,361-364 (1966). 18. W. GO-NER, Projective land use model, BATSC Technical Report 219, Bay Area Transportation Study Commission, Berkeley (1968). 19. B. w, (editor), Urban development models, J. Am. Inst. Plann. 31,90-184 (1965). 20. J. D. Haaaaa~ and B. STEVENS, A model for the distribution of residential activity in urban areas, J. Reg. Sci. 2, 21-36 (1960). 21. D. M. HILL, A growth allocation model for the Boston region, J. Am. Inst. Plann. 31, 111-120 (1965). 22. G. M. HYMAN,The calibration of trip distribution models, Envir. Plann. 1, 105-112 (1969). 23. J. JOHNSTON, Econometric Methoak McGraw-Hill (1963). 24. J. F. RUN and J. M. QUKXEY,Evaluating the quality of the residential environment, Envir. Plann. 2,23-32 (1970). 25. I. S. LOWRY,Location parameters in the Pittsburgh model, Pap. Proc. Reg. Sci. Ass. 11,145-

165(1963). 26. I. S. LOWRY, A Model of Metropolis, RM-4035-RC RAND Corporation, Santa Monica

(1964). 27. I. S. LOWRY,A short course in model design, J. Am. Inst. Phn. 31,158-165 (1965). 28. M. L. MANHLUM, Problem-solving processes in planning and design, P67-3, Department of Civil Enaineerina. M.I.T.. Cambridae. Mass. (1967). 29. G. STEELXY: Analy& by the Garin-L&&y Model, in~Centre for Environmental Studies, Papers from the Seminar on the Notts.-Derbys. Subregional Study, CES-IP-I 1, London (1970). 30. E. L. ULLMANand M. F. DAC~Y, The minimum requirements approach to the urban economic base, LundStud. Geogr. Series B, 24,121~143 (1962). 31. L. WINOO.JR.. Transuortationand Urban Lund. Resources for the Future. Washington. . D.C. (196i). ’ 32. A. G. Wuao~, A statistical theory of spatial distribution models, Transp. Res. 1,253-269 (1967).

33. A. G. W-N, Developments of some elementary residential location models, J. reg. Sci. 9,377-385 (1969). 34. A. G. W~SON,A. F. HAWKPIS,G. J. HILL and D. J. WAGON,Calibration and testing of the SELNEC transport model, Reg. Stud. 3,337-350 (1969). 35. H. B. WOLFE,Model of San Francisco housing market, Socio-Econ. Ph. Sci. 1, 71-95

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RAND Corporation, Santa Monica (1962).

123

124

MICHAELBATTY

This this

appendix

provides

a summary

may be of use if the reader

wishes

of the notation to refer

which

to the notation

is introduced

in this paper;

of any equation

quickly.

APPENDIX Z 2 Z2 Z3

= = = = =

2 4i

1 =

k

=

= = s x, y, z = = Lj L;h

=

L;b

=

Lj’

=

L,u

=

P Pj

=

Mi

=

si

=

=

Ei = = S Dj = Sij = Tij = Fi, Fj = Ai, Bj = Ki, Rj =

Cij

=

dij ti. tj tij Yj

= = = =

_xi

=

C

s

=

;

1

Sj

=

9,4

=

I, le

zz

i

= =

I*

=

min

=

the total set of zones in the subregion; a subscript referring to a subset of zones Z,; a subset of zones with no constraints on location; a subset of zones with only constraints on residential location; a subset of zones with only constraints on service centre location; a subset of zones with constraints on both residential and service centre location; the empty set; subscripts referring to zones of the subregion; a superscript referring to different classes of service employment ; an index appended to certain variables describing the number of inner iterations of the model; an index appended to certain variables describing the number of outer iterations of the model; superscripts defining different values of the model’s parameters: total land inj; land used for residential nuruoses in i: land used for basic industry in j; _’ land used for service industry in j; unusable land in j; total population in the subregion ; population in j; basic employment in i; service employment in i; total employment in i; total service employment in the subregion; service employment demanded by population at j; service employees working at i and demanded by population at j; employees working at i and living at j; the locational attraction of zone i, or zone j; factors which normalise the production and attraction of worktrips at i and j respectively; factors which normalise the attraction and production of trips to service centres at i and j respectively ; generalised cost of travel between i and j; total travel time between i and j; terminal times for parking at i and j; travel time between i and j excluding terminal times; the ratio of predicted to observed population in j; the ratio of undeveloped land to total land in j; the mean work trip length; the mean service trip length; the inverse activity rate, i.e. the ratio of total population to total employment; the population-serving ratio, i.e. the ratio of total service employment to total population; the maximum population density at j; coefficients describing the relative importance of population and employment in service centre location ; the amount of land per service employee; predetermined limits on the generation by the model of population and employment respectively; a parameter of the residential location model; a parameter of the service centre location model ; the minimum value of a variable.