A paradigm for location and multidimensional problems in planning

SocieEcon. P&n. Sci. Vol. 18, No. 3, pp. 159-W Printed in the U.S.A.

A PARADIGM

1984 0

0038-0121/84 S3.00 + .OO 1984 Pergamon plrss Ltd.

FOR LOCATION AND MULTIDIMENSIONAL PROBLEMS IN PLANNING A. R. BANAEKASHANI

Graduate Department of City and Regional Planning, Memphis State University, Memphis, TN 38152, U.S.A. (Received 12 October 1983) theories, concepts, methods, or alternatively, “paradigms” have been suggested for the explanation and prediction of the location behavior of urban households. Increasingly, hwoever, “behavioral” approaches to the explanation of the dimensions of “choice” and/or exploration of

Ah&Pet-Several

alternative hypotheses have been cumbersome in the “mechanistic” paradigms of the social system and its related subsystems. An alternative paradigm of Analytic Hierarchy Process (AHP) is proposed to explore alternative structural specification hypotheses (sequential versus simultaneous) on the grouping and changing relative importance of “instrumental” and “non-instrumental” factors affecting location decision-making. AHP estimates of the “locational shares” of an urban (corridor) zonal population provide a paradigmatic basis for behavioral, vis-a-vis environmental, explanation of lcoation decision-processes in a methodologically efficient, robust and theoretically inclusive framework of hierarchy systems. This paradigm is proposed for locational analysis requiring an effective integration of multilevel, environmental (contextual)‘and behavioral measures of relative importance, with limited data, and, for multidimensional problems in planning and policy-making, formidably requiring the integration of positive with normative analysis of systems.

In a period of “paradigmatic crisis”, “disrepute”, beginning in the mid- 1960s through the mid- 1970s [ 11, the critics posed the question of the adequacy of urban models and problem-solving paradigms and the contention that the “quantitative revolution”, which started in the late 1950s with transportation studies, had begun to run its course[2]. The wisdom of the theories, concepts, relationships, or altematively, “paradigms”[2,3] were being questioned and reconsidered in the positive and normative planning theoretical discourse. Rittle and Webber[22] and others contended that the nature of planning problems are “wicked”, or alternatively, problems that are “illdefined”. Whereas there exist many examples of ill-defined problems in planning, quantitative models of planning and regional science have formidably dealt with “tame” problems, or problems that are “stripped of their ill-defined nature though simplifying assumptions”[4]. Still others characterized planning problems as “messes”, or dynamic situations that consist of complex structures of changing problems that interact with each other[5]. Such a characterization, of course, presupposes the rationality of the paradigm of systems: A conceptualization of an entity as a part of one or more larger wholes, not as a whole to be taken apart[5]. The use of analogies, metaphors, models and the paradigmatic application of the natural and physical science concepts in the social sciences have recently raised questions of the appropriateness of the “framework” or the underlying premises of paradigms [6] the critical scrutiny of the “value’‘-laden, methodological assumptions (Faludi[23]; Los[7]), and the “contradictory” or dilemmatic dimensions of the competing paradigms[ 51. The criticism of paradigms, however, is more inclu159

sively posed as the epistemic inquiring problem in the “dialectic” of the “object” and “subject” of paradigms. In their recent paper, Batty and Spooner[8] contend that urban systems analysis “has been developed more analogous to the physical sciences where the object or system of interest is regarded as being independent of its social context”. The paradigm of “social physics” is hence criticized on “social phychological” grounds, the paradigm of “systems” on contextual or “environmental” groundsp], the paradigm of “optimization” on “behavioral”, “adaptive”, “learning” and “decision-making” grounds[5]. The epistemic “object-subject” interrelationship is here posited in relation to an “information theoretic” paradigm suggested by Simon[lO]. The paradigm is based on the “state of information” which is regarded as a “characteristic of the decision-maker” as well as “characteristic of the environment”. The Simon paradigm hence, suggested inquiries into the attributes of the “choosing organism” and the “environment of choice”. Hence, also, the implied fallacy or the inadequacy of paradigmatic and methodological inquiries in which variant “behavioral” and/or “contextual” (environmental) dimensions have been assumed “given”. Posited as a concomitant information theoretic, systems paradigm, this paper develops an alternative procedure for lcoation simulation via Analytic Hierarchy Process[l 1,121. An approach is developed to integrate the characteristics of locators with the attributes of locations represented jointly by the variant dimensions of a hierarchy system in which their interrelationship can be simultaneously analyzed. The proposed procedure is contrasted with the “mechanistic” simulations of location in which the “behavioral” basis for location decision-making has been implicitly measured or assumed given.

160

A. R. BANAI-KASHANI

It has been hypothesized that the “choice” of a location by the urban household is made on the basis of a hierarchy of decisions involving the “comparative assessment” of the (composite) locational systems attributes. Given this hypothesis, the estimates of locational shares of the various zonal populations are given by the measurement of the relative priority (“trade-offs”) of the factors affecting location decision-making. In contrast to the urban econometric and simulation models of location and the increasing demand of these models for data (quality) input, processing, computation and calibration requirements, the proposed approach is methodologically efficient and theoretically inclusive in accounting for the multiple dimensions of behavior-environment interactions. The inclusion of a host of diverse variables to capture the variant dimensions of location decision-making has also posed certain difficulties for empirical models which rely on the availability and accuracy of data. The location models that have encountered difficulties due to often imperfect or unreliable data have also encountered concomitant problems of explanation and prediction. The exploration of alternative behavioral and/or environmental hypotheses have been increasingly cumbersome in the paradigmatic and methodological inquiries restrained by functional form, empirical adata and calibration requirements. A paradigmatic problem is posed for “deterministic”, “closed system” models in which “instrumental” and “non-instrumental” system variables have been specified with a conception of systems and/or non-instruvariables (instrumentality mentality) as exclusively absolute rather than inclusively relative concepts. As pointed out by Ackoff[S], this is a concomitant problem in which intrinsic (or non-instrumental) value of means and the extrinsic (instrumental) value of ends is unaccounted for or is being confounded. Conceptually, and, in contrast, the addition of dimensions in a hierarchy system results in the previously assumed non-instrumental variables, alternatively as instrumental variables; hence, variables can be specified with the conception of the interrelationships as relative rather than absolute, and while hierarchy systems remain inclusive in the basic structure. AHP provides “functional values”, without specifying a function[l2], which can be robustly used for the assessment of the relative importance of the “instrumental” vs “non-instrumental” system elements, i.e. the exploration of the causal order as well as the order of the magnitude of the elements of a

I:

II: III: IV: V:

Location System

non-17 IV (or non-IV) IV Location Alternatives

“location system” and the formulation and verification of alternative behavioral hypotheses, given alternative specification of the location system elements. The reason “robustness’ in measurement is emphasized is that the elements (or components) of a locational system are not “perfectly orthogonal”, reflecting some ambiguity in the studies of location and/or relocation. “Accessibility”, for example, can be studied as a component of “neighborhood characteristics”, but measured by the importance of the different types of accessibility (to work place and non-work place), it is listed separately as specific to household rather than to neighborhoods[l3]. The measurement of interactions of the location variables have been exacerbated, however, with the choice theory assumptions in the analysis of interrelationships using statistical and econometric models of locational choice. The “structural” properties of AHP can be effectively utilized to explore location decision processes (sequential vs simultaneous) via the decomposition and the composition of the elements represented by the variant dimensions of a hierarchy system. The advantage of hierarchical decmposition and/or composition (or clustering), however, is that impacts of possible changes associated with (small or large) perturbation of the weight of the individual components on the rest of the interacting (and noninteracting, or “independent”) components can be sensibly and efficiently discerned. This approach is contrasted to a single, simultaneous model that includes all the choice dimensions, however, methodologically unmanageable and hypothetically implausible [ 141. Hence, alternative behavioral hypotheses to structure the dimensions of choice have been made hierarchically as in the models of transportation-location interactions, or in the models of urban travel demand (Manheim et aL[24]; 114, 151). The proposed procedure emphasizes the importance of capturing homogeneous and/or heteragenous locators’ characteristics (behavior) vis-a-vis locational attributes (environment) by “clustering” of the factors that are being compared. The changing weights of the components that belong to variant clusters suggest corresponding changes associated with the socioeconomic characteristics or the spatial attributes of the locations jointly affecting the pattern of location. The composite value of the two types of measures, reflecting in the estimated values of the “locational shares”, provide a basis for inferences to be made at the interface: the “intrinsic” and “ex-

A paradigm for location and multidimensional problems in planning trinsic” aspects of behavior and environment are relationally investigated thereby simulating the “contained” vis-a-vis the “containing” system. Systems thinking is particularly plausible for the (simulation) study of location using the notion of “interaction” in favor of “independence” of location elements. Consider the following properties of a systems oriented paradigm of AHP[l 1,121. Suppose a set of n objects, A,, . . . , A, are compared in pair according to their known weights W,, . . . , w,. Pairwise comparison of the objects by their relative weights are given in matrix A. This matrix of ratios, A, is reciprocal, aij = l/u,, and additionally, a, = 1. The (post) multiplication of A by the vector of weights w results in Aw = nw, a system of homogeneous linear equations: (A - nZ)w = 0. A,

A, A=

i

A”

. . . A, WI

WI

ilI =n

f

=

nw

W”

W”

It is initially assumed that the weights w are known. However, if only the matrix A is known and the weights w are not known, the system (A - nZ) w = 0 can be solved to recover the unknown w. This system has a non-zero solution if the only if n is an eigenvalue of A, i.e. it is a root of the characteristic equation of A. Since A has unit rank (there is only one independent row or column of A, or altematively, any row or column can be found by a constant multiple of the tirst row or column) all the eigen, n of A are zero except one. values 1, i=l,... Furthermore, since the sum of the diagonal elements is the trace of matrix, for matrix A, l! li = tr (A) = n. i=l

given A, only one Izi= n which is called & # li = 0. The solution w of this problem is any column of A all of which differ by a multiplicative constant. If this solution is normalized, the result is a unique solution no matter which column is used. The scale is recovered from the matrix of ratios. The condition of consistency, al. ajk = ati, holds for A so that the entire matrix can be constructed from a set of n elements. Since in general, the estimate of the coefficients of A is not precise, small perturbation in the value of the coefficients implies small perturbations of the eigenvalues. The system Aw = nw becomes A’w’ = X-w’, where A’, w’, X,, are perturbations of A, w and n, respectively. A problem of inconsistency is implied which suggests the revising of the estimates in A in order to improve the solution vector w’. For values that are near consistency, X,, are close to n. Hence, deviation from consistency is measured by (& - n)/(n - l), or the “consistency index”, u. Normally, this value would be compared with its average value for a randomly generated reciprocal matrix and estimates are revised by incorporating better information, if the ratio of the two indices, “consistency ratio”, exceeds 0.10. Fractional values would not change the eigenvector of a reciprocal matrix significantly, noting further that the Thus,

perturbation

161

problem requires taking n(n - 1) n

judgements to improve consistency rather than n. Saaty [ 121suggests the following scale for the pairwise comparison of the elements in the matrix of ratio estimates: (1) Equal importance-Two activities contribute equally to the objective. (3) Weak importance of one over anotherExperience and judgement slightly favor one activity over another. (5) Essential or strong importance-Experience and judgement strongly favor one activity over another. (7) Demonstrated importance--An activity is strongly favored over another and its dominance is demonstrated in practice. (9) Absolute importance-The evidence favoring one activity over another is of the highest possible order of affirmation. The conception of hierarchical elements in the urban activity system is made operational by forming various clusters of the elements which comprise variant “nested” structures: A transport-landuse system, or alternatively, a mode-location housing “choicehierarchy”. The mesurement problem becomes one of estimating the conditional or composite value of the weights of the elements to investigate both the causal order and the relative priority of the factors affecting location decision-making. A “synthetic”, sample data for a corridor of a metropolitan area (Chicago, Southwest) reported by Anas[ 161 for a housing market simulation is referenced here for a location decision simulation by the procedure of AHP. This data along with (1970) census block and tract data provide information on spatial and physical attributes of zones, housing, neighborhood status and accessibility, as well as socioeconomic data on household characteristics. From the zones in the corridor (Table 4) three zonal clusters are formed in the simulation by AHP. The three clusters are locations at rings: (R,, R,, R,); (R,,, R,,, R,,); and (R,,, R,,, R,,) with the subscript of R indicating the ring distance from the C.B.D. In the simulation are city, suburban and suburban fringe locations. The design of the zoning system and the approach of clustering provide more explicit account of metropolitan, corridor-wide measures of spatial and non-spatial variation. In some metropolitan areas, even block-by-block variation in the stock of housing (by type), neighborhood quality and accessibility can be observed requiring “qualitative” as well as “quantitative” measurement of spatial (locational) disparity. Hence, the AHP simulation of location is contrasted to approaches using highly aggregated and indiscriminant spatial data. This zoning system provides a basis for the comparison of the locational attributes. Factors which are included for comparison are: (a) accessibility (AC), subfactored into “access to employment” and “access to shopping and amenities”; (b) housing unit (HU), subfactored into housing unit “cost” and “space”; and (c) neighborhood status (NS). The methodology consists of constructing matrices

A. R. BANAI-KASHANI

162

Zone

.4HS Structure for Location Analysis.

in a two-staged process of the comparison of (a) factors, and (b) locations, given individual factors. HU

NS

CAM

c

snse

C

w, -

...

HU:

;

CSM R, R2

a WI -

WI ;

WI8

W” ... WI R, R2 R,

-Wll W”

NS:; AC:

AC

ml

mi

Z,

‘+‘I --- ‘+‘I “‘I WI wz w3

zi,

Wz

W2

W2

z,~

WI

w2

w3

---

mi

R3

“‘3 W3 “‘3 zi3 --WI w2 w3 Example: R,, R2, R3

In the first group of matrices (CAM), Accessibility, Housing Unit and Neighborhood Status are compared to assess their relative dominance as factors affecting the spatial organization of the corridor. In the second group of matrices (CSM), the same factors are compared from a different vantage point; as and/or non-instrumental factors instrumental affecting the “choice” of location by households. Locations are compared in a process of “mapping” socioeconomic characteristics of households with the spatial and physical attributes of loations. The composite weight of the elements compared in CAM and CSM results in the values of “locational shares”, Rj, which are determined by the following procedure: R, = i miZ, i=l

where; Rj = share of population at jth location, marginal valuation of the ith variable, Z, = share of ith variable for jth location mi =

i j=1

Rj= 1,

j=l,...,J.

The two stages in the proposed procedure are posited as concomitant conceptual and methodological assumptions in the “choice-abstract” and “choice-specific” models, respectively. The hypothesis in the so-called choice-abstract states that choice attributes are perceived and, hence, modeled independently from the choice context. Models of abstract travel modal choice, e.g. compare and analyze travel modal attributes independently from the attributes of the travel facilities on which modes operate[l7, 181, whereas in the “choice-specific” approach, choice attributes are analyzed relationally, given the environment (or context) of choice. Hierarchy systems provide a more inclusive concept [ 141to capture variant dimensions of “choice” via “complete” and/or “incomplete” hierarchy systems [ 111. The treatment of the “choice-abstract” and “choice-specific” concepts for the analysis of the extent of interaction (vs independence) is hence conceptually mutually exhaustive in hierarchy systems. The proposed procedure, hence, sets out to integrate two, otherwise, separate approaches represented inclusively as “incomplete” or “complete” hierarchy system(s) of “choice-dimensions”. Alternative specification and scaling of the elements within a hierarchical structure of location result in the correspondingly changing (composite) value of the “share” of locations which can be compared with the observed zonal population to verify the simulation hypothesis. The resulting composite solutions can be interpreted for the corridor population simulated. The eigenvector solution resulting from the tirst stage of comparisons is interpreted as indicating the relative importance of factors affecting locational choice. In decreasing order of their magnitude, they include (a) accessibility (“access” to employment” and “access to shopping and amenities”); (b) housing unit (housing unit “cost” and “space”); and (c) neighborhood status. Since the corridor zones have been disaggregated and represented as clusters of zones with variant distance from C.B.D., and given multiple center(s) of employment in the corridor, the relative importance of “accessibility” varies by the distance to C.B.D. from the zonal groups that were compared. The relative importance of accessibility, as a determinant of location, increases with the zones distant to the city center (0.49, 0.51, 0.50) for the three group of zones,

A paradigm for location and multidimensional problems in planning respectively indicating the importance of the C.B.D., relative to other “centers” of employment, for the distant group of zones (&, &, R,,). Furthermore, the resulting weight of “access to employment” is much greater than the weight of the second component of accessibility, “access to shopping and amenities”. Although factors have been prioritized using an absolute number scale, the normalized eigenvector solution (CAM) can be directly compared with an attitudinal survey data[ 191 which reports fractional values for the importance of factors surveyed. This survey data reports on a (1975) household attitudinal survey conducted for a group of metropolitan regions on factors that households consider in the choice of housing and neighborhood location. The estimated weight for access to shopping and amenities for the three groups of zones are (0.05; 0.05; 0.04). The survey data reports a value of (0.049). The relative importance of “housing unit cost” for the three zonal clusters are (0.29); 0.30; 0.22) in the AHP, compared to the survey data reporting (0.218) for the same factor. Next are the weights indicating the importance of “neighborhood status” (0.12; 0.11; 0.12) which although not as significant as the weight of accessibility, emphasize the consideration of neighborhoods as a factor in residential location decision. The weight for this factor compares very closely with the survey data reporting (0.139).

The resulting composite value of the weights provides AHP estimates that approximate the locational distribution of the zonal population (Table 4). Further investigation of locational hypotheses, including, in particular, the effects of the “life cycle” on location, unaccounted for in the spatial interaction type models of location, and alternative specification of the elements and the dimensions of a hierarchy system, contrasting simultaneous versus sequential decision-processes given location, housing, neighborhood and accessibility interrelationships, can be made via the “structural” properties of AHP. Whereas macro models of spatial interaction variety have provided adequate metropolitan-wide, regional projection and analysis, procedures of the sort proposed here appear more suitable for the behavioral analysis of household decision-process with a small number of disaggregated (or aggregated) zones. The behavioral decision processes have been obscured in models using highly aggregated data, and the application of the macro models of the spatial interaction variety does not appear suitable when the number of zones is small. The census reports, attitudinal surveys and “synthetic” sample data, provide a variety of extraneous information which has not been advantageously utilized in models, since a model structure can impose strict restrictions as to the type of data which is inputed. The proposed procedure provides an alter-

Table 1. Dimensions, eigendata and composition

R,, R,, R,

1.5.5011001 a23 bfws-d*fcw= 3 3

1 .33 .25 0 1 3 0 0 1 a25 nx~~e.xmk 3 3

LCIMrm\=

5.54

co14s,

rax,s

0.14

WEIGHTS: 1 2 3 4 5 LEi!EL 2 MaTR*X LAhmGa CONS ID:< WEXGHTS 1 2 3

: : : :

163

0.39 0.05 0.12 0.49 0.05 i 3.16 0.13

2 3.00 0.00

3 3.02 0.01

4 3.21 0.11

5 3.01 0.00

0.30 0.16 0.54

0.20 0.40 0.40

0.12 0.32 0.56

0.12 0.58 0.30

0.16 0.30 0.54

CDMPOSITION 0.18 0.40 0.42

164

A. R. BANA&A.SHANI Table 2. Dimensions, eigendata and composition R,,, R,,, R,,

1 bE_vEL 2

7 7 .25 7 0

1 ,142

,125

.15

0 0 1 .14250001700001

1 1 .33 0 1 .33 0 0 1 EtiITEFia33 DTME,.~STO,4= 3 3 0: 1 .5 .33 0 1 .5 0 0 1 ENTER a33 DIME,-ISID,+= 3 3 0: 9 .5 .5 0 1 1 0 0 1 ENTER II: ENTER 0:

a24

DI'ME,-LSIOz+=3 3

123012001 025 DIMENSTclN=

3 3

111011001 RESULTS DF EltrEt+DfaTB ClPIDCDHFOSIT~ON LEVCL 1 LclHDL)= 5.94 CWE., TDX,= 0.24 WEIGHTS: 1 0.30 2 0.03 3 0,ll 4 0.51 5 0.05 LEVEC; 2 4 HLITRTX : 1 2 3 3.01 LAMno : 3.00 3.01 3.00 0.00 cot,5 In:.:: 0.00 0.00 0.00 WEIGHTS : 0.54 1 0.20 0.16 0.20 0.30 3 0.20 0.30 0.40 0.16 3 0.60 0.54 0.40

native to the empirical modeling approaches that are critically dependent on the availability and quality of the data used to formulate or verify hypotheses. More importantly however, procedures of the sort proposed here can be effectively utilized to assess the “data” itself, given the simulation problem. For a large class of existing models, data is so instrumental that often analysis can not proceed unless data has been obtained. When the “revealed” characteristics do become known, observational data, however necessary for the study of the revealed characteristics, does not provide a sufficient basis for normative analysis and planning. This suggests the priority given to alternative approaches providing both analytic (positive) as well as synthetic (normative) measurements of the social system and its related subsystems. The use of AHP as a procedure for both analysis and design has shown minimal dependence on data. The inadequacy and/or bias in data can be compensated by incorporating “consistent” analytical and synthesizing judgement. The consistency of the judgements, however, can be assessed in AHP so that ratio estimates, as a sample collection, are closer to being

5 3.00 0.00 0.33 0.33 0.33

COYFDSlTION 0.38 0.18 0.34

logically related rather than randomly chosen [20]. The importance of the role of judgement is realized given that decision/policy-making and planning are conducted even in the absence of observational data. The role of judgement in “deterministic” models is in contrast to the data which is randomly processed in probabilistic models and this contrast suggests a criticism made of both deterministic and probabilistic simulation models. Computer-based simulation models have sometimes been characterized as the “black box”, hence, a criticism that an otherwise, useful “process information”, as in the “sketch” planning procedure, has not been advantageously incorporated as a part of both analysis and synthesis. It is important to raise the question of the correspondence of a model (structure) with the “reality” that a model is presumed to signify. In light of recent contentions in favor of dynamics and non-equilibrium systems[6] and, the limitation of optimization processes that produce (optimal) solutions that deteriorate with change[5], robust alternatives that do not require the conventional, empiricist wisdom of imposing simplifying restrictions upon the properties of a behavioral sys-

A paradigm for location and multidimensional problems in planning

165

Table 3. Dimensions, eigendata and composition T,,, R,,, R,,

LEVEb

2

MCITRIX.

:

LPHDP

:

cows xn:: : WEIGHTS : 1 2 3

1 3.09 0.04

3.05 0.03

3 3.3i 0.19

4 3.0: 0.00

S 3.05 0.03

0.09 0.63 0.38

0.26 0.33 0.41

0.22 0.41 0.37

0.54 0.30 0.16

0.20 0.31 0.49

E:15,W16,F.17

2

COHCOSXTIOE~ 0.38 0.38 0.23

HIEF:l

tern are found more plausible. Approaches that can trace the dynamics of a system in a process of capturing the changing parametrical values as well as functional relationships provide an alternative conception of social systems. AHP is to be considered as a means towards this aim providing a process in which functional values can be generated and investigated without having specified the function itself. The changing relative weight of the elements in Hollarchy Systems[l2] is a process in which the sensitivity of the system elements can be investigated to make inferences regarding the dynamic performance of the “decision-making” systems. It should be pointed out, however, that some, among systems approachers, actually despise the notion of hierarchy as a particularly useful typology of systems. Yet, when an intrinsically systemic problem in the specification of variables has been considered, i.e. the causality problem, or, the problematic of exogeneous and endogenous variable specifications, the inclusiveness of hierarchy systems is realized as a highly plausible system conception. To emphasize, with the addition of the dimensions to a hierarchy system, the elements comprising of the preceding dimensions, considered instruumental at a given instance, assume alternatively, non-instrumental functional specification in another instance. Hierarchy systems, in other words, have the property of “open” systems. Furthermore, the variant dimensions of a

hierarchy system can robustly represent actors, actions, policies, decisions, and choice [ 111which can be effectively used for the systemic modeling of the diverse aspects of the locational problems which are multi-faceted. Furthermore, the changing relative weight of the elements, in their corresponding dimensions, suggests the dynamics in the structure of the interrelationship of the specified elements. The notions of dynamics and interaction also hold epistemically when viewed systemically. Epistemically, the processes of the production of paradigms suggest a system of concepts, methods, theories that can be relationally evaluated. An “epistemic” system, furthermore, can be denoted by a hierarchy system in which a hierarchy of concepts is implied. The contradictory or dilemmatic dimensions of competing alternative paradigms can be assessed more adequately within such an epistemic system. To state that a system of concepts can be represented by an epistemic, hierarchy system, however is not to state that the process of the production of concepts are necessarily hierarchial as is commonly believed[21]. The dynamics of such an epistemic system, furthermore suggests the changing structure of the relationship between concepts, theories and methods affecting the processes of the affirmation or refutation of alternative paradigms. It is through the study of the dilemmatic dimensions of paradigms, vis-a-vis problem dimensions, that the adequacy or deficiency

166

A. R.

BANAI-KAWANI

Table 4. Location data and AI-IS estimate 1

2

croup Ring

Observed Pop. (1970)

I

Nun. ROOM

SUPPlY sq. Hile

Observed Pop. Norm

Observed

Inv. Area

Iknaity

2

1

nousing Zone by

Pop,

Observed POP. NOIt

Pop.

Estimate AIK

Rl

16451

3.75

9520

.017

.375

22.05

.75

.2lS

.224

R2

27759

4.00

8040

.n20

.750

37.5

1.00

.36S

.382

.40

R3

31226

4.25

6240

.026

30.46

1.20

.413

.392

.42

Total

75436

RIO

16625

6.00

1480

.lOS

1.75

16.18

1.90

.362

.438

.38

RI1

14025

6.25

1220

.131

1.75

13.35

2.00

.305

.28

R12

15215

6.50

1020

.157

2.25

14.28

2.00

.331

.302 .325

Total

45865

1

. IS

II

III

.34

Rl5

9135

7.25

560

.286

2.25

7.65

2.10

.391

.411

.38

R16

7709

7.25

440

.400

2.25

5.62

2.10

.330

.294

.3S

R17

6503

7.25

360

.400

2.25

5.62

2.15

.278

-294

.23

Total

2347

u

Quarter

0

Work

squsre

flilc

Zones

Place

city

limit

t

,

RIR2

R3 OiBt.

from

city

center-R,0

RIG R12

4 Spaclnp, of

So,,rcc:

Anas,

of the alternative problem-solving paradigms can be fully assessed, including, of course, the proposed paradigm of AHP. REFERENCES

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Peterson.

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Spacing

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adjustment

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