Application of fuzzy measures in multi-criteria evaluation in GIS

int. j. geographical information science, 2000, vol. 14, no. 2, 173± 184

Research Article Application of fuzzy measures in multi-criteria evaluation in GIS HONG JIANG Department of Geography, University of Wisconsin at Madison, Madison, WI 53706, USA

J. RONALD EASTMAN Graduate School of Geography, Clark University, Worcester, MA 01610, USA (Received 8 February 1997; accepted 23 June 1999 ) Abstract. Multi-criteria evaluation (MCE) is perhaps the most fundamental of decision support operations in geographical information systems (GIS). This paper reviews two main MCE approaches employed in GIS, namely Boolean and Weighted Linear Combination (WLC), and discusses issues and problems associated with both. To resolve the conceptual di€ erences between the two approaches, this paper proposes the application of fuzzy measures, a concept that is broader but that includes fuzzy set membership, and argues that the standardized factors of MCE belong to a general class of fuzzy measures and the more speciŽ c instance of fuzzy set membership. This perspective provides a strong theoretical basis for the standardization of factors and their subsequent aggregation. In this context, a new aggregation operator that accommodates and extends the Boolean and WLC approaches is discussed: the Ordered Weighted Average. A case study of industrial allocation in Nakuru, Kenya is employed to illustrate the di€ erent approaches.

1.

Introduction Increasingly, geographical information systems (GIS) are used as an important aid for spatial decision making (Carver 1991, Pereira and Duckstein 1993 ). Recent developments in GIS have led to signiŽ cant improvements in its capability for decision making processes in land allocation and environmental management, among which Multi-Criteria Evaluation (MCE) is one of the most important procedures (Janssen and Rietveld 1990, Burrough et al. 1992, Jankowski 1995 ). In the context of GIS, two procedures are common for MCE. The Ž rst involves Boolean overlay whereby all criteria are assessed by thresholds of suitability to produce Boolean maps, which are then combined by logical operators such as intersection (AND) and union (OR). The second is known as weighted linear combination, wherein continuous criteria ( factors) are standardized to a common numeric range, and then combined by weighted averaging. The result is a continuous mapping of suitability, that may then be masked by one or more Boolean constraints to accommodate qualitative criteria, and Ž nally thresholded to yield a Ž nal decision (Hall et al. 1992, present a similar comparison). A comparison of the two approaches International Journal of Geographical Information Science ISSN 1365-8816 print/ISSN 1362-3087 online © 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals/tf/13658816.html

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is illustrated in ® gure 1 using industrial allocation in Nakuru, Kenya as an example; details are discussed in a later section. Despite the very common use of these procedures, there are some fundamental problems associated with their use. The ® rst problem has to do with di€ erent aggregation methods employed in decision making. Despite a casual expectation that the Boolean and weighted linear combination methods should yield similar results, they very often do not (see ® gure 1 ). The reason has to do with their logic of aggregation, particularly with tradeo€ . For example, Boolean intersection results in a very hard AND; a region will be excluded from the result if any single criterion fails to meet its threshold. Conversely, the Boolean union operator implements a very liberal mode of aggregation: a region will be chosen in the result as long as a single criterion meets its threshold. Weighted Linear Combination is quite di€ erent from these Boolean options. Here a low score on one criterion can be compensated by a high score on anotherÐ a feature known as tradeo€ or substitutability . The second problem with MCE has to do with the standardization of factors in weighted linear combination. The most common approach is to rescale the range to a common numerical basis by simple linear transformation. However, the rationale for doing so is unclear (Voogd 1983, Eastman et al. 1993 ). Indeed, there are many instances where it would seem logical to rescale values within a more limited range. For example, if proximity to roads is a factor in determining suitability for industrial development, it may be that a location 10 km away is no less suitable than one that is 5 km awayÐ they are both simply too far to consider. In some cases, a non-linear scaling may seem appropriate. Since the recast criteria really express suitability , there are many cases where it would seem appropriate that criterion scores asymptotically approach the maximum or minimum suitability level. The third problem concerns decision risk. Decision risk may be considered as the likelihood that the decision made will be wrong (Eastman 1996 ). For the Boolean procedure, decision risk can be estimated by propagating measurement error through the decision rule, whereby determining the risk that the decision made for a given location is wrong. In comparison, the continuous criteria of weighted linear combination would appear to express a further uncertainty that is not so easily estimated with stochastic methods. The standardized factors of weighted linear combination each express a perspective of suitability: the higher the score, the more suitable the location is for the intended land use. There is no real threshold, however, that allows de® nitive allocation of areas to be chosen and areas to be excluded. How are these uncertainties to be accommodated in expressions of decision risk? If these criteria really express uncertainties, why are they combined through an averaging process? It is the contention here that all these questions can be resolved by considering decision making as a set problem and through the application of fuzzy measures in MCE. 2.

Decision making and fuzzy measures A decision is a choice between alternative actions, hypotheses, locations, and so on (Eastman et al. 1993 ). Taking each possible combination of alternative as a subset, the decision process is essentially one of allocating alternatives to the subset that will be chosen or acted upon. In the case of single-objective land allocation, a decision set contains two subsets: suitable (for allocation), and not suitable (for allocation). A decision is then derived from an assessment of suitability, the degree to which a location belongs to the set `suitable’ (in most cases, fuzzy membership in

Fuzzy measures in multi-criteria evaluation

Figure 1.

GIS procedures for multi-criteria evaluation (MCE).

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the set `not suitable’ is assumed to be the complement of `suitable’ set membership). In most decision making processes, multiple criteria are considered to assess the degree of suitability each location bears to the allocation under consideration. Thus suitability is commonly not Boolean in character, but expresses varying degrees of set membership, i.e. a fuzzy set. Each criterion chosen by the analyst thus constitutes direct or indirect evidence, based on which fuzzy set membership (or suitability) of a location can be evaluated. We recognize two kinds of criteria, factors and constraints , with a factor signifying a continuous degree of fuzzy membership (in the range of 0± 1 ), and constraints acting to limit the alternatives altogether (i.e. fuzzy membership is either 0 or 1 ) (Eastman et al. 1993 ). The latter is a special case of fuzzy sets. Fuzzy set theory has not uncommonly been applied in multi-criteria decision making (Burrough 1989, Wang et al. 1990, Smith 1992, Xiang et al. 1992, Banai 1993 ). In the GIS context, in decision making regarding land allocation, suitability is considered a fuzzy concept expressed as fuzzy set membership (Burrough et al. 1992, Hall et al. 1992 ). There are a variety of reasons why the application of fuzzy set membership in criteria standardization is highly appealing. First, it provides a very strong logic for the process of standardization. The process of criteria standardization can be seen as one of recasting values into a statement of set membership, the degree of membership in the ® nal decision set. Compared with linear scaling, standardization using fuzzy set membership represents a speci® c relation between the criterion and decision set. This clearly opens the way for a broader family of set membership functions than that of linear rescaling. For example, the commonly used sigmoidal function provides a simple logic for cases where a function is required that is asymptotic to 0 and 1. Second, the logic of fuzzy sets bridges a major gap between Boolean assessment and continuous scaling in weighted linear combination. Boolean overlay is a classical set problem and assumes a crisp boundary and certainty. By considering the process of criteria evaluation as one of fuzzy sets, the continuity and uncertainty in the relation between criteria and decision set are recognized. When set membership values are reduced to 0 and 1 (i.e. certainty), the set becomes crisp (a special case of fuzzy set ) and the results will be identical to those of Boolean overlay. In this paper, we expand the application of fuzzy sets in MCE to a broader concept: fuzzy measures. The term F uzzy Measure refers to any set function which is monotonic with respect to set membership (Dubois and Prade 1982 ). As an inclusive concept, fuzzy measures include Bayesian probabilities , the beliefs and plausibilities of Dempster-Shafer theory, and the possibilities of fuzzy sets, thus providing a uni® ed framework for the methodologies of uncertainty studies (Bonissone and Decker 1986 ). Although an important concept, fuzzy measure has not been widely recognized and applied in decision making. Since the likely mistake of reducing fuzzy measure to fuzzy set will limit the exploration of this concept, we want to emphasize here that fuzzy measure is a broader concept than fuzzy set, although our exploration of its application in MCE is based largely on the experience of fuzzy set application. Considering standardized criteria as fuzzy measures, as will be shown in later discussions, expands our understanding of MCE procedures, allows more ¯ exible MCE operations, and helps to resolve the problems associated with Boolean and weighted linear operations as discussed earlier. When the standardization of a single criterion is considered a fuzzy measure, MCE concerns the manner in which multiple fuzzy measures derived from di€ erent

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evidence are combined into a single statement of set membership, which represents the ® nal degree of suitability in a location. The combination approach chosen represents an important decision rule. This leads us to explore the properties of fuzzy measures and approaches in their combination. 3.

Properties of fuzzy measures A common trait of fuzzy measures is that they follow DeMorgan’s Law in the construction of the intersection and union operators (Bonissone and Decker 1986 ). DeMorgan’s Law establishes a triangular relationship between the intersection, union and negation operators such that: T (a,b)= ~ S(~ a,~ b )

where T = Intersection (AND) = T -Norm, S = Union (OR) = T -Conorm, and ~ = Negation (NOT ). The intersection operators in this context are known as triangular norms , or simply T -Norms, while the union operators are known as triangular co-norms , or T -Conorms. A T -Norm can be de® ned as (Yager 1988 ): a mapping T: [0,1] * [0,1]  [0,1] such that: T (a,b)= T (b,a ), T (a,b)> = T (c,d ) if a > = c and b > = d T (a,T (b,c)) = T (T (a,b),c) T (1,a)= a

(commutative) (monotonic) (associative) (boundary)

Some examples of T -Norms include: min(a,b) ab 1Õ min( 1,(( 1Õ max( 0, a +b Õ

(the intersection operator of fuzzy sets) (the intersection operator of probabilities) a ) 9 p +(1Õ b ) 9 p ) 9 (1/ p )) (for p > = 1) (Yager’s T -Norm operator) 1) (bounded di€ erence)

Conversely, a T -Conorm is de® ned as: a mapping S: [0,1] * [0,1]  [0,1] such that: S(a,b)= S(b,a) S(a,b)> = S(c,d ) if a > = c and b > = d S(a,S(b,c)) = S(S(a,b),c) S(0,a)= a

(commutative) (monotonic) (associative) (boundary)

Some examples of T -Conorms include: max(a,b) a +b Õ ab min( 1,(a 9 p +b 9 p) 9 (1 / p )) (for p > = 1 ) min( 1, a +b )

(the union operator of fuzzy sets) (the union operator of probabilities) (Yager’s T -Conorm operator) (bounded sum)

In the above equations, a, b, c and d are fuzzy measures, and p is a power term. These examples show that a very wide range of operations are available for fuzzy measure aggregation, therefore multi-criteria aggregation in decision making

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processes. Among the di€ erent operators, the most signi® cant is MIN for T -Norm, and MAX for T -Conorm, operations employed widely in fuzzy set membership problems including applications in MCE. These two operators also signify the upper and lower boundaries of T -Norm and T -Conorm, respectively: 0 < = T (a,b)< = min (a,b ) 1 > = S(a,b)> = max (a,b ) The aggregation approaches of fuzzy measures release us from the limitations of fuzzy set integrative operations (i.e. MIN and MAX), and lead us to believe that integration methods of multi-criteria need to go beyond the common approaches of union, intersection, and weighted linear combination. Furthermore, from the boundary conditions of T -Norm and T -Conorm, the range between min(a,b ) and max (a,b) are left un-represented, where weighted linear combination is found. It is this unexplored range in the aggregation of fuzzy measures that is examined in the later part of this paper in the design of MCE procedures, following Yager (1988 ). Thus, fuzzy measures provide a theoretical base from which to explore an expanded understanding of MCE processes and the design of new aggregation operators. 4.

Weighted linear combination as a fuzzy operator While MIN and MAX operators are used by some for Boolean overlay and fuzzy membership aggregation in MCE, the averaging operator of weighted linear combination is also widely used by others (Buckley 1988 ). There is a lack of association between the two traditions, as the latter is not considered as a fuzzy logical operator because it lacks the property of associativity (Bonissone and Decker 1986 ). Can the aggregation process of weighted linear combination be seen as a fuzzy measure aggregation? The answer is yes, and the reason lies in the practices in decision making. Bonissone and Decker (1986 ) also indicate that although lacking certain properties of T -Norms and T -Conorms, the averaging operator is yet another basic function that satis® es the condition of tradeo€ . Both MIN/ MAX and averaging traditions are well known in human experience. The minimum operator commonly represents a form of limiting factor analysis. Here the intent is one of risk aversion, by characterizing the suitability of a location in terms of its worst quality. The maximum operator is the opposite, and can thus be thought of as a very optimistic aggregation operator: an area will be suitable to the extent of its best quality. Interestingly, the averaging operator falls mid-way between the extreme cases of the T -Norm (the minimum operator) and its corresponding T -Conorm (the maximum operator). It is neither an AND nor an OR operator, but rather, one that lies halfway in between: in essence, a perfect ANDOR operator representing attitudes at a middle point. It is, therefore, not suprising that the averaging operator is used more often in decision making processes. This echoes Yager (1988 ), who maintains that in many cases the type of aggregation operator desired by decision makers lies somewhat between `the pure ``anding’’ of the t-norm with its complete lack of compensation’ and `the pure ``oring’’ of the S operator with its complete submission to any good satisfaction as well as its indi€ erence to the individual criteria (p. 184 )’. The combination of the min/ max and weighted linear traditions becomes increasingly attractive (Yager 1988, Eastman and Jiang 1996 ). To combine the above two traditions, it is highly appealing to consider weighted linear combination as a fuzzy membership operator, together with the MIN and

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MAX operators, in the framework of fuzzy measures. It provides a strong logic to bridge the two traditions in both decision science (i.e. MIN/ MAX and weighted linear combination) and in GIS (i.e. Boolean operation in Vector GIS and averaging operation in Raster GIS). Perhaps most importantly, it allows us to explore a variety of operators that fall between the MIN (AND) and MAX (OR) operators, and as we shall see, this allows us to control not only the degree of ANDORness but also the degree of tradeo€ in MCE. 5.

The Ordered Weighted Averaging (OWA) The Ordered Weighted Averaging (OWA) operator provides continuous fuzzy aggregation operations between the fuzzy intersection (MIN or AND) and union (MAX or OR), the upper boundary of T -Norm and lower boundary of T -Conorm, with weighted linear combination falling midway in between. It adopts the logic of Yager (1988 ) and can achieve continuous control over the degree of ANDORness of the operator and the degree of tradeo€ between criteria. In Yager’s implementation, criteria are weighted on the basis of their rank order rather than their inherent qualities. For example, we might decide to apply weights of 0.5, 0.3, 0.2 to a set of factors A, B and C based on their rank order. If at one location the criteria are ranked BAC (from lowest to highest ), the weighted combination would be 0.5 B +0.3 A +0.2 C. However, if at another location the factors are ranked CBA, the weighted combination would be 0.5 C +0.3 B +0.2 A. In our implementation of Yager’s concept (Eastman and Jiang 1996 ), we have retained the concept of weights that apply to speci® c criteria (factors) as in traditional weighted linear combination techniques, yielding two sets of weightsÐ criterion weights that apply to speci® c criteria and order weights that apply to the ranked criteria after the application of the criterion weights. In traditional weighted linear combination, criteria weights determine how factors tradeo€ relative to each other. However, the level of tradeo€ is not adjustable, and full tradeo€ is always assumed. Here, criteria weights are adjusted according to the level of tradeo€ , such that they retain their full signi® cance when full tradeo€ is chosen and gradually lose their meaning (i.e. criteria weights become equal ) as one approaches no tradeo€ . While criteria weights are associated with the relative signi® cance of a particular criterion for the decision set as well as the way criteria compensate for each other, order weights control the position of the aggregation operator on a continuum between the extremes of MIN and MAX, as well as the degree of tradeo€ . We use two parameters, ANDORness and TRADEOFF, to characterize the nature of an OWA operation: 1 ))ž

ANDness = (1 / (n Õ ORness = 1 Õ T RADEOFF = 1Õ

((n Õ

i)W orde r i )

ANDness

S nž

(W orde r i Õ 1/ n )2 nÕ 1

(1 ) (2 ) (3 )

in which n is the total number of factors, i is the order of factors, and W orde r i is the weight for the factor of the i th order. From the equation, ANDness or ORness is governed by the amount of skew in the order weights and tradeo€ is controlled by the degree of dispersion in the order weights. ANDORness re¯ ects an attitude toward risk in decision making. It is

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represented by ANDness, the degree to which an operator is close to the MIN (AND) operation; and ORness, the degree to which an operator is close to the MAX (OR) operation. ANDness and ORness are complimentary to each other and their sum yields a value of 1. At one extreme, absolute ANDness (where ANDness equals 1 ) represents the most risk-aversion position, and at the other, absolute ORness (where ORness equals 1 ) represents the most risk-taking attitude. The second parameter, TRADEOFF, represents the degree to which di€ erent factors are allowed to tradeo€ with one another. The interesting feature of OWA is that it is possible to control a continuous degree of ANDORness and tradeo€ (® gure 2 ). For example, using order weights of [1 0 0] yields the minimum operator of fuzzy sets with full ANDness (totally skewed ) and no tradeo€ (not dispersed ). Using order weights of [0 0 1] yields the maximum operator of fuzzy sets with full ORness (skewed in the opposite direction) and no tradeo€ . Using order weights [0.33 0.33 0.33] yields the traditional averaging operator of MCE with intermediate ANDness and ORness, and full tradeo€ (totally dispersed ). Order weights of [0 1 0] would yield an operator with intermediate ANDness and ORness, but no tradeo€ , while the original example with order weights of [0.5 0.3 0.2] would yield an operator with substantial tradeo€ and a moderate degree of ANDness. The OWA procedure has been incorporated in the IDRISI for Window GIS software package. In use, the OWA procedure can be applied in a single operation, or as part of a hybrid design. For example, if one had ® ve factors, where it was desired that three should trade o€ but that two should not, the problem can be solved in two stages. In the ® rst stage, OWA would be run using order weights with the desired degree of tradeo€ , followed by a second run without tradeo€ in which the result of the ® rst run was included as an additional factor. Criterion weights in the two runs can easily be adjusted to maintain the original weights of all factors. 6.

Case study: industrial allocation in Nakuru, Kenya Nakuru town is a rapidly growing urban area in Kenya, where industry is also expanding. This case study requires that certain areas of the most suitable land be selected for industrial development. Within the study area lies Lake Nakuru National Park, which is designated as a wildlife reserve of international importance. In discussions with local groups and decision makers, the following factors were identi® ed to be important in industrial land allocation: proximity to road, because of the need for transporting raw products and ® nished materials; proximity to the town, where the labor source is located; slope gradients, which a€ ects construction cost; and distance from Lake Nakuru Park, which needs to be preserved as both a wildlife refuge and a tourist attraction. The above factors serve to either enhance or distract

Figure 2. Decision strategy space in OWA.

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the suitability of industrial allocation. One constraint is identi® ed: land within the park boundaries is prohibited from industrial development. To explore di€ erent MCE approaches, we ® rst employ Boolean and weighted linear combination techniques, and the results are shown in ® gure 1. The Boolean method results in 1670 ha of land suitable for industrial development. In the right column is weighted linear combination example, where each factor has been standardized using a FUZZY sigmoid-shaped function, and factor weights, generated using pairwise comparison method of the Analytical Hierarchy Process (AHP) (Saaty 1980, Eastman et al. 1993 ) have been applied. The factor weights for the four factors are 0.3770, 0.0979, 0.3605, and 0.1647, respectively. For the purpose of comparison, we have allocated the same area of land (i.e. 1670 ha) as a result of weighted linear combination. It is apparent from ® gure 1 that the results from the Boolean and weighted linear combination traditions are very di€ erent. With the same standardized factor and set of factor weights, we apply di€ erent OWA approaches in MCE aggregation, and the results for industrial suitability are shown in ® gure 2. Table 1 shows the order weights employed to yield the results in ® gure 3. These weights determine the position of the aggregation operation along the continuum between the MIN and MAX operators, and their corresponding ANDness, ORness and TRADEOFF. The di€ erences in the MCE results are apparent! From ® gure 2, as the ORness increases from MIN to MAX, suitability scores increase (represented by the dark green and green colours). After the suitability images are ranked and the same size of highly suitable area selected, the spatial di€ erences among di€ erent OWA procedures are also signi® cant. For example, when 1670 ha is selected for industrial allocation for all OWA procedures, between the MIN and AVERAGE operators, the spatial consistency (areas chosen by both operators) is 67%, and between the AVERAGE and MAX operators, the consistency is only 34%. Do the varying results of MCE make decision making easier or more di cult? What should be the ® nal decision? These questions remain to be answered by decision makers. Selection of any criteria-aggregation approach without scrutiny is clearly questionable. The aim of this study is to point out the strong dependency between the manner in which the decision rule is constructed and the character of the results, as are shown in ® gures 1, 2 and 3. The results from di€ erent OWA operators demonstrate the importance of examining di€ erent aggregation approaches, and of justifying a rationale for the one that is adopted. OWA provides a tool that allows for the exploration of a range of aggregation rules using di€ erent degrees of ANDORness and TRADEOFF. Decision makers can then carefully evaluate each decision scenario and choose the one that can best suit their needs Table 1. Di€ erent OWA procedures in ® gure 2. OWA operator MIN Average MAX

Order weights

ANDness

ORness

Tradeo€

1, 0, 0, 0 0.6, 0.2, 0.15, 0.05 0.4, 0.3, 0.2, 0.1 0.25, 0.25, 0.25, 0.25 0.1, 0.2, 0.3, 0.4 0.05, 0.15, 0.2, 0.6 0, 0, 0, 1

1 0.7833 0.6667 0.5 0.3333 0.2167 0

0 0.2167 0.3333 0.5 0.6667 0.7833 1

0 0.517 0.7418 1 0.7418 0.517 0

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Figure 3.

OWA results with di€ erent order weights.

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and expectations. Apparently, the selection process is wrought with subjectivity, which is an important message this study tries to convey. The application of the OWA operator in MCE has strong implications for the decision making process and decision risk. Because di€ erent aggregation approaches can yield strikingly di€ erent results, the decision rule is clearly a source of uncertainty in the decision making process (in addition to measurement error), and has to be considered in the process of decision making and the evaluation of decision risk. 7.

Conclusion This paper adopts the concept of fuzzy measure and applies it to MCE procedures. As has been demonstrated here, the standardized criteria of MCE belong to the general class of fuzzy measures. Therefore, integration of multiple criteria follows the broad approaches in fuzzy measure integration. By considering weighted linear combination as a form of fuzzy measure integration, we expand the logical fuzzy measure operators of T -Norm and T -Conorm to include the unexplored range between the MIN and MAX operators, in which weighted linear combination is found at the midpoint. Furthermore, we have designed a GIS implementation of the Ordered Weighted Average (OWA) operator that allows the exploration of a full range of operators between MIN and MAX, with varying degrees of ANDORness and tradeo€ . The OWA operator provides a consistent theoretical link between the two common MCE logics of Boolean overlay and weighted linear combination, and opens up the possibilities for aggregation of criteria. To this end, OWA is presented as an extension to the traditional MCE aggregation operators, and is demonstrated through a case study of industrial land allocation in Nakuru, Kenya. As a relatively new topic of research, the application of fuzzy measures in MCE in general and OWA in particular require further research. Although theoretically appealing and capable of a wide range of MCE integrations, OWA is yet to be tested in a variety of decision making processes and contexts to determine how it may help decision makers in the real world; guidelines as to how decision makers may choose the appropriate ANDORness (re¯ ecting risk attitude) and degree of tradeo€ remain to be formulated under di€ erent contexts. References Banai, R., 1983, Fuzziness in geographical information systems: contributions from the analytical hierarchy process. International Journal of Geographical Information Systems, 7, 315± 329. Bonissone, P. P., and Decker, K., 1986, Selecting uncertainty calculi and granularity: an experiment in trading-o€ precision and complexity. In Uncertainty in Arti® cial Intelligence, edited by L. N. Kanal and J. F. Lemmer (North Holland: Elsevier Science), pp. 217± 247. Buckley, J. J., 1984, The multiple judge, multiple criteria ranking problem: a fuzzy set approach. Fuzzy Set and Systems, 13, 25± 37. Burrough, P. A., 1989, Fuzzy mathematical methods for soil survey and land evaluation. Journal of Soil Science, 40, 477± 492. Burrough, P. A., MacMillan, R. A., and van Deursen, W., 1992, Fuzzy classi® cation methods for determining land suitability from soil pro® le observations and topography. Journal of Soil Science, 43, 193± 210. Caver, S. J., 1991, Integrating multi-criteria evaluation with geographical information systems. International Journal of Geographical Information Systems, 5, 321± 339. Dubois, D., and Prade, H., 1982, A class of fuzzy measures based on triangular norms. International Journal of General Systems, 8, 43± 61.

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