This file was created by scanning the printed publication. Errors identified by the software have been corrected; however, some errors may remain. Fuzzy Measures in Multi-Criteria Evaluation J. Ronald ~ a s t m a nand ' Hong ~ i a n g ~ Abstract. -- A consideration of Multi-Criteria Evaluation (MCE) in GIs leads to the conclusion that the standardized factors of MCE belong to a general class of fuzzy measures and the more specific instance of fuzzy sets: a perspective that provides a strong theoretical basis for the standardization of factors and their subsequent aggregation. In this context, a new aggregation operator that permits continuous variation of ANDORness and tradeoff is discussed: the Ordered Weighted Average. INTRODUCTION Multi-Criteria Evaluation (MCE) is perhaps the most fundamental of decision support operations in GIs. MCE is most commonly achieved by one of two procedures. The first involves boolean overlay whereby all criteria are reduced to logical statements of suitability and then combined by means of one or more logical operators such as intersection or union. The second is known as weighted linear combination wherein continuous criteria @ctors) are standardized to a common numeric range, and then combined by means of a weighted average. The result is a continuous mapping of suitability that may then be masked by one or more boolean constraints to accommodate qualitative criteria, and finally thresholded to yield a final decision (Figure 1). Despite the very common use of these procedures, there are some fundamental problems associated with their use. First, despite a casual expectation that the two procedures should yield similar results, they very often do not (Figure 1). The reason clearly has to do with tradeoff, but also with the logic of the aggregation operation. For example, boolean intersection results in a very hard AND -- a region will be excluded from the result if any single criterion fails to exceed its threshold. Conversely, the boolean union operator implements a very liberal mode of aggregation -- a region will be included in the result even if only a single criterion meets its threshold. Weighted linear combination is quite unlike these options. Here a low score on one criterion can be compensated by a high score on another -- a feature known as tradeoffor substitutability. ' ' Director, The Clark Labs for Cartographic Technology and Geographic AnalysisIIDRISI Project, and Professor of Geography, Clark University, Worcester, MA 0 1610, USA. Research Analyst, The IDRISI Project, Clark University, Worcester, MA 01610, USA. Figure 1.-GIs procedures for Multi-Criteria Evaluation (MCE) Left: Boolean evaluation for areas suitable for industrial development based on four criteria: proxmity to road, proximity to the main town, slope gradient and distance from a wildlife reserve. The bottom is the result for industrial allocation produced by the intersection operator. Right: The same criteria as continuous factors, weighted linear com bination, and at the bottom industrial allocation with an area equal to that of the boolean result. Center: At the top is the constraint image with the area that precludes industrial development. On the bottom is the cross-tabulation image of the boolean and weighted linear results. 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 krn away -- they are both simply too far to consider. Furthermore, there are cases where 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. A third problem concerns decision risk. Decision risk may be considered as the likelihood that the decision made will be wrong (Eastman, 1996). For both procedures it is a fairly simple matter to propagate measurement error through the decision rule and subsequently to determine the risk that a given location will be assigned to the wrong set (i.e., the set of selected alternatives or the set of those not to be included). However, the continuous criteria of weighted linear combination would appear to express a further uncertainty that is not so easily accommodated. The standardized factors of weighted linear combination each express a perspective of suitability -- the higher the score, the more suitable. However, there is no real threshold that can definitively allocate locations to one of the two sets involved (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 these and all of the issues raised here can be resolved through a consideration of fuzzy measures. FUZZY MEASURES The termfuzzy measure refers to any set function which is monotonic with respect to set membership (Dubois and Prade, 1982). Notable examples of fuzzy measures include probabilities, the beliefs and plausibilities of Dempster-Shafer theory, and the possibilities of fuzzy sets. Interestingly, if we consider the process of standardization in MCE to be one of transforming criterion scores into set membership statements (i.e., the set of suitable choices), then standardized criteria are 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: where T = and S = and = - Intersection (AND) Union (OR) Negation (NOT) - T-Norm T-CoNorm 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. T-NORMS/T-CONORMS AND THE AVERAGING OPERATOR A T-Norm can be defined as (Yager, 1988): a mapping T: [O, 11 * [O,l] -> [O,11 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) Some examples of T-norms include: min(a,b) (the intersection operator of fuzzy sets) (the intersection operator of probabilities) a*b (for p>= 1) 1 - mh(1 ,((I-alAp + (1-bIAp)A(l/p)) max(O,a+b- 1) Conversely, a T-CoNorm is defined as: a mapping S: [O, 11 * [O, 11 -> [O,11 such that : S(a,b) = T(b,a) S(a,b) >= T(c,d) if a >= c and b >= d S(a,T(b,c)) = T(T(a,b),c) S(0,a) = a (commutative) (monotonic) (associative) Some examples of T-CoNorms include: (the union operator of fuzzy sets) max(a,b) (the union operator of probabilities) a+b-a*b (for p>= 1) min(1 ,(aAp+ bAp)"(l /p)) min(1,a+b) Interestingly, while the intersection and union operators of boolean overlay represent a T-NodT-CoNorm pair, the averaging operator of weighted linear combination cannot because it lacks the property of associativity (Bonissone and Decker, 1986). Rather, it has been determined (Bonissone and Decker, 1986) that the averaging operator falls mid-way between the extreme cases of the T-Norm of fuzzy sets (the minimum operator) and its corresponding T-CoNorm (the maximum operator) in the sense that it is neither an AND nor an OR operator, but rather, one that lies halfway in between -- in essence, a perfect ANDOR operator. This quality has interested some in the Decision Science field (e.g., Yager, 1988) because of the recognition that in human perception of decision logics, it is not uncommon to wish to combine criteria with something less extreme than the hard operations of union or intersection. What is particularly interesting here, however, is that the averaging operator falls along a continuum of ANDORness between the intersection and union operators of fuzzy sets. Can the aggregation process of weighted linear combination be seen as fuzzy set membership calculation? Given that it has already been determined that standardized criteria qualify as fuzzy measures, and that fuzzy sets are fuzzy measures, the concept is plausible. WEIGHTED LINEAR COMBINATION AND FUZZY SETS There are a variety of reasons why consideration of the weighted linear combination process as a fuzzy set membership aggregation operator is highly appealing. First, it provides a very strong logic for the process of standardization. In this context, the process of standardizing a criterion can be seen as one of recasting values into a statement of set membership -- the degree of membership in the final decision set. Second, 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. Third, the logic of fuzzy sets bridges a major gap between that of boolean overlay and that of the weighted linear combination. Boolean overlay is very clearly a classical set problem. By considering the process as one of fuzzy sets, the process of weighted linear combination can also be seen as a set problem. Fourth, in cases where set membership approaches certainty, the results from fuzzy sets will be identical to those of boolean overlay. If the problem can indeed be recast into the framework of fuzzy sets, is it reasonable that the minimum and maximum operators should function as the intersection and union operators? As it happens, both are well known in human experience. The minimum operator is commonly used in GIs applications, and represents a form of limitingfactor 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. That the averaging operator should fall half way between these extremes is not surprising. From left to right and top to bottom, ORness increases and AN Dness decreases. The order weights are [ I 0 0 01, [.6 .2 . I 5 ,051, [.4 .3 .2 .I], [.25 .25 .25 .25], [.I .2 .3 .4], [.05 .15 .2 .6], and [0 0 0 I],respectively. At the two extremes are the MIN and MAX operators Figure 2.-OWA results with different order weights THE ORDERED WEIGHTED AVERAGE With the recognition that the averaging function represents an ANDOR operator midway between the fuzzy intersection and union operators, and that it also provides full tradeoff between criteria, the question arises of the extent to which these two qualities might be varied independently. Looking to the field of Decision Science, a very interesting approach is that of Yager (1988). Yager has proposed a variant of the averaging operator, called an Ordered Weighted Average, that can achieve continuous control over the degree of ANDORness of the operator and the degree of tradeoff between criteria. In his implementation, the criteria are weighted on the basis of their rank order rather than their inherent qualities. Thus, for example, we might decide to apply weights of 0.5, 0.3, 0.2 to weight a set of factors A, B and C based on their rank order. Thus if at one location the criteria are ranked BAC (from lowest to highest), the weighted combination would be 0S*B + 0.3*A + 0.2 *C. However, if at another location the factors are ranked CBA, the weighted combination would be 0S*C + 0.3*B + 0.2 *A. In our implementation of Yager's concept, we have retained the concept of weights that apply to specific factors, yielding two sets of weights -- criterion weights that apply to specific criteria and order weights that apply to the ranked criteria, after application of the criterion weights. The interesting feature of the Ordered Weighted Average is that it is possible to control the degree of ANDORness and tradeoff between factors within limits. For example, using order weights of [I 0 0] yields the minimum operator of fuzzy sets, with full ANDness and no tradeoff. Using order weights of [0 0 11 yields the maximum operator of fuzzy sets with full ORness and no tradeoff. Using weights of [0.33 0.33 0.331 yields the traditional averaging operator of MCE with intermediate ANDness and ORness, and full tradeoff. Tradeoff is thus controlled by the degree of dispersion in the weights while ANDness or ORness is governed by the amount of skew. Thus, for example, order weights of [0 1 0] would yield an operator with intermediate ANDness and ORness, but no tradeoff, while the original example with order weights of [0.5 0.3 0.21 would yield an operator with substantial tradeoff and a moderate degree of ANDness. We have recently implemented the Ordered Weighted Average as a new module (called OWA) to be introduced with the next release of the IDRISI GIs software system. Figure 2 illustrates the output from this module using varying order weights. The results are strikingly different, pointing out the strong dependency between the manner in which the decision rule is constructed and the character of the result. In use, the OWA module can be used in a single operation, or as part of a hybrid design. For example, if one had five factors, where it was desired that three should trade off, but that two should not, the problem can be solved in two stages. In the first stage, OWA would be run using order weights with the desired degree of tradeoff, followed by a second run without tradeoff in which the result of the first 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. CONCLUSIONS It has been demonstrated here that the standardized criteria of MCE belong to the general class of fuzzy measures, and it has been fbrther proposed that they belong to the more specific instance of fuzzy sets. The proposal is appealing in that it provides a consistent theoretical link between the MCE logics of boolean overlay and weighted linear combination: the former representing set membership relations between crisp sets and the latter between fuzzy sets. It also provides a theoretical basis for the standardization of criteria: the process is seen as a recasting of criterion values into set membership statements. Furthermore, the simple monotonicity requirement of fuzzy measures permits a much richer range of standardization functions than that of linear rescaling, including sigmoidal and jshaped forms. However, perhaps the most appealing outcome of this perspective is the manner in which it opens up the possibilities for aggregation of criteria. To this end, the Ordered Weighted Average was demonstrated as an extension to the traditional aggregation operator that allows continuous control over the degree of ANDORness and tradeoff between factors. REFERENCES Bonissone, P.P. and Decker, K., (1986) "Selecting uncertainty calculi and granularity: an experiment in trading-off precision and complexity. In L.N.Kana1 and J.F. Lemmer eds., Uncertainty in Artijicial Intelligence (North-Holland: Elsevier Science). Dubois, D., and Prade, H., (1982) "A Class of Fuzzy Measures based on Triangular Norms", International Journal of General Systems, 8,43-6 1. 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