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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.
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