Possibility and necessity in weighted aggregation ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
robert.fuller@abo.fi
Szvetlana Fullér
sfuller@ra.abo.fi
Abstract
Yager [14] discussed the issue of weighted min and max aggregations and
provided for a formalization of the process of importance weighted transformation. Generalizing Yager’s principle we suggest the use of fuzzy implication
operators for importance weighted transformation.
1
Definitions
In this section we provide the definitions of terms needed in the process of weighted
aggregation. Triangular norms were introduced by Schweizer and Sklar [8] to model
the distances in probabilistic metric spaces. In fuzzy sets theory triangular norms are
extensively used to model logical connective and. Triangular conorms are extensively
used to model logical connective or.
Definition 1.1 A mapping
T : [0, 1] × [0, 1] → [0, 1]
is a triangular norm (t-norm for short) iff it is symmetric, associative, non-decreasing
in each argument and T (a, 1) = a, for all a ∈ [0, 1].
Definition 1.2 A mapping
S: [0, 1] × [0, 1] → [0, 1]
is a triangular co-norm (t-conorm for short) if it is symmetric, associative, nondecreasing in each argument and S(a, 0) = a, for all a ∈ [0, 1].
∗
The final version of this paper appeared in: R.R.Yager and J.Kacprzyk eds., The ordered weighted
averaging operators: Theory, Methodology, and Applications, Kluwer Academic Publishers, Boston,
[ISBN 0-7923-9934-X], 1997 18-28.
1
If T is a t-norm then the equality
S(a, b) := 1 − T (1 − a, 1 − b)
(1)
defines a t-conorm and we say that S is derived from T . The basic t-norms and
t-conorms pairs are
• minimum/maximum:
M IN (a, b) = min{a, b} = a ∧ b,
M AX(a, b) = max{a, b} = a ∨ b
• L
, ukasiewicz:
LAN D(a, b) = max{a + b − 1, 0},
• probabilistic: P AN D(a, b) = ab,
LOR(a, b) = min{a + b, 1}
P OR(a, b) = a + b − ab
• weak/strong:
W EAK(a, b) =
min{a, b} if max{a, b} = 1
0
ST RON G(a, b) =
max{a, b} if min{a, b} = 0
1
• Hamacher:
HAN Dγ (a, b) =
HORγ (a, b) =
otherwise
otherwise
ab
,
γ + (1 − γ)(a + b − ab)
a + b − (2 − γ)ab
, γ≥0
1 − (1 − γ)ab
• Yager:
Y AN Dp (a, b) = 1 − min{1,
Y ORp (a, b) = min{1,
p
√
p
(1 − a)p + (1 − b)p },
ap + bp }, p > 0
Definition 1.3 Let A and B be two fuzzy predicates defined on the real line IR. Knowing that ’X is B’ is true, the degree of possibility that the proposition ’X is A’ is true,
Π[A|B], is given by
Π[A|B] = sup{A(t) ∧ B(t)|t ∈ IR},
(2)
the degree of necessity that the proposition ’X is A’ is true, N [A|B], is given by
N [A|B] = 1 − Π[¬A|B],
where A and B are the possibility distributions (for simplicity we write A instead of
µA ) defined by the predicates A and B, respectively, and
(¬A)(t) = 1 − A(t)
for any t. We can use any t-norm T in (2) to model the logical connective and:
Π[A|B] = sup{T (A(t), B(t))|t ∈ IR}.
2
There are three important classes of fuzzy implication operators:
• S-implications: defined by
x → y = S(n(x), y)
(3)
where S is a t-conorm and n is a negation on [0, 1]. These implications arise
from the Boolean formalism p → q = ¬p ∨ q. We shall use the following Simplications: x → y = min{1 − x + y, 1} (,Lukasiewitz) and x → y = max{1 −
x, y} (Kleene-Dienes).
• R-implications: obtained by residuation of continuous t-norm T , i.e.
x → y = sup{z ∈ [0, 1] | T (x, z) ≤ y}
These implications arise from the Intutionistic Logic formalism. We shall use
the following R-implication: x → y = 1 if x ≤ y and x → y = y if x > y
(Gödel), x → y = min{1 − x + y, 1} (,Lukasiewitz)
• t-norm implications: if T is a t-norm then
x → y = T (x, y)
Although these implications do not verify the properties of material implication
they are used as model of implication in many applications of fuzzy logic. We
shall use the minimum-norm as t-norm implication (Mamdani).
Consider again the definition of t-norm-based possibility
Π[A|B] = sup{T (A(t), B(t))|t ∈ IR},
where T is t-norm. Then for the measure of necessity of A, given B we get
N [A|B] = 1 − Π[¬A|B] = 1 − sup T (1 − A(t), B(t))
t
Let S be a t-conorm derived from T by (1), then
1 − sup T (1 − A(t), B(t)) = inf {1 − T (1 − A(t), B(t))} =
t
t
inf {S(1 − B(t), A(t))} = inf {B(t) → A(t)}
t
t
where the implication operator is defined in the sense of (3). That is,
N [A|B] = inf {B(t) → A(t)}.
t
Let A and W be discrete fuzzy sets in the unit interval, such that
A = a1 /(1/n) + a2 /(2/n) + · · · + an /1
and
W = w1 /(1/n) + w2 /(2/n) + · · · + wn /1
3
(4)
where n > 1, and the terms aj /(j/n) and wj /(j/n) signify that aj and wj are the
grades of membership of j/n in A and W , respectively, i.e.
A(j/n) = aj ,
W (j/n) = wj
for j = 1, . . . , n, and the plus sign represents the union. Then we get the following
simple formula for the measure of necessity of A, given W
N [A|W ] = min {W (j/n) → A(j/n)} = min {wj → aj }
j=1,...,n
j=1,...,n
(5)
and we use the notation
N [A|W ] = N [(a1 , a2 , . . . , an )|(w1 , w2 , . . . , wn )]
2
Weighted aggregations
A classical MADM problem can be expressed in a matrix format. A decision matrix
is an m × n matrix whose element xij indicates the performance rating of the i-th
alternative, xi , with respect to the j-th attribute, cj :
c1
x1
x2
..
.
xm
c2
...
cn
x11 x12 . . . x1n
x21 x22 . . . x2n
..
..
..
.
.
.
xm1 xm2 . . . xmn
In fuzzy case the values of the decision matrix are given as degrees of ”how an alternative satisfies a certain attribute”. Let x be an alternative such that for any criterion
Cj (x) ∈ [0, 1] indicates the degree to which this criterion is satisfied by x. So, in fuzzy
case we have the following decision matrix
C1
x1
x2
..
.
xm
C2
...
Cn
a11 a12 . . . a1n
a21 a22 . . . a2n
..
..
..
.
.
.
am1 am2 . . . amn
where aij = Cj (xij ), for i = 1, . . . , m and j = 1, . . . , n.
Let x be an alternative and let
(a1 , a2 , . . . , an )
denote the degrees to which x satisfies the criteria, i.e.
aj = Cj (x), i = 1, . . . , n.
4
In many applications of fuzzy sets as multi-criteria decision making, pattern recognition, diagnosis and fuzzy logic control one faces the problem of weighted aggregation.
The issue of weighted aggregation has been studied by Carlsson and Fullér [1], Dubois
and Prade [2, 3, 4], Fodor and Roubens [5, 6] and Yager [9, 10, 11, 12, 13, 14, 15, 16].
Assume associated with each fuzzy set Cj is a weight wj ∈ [0, 1] indicating its
importance in the aggregation procedure, j = 1, . . . , n. The general process for the
inclusion of importance in the aggregation involves the transformation of the fuzzy
sets under the importance. Let Agg indicate an aggregation operator, max or min,
to find the weighted aggregation. Yager [14] first transforms each of the membership
grades using the weights
g(wi , ai ) = âi ,
for i = 1, . . . , n, and then obtain the weighted aggregate
Aggâ1 , . . . , ân .
The form of g depends upon the type of aggregation being performed, the operation
Agg.
As discussed by Yager [14] in incorporating the effect of the importances in the
min operation we are interested in reducing the effect of the elements which have low
importance.
In performing the min aggregation it is the elements with low values that play
the most significant role in this type of aggregation, one way to reduce the effect of
elements with low importance is to transform them into big values, values closer to
one. Yager introduced a class of functions which can be used for the inclusion of
importances in the min aggregation
g(wi , ai ) = S(1 − wi , ai )
where S is a t-conorm, and then obtain the weighted aggregate
min{â1 , . . . , ân } = min{S(1 − w1 , a1 ), . . . S(1 − wn , an )}
(6)
We first note that if wi = 0 then from the basic property of t-conorms it follows that
S(1 − wi , ai ) = S(1, wi ) = 1
Thus, zero importance gives us one. Yager notes that the formula can be seen as a
measure of the degree to which an alternative satisfies the following proposition:
All important criteria are satisfied
Example 2.1 Let
(0.3, 0.2, 0.7, 0.6)
be the vector of weights and let
(0.4, 0.6, 0.6, 0.4)
be the vector of aggregates. If g(wi , ai ) = max{1 − wi , ai } then we get
g(w1 , a1 ) = (1 − 0.3) ∨ 0.4 = 0.7, g(w2 , a2 ) = (1 − 0.2) ∨ 0.6 = 0.8
5
g(w3 , a3 ) = (1 − 0.7) ∨ 0.6 = 0.6, g(w4 , a4 ) = (1 − 0.6) ∨ 0.4 = 0.4
That is
min{g(w1 , a1 ), . . . , g(w4 , a4 )} = min{0.7, 0.8, 0.6, 0.4} = 0.4
As for the max aggregation operator: Since it is the large values that play the most
important role in the aggregation we desire to transform the low importance elements
into small values and thus have them not play a significant role in the max aggregation.
Yager suggested a class of functions which can be used for importance transformation
in max aggregation
g(wi , ai ) = T (wi , ai )
where T is a t-norm. We see that if wi = 0 then T (wi , ai ) = 0 and the element plays
no rule in the max.
Let Agg indicate any aggregation operator and let
(a1 , a2 , . . . , an )
denote the vector of aggregates. We define the weighted aggregation as
Agg g(w1 , a1 ), . . . , g(wn , an ).
where the function g satisfies the following properties
• if a > b then g(w, a) ≥ g(w, b)
• g(w, a) is monotone in w
• g(0, a) = id, g(1, a) = a
where the identity element, id, is such that if we add it to our aggregates it doesn’t
change the aggregated value.
3
Extensions
Let us recall formula (5)
N [(a1 , a2 , . . . , an )|(w1 , w2 , . . . , wn )] = min{w1 → a1 , . . . , wn → an }
(7)
where
A = a1 /(1/n) + a2 /(2/n) + · · · + an /1
is the fuzzy set of performances and
W = w1 /(1/n) + w2 /(2/n) + · · · + wn /1
is the fuzzy set of weights; and the formula for weighted aggregation by the minimum
operator
min{â1 , . . . , ân }
where
âi = g(wi , ai ) = S(1 − wi , ai )
6
and S is a t-conorm.
It is easy to see that if the implication operator in (7) is an S-implication then
from the equality
wj → aj = S(1 − wj , aj )
it follows that the weighted aggregation of the ai ’s is nothing else, but
N [(a1 , a2 , . . . , an )|(w1 , w2 , . . . , wn )],
the necessity of performances, given weights.
This observation leads us to a new class of transfer functions (which contains
Yager’s functions as a subset):
âi = g(wi , ai ) = wi → ai
(8)
where → is an arbitrary implication operator. Then we combine the âi ’s with an
appropriate aggregation operator Agg.
However, we first select the implication operator, and then the aggregation operator Agg to combine the âi ’s. If we choose a t-norm implication in (8) then we will
select the max operator, and if we choose an R- or S-implication then we will select
the min operator to aggregate the âi ’s.
Example 3.1 Let
(0.3, 0.2, 0.7, 0.6)
be the vector of weights and let
(0.4, 0.6, 0.6, 0.4)
be the vector of aggregates. If g(wi , ai ) = min{1, 1 − wi + ai } is the L
( ukasiewicz
implication then we compute
g(w1 , a1 ) = 0.3 → 0.4 = 1, g(w2 , a2 ) = 0.2 → 0.6 = 1
g(w3 , a3 ) = 0.7 → 0.6 = 0.9, g(w4 , a4 ) = 0.6 → 0.4 = 0.8
That is
min{g(w1 , a1 ), . . . , g(w4 , a4 )} = min{1, 1, 0.9, 0.8} = 0.8
If g(wi , ai ) is implemented by the Gödel implication then we get
g(w1 , a1 ) = 0.3 → 0.4 = 1, g(w2 , a2 ) = 0.2 → 0.6 = 1
g(w3 , a3 ) = 0.7 → 0.6 = 0.6, g(w4 , a4 ) = 0.6 → 0.4 = 0.4
That is
min{g(w1 , a1 ), . . . , g(w4 , a4 )} = min{1, 1, 0.6, 0.4} = 0.4
If g(wi , ai ) = wi ai is the Larsen implication then we have
g(w1 , a1 ) = 0.3 × 0.4 = 0.12, g(w2 , a2 ) = 0.2 × 0.6 = 0.12
g(w3 , a3 ) = 0.7 × 0.6 = 0.42, g(w4 , a4 ) = 0.6 × 0.4 = 0.24
That is
max{g(w1 , a1 ), . . . , g(w4 , a4 )} = max{0.12, 0.12, 0.42, 0.24} = 0.42
7
It should be noted that if we choose an R-implication in (8) then the equation
min{w1 → a1 , . . . , wn → an } = 1
holds iff wi ≤ ai for all i, i.e. when each performance rating is at least as big as its
associated weight. In other words, if a performance rating with respect to an attribute
exceeds the value of the weight of this attribute then this rating does not matter in
the overall rating. However, ratings which are well below of the corresponding weights
play a significant role in the overall rating. Thus the formula (7) with an R-implication
can be seen as a measure of the degree to which an alternative satisfies the following
proposition:
All scores are bigger than or equal to the importances
It should be noted that the min aggregation operator does not allow any compensation,
i.e. a higher degree of satisfaction of one of the criteria can not compensate for a
lower degree of satisfaction of another criteria. Averaging operators realize trade-offs
between objectives, by allowing a positive compensation between ratings.
Another possibility is to use an andlike or an orlike OWA-operator to aggregate
the elements of the bag
w1 → a1 , . . . , wn → an .
Let A and W be discrete fuzzy sets in [0, 1], where A(t) denotes the performance
rating and W (t) denotes the weight of a criterion labeled by t. Then the weighted
aggregation of A can be defined by,
• a t-norm-based measure of necessity of A, given W :
N [A|W ] = min{W (t) → A(t)}
t
For example, the Kleene-Dienes implication opeartor,
wi → ai = max{1 − wi , ai },
implements Yager’s approach to fuzzy screening [11].
• a t-norm-based measure of possibility of A, given W :
Π[A|W ] = max{T (A(t), W (t))}
t
• an OWA-operator defined on the bag
W (t) → A(t) | t
Other possibility is to take the value
1
0
A(t) ∧ W (t) dt
1
0 W (t) dt
for the overall score of A. If A(t) ≥ W (t) for all t ∈ [0, 1] then the overall score of A
is equal to one. However, the bigger the set {t ∈ [0, 1]|A(t) ≤ W (t)} the smaller the
overall rating of A.
8
4
Summary and Conclusions
Generalizing Yager’s principles for weighted min and max aggregations we introduce
fuzzy implication operators as a means for importance weighted transformation.
Weighted aggregations are important in decision problems where we have multiple
attributes to consider and where the outcome is to be judged in terms of attributes
which are not equally important for the decision maker. The importance is underscored if there is a group of decision makers with varying value judgments on the
attributes and/or if this group has factions promoting some subset of attributes.
The results shown here can easily be implemented with a number of software tools.
5
Acknowledgments
The second author, who is presently a Donner Visiting Professor at Institute for
Advanced Management Systems Research, Åbo Akademi University, has been partially
supported by OTKA T 14144 and T 019455.
References
[1] C. Carlsson and R. Fullér, On fuzzy screening system, in: Proceedings of
EUFIT’95 Conference, August 28-31, 1995 Aachen, Germany, Verlag Mainz,
Aachen, 1995 1261-1264.
[2] D.Dubois, H.Prade, Criteria aggregation and ranking of alternatives in the
framework of fuzzy set theory, TIMS/Studies in the Management Sciences,
20(1984) 209-240.
[3] D. Dubois and H. Prade, A review of fuzzy sets aggregation connectives, Information Sciences, 36(1985) 85-121.
[4] D. Dubois and H. Prade, Weighted minimum and maximum operations in fuzzy
sets theory, Information Sciences, 39(1986) 205-210.
[5] J.C. Fodor and M. Roubens, Aggregation and scoring procedures in multicriteria
decision-making methods, Proceedings of the IEEE International Conference on
Fuzzy Systems, San Diego, 1992 1261–1267.
[6] J.C. Fodor and M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Theory and Decision Library, Series D, system Theory, Knowledge
Engineering and Problem Solving, Vol. 14, Kluwer, Dordrecht, 1994.
[7] J. Kacprzyk and R.R. Yager, Using fuzzy logic with linguistic quantifiers in
multiobjective decision making and optimization: A step towards more humanconsistent models, in: R.Slowinski and J.Teghem eds., Stochastic versus Fuzzy
Approaches to Multiobjective Mathematical Programming under Uncertainty,
Kluwer Academic Publishers, Dordrecht, 1990 331-350.
9
[8] B.Schweizer and A.Sklar, Associative functions and abstract semigroups, Publ.
Math. Debrecen, 10(1963) 69-81.
[9] R.R. Yager, Fuzzy decision making using unequal objectives, Fuzzy Sets and
Systems, 1(1978) 87-95.
[10] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems, Man and Cybernetics
18(1988) 183-190.
[11] R.R.Yager, Fuzzy Screening Systems, in: R.Lowen and M.Roubens eds., Fuzzy
Logic: State of the Art, Kluwer, Dordrecht, 1993 251-261.
[12] R.R.Yager, Families of OWA operators, Fuzzy Sets and Systems, 59(1993) 125148.
[13] R.R.Yager, Aggregation operators and fuzzy systems modeling, Fuzzy Sets and
Systems, 67(1994) 129-145.
[14] R.R.Yager, On weighted median aggregation, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 1(1994) 101-113.
[15] R.R.Yager, Constrained OWA aggregation, Technical Report, #MII-1420,
Mashine Intelligence Institute, Iona College, New York, 1994.
[16] R.R.Yager, Quantifier guided aggregation using OWA operators, Technical Report, #MII-1504, Mashine Intelligence Institute, Iona College, New York, 1994.
10