A new look at linguistic importance weighted aggregations∗
Christer Carlsson
Robert Fullér†
e-mail: ccarlsso@ra.abo.fi
e-mail: rfuller@cs.elte.hu
Abstract
We extend our earlier results [1; 2; 4] on
the issue of weighted aggregations to linguistic importance weighted aggregations when
both the importances (interpreted as benchmarks) and the ratings are given by symmetric triangular fuzzy numbers. We show that
(unlike the case with finite term sets) small
changes in the membership functions of the
weights can cause only small variations in the
(crisp) weighted aggregate.
Keywords: Possibility, Necessity, Triangular fuzzy
number, Linguistic variables, Sensitivity analysis,
OWA operators, Fuzzy implications.
1
Introduction
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 extensively by Carlsson and Fullér [1;
2; 3], Delgado et al [5; 6], Dubois and Prade [7; 8;
9], Fodor and Roubens [10] Herrera et al [12; 13; 14;
15; 16] and Yager [20; 21; 22; 23; 24; 25].
Unlike Herrera and Herrera-Viedma [15] who perform
direct computation on a finite and totally ordered
term set, we use the membership functions to aggregate the values of the linguistic variables rate and
importance. The main problem with finite term sets
is that the impact of small changes in the weighting
vector can be too large on the weighted aggregate
(because the set of possible output values is finite,
but the set of possible weight vectors is a subset of
∗
The final version of this paper appeared in: Cybernetics and Systems ’98, Proceedings of the Fourteenth European Meeting on Cybernetics and Systems Research, Austrian Society for Cybernetic Studies, Vienna, [ISBN 385206-139-3], 1998 169-174 (Best Paper Award).
†
The preparation and presentation of this paper was
supported by the International Federation for Systems
Research.
IRn ). For example, the rounding operator in the convex combination of linguistic labels, defined by Delgado et al. [5], is very sensitive to the values around
half (round(0.499) = 0 and round(0.501) = 1). In this
paper we consider the process of importance weighted
aggregation when both the the aggregates and importances are given by an infinite term set, namely by the
values of the linguistic variables rate and importance.
2
Possibility and necessity in
linguistic weighted aggregations
Definition 2.1 A fuzzy set A is called a symmetric
triangular fuzzy number with center a and width α > 0
if its membership function has the following form
1 − |a − t|/α if |a − t| ≤ α
A(t) =
0
otherwise
and we use the notation A = (a, α).
We will use symmetric triangular fuzzy numbers to
represent the values of linguistic variables [26] rate and
importance in the universe of discourse I = [0, 1]. The
set of all symmetric triangular fuzzy numbers in the
unit interval will be denoted by F(I).
Definition 2.2 Let A = (a, α) and B = (b, β). The
degree of possibility that the proposition ”A is less than
or equal to B” is true, denoted by Pos[A ≤ B], is
defined as
Pos[A ≤ B] = sup min{A(x), B(y)},
x≤y
and computed by


 1
1−
Pos[A ≤ B] =


0
if a ≤ b
a−b
α+β
otherwise
(1)
if a ≥ b + α + β
It is easy to see that
Pos[A ≤ B] = sup(A − B)(t)
t≤0
(2)
A
Pos[A≤B]=1
a
A
B
a
b
Figure 1: Pos[A ≤ B] = 1, because a ≤ b.
B
Definition 2.5 R-implications are obtained by residuation of a continuous t-norm T , i.e.
x → y = sup{z ∈ [0, 1] | T (x, z) ≤ y}.
Pos[A≤B]
These implications arise from the Intutionistic Logic
formalism. We shall use the following R-implication:
1 if x ≤ y
x→y=
(Gödel)
(4)
y otherwise
a
Figure 2: Pos[A ≤ B] < 1, because a > b.
The degree of necessity that the proposition ”A is less
than or equal to B” is true, denoted by Pos[A ≤ B],
is defined as
Nes[A ≤ B] = 1 − Pos[A ≥ B],
and computed by
Nes[A ≤ B] =


 1


if a ≤ b − α − β
b−a
α+β
otherwise
0
if a ≥ b
(3)
Definition 2.3 The Hausdorff distance of A and B,
denoted by D(A, B), is defined by [11]
D(A, B) = max max {|a1 (θ) − b1 (θ)|, |a2 (θ) − b2 (θ)|}
θ∈I
where [a1 (θ), a2 (θ)] and [b1 (θ), b2 (θ)] denote the θ-level
sets of A and B, respectively.
Definition 2.4 Let φ: IRn → [0, 1] be an arbitrary
continuous function. Then for any θ ≥ 0 we define
ωφ (θ), the modulus of continuity of φ as
ωφ (θ) =
max |φ(u) − φ(v)|.
||u−v||≤θ
where ||u − v|| = maxi |ui − vi |.
b
Figure 4: Nes[A ≤ B] = 1, (a < b and A ∩ B = ∅).
A
b
Nes[A≤B]=1 B
x → y = min{1 − x + y, 1}
(2Lukasiewitz)
(5)
Let A be an alternative with ratings (A1 , A2 , . . . , An ),
where Ai = (ai , αi ) ∈ F(I), i = 1, . . . , n. For example, the symmetric triangular fuzzy number Aj =
(0.8, α) when 0 < α ≤ 0.2 can represent the property
”the rating on the j-th criterion is around 0.8”
and if α = 0 then Aj = (0.8, α) is interpreted as
”the rating on the j-th criterion is equal to 0.8”
and finally, the value of α can not be bigger than 0.2
because the domain of Aj is the unit interval. Assume
associated with each criterion is a weight Wi = (wi , γi )
indicating its importance in the aggregation procedure, i = 1, . . . , n.
For example, the symmetric triangular fuzzy number
Wj = (0.5, γ) ∈ F(I) when 0 < γ ≤ 0.5 can represent
the property
”the importance of the j-th criterion is around 0.5”
and if γ = 0 then Wj = (0.5, γ) is interpreted as
”the importance of the j-th criterion is equal to 0.5”
and finally, the value of γ can not be bigger than 0.5
becuase the domain of Wj is the unit interval. The
general process for the inclusion of importance in the
aggregation involves the transformation of the ratings
under the importance. In this paper we suggest the
use of the transformation function
g: F(I) × F(I) → [0, 1],
B
Nes[A≤B]
A
Pos[B≤A]
a
b
Figure 3: Nes[A ≤ B] < 1, (a < b, A ∩ B = ∅).
where,
g(Wi , Ai ) = Pos[Wi ≤ Ai ],
for i = 1, . . . , n, and then obtain the weighted aggregate,
φ(A, W ) =
AggPos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ].
(6)
where Agg denotes an aggregation operator.
For example if we use the min function for the aggregation in (6), that is,
for i = 1, . . . , n, and then obtain the weighted aggregate,
φ(A, W ) =
φ(A, W ) =
min{Nes[W1 ≤ A1 ], . . . , Nes[Wn ≤ An ]}.
min{Pos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ]}
(7)
(8)
If we do allow a positive compensation between ratings
then we can use OWA-operators in (8). That is,
then the equality
φ(A, W ) =
φ(A, W ) = 1
holds iff wi ≤ ai for all i, i.e. when the mean value
of each performance rating is at least as big as the
mean value of its associated weight. In other words,
if a performance rating with respect to an attribute
exceeds the value of the weight of this attribute with
possibility one, then this rating does not matter in the
overall rating. However, ratings which are well below
the corresponding weights (in possibilistic sense) play
a significant role in the overall rating (it is why the
weight can be considered as benchmark or reference
level for the performance). Thus formula (6) with the
min operator can be seen as a measure of the degree
to which an alternative satisfies the following proposition: ”All ratings are bigger than or equal to their
importance”. 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 criteria, by allowing a positive compensation
between ratings. We can use then an andlike or orlike
OWA-operator [22] to aggregate the elements of the
bag
Pos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ].
In this case (6) becomes,
φ(A, W ) =
OWANes[W1 ≤ A1 ], . . . , Nes[Wn ≤ An ].
The following theorem shows that if we choose the
min operator for aggregation (7) then small changes
in the membership functions of the weights can cause
only a small change in the weighted aggregate, i.e.
the weighted aggregate depends continuously on the
weights.
Theorem 2.1 [4] Let Ai = (ai , α) ∈ F(I), αi >
0, i = 1, . . . , n and let δ > 0 such that
δ < α := min{α1 , . . . , αn }
If Wi = (wi , γi ) and Wiδ = (wiδ , γ δ ) ∈ F(I), i =
1, . . . , n, satisfy the relationships
max D(Wi , Wiδ ) ≤ δ
(9)
i
then the following inequalities hold,
max |Pos[Wi ≤ Ai ] − Pos[Wiδ ≤ Ai ]| ≤
i
and
|φ(A, W ) − φ(A, W δ )| ≤
δ
α
δ
α
(10)
(11)
where φ(A, W ) is defined by (7) and
φ(A, W δ ) = min Pos[Wiδ ≤ Ai ].
i
OWAPos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ],
Theorem 2.1 can be extended to any continuous Agg
operator in (6). Namely,
where OWA denotes an Ordered Weighted Averaging
Operator introduced by Yager in [21].
Formula (6) does not make any difference among alternatives whose performance ratings exceed the value
of the importances with respect to all attributes with
possibility one: the overall rating will always be equal
to one. Penalizing ratings that are ”bigger than the
associated importance, but not big enough” (that is,
their intersection is not empty) we can modify formula
(6) to measure the degree to which an alternative satisfies the following proposition: ”All ratings are essentially bigger than their importance”.
In this case the transformation function can be defined
as
If Wi = (wi , γi ) and Wiδ = (wiδ , γ δ ) ∈ F(I), i =
1, . . . , n, satisfy the relationship (9) then the following inequality holds,
g(Wi , Ai ) = Nes[Wi ≤ Ai ] = 1 − Pos[Wi ≥ Ai ],
Agg{Pos[W1δ ≤ A1 ], . . . , Pos[Wnδ ≤ An ]}.
Theorem 2.2 Let Ai = (ai , α) ∈ F(I), αi > 0, i =
1, . . . , n and let δ > 0 such that
δ < α := min{α1 , . . . , αn }
|φ(A, W ) − φ(A, W δ )| ≤ ωG (δ/α)
(12)
where φ(A, W ) is defined by (6), ωG denotes the modulus of continuity of Agg and φ(A, W δ ) is defined as
Proof. Using (10) and the definition of modulus of
continuity we immediately get
|Agg{Pos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ]}−
holds iff wi ≤ ai for all i, i.e. when the value of each
performance rating is at least as big as the value of its
associated weight. However, the crucial question here
is: does the
Agg{Pos[W1δ ≤ A1 ], . . . , Pos[Wnδ ≤ An ]}| ≤
lim φ(a, wδ ) = φ(a, w), ∀a ∈ I,
wδ →w
ωG (δ/α).
Which ends the proof.
Remark 2.1 From (9) and (12) it follows that
lim φ(A, W δ ) = φ(A, W )
δ→0
for any A, which means that if δ is small enough then
φ(A, W δ ) can be made arbitrarily close to φ(A, W ).
As an immediate consequence of (11) we can see that
Theorem 2.1 remains valid for the case of crisp weighting vectors, i.e. when γi = 0, 1 = 1, . . . , n. In this
case

if wi ≤ ai

 1
A(wi ) if 0 < wi − ai < αi
Pos[w̄i ≤ Ai ] =


0
otherwise
where w̄i denotes the characteristic function of wi ∈
[0, 1]; and the weighted aggregate, denoted by φ(A, w),
is computed as
φ(A, w) = Agg{Pos[w̄1 ≤ A1 ], . . . , Pos[w̄n ≤ An ]}
If Agg is the minimum operator then we get
φ(A, w) =
min{Pos[w̄1 ≤ A1 ], . . . , Pos[w̄n ≤ An ]}
(13)
If both the ratings and the importances are given by
crisp numbers (i.e. when γi = αi = 0, 1 = 1, . . . , n)
then Pos[w̄i ≤ āi ] implements the standard strict implication operator, i.e.,
1 if wi ≤ ai
Pos[w̄i ≤ āi ] = wi → ai =
0 otherwise
It is clear that whatever is the aggregation operator
in
φ(a, w) = Agg{Pos[w̄1 ≤ ā1 ], . . . , Pos[w̄n ≤ ān ]},
the weighted aggregate, φ(a, w), can be very sensitive
to small changes in the weighting vector w.
However, we can still sustain the benchmarking character of the weighted aggregation if we use an Rimplication operator to transform the ratings under
importance [1; 2]. For example, for the operator
φ(a, w) = min{w1 → a1 , . . . , wn → an }.
(14)
where → is an R-implication operator, the equation
φ(a, w) = 1,
relationship remain still valid for any R-implication?
The answer is negative. φ will be continuous in w
if and only if the implication operator is continuous.
For example, if we choose the Gödel implication in
(14) then φ will not be continuous in w, because the
Gödel implication is not continuous. To illustrate the
sensitivity of φ defined by the Gödel implication (4)
consider (14) with n = 1, a1 = w1 = 0.6 and w1δ =
w1 + δ. In this case
φ(a1 , w1 ) = φ(w1 , w1 ) = φ(0.6, 0.6) = 1,
but
φ(a, w1δ ) = φ(w1 , w1 + δ) =
φ(0.6, 0.6 + δ) = (0.6 + δ) → 0.6 = 0.6,
that is,
lim φ(a1 , w1δ ) = 0.6 = φ(a1 , w1 ) = 1.
δ→0
But if we choose the (continuous) L
J ukasiewitz implication in (14) then φ will be continuous in w, and
therefore, small changes in the imporances can cause
only small changes in the weighted aggregate. Thus,
the following formula
φ(a, w) =
min{(1 − w1 + a1 ) ∧ 1, . . . , (1 − wn + an ) ∧ 1}. (15)
not only keeps up the benchmarking character of φ,
but also implements a stable approach to importance
weighted aggregation in the nonfuzzy case.
If we do allow a positive compensation between ratings
then we can use an OWA-operator for aggregation in
(15). That is,
φ(a, w) =
OWA (1 − w1 + a1 ) ∧ 1, . . . , (1 − wn + an ) ∧ 1. (16)
Taking into consideration that OWA-operators are
usually continuous, equation (16) also implements a
stable approach to importance weighted aggregation
in the nonfuzzy case.
3
Illustrations
In this section we illustrate our approach by several
examples.
• Crisp importances and crisp ratings
4
Consider the aggregation problem with




0.7
0.8
 0.5 
 0.7 



a=
 0.8  and w =  0.9  .
0.9
0.6
Using (15) for the weighed aggregate we find
φ(a, w) =
min{0.8 → 0.7, 0.7 → 0.5, 0.9 → 0.8, 0.6 → 0.9} =
We have introduced a possibilistic approach to the
process of importance weighted transformation when
both the importances and the aggregates can be given
by triangular fuzzy numbers. We have shown that
if the aggregation function is continuous then small
changes in the membership functions of the weights
can cause only small variations in the (crisp) weighted
aggregate.
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min{0.9, 0.8, 0.9, 1} = 0.8
• Crisp importances and fuzzy ratings
Consider the aggregation problem with



(0.7, 0.2)
0.8
 (0.5, 0.3) 
 0.7


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
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The essential reason for the low performance of
this object is that it performed low on the second
criteria which has a high importance. If we allow
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then we find
φ(A, w) = OWA1/2, 1/3, 1/2, 1 =
1/6 + 1/2 × (1/3 + 1/6) + 1/3 × 1/3 =
19/36 ≈ 0.5278
• Fuzzy importances and fuzzy ratings
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



(0.7, 0.2)
(0.8, 0.2)
 (0.5, 0.3) 
 (0.7, 0.3) 



A=
 (0.8, 0.2)  and W =  (0.9, 0.1)  .
(0.9, 0.1)
(0.6, 0.2)
Using (7) for the weighed aggregate we find
φ(A, W ) = min{3/4, 2/3, 2/3, 1} = 2/3.
The reason for the relatively high performance of
this object is that, even though it performed low
on the second criteria which has a high importance, the second importance has a big tolerance
level, 0.3.
Summary
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