Benchmarking and linguistic importance weighted
aggregations ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
rfuller@abo.fi
Abstract
In this work we concentrate on the issue of weighted aggregations and provide a possibilistic approach to the process of importance weighted transformation when both the importances (interpreted
as benchmarks) and the ratings are given by symmetric triangular fuzzy numbers. We will show that
using the possibilistic approach (i) small changes in the membership function of the importances can
cause only small variations in the weighted aggregate; (ii) the weighted aggregate of fuzzy ratings remains stable under small changes in the nonfuzzy importances; (iii) the weighted aggregate of crisp
ratings still remains stable under small changes in the crisp importances whenever we use a continuous
implication operator for the importance weighted transformation. We illustrate the proposed method
by a simple example.
1
Introduction
In many applications of fuzzy sets such 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 [4, 5], Dubois and Prade
[6, 7, 8], Fodor and Roubens [9] Herrera et al [12, 13, 14, 15, 16] and Yager [20, 21, 22, 23, 24, 25, 26, 27,
28, 29]. 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 disproportionately 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 Rn ). For example, the rounding operator in
the convex combination of linguistic labels, defined by Delgado et al. [4], is very sensitive to the values
around 0.5 (round(0.499) = 0 and round(0.501) = 1).
In this paper we consider the process of importance weighted aggregation when both the aggregates and
the importances are given by an infinite term set, namely by the values of the linguistic variables ”rate” and
”importance”. In our approach the importances are considered as benchmark levels for the performances,
i.e. an alternative performs well on all criteria if the degree of satisfaction to each of the criteria is at least
as big as the associated benchmark. The proposed ”stable” method ranks the alternatives by measuring the
degree to which they satisfy the proposition: ”All ratings are larger than or equal to their importance”.
We will also use OWA operators to measure the degree to which an alternative satisfies the proposition:
”Most ratings are larger than or equal to their importance”, where the OWA weights are derived from a
well-chosen lingusitic quantifier.
∗
The final version of this paper appeared in: C. Carlsson and R. Fullér, Benchmarking and linguistic importance weighted
aggregations, Fuzzy Sets and Systems, 114(2000) 35-41. doi: 10.1016/S0165-0114(98)00047-5
1
2
Benchmarking and 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

|a − t|

1−
if |a − t| ≤ α
A(t) =
α

0
otherwise
and we use the notation A = (a, α). If α = 0 then A collapses to the characteristic function of {a} ⊂ R
and we will use the notation A = ā.
We will use symmetric triangular fuzzy numbers to represent the values of linguistic variables [30] 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 R(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
if a ≤ b




a−b
Pos[A ≤ B] =
if 0 < a − b < α + β
1−

α+β



0
otherwise
(1)
It is easy to see that
Pos[A ≤ B] = sup min{A(x), B(y)} = sup(A − B)(t)
x−y≤0
t≤0
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.
Figure 1: Pos[A ≤ B] < 1, because a > b.
2
(2)
Lemma 2.1 [10] Let δ > 0 be a real number and let A = (a, α), B = (b, β) be symmetric triangular
fuzzy numbers. Then from the inequality
D(A, B) ≤ δ
it follows that
sup |A(t) − B(t)| ≤ max{1/α, 1/β}δ
(3)
t∈R
Definition 2.3 R-implications are obtained by residuation of a 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:
(
1 if x ≤ y
x→y=
(Gödel implication)
(4)
y otherwise
x → y = min{1 − x + y, 1}
(Łukasiewitz implication)
(5)
Let A be an alternative with ratings (A1 , A2 , . . . , An ), where Ai = (ai , αi ) ∈ R(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 that 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) ∈ R(I) when 0 < γ ≤ 0.5 can
represent the property
”the importance of the j-th criterion is approximately 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 : R(I) × R(I) → [0, 1],
where,
g(Wi , Ai ) = Pos[Wi ≤ Ai ],
for i = 1, . . . , n, and then obtain the weighted aggregate,
φ(A, W ) = AgghPos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ]i.
where Agg denotes an aggregation operator.
3
(6)
For example if we use the min function for the aggregation in (6), that is,
φ(A, W ) = min{Pos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ]}
(7)
then the equality
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 large as the
mean value of its associated weight. In other words, if a performance rating with respect to a criterion
exceeds the importance of this criterion with possibility one, then this rating does not matter in the overall
rating. However, ratings which are well below the corresponding importances (in possibilistic sense) play
a significant role in the overall rating. In this sense the importance 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 larger 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
criterion.
Averaging operators realize trade-offs between criteria, by allowing a positive compensation between
ratings. We can use an andlike or an orlike OWA-operator [23] to aggregate the elements of the bag
hPos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ]i.
In this case (6) becomes,
φ(A, W ) = OWAhPos[W1 ≤ A1 ], . . . , Pos[Wn ≤ An ]i,
where OWA denotes an Ordered Weighted Averaging Operator introduced by Yager in [21].
Remark 2.1 Formula (6) does not make any difference among alternatives whose performance ratings
exceed the value of their importance with respect to all criteria with possibility one: the overall rating will
always be equal to one. Penalizing ratings that are ”larger than the associated importance, but not large
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 larger than their importance”.
In this case the transformation function can be defined as
g(Wi , Ai ) = Nes[Wi ≤ Ai ] = 1 − Pos[Wi > Ai ],
for i = 1, . . . , n, and then obtain the weighted aggregate,
φ(A, W ) = min{Nes[W1 ≤ A1 ], . . . , Nes[Wn ≤ An ]}.
(8)
If we do allow a positive compensation between ratings then we can use OWA-operators in (8). That is,
φ(A, W ) = OWAhNes[W1 ≤ A1 ], . . . , Nes[Wn ≤ An ]i.
The following theorem shows that if we choose the min operator for Agg in (6) 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.
4
Theorem 2.1 Let Ai = (ai , α) ∈ R(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 ) ∈ R(I), i = 1, . . . , n, satisfy the relationship
max D(Wi , Wiδ ) ≤ δ
(9)
i
then the following inequality holds,
|φ(A, W ) − φ(A, W δ )| ≤
δ
α
(10)
where φ(A, W ) is defined by (7) and
φ(A, W δ ) = min{Pos[W1δ ≤ A1 ], . . . , Pos[Wnδ ≤ An ]}.
Proof. It is sufficient to show that
|Pos[Wi ≤ Ai ] − Pos[Wiδ ≤ Ai ]| ≤
δ
α
(11)
for 1 ≤ i ≤ n, because (10) follows from (11). Using the representation (2) we need to show that
| sup(Wi − Ai ) − sup(Wiδ − Ai )| ≤
t≤0
t≤0
δ
.
α
Using (9) and applying Lemma 2.1 to
Wi − Ai = (wi − ai , αi + γi ) and Wiδ − Ai = (wi − ai , αi + γiδ ),
we find
D(Wi − Ai , Wiδ − Ai ) = D(Wi , Wiδ ) ≤ δ,
and
| sup(Wi − Ai )(t) − sup(Wiδ − Ai )(t)| ≤ sup |(Wi − Ai )(t) − (Wiδ − Ai )(t)| ≤
t≤0
t≤0
sup |(Wi − Ai )(t) −
t∈R
t≤0
(Wiδ
1
1
− Ai )(t)| ≤ max
,
αi + γi αi + γiδ
×δ ≤
δ
.
α
Which ends the proof.
Remark 2.2 From (9) and (10) 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 (10) we can see that Theorem 2.1 remains valid for the case of crisp
weighting vectors, i.e. when γi = 0, i = 1, . . . , n. In this case

if wi ≤ ai

 1
A(wi ) if 0 < wi − ai < αi
Pos[w̄i ≤ Ai ] =


0
otherwise
5
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 ]}
(12)
If both the ratings and the importances are given by crisp numbers (i.e. when γi = αi = 0, i = 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
R-implication operator to transform the ratings under importance [1, 2]. For example, for the operator
φ(a, w) = min{w1 → a1 , . . . , wn → an }.
(13)
where → is an R-implication operator, the equation
φ(a, w) = 1,
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
lim φ(a, wδ ) = φ(a, w), ∀a ∈ I,
wδ →w
relationship still remain 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 (13) 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 (13) 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 6= φ(a1 , w1 ) = 1.
δ→0
But if we choose the (continuous) Łukasiewitz implication in (13) then φ will be continuous in w, and
therefore, small changes in the importance 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}.
6
(14)
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 (14). That is,
φ(a, w) = OWA h(1 − w1 + a1 ) ∧ 1, . . . , (1 − wn + an ) ∧ 1i.
(15)
Taking into consideration that OWA-operators are usually continuous, equation (15) also implements a
stable approach to importance weighted aggregation in the nonfuzzy case.
3
Illustration
In this section we illustrate our approach by several examples.
• Crisp importance and crisp ratings
Consider the aggregation problem with

0.7
 0.5
a=
 0.8
0.9



 and


0.8
 0.7 

w=
 0.9  .
0.6
Using formula (14) for the weighted aggregate we find
φ(a, w) = min{0.8 → 0.7, 0.7 → 0.5, 0.9 → 0.8, 0.6 → 0.9} =
min{0.9, 0.8, 0.9, 1} = 0.8
• Crisp importance and fuzzy ratings
Consider the aggregation problem with

(0.7, 0.2)
 (0.5, 0.3)
a=
 (0.8, 0.2)
(0.9, 0.1)


 and



0.8
 0.7 

w=
 0.9  .
0.6
Using formula (12) for the weighted aggregate we find
φ(A, w) = min{1/2, 1/3, 1/2, 1} = 1/3.
The essential reason for the low performance of this object is that it performed low on the second
criterion which has a high importance. If we allow positive compensations and use an OWA operator
with weights, for example, (1/6, 1/3, 1/6, 1/3) then we find
φ(A, w) = OWAh1/2, 1/3, 1/2, 1i =
1/6 + 1/2 × (1/3 + 1/6) + 1/3 × 1/3 = 19/36 ≈ 0.5278
7
• Fuzzy importance and fuzzy ratings
Consider the aggregation problem with

(0.7, 0.2)
 (0.5, 0.3)
A=
 (0.8, 0.2)
(0.9, 0.1)

(0.8, 0.2)



 and W =  (0.7, 0.3)  .

 (0.9, 0.1) 
(0.6, 0.2)


Using formula (7) for the weighted 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 criterion which has a high importance, the second importance has a relatively large
tolerance level, 0.3.
4
Summary
We have introduced a possibilistic approach to the process of importance weighted transformation when
both the importances and the aggregates are given by triangular fuzzy numbers. In our approach the
importances have been considered as benchmark levels for the performances, i.e. an alternative performs
well on all criteria if the degree of satisfaction to each of the criteria is at least as big as the associated
benchmark. We have suggested the use of measure of necessity to be able to distinguish alternatives with
overall rating one (whose performance ratings exceed the value of their importance with respect to all
criteria with possibility one). We have shown that using the possibilistic approach (i) small changes in the
membership function of the importances can cause only small variations in the weighted aggregate; (ii) the
weighted aggregate of fuzzy ratings remains stable under small changes in the nonfuzzy importances; (iii)
the weighted aggregate of crisp ratings still remains stable under small changes in the crisp importances
whenever we use a continuous implication operator for the importance weighted transformation. These
results have further implications in several classes of multiple criteria decision making problems, in which
the aggregation procedures are rough enough to make the finely tuned formal selection of an optimal
alternative meaningless. This will be the topic of a forthcoming paper.
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5
Follow ups
The results of this paper have been improved and generalized later in the following works:
in journals
A17-c39 Hong-Bin Yan, Van-Nam Huynh, Yoshiteru Nakamori, Tetsuya Murai, On prioritized weighted
aggregation in multi-criteria decision making, EXPERT SYSTEMS WITH APPLICATIONS, 38(2011),
pp. 812-823. 2011
http://dx.doi.org/10.1016/j.eswa.2010.07.039
Remark. It is of interest noting that in a different context, Carlsson and Fullér (2000)
concentrated on the issue of linguistic importance weighted aggregations, where the importance is interpreted as benchmarks. In their study, when both importance weights and
ratings of criteria are given as crisp numbers, Łukasiewicz implication is also used to
compute the benchmark achievement. In case of fuzzy numbers, a possibilistic approach
is presented to compute the benchmark achievement. In addition, Carlsson and Fullér’s
method only focus on symmetric triangular fuzzy numbers. The benchmark used in our
aggregation operator has similar but different meaning with Carlsson and Fullér’s method
(Carlsson & Fullér, 2000). Firstly, as there are various types of fuzzy numbers, specifying
only symmetric triangular fuzzy numbers will not be appropriate in practical applications.
10
Secondly, instead of the possibility interpretation, the benchmark has a probability interpretation lying in the philosophical root of Simon’s bounded rationality (Simon, 1955)
as well as represents the S-shaped value function (Kahneman & Tversky, 1979). Finally,
benchmark in Carlsson and Fullér’s work (Carlsson & Fullér, 2000) and ours has different meanings. Carlsson and Fullér viewed importance weight as benchmark of criteria
satisfaction, whereas our method considered DM’s requirements. (page 820)
A17-c38 Xiaohan Yu, Zeshui Xu, Xiumei Zhang, Uniformization of multigranular linguistic labels and
their application to group decision making, JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS
ENGINEERING, 19(2010), number 3, pp. 257-276. 2010
http://dx.doi.org/10.1007/s11518-010-5137-7
In some multiple attribute decision making problems of real world, decision makers are
accustomed to provide their preferences over alternatives expressed in linguistic forms,
such as ”good”, ”fair”, ”poor”, etc., because of the complexity and uncertainty of practical
things, and fuzzy human thinking. In order to assess alternatives using linguistic information conveniently, linguistic label sets have been researched in lots of papers (Yager
1995, 1998, Carlsson & Fullér 2000, Herrera & Herrera-Viedma 1995, Torra 1996, Xu
2004a, etc.). In the following, we briefly review some basic knowledge about linguistic
label sets. (page 259)
A17-c37 Zhi-Ping Fan, Bo Feng, Wei-Lan Suo, A fuzzy linguistic method for evaluating collaboration
satisfaction of NPD team using mutual-evaluation information, INTERNATIONAL JOURNAL OF
PRODUCTION ECONOMICS, 122(2009), Issue 2, December 2009, Pages 547-557. 2009
http://dx.doi.org/10.1016/j.ijpe.2009.05.018
In the former two categories of methods, the results usually do not exactly match any
of the initial linguistic terms, and then an approximation process must be developed to
express the result in the initial expression domain. This produces the consequent loss
of information and hence the lack of precision (Carlsson and Fuller, 2000), whereas, the
third category of methods overcome the above limitations. The main advantage of this
representation model is to be continuous in its domain. It can express any counting of
information in the universe of the discourse. Therefore, the third one is more convenient
and precise to deal with linguistic terms. (pages 550-551)
A17-c36 Ching Min Sun, Cynthia H.F. Wu, To choose or not? That is the question of memberships:
Fuzzy statistical analysis as a new analytical approach in child language research, JOURNAL OF
MODELLING IN MANAGEMENT, 4(2009), pp. 55-71. 2009
http://dx.doi.org/10.1108/17465660910943757
A17-c35 S. Alonso, F.J. Cabrerizo, F. Chiclana, F. Herrera, E. Herrera-Viedma, Group decision making
with incomplete fuzzy linguistic preference relations, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 24(2009), pp. 201-222. 2009
http://dx.doi.org/10.1002/int.20332
A17-c34 Zeshui Xu, An Interactive Approach to Multiple Attribute Group Decision Making with Multigranular Uncertain Linguistic Information, GROUP DECISION AND NEGOTIATION, 18(2009)
pp. 119-145. 2009
http://dx.doi.org/10.1007/s10726-008-9131-0
11
A17-c33 Sadiq R, Tesfamariam S, Developing environmental indices using fuzzy numbers ordered weighted
averaging (FN-OWA) operators, STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK
ASSESSMENT, 22: (4), pp. 495-505. 2008
http://dx.doi.org/10.1007/s00477-007-0151-0
A17-c32 JIN Wei; FU Chao, A group decision making model with phased feedback based on 2-tuple
and T-OWA operator, JOURNAL OF HEFEI UNIVERSITY OF TECHNOLOGY (NATURAL SCIENCE), 32(2008), number 6, pp. 851-856 (in Chinese). 2008
http://d.wanfangdata.com.cn/Periodical_hfgydxxb200906020.aspx
A17-c31 Wang, J.-H., Hao, J., An approach to computing with words based on canonical characteristic
values of linguistic labels, IEEE TRANSACTIONS ON FUZZY SYSTEMS, 15(4), pp. 593-604.
2007
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In such approach, it will either result in large-scale increases in computational complexity
[14] or make the results do not exactly match any of the initial linguistic terms (and then
an approximation process must be developed to express the result in the initial expression
domain which induces the consequent loss of information and hence the lack of precision
[A17]). (page 593)
A17-c30 Yeh, D.-Y., Cheng, C.-H., Chi, M.-L., A modified two-tuple FLC model for evaluating the performance of SCM: By the Six Sigma DMAIC process, APPLIED SOFT COMPUTING JOURNAL,
7(3), pp. 1027-1034. 2007
http://dx.doi.org/10.1016/j.asoc.2006.06.008
A17-c29 Pei, Z., Ruan, D., Xu, Y., Liu, J., Handling linguistic Web information based on a multi-agent
system, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 22 (5), pp. 435-453. 2007
http://dx.doi.org/10.1002/int.v22:5
A17-c28 Herrera-Viedma, E., Lopez-Herrera, A.G., Luque, M., Porcel, C., A fuzzy linguistic IRS model
based on a 2-tuple fuzzy linguistic approach, INTERNATIONAL JOURNAL OF UNCERTAINTY
FUZZINESS AND KNOWLEDGE-BASED SYSTEMS, 15 (2), pp. 225-250. 2007
http://dx.doi.org/10.1142/S0218488507004534
A17-c27 Lin, Y.-H., Wu, B., The comparisons and applications of traditional mode and fuzzy mode in
quantitative research, WSEAS Transactions on Mathematics, 6 (2007), number 1, pp. 145-150. 2007
A17-c26 Jiang, Y.-P., Xing, Y.-N., Consistency analysis of two-tuple linguistic judgement matrix, JOURNAL OF NORTHEASTERN UNIVERSITY (NATURAL SCIENCE) , 28 (1), pp. 129-132 (in Chinese). 2007
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A17-c25 Xu ZS, A practical procedure for group decision making under incomplete multiplicative linguistic preference relations GROUP DECISION AND NEGOTIATION 15 (6): 581-591 NOV 2006
http://dx.doi.org/10.1007/s10726-006-9034-x
12
A17-c24 Xu ZS, A note on linguistic hybrid arithmetic averaging operator in multiple attribute group
decision making with linguistic information, GROUP DECISION AND NEGOTIATION 15 (6):
593-604 NOV 2006
http://dx.doi.org/10.1007/s10726-005-9008-4
A17-c23 Wang JH, Hao JY A new version of 2-tuple. fuzzy linguistic, representation model for computing with words IEEE TRANSACTIONS ON FUZZY SYSTEMS 14 (3): 435-445 JUN 2006
http://dx.doi.org/10.1109/TFUZZ.2006.876337
In the former two approaches, the results usually do not match any of the initial linguistic
terms, then an approximation process must be developed to express the result in the initial
expression domain. This produces the consequent loss of information and hence the lack
of precision [A17]. (page 435)
A17-c22 Xu ZS, A direct approach to group decision making with uncertain additive linguistic preference
relations, FUZZY OPTIMIZATION AND DECISION MAKING, 5 (1), pp. 21-32. 2006
http://dx.doi.org/10.1007/s10700-005-4913-1
A17-c21 Xu ZS, Deviation measures of linguistic preference relations in group decision making, OMEGAINTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE 33 (3): 249-254 JUN 2005
http://dx.doi.org/10.1016/j.omega.2004.04.008
A17-c20 Huynh VN, Nakamori Y, A satisfactory-oriented approach to multiexpert decision-making with
linguistic assessments, IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS PART
B-CYBERNETICS 35 (2): 184-196 APR 2005
http://dx.doi.org/10.1109/TSMCB.2004.842248
The issue of weighted aggregation has been studied extensively in, e.g., [A17], [10], [12],
[22]-[24], [48], [51], and [52]. (page 184)
A17-c19 Xu ZS, EOWA and EOWG operators for aggregating linguistic labels based on linguistic preference relations, INTERNATIONAL JOURNAL OF UNCERTAINTY FUZZINESS AND KNOWLEDGEBASED SYSTEMS 12 (6): 791-810 DEC 2004
http://dx.doi.org/ 10.1142/S0218488504003211
A17-c18 Fan, Z.-P., Jiang, Y.-P., Judgment method for the satisfying consistency of linguistic judgment
matrix, CONTROL AND DECISION, 19(2004), number 8, pp. 903-906 (in Chinese). 2004
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A17-c17 Luo XD, Lee JHM, Leung HF, et al. Prioritised fuzzy constraint satisfaction problems: axioms,
instantiation and validation, FUZZY SETS AND SYSTEMS, 136 (2): 151-188 JUN 1 2003.
http://dx.doi.org/10.1016/S0165-0114(02)00385-8
A17-c16 F. Herrera, E. Lopez and M.A. Rodrı́guez A linguistic decision model for promotion mix management solved with genetic algorithms, FUZZY SETS AND SYSTEMS, 131(2002) 47-61. 2002
http://dx.doi.org/10.1016/S0165-0114(01)00254-8
13
A17-c15 Herrera F, Martinez L, A 2-tuple fuzzy linguistic representation model for computing with
words, IEEE TRANSACTIONS ON FUZZY SYSTEMS, 8(6): 746-752 DEC 2000.
http://ieeexplore.ieee.org/iel5/91/19275/00890332.pdf?arnumber=890332
These computational techniques are as follows.
• The first one is based on the extension principle [2], [6]. It makes operations on the
fuzzy numbers that support the semantics of the linguistic terms.
• The second one is the symbolic method [5]. It makes computations on the indexes
of the linguistic terms.
In both approaches, the results usually do not exactly match any of the initial linguistic
terms, then an approximation process must be developed to express the result in the initial
expression domain. This produces the consequent loss of information and hence the lack
of precision [A17]. (page 746)
in proceedings and in edited volumes
A17-c10 Jiafeng Ji; Zheng Pei, Obtaining complex linguistic rules from decision information system
based genetic algorithms, IEEE International Conference on Granular Computing, 17-19 August
2009, Lushan mountain/Nanchang, China, art. no. 5255114, pp. 268-273. 2009
http://dx.doi.org/10.1109/GRC.2009.5255114
A17-c9 Van-Nam Huynh; Yoshiteru Nakamori, Group decision making with linguistic information using
a probability-based approach and OWA operators, IEEE International Conference on Systems, Man
and Cybernetics, 7-10 Oct. 2007, [doi 10.1109/ICSMC.2007.4413915], pp. 570-575. 2007
http://www.ieeexplore.ieee.org/iel5/4413560/4413561/04413915.pdf?
A17-c8 Rakus-Andersson, E. Approximation of clock-like point sets, in: Fuzzy and Rough Techniques
in Medical Diagnosis and Medication, Studies in Fuzziness and Soft Computing, Springer, vol. 212,
pp. 155-181+183-189. 2007
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A17-c7 Zheng, P., Liangzhong, Y. A new aggregation operator of linguistic information and its properties,
2006 IEEE International Conference on Granular Computing, art. no. 1635846, pp. 486-489. 2006
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1635846
The management of linguistic information implies the use of operators of comparison and
aggregation. Many researchers have studied operators of comparison and aggregation
[7] - [A17]. In [7], the linguistic weighted averaging (LWA) operator is presented as a
tool to aggregate linguistic weighted information: namely, linguistic information which
has associated different linguistic importance degrees. Nowadays, the fuzzy linguistic
approach has been successfully applied to many different problems such as decision,
information retrieval, medicine, and education etc [A17] - [15]. (page 486)
14
A17-c6 Yuan-Horng Lin and Berlin Wu, Fuzzy mode and its applications in survey research, in: G. R.
Dattatreya ed., Proceedings of the 10th WSEAS International Conference on Applied Mathematics,
2006, pp. 286-291. 2006
A17-c5 Van-Nam Huynh and Yoshiteru Nakamori, Multi-Expert Decision-Making with Linguistic Information: A Probabilistic-Based Model, in: Proceedings of the 38th Hawaii International Conference
on System Sciences, [file name: 22680091c]. 2005
http://dx.doi.org/10.1109/HICSS.2005.448
A17-c4 Pei Z, Du YJ, Yi LZ, et al. Obtaining a complex linguistic data summaries from database based
on a new linguistic aggregation operator, in: Computational Intelligence and Bioinspired Systems,
8th International Workshop on Artificial Neural Networks, IWANN 2005, LECTURE NOTES IN
COMPUTER SCIENCE, vol. 3512, pp. 771-778. 2005
http://dx.doi.org/10.1007/11494669_94
A17-c3 Huynh, V.N., Nguyen, C.H., Nakamori, Y. MEDM in general multi-granular hierarchical linguistic contexts based on the 2-tuples linguistic model, 2005 IEEE International Conference on Granular
Computing, 2005, art. no. 1547338, pp. 482-487. 2005
http://dx.doi.org/10.1109/GRC.2005.1547338
In linguistic decision analysis, irrespective of the membership function based semantics
or ordered structure based semantics of the linguistic term set, one has to face the problem
of weighted aggregation of linguistic information. The issue of weighted aggregation has
been studied extensively in, e.g., [A17], [4], [14]. (page 482)
A17-c2 Xu, Z.-S. An ideal point based approach to multi-criteria decision making with uncertain linguistic information in:) Proceedings of 2004 International Conference on Machine Learning and
Cybernetics, 4, pp. 2078-2082. 2004
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A17-c1 Peneva, V., Popchev, I. Fuzzy decisions in soft computing, 2nd International IEEE Conference on
Intelligent Systems - Proceedings, vol. 2, pp. 606-609. 2004
http://dx.doi.org/10.1109/IS.2004.1344821
. . . aggregation operators of linguistic weighted information [2], [21], e.g. the Linguistic
Weighted Averaging (LWA) operator, the Weighted Min and Weighted Max operators
[19], [A17]; (page 607)
15