Fourth International Conference on Natural Computation
Harmonic Mean Operators for Aggregating Linguistic Information Zeshui Xu College of Economics and Management Southeast University, Nanjing, Jiangsu 210096, China E-mail:
[email protected] In the real-life world, there are many situations, such as selecting applications for different kinds of scholarships and selecting projects for different kinds of funding policies, and evaluating the “speed”, “comfort” or “design” for different kinds of cars, in which a more realistic approach may be to use linguistic assessments instead of numerical values [8]. Some authors [8-17] extended the weighted averaging operator, the ordered weighted averageing operator, the weighted geometric mean operator, and the ordered weighted geometric mean operator to accommodate the situations where the input data are expressed in linguistic labels, and applied the extended operators for solving various decision making problems under linguistic environments. In this paper, we extend the well-known harmonic mean to accommodate linguistic situations, and develop some linguistic harmonic mean aggregation operators, such as the linguistic weighted harmonic mean (LWHM) operator, the linguistic ordered weighted harmonic mean (LOWHM) operator, and the linguistic hybrid harmonic mean (LHHM) operator for aggregating linguistic information. The characteristics of these developed operators are also analyzed.
Abstract Harmonic mean is widely used to aggregate central tendency data, which is usually expressed in exact numerical values. In this paper, we investigate the situations where the input data are given in the form of linguistic labels, and develop some linguistic harmonic mean aggregation operators, such as the linguistic weighted harmonic mean (LWHM) operator, the linguistic ordered weighted harmonic mean (LOWHM) operator, and the linguistic hybrid harmonic mean (LHHM) operator for aggregating linguistic information. Some examples are given to illustrate the developed operators.
1. Introduction Over the last decades, many operators have been developed for aggregating numerical data information [1], where the weighted averaging operator, the ordered weighted averageing operator, the weighted harmonic mean operator, the weighted geometric mean operator, and the ordered weighted geometric mean operator are five of the most common aggregation operators in the information fusion literature [2-7]. The weighted averaging operator [2] and the weighted geometric mean operator [5] are the aggregation techniques which take into account all the given numerical data together with their weights. The ordered weighted averaging (OWA) operator [3] was introduced to provide a method for aggregating several inputs that lie between the max and min operators, whose fundamental aspect is the re-ordering step, that is, the OWA operator first reorders all the given data in descending order, and then aggregates the reordered data together with the weights of their positions, the ordered weighted geometric mean operator [6,7] is based on the OWA operator and on the geometric mean, which has some similar characteristics with the OWA operator. The weighted harmonic mean operator [4] is a conservative average, which is widely used to aggregate central tendency data.
978-0-7695-3304-9/08 $25.00 © 2008 IEEE DOI 10.1109/ICNC.2008.887
2. Preliminaries For convenience, we first recall some basic notions and operations related to linguistic information, which will be used in this paper: Linguistic labels are a basic tool used to describe the qualitative aspects of a problem. In [18], Xu defined S = {sα | α = 1 t ,...,1 2 , 1, 2,..., t} as a linguistic label set with odd cardinality. Any label, sα , represents a possible value for a linguistic variable, and it is required that the linguistic label set should satisfy the following characteristics: 1) sα > sβ iff α > β ; 2) There is the reciprocal operator: rec( sα )= s β such that αβ = 1 . Especially, rec( s1 )= s1 , where s1 is the mid
204
linguistic label represents an assessment of “indifference”, and with the rest of the linguistic labels being placed reciprocally around it. In particular, s 1 t and s t indicate
LWHM ( sα1 , sα 2 ,..., sα n )
(
= w1 ( sα1 ) −1 ⊕ w2 ( sα 2 ) −1 ⊕ " ⊕ wn ( sαn ) −1
( = (s
the lower and upper limits of the linguistic labels in S , respectively, t is a positive integer, and the cardinality of S is 2t − 1 . For example, a set of nine linguistic labels could be as follows:
= w1s1 α1 ⊕ w2 s1 α 2 ⊕ " ⊕ wn s1 α n ⊕ " ⊕ s wn
α2
αn
)
−1
−1
(2)
⎛
where α = ⎜ ⎜
s1 2 = slightly low, s1 = medium, s2 = slightly high,
⎝
s3 = high, s4 = very high, s5 = perfect }
n
∑
j =1
−1
wj ⎞ ⎟ . a j ⎟⎠
By Definition 1, we have the following conclusions for some special cases: 1) If wi = 1 , w j = 0 , j ≠ i , then
Xu [185] extended the discrete linguistic label set S to a continuous label set S = {sα | α ∈[1 q , q]} so as to preserve all the given information, where q ( q > t ) is a
LWHM ( sα1 , sα 2 ,..., sα n ) = sα i
sufficiently large positive integer. If sα ∈ S , then sα is
(3)
2) If w = (1 n ,1 n ,...,1 n)T , then the LWHM operator is reduced to the linguistic harmonic mean (LHM) operator:
termed an original linguistic label, otherwise, sα is termed a virtual linguistic label. In general, the virtual linguistic labels can only appear in calculations. If sα , s β ∈ S ,
LWHM ( sα1 , sα 2 ,..., sα n )
λ ∈ [ 0,1] , then their operational laws were defined as
(
follows [18]: 1) sα ⊕ sβ = sα + β ;
= w1 ( sα1 ) −1 ⊕ w2 ( sα 2 ) −1 ⊕ " ⊕ wn ( sαn ) −1
(
= n ( sα1 ) −1 ⊕ ( sα 2 ) −1 ⊕ " ⊕ ( sα n ) −1
2) λ sα = sλ α ;
)
)
−1
−1
= sα = LHM ( sα1 , sα 2 ,..., sα n )
3) ( sα ) −1 = s1 α .
(4)
−1
⎛ n 1 ⎞ . where α = n ⎜ ∑ ⎜ j =1 a j ⎟⎟ ⎝ ⎠
3. Linguistic harmonic mean aggregation operators
Moreover, the LWHM operator is a bounded operator which provides for aggregation lying between the linguistic max operator and the linguistic min operator:
Based on the well-known harmonic mean [4], in the following, we develop some operators for aggregating linguistic information:
min{sα j } ≤ LWHM ( sα1 , sα 2 ,..., sα n ) ≤ max{sα j } j
Definition 1. Let LWHM : S n → S , and (sα1 , sα 2 ,..., sαn )
LWHM ( sα1 , sα 2 ,..., sα n ) −1
= w1 ( sα1 ) ⊕ w2 ( sα2 ) ⊕ " ⊕ wn ( sαn )
−1
)
−1
(1)
(
= w1 ( sα1 ) −1 ⊕ w2 ( sα2 ) −1 ⊕ " ⊕ wn ( sαn )−1
(
∑w
j
)
−1
≤ w1 (max{sα j })−1 ⊕ w2 (max{sα j })−1 ⊕ " ⊕ wn (max{sα j })−1 j
j
(
the weight vector associated with the linguistic labels sα j
( j = 1, 2,..., n) , w j ≥ 0 , j = 1, 2,..., n , and
(5)
LWHM ( sα1 , sα 2 ,..., sα n )
then LWHM is called a linguistic weighted harmonic mean (LWHM) operator, where w = ( w1 , w2 ,..., wn )T is n
j
In fact, by the operational laws of the linguistic labels, we have
be a collection of n linguistic labels. If
(
⊕ s w2
−1
= sα
S = {s1 5 = none, s1 4 = very low, s1 3 = low,
−1
w1 α1
)
)
j
= ( w1 + w2 + " + wn )(max{sα j }) −1
= 1.
j
= max{sα j } j
j =1
and
Based on the operational laws of the linguistic labels, we can transform (1) into the following:
LWHM (sα1 , sα 2 ,..., sα n )
205
)
−1
)
−1
(
= w1 ( sα1 )−1 ⊕ w2 ( sα 2 ) −1 ⊕ " ⊕ wn ( sα n ) −1
( = ( ( w + w + " + w ) (min{s
)
−1
−1
≥ w1 (min{sα j })−1 ⊕ w2 (min{sα j })−1 ⊕"⊕ wn (min{sα j })−1 j
j
1
n
2
j
αj
j
}) −1
= min{sα j }
)
)
⎛ n ωj ⎞ . where α = ⎜ ∑ ⎜ j =1 σ (α ) ⎟⎟ j ⎠ ⎝ The associated vector ω = (ω1 , ω2 ,..., ωn )T can be
−1
−1
determined by using some weight determining methods, for example, O’Hagan [19] developed a procedure to generate the weights that have a predefined degree of orness and maximize the entropy of the weights. Filev and Yager [20] developed two procedures, based on the exponential smoothing, to obtain the weights. Yager [21] introduced a basic unit-interval monotonic (BUM) function based approach to determining the weights. Xu [22] made a survey of the weight determining methods and then developed a normal distribution based method, which can relieve the influence of the unfair arguments by assigning low weights to those ‘‘false’’ or ‘‘biased’’ ones.
j
thus, (5) holds. The LWHM operator aggregates all the given linguistic labels together with their associated weights. It is a conservative average, which is very suitable to be used as a tool to aggregate central tendency data. Example 1. Given a collection of five linguistic labels: sα 1 = s 2 , sα 2 = s1 3 , sα3 = s4 , sα 4 = s1 2 , sα 4 = s1 , let
w = (0.3, 0.2, 0.1, 0.3, 0.1)T be the weight vector of sα j
By Definition 2, we have 1) If ω = (1, 0,..., 0) T , then
( j = 1, 2, 3, 4, 5) , then by (1), we have
LOWHM (sα1 , sα2 ,..., sαn ) = sσ (α1 ) = max{sα j } (8)
LWHM (sα1 , sα2 , sα3 , sα4 , sα5 )
( = ( 0.3 × (s ) −1
−1
−1
−1
−1
= w1(sα1 ) ⊕ w2 (sα2 ) ⊕ w3 (sα3 ) ⊕ w4 (sα4 ) ⊕ w5 (sα5 ) 2
−1
)
j
−1
2) If ω = (0, 0,...,1) T , then
⊕ 0.2 × (s1 3 )−1 ⊕ 0.1× (s4 )−1 ⊕ 0.3 × (s1 2 )−1
⊕ 0.1× ( s1 )−1 )
LOWHM ( sα1 , sα 2 ,..., sα n ) = sσ (αn ) = min{sα j } j
−1
3) If ω j ≥ 0 ,
= s 0.678
n
j
LOWHM (sα1 , sα2 ,..., sαn ) = LHM (sα1 , sα2 ,..., sαn ) (11) Furthermore, the LOWHA operator has some desirable properties similar to those of the OWA operator [3]: 1) (Idempotency): Let (sα1 , sα2 ,..., sαn ) be a collection
= 1 . Moreover,
of n linguistic labels, if all sα j ( j = 1, 2,..., n) are equal,
j =1
i.e., sα j = sα , for all j , then
LOWHM (sα1 , sα2 ,..., sαn )
(
= ω1(sσ (α1) ) ⊕ ω2 (sσ (α2 ) ) ⊕"⊕ωn (sσ (αn ) ) −1
(10)
4) If ω = (1 n ,1 n ,...,1 n)T , then the LOWHM operator is reduced to the LHM operator:
LOWHM : S n → S , that has an associated vector ω = (ω1 , ω2 ,..., ωn )T such that ω j ≥ 0 , j = 1, 2,..., n , and
∑ω
ωi = 0 , i ≠ j , then
LOWHM ( sα1 , sα 2 ,..., sα n ) = sσ (α j )
Motivated by the ordered weighted averaging (OWA) operator [3], here we define a linguistic ordered weighted harmonic mean (LOWHM) operator: Definition 2. A linguistic ordered weighted harmonic mean (LOWHM) operator of dimension n is a mapping
(9)
−1
−1
)
−1
LOWHM ( sα1 , sα 2 ,..., sα n ) = sα
(6)
2) (Boundary): Let ( sα1 , sα 2 ,..., sα n ) be a collection of
n linguistic labels, then
where ( sα1 , sα 2 ,..., sα n ) be a collection of n linguistic
min{sσ (α j ) } ≤ LOWHM (sα1 , sα2 ,..., sαn ) ≤ max{sσ (α j ) } (13)
labels. (σ (α 1 ), σ (α 2 ),..., σ (α n )) is a permutation of
j
j
3) (Monotonicity): Let ( sα1 , sα 2 ,..., sα n ) and ( sα* 1 , sα* 2 ,
(α 1 , α 2 ,..., α n ) such that sσ (α j−1 ) ≥ sσ (α j ) , for all j .
..., sα* n ) be two collections of linguistic labels, if
Based on the operational laws of the linguistic labels, (6) can be further transformed into the following:
LOWHM ( sα1 , sα 2 ,..., sα n ) = sα
(12)
sα j ≤ sα* j , for all j , then
(7)
LOWHM (sα1 , sα2 ,..., sαn ) ≤ LOWHM (sα*1 , sα*2 ,..., sα*n )
206
(14)
4)
(Commutativity):
Let ( sα1 , sα 2 ,..., sα n )
be
where sσ (α j ) is the j th largest of the weighted linguistic
a
collection of n linguistic labels, then
'
'
labels sα j
'
LOWHM (sα1 , sα2 ,..., sαn ) = LOWHM(sα1 , sα2 ,..., sαn )
(15)
j
The LOWHM operator also provides a technique for fusing the given linguistic labels that lie between the max and min linguistic aggregation operators. Similar to the Yager’ OWA operator, the fundamental aspect of the LOWHM operator is the re-ordering step, that is, the LOWHM operator first reorders all the given linguistic labels in descending order, and then aggregates the reordered linguistic labels together with the weights of their positions. Example 2. Given a collection of four linguistic labels: sα 1 = s 3 , sα 2 = s1 4 , sα 3 = s 2 , sα 4 = s1 2 . We first
j = 1, 2 , ..., n , and
sα 4 = s1 2 , sα5 = s4 , and sα6 = s2 be a collection of six linguistic
and let w = (0.20,0.15,0.10,0.20,0.10,0.25) be the weight vector of sα j ( j = 1,2,...,6) , then the weighted linguistic labels
sα j ( sα j = 6wj sα , j = 1, 2,..., 6 ) are as follows: j
sα1 = 6 × 0.20 × s1 4 = s0.3 , sα 2 = 6 × 0.15 × s1 = s0.9 ,
LOWHM (sα1 , sα2 , sα3 , sα4 ) = ω1 (sσ (α1 ) )−1 ⊕ ω2 (sσ (α2 ) )−1 ⊕ ω3 (sσ (α3 ) )−1 ⊕ ω4 (sσ (α4 ) )−1
)
sα 3 = 6 × 0.10 × s3 = s1.8 , sα 4 = 6 × 0.20 × s1 2 = s0.6
−1
⊕0.345×(s2 )−1 ⊕0.345×(s1 2 )−1 ⊕0.155×(s1 4 )−1 )
−1
= s0.652
We rank the weighted linguistic labels sα j ( j = 1,2,...,6 )
sσ (α1 ) = sα5 = s3 , sσ (α2 ) = sα4 = s2.4 , sσ (α3 ) = sα3 = s1.8 sσ (α 4 ) = sα2 = s0.9 , sσ (α5 ) = sα 4 = s0.6 , sσ (α6 ) = sα1 = s0.3 Let ω = (0.09, 0.17, 0.24, 0.24, 0.17, 0.09) T be the associated vector of the LHHM operator derived from the normal distribution based method [29], then by (16), we have
Definition 3. A linguistic hybrid harmonic mean (LHHM) operator of dimension n is a mapping LHHM :
S n → S , which has an associated vector ω = (ω1, ω2 ,..., ωn )T with ω j ≥ 0 , j = 1, 2,..., n , and
LHHM ( sα1 , sα 2 ,..., sα 6 )
( = ( 0.09× (s )
= 1 , such that
= ω1 ( sσ (α1 ) ) −1 ⊕ ω2 ( sσ (α2 ) ) −1 ⊕ " ⊕ ω6 ( sσ (α 6 ) ) −1
j =1
LHHM (sα1 , sα2 ,..., sαn )
(
sα 5 = 6 × 0.10 × s 4 = s2.4 , sα 6 = 6 × 0.25 × s2 = s3 in descending order:
In what follows, we introduce another linguistic harmonic mean aggregation operator—linguistic hybrid harmonic mean (LHHM) operator, which is based on the LWHM and LOWHM operators:
= ω1 (sσ (α1 ) )−1 ⊕ ω2 ( sσ (α 2 ) )−1 ⊕ " ⊕ ωn (sσ (αn ) )−1
labels,
T
Let ω = (0.155, 0.345, 0.345, 0.155)T be the associated vector of the LOWHM operator derived from the normal distribution based method [22], then by (6), we have
j
= 1 . Especially, if w =
Example 3. let sα 1 = s1 4 , sα 2 = s1 , sα3 = s3 ,
sσ (α3 ) = sα4 = s1 2 , sσ (α4 ) = sα2 = s1 4
∑ω
j
this case, the LHHM operator is reduced to the LOWHM operator; if ω = (1 n ,1 n ,...,1 n)T , then the LHHM operator is reduced to the LWHM operator. Obviously, the LHHM operator generalizes the LOWHM and LWHM operators, and reflects the importance of both the given linguistic labels and their ordered positions.
sσ (α1 ) = sα1 = s3 , sσ (α2 ) = sα3 = s2
n
∑w
(1 n ,1 n ,...,1 n) T , then sα j = sα j , j = 1,2,..., n , in
order:
3
n
j =1
rank the linguistic labels sα j ( j = 1, 2,3, 4) in descending
−1
j = 1, 2 ,..., n ), w =
( w1 , w2 ,..., wn )T is the weight vector of the linguistic labels sα ( j = 1, 2 ,..., n ) , with w j ≥ 0 ,
where (sα' 1 , sα' 2 ,..., sα' n ) is any permutation of (sα1 , sα2 ,..., sαn ) .
( = ( 0.155×(s )
( sα j = nw j sα j ,
3
)
−1
−1
= s1.084
207
−1
⊕0.17×(s2.4 )−1 ⊕0.24×(s1.8 )−1 ⊕0.24×(s0.9 )−1
⊕ 0.17 × ( s0.6 ) −1 ⊕ 0.09 × ( s0.3 ) −1 )
(16)
)
−1
4. Conclusions
[8]
Harmonic mean is one of the most commonly-used measure of central tendency. Consider that, in real-life situations, the input data are usually expressed in linguistic labels instead of numerical values, in this paper we have extended the traditional harmonic mean to deal with linguistic information. We have developed some linguistic harmonic mean aggregation operators, including the linguistic weighted harmonic mean (LWHM) operator, the linguistic ordered weighted harmonic mean (LOWHM) operator, and the linguistic hybrid harmonic mean (LHHM) operator for aggregating linguistic information. Moreover, we have analyzed the characteristics of the developed operators, i.e., the LWHM operator directly aggregates all the given linguistic labels together with their associated weights; the LOWHM operator first reorders all the given linguistic labels in descending order, and then aggregates the reordered linguistic labels together with the weights of their positions, the fundamental aspect of the LOWHM operator is the re-ordering step; while the LHHM operator generalizes both the LWHM and LOWHM operators, and emphasizes the importance of both the given linguistic labels and their ordered positions. In the future research, these linguistic harmonic mean aggregation operators can be applied in many fields such as decision making, medical diagnosis, data mining, and soft computing, etc.
[9] [10]
[11]
[12] [13]
[14] [15]
[16]
Acknowledgements [17]
The work was supported by the National Natural Science Foundation of China (No.70571087) and the National Science Fund for Distinguished Young Scholars of China (No.70625005).
[18]
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