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symbolic aggregation operation. β ∈[ 0, g]
and g+1 is
the cardinality of S. Let i = round(β ) and α = β − i
be two values, such that, i ∈ [ 0, g]
and
α ∈[−0.5,0.5] then α is called a Symbolic
Translation.
Let S={s0, s1, …,sg} be a linguistic term set and
β ∈[ 0, g]
be a value representing the result of a
symbolic aggregation operation, then the 2-tuple that
expresses the equivalent information to β is obtained
with the following function:
∇ :[0, g]
→ S × [ −0.5,0.5]
⎧ si
, i = round(
β)
(1)
∇(
β)
= ( si
, α),
with⎨
⎩α
= β − i,
α ∈[
−0.5,0.5]
Where round(.) is the usual round operation, si had the
closest index label to β .
Proposition 1[14] Let S={s0, s1, …,sg}be a
linguistic term set and ( s i
, α)
be a 2-tuple. There is
−1
always a ∇ function, such that, from a 2-tuple it
returns its equivalent numerical value β ∈[ 0, g]
, which
is:
−1
∇ : S × [ −0.5,0.5]
→ [0, g]
(2)
−1
∇ ( si , α ) = i + α = β
⑴ A linguistic 2-tuple negation operator [14]
−1
Neg((
si
, α))
= ∇(
g − ( ∇ ( si
, α)))
(3)
⑵ Linguistic 2-tuple aggregation operators
Let ( s1,
α1),(
s1,
α
2
), L ,( s n
, α
n
) be a set with n
linguistic 2-tuples and ω = ( ω1,
ω2
, L,
ωn
) be the
n
related weighted vector with ∑ω
i
= 1 , then the
i=
1
weighted average operator of linguistic 2-tuples
ξ ω
= ∇(
(( s , α ),( s
1
n
∑
i=
1
1
∇
−1
2
, α ), L,(
s
2
( s , α ) ω )
i
i
i
n
ω
ξ is
, α )) = (ˆ, s ˆ) α
Let P ( pij, α
ij
)
n × n
comparison matrix and the element ( p , α )
n
(4)
= be a linguistic 2-tuple
ij
ij
represent
the result of comparing two solutions. If the following
propositions are right.
(1) p ∈S; α ∈[ −0.5,0.5]
ij ij
−1
(2) ∇ ( pii, αii
) = g/ 2
−1 −1
ij
αij ji
α
ji
(3) ∇ ( p , ) +∇ ( p , ) = g
then P is called a linguistic 2-tuple judgment matrix.
(5)
Let P= ( pij , α
ij
)
n × n
be a linguistic 2-tuple judgment
matrix, if ∀ i, j,
k ∈ I , elements in P has the properties
of the formula (6), then P is called a linguistic 2-tuple
judgment matrix with additive consistency.
−1 −1 −1
∇ ( p , α ) +∇ ( p , α ) =∇ ( p , α ) + g/2, ∀i, j,
k∈ I (6)
ij ij jk jk ik ik
Ⅲ. CALCULATE THE ELEMENT CONSISTENCY LEVEL
If the judgment matrix given by an expert is additive
consistent, the comparison of each pair of alternative is
identical with the indirect value based on additive
consistency. In real decision making environment,
however, one element in the given judgment matrix may
have high similarity to its indirect value and another
element may have low similarity to its indirect value.
Therefore, it is unreasonable to give the expert a fixed
weight when the judgment matrices are aggregated into
group decision judgment matrix. To measure the
similarity between an element and its indirect valued, the
concept of element consistency level was introduced .
If a linguistic 2-tuple representation judgment matrix P
P = ( , α ) ) is additive consistent, there exists
p ij
( [
ij
] n × n
∇
−1
( p
−1
−1
, α ) = ∇ ( p , α ) − ∇ ( p , α ) g / 2 .
ij ij
kj kj
ki ki
+
The property can be used to compute the indirect value of
an element in the judgment matrix.
Based on the two properties of an additive consistent
linguistic 2-tuple judgment matrix
−1
−1
−1
∇ ( pij , α
ij
) = ∇ ( pkj
, α
kj
) − ∇ ( pki
, α
ki
) + g / 2
−1
−1
and ∇ ( pij , α
ij
) + ∇ ( p
ji
, α
ji
) = g , the following
formulas can be reasoned out.
−1
∇ ( p , α ) =∇
ij
ij
ij
ij
ij
ij
−1
−1
∇ ( p , α ) =∇
−1
−1
∇ ( p , α ) =∇
−1
( p
( p
kj
( p
ik
ik
, α ) +∇
ik
, α ) −∇
kj
, α ) −∇
ik
−1
−1
−1
( p
( p
ki
( p
kj
, α ) −g/2
, α ) + g/2
jk
kj
ki
, α ) + g/2
Therefore, the indirect valued of an element in a
linguistic 2-tuple judgment with additive consistency can
be calculated through the neighbor elements. There are
different elements in the judgment matrix can be used to
compute an element’s indirect value, thus, to assessment
the indirect values comprehensively, the RMM( Row
Mean Method) is used to calculate the indirect value of
an element. The following formulas is induced from the
above formula
cp
cp
p1
ij
p2
ij
cp
=
n
∑
k = 1
k ≠i,
j
=
p3
ij
( ∇
n
∑
k=
1
k≠i,
j
=
−1
( ∇
n
∑
k=
1
k≠i,
j
( p
−1
( ∇
p
ik
( p
−1
p
, α
p
kj
( p
ik
p
, α
p
ik
) + ∇
−1
n − 2
kj
p
, α
( p
) −∇
−1
n − 2
ik
p
kj
( p
) −∇
−1
n − 2
p
, α
p
ki
( p
kj
jk
) − g / 2)
p
, α
p
jk
ki
) + g /2)
p
, α
ik
) + g /2)
(7)
(8)
(9)
(10)
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