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The preceding three formulas can be used to calculate
the indirect value of an element in an linguistic 2-tuple
judgment matrix, the indirect valued can be expressed as
follows.
p1 p2
p3
cpij
+ cp
p
ij
+ cpij
cpij
= (11)
3
If the given judgment matrix given by an expert was
not additive consistent, the indirect value of an element
calculated by formula may not be in the scope of [0,g].
The formula(14) should be revised as the following
formula.
cp
p
ij
cp
max(min(
p1
ij
+ cp
p2
ij
3
+ cp
p3
ij
, g),0)
= (12)
The indirect value of an element in a linguistic 2-tuple
judgment matrix is calculated based on its additive
consistency. The element similarity between the element
and its indirect value can be got through the distance
between the element and its indirect value. And element
similarity between the element and its indirect value is
simplified as the consistent level of an element, which
can be denoted as cl .
cl
p
ij
p p
CL = clij
)
n×
n
p
ij
p −1 p
| cpij
− ∇ (&
r
ij
) |
= 1−
(13)
g
( is the element consistent level matrix
of a linguistic 2-tuple judgment matrix, which can be
used to designate the weight in aggregating the experts’
options into group decision.
Ⅳ. AGGREGATE THE EXPERTS’ OPINIONS BASED ON
ELEMENT CONSISTENCY LEVEL
The aggregating method based on 2-tuple IOWA give
the expert higher weight if he/she gets high element
consistency level. It does not think of the distance of the
element consistency level between the experts, while
the element consistency level of an expert may be almost
same with the another expert’s element consistency level,
while the weights designate to them are different. To
resolve this question, a aggregating method based on
element consistency level was put forward.
The expert’s weight to aggregate his/her options on the
pair of alternatives can be obtained based on element
consistency level.
k
cp
k
ij
wij
= , i ≠ j
(21)
m
∑
k
cp
k = 1 ij
As the weight is assigned based on element
consistency level, the experts’ preferences on alternatives
can be aggregated into group preference.
m
n× n ij ∑ ij
⋅
k = 1
k k
GP = ( gp ) , p = ∇(
p w ) (22)
ij
The method shows the idea that the expert gets higher
weight to aggregate his/her preference over the pair of
alternatives if the related element gets higher consistency
ij
level. It also thinks of the difference between the element
consistency level of experts. If the element consistency
level of two experts are similar, the weight assigned to
them are also similar.
Ⅴ DEMONSTRATED EXAMPLE
A project to build the non-financial performance of
listed companies in SME board was carried last year. The
organization capability, customer relation management,
quality of employee, the sustainability were used to
assess the non-financial performance of listed companies
and denoted them as X={X1,X2,X3,X4}. And a
dean of Loan Department in China Bank, a CFO in a
company, a CFO in a Security Agency, and a senior
professor researching on enterprise performance were
invited to give their preference on the above indicators.
And used the mentioned method to compute the weight of
the indicators. The calculation process can be illustrated
as follows:
(1)The preference on the non-financial performance
indicators were expressed by using fuzzy terms, the fuzzy
terms set S={s 0 =absolutely worse, s 1 =extremely worse,
s 2 =much worse, s 3 =worse, s 4 =no difference, s 5 =better,
s 6 =much better, s 7 =extremely better, s 8 =absolutely better},
and got the following judgment matrices.
⎡ − ( s3,0)
( s7,0)
⎢
⎢
( s5,0)
− ( s ,0)
1
6
P =
⎢(
s1,0)
( s2,0)
−
⎢
⎣(
s0,0)
( s1,0)
( s3,
−0.1)
⎡ − ( s5,0)
( s7
,0)
⎢
⎢
( s3,0)
− ( s ,0)
2
5
P =
⎢(
s1,0)
( s3,0)
−
⎢
⎣(
s6,0)
( s5,0)
( s6,0)
⎡ − ( s5,0)
( s7,0)
⎢
⎢
( s3,0)
− ( s ,0)
3
5
=
⎢(
s1,0)
( s3,0)
−
⎢
⎣(
s3,0)
( s5,0)
( s6,0)
⎡ − ( s2,0)
( s5,0)
⎢
⎢
( s6,0)
− ( s ,0)
4
2
P =
⎢(
s3,0)
( s6,0)
−
⎢
⎣(
s5,0)
( s5,0)
( s6,0)
( s8,0)
⎤
( s
⎥
7,0)
⎥
( s ,0.1)) ⎥
5
⎥
− ⎦
( s2,0)
⎤
( s
⎥
3,0)
⎥
( s ,0) ⎥
2
⎥
− ⎦
( s5,0)
⎤
( s
⎥
3,0)
⎥
( s ,0) ⎥
2
⎥
− ⎦
( s3,0)
⎤
( s
⎥
3,0)
⎥
( s ,0) ⎥
2
⎥
− ⎦
P ,
( 2 ) calculate the similarity between the given
element and the consistency in a judgment matrix based
on additive consistency.
(3)Using the similarity between the element and
indirect consistency to aggregate the experts’ opinion
into group decision. The group decision matrix followed.
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