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combinations is minimum number of combinations to
be able to calculate all evaluations.
All order, whether in human thinking, involves
proportionality among the parts. Thus, to create pos-
itive matrix or negative matrix, we must use ratio
scales to capture and synthesize the relations inherent
in that order. If, for example, to create positivematrix,
decision-maker is comparing each element according
to wight we ask: ”How much important is the ele-
ment i than the element j in proportion ?” or ”How
much inferior is the element i than the element j in
proportion ?” If the answer, ” i is more important than
j in proportion of 8 parts to 2” is gotten, the score of
pairwise comparison is ω
i
/ω
j
= 8/2. In positive and
negative evaluations, it is better to compare pair of el-
ements as differently as possible. Thus, the contradic-
tion between positive question and negative question
can be made easy to discover.
3.3 Consistency
In step 2, we must judge the consistency for the con-
structed matrix. There are two kinds of consistency.
One is the consistency of each matrix. Another is the
consistency of duality matrices. In this paper, Saaty’s
method (T.L.Saaty, 1980) is adopted for the former
and new method is proposed for the latter.
3.3.1 Consistency for Each Matrix
Next, consistency of each matrix is checked. The pro-
posed positive matrix and negative matrix are consis-
tent because their values are consisted by n−1 of ratio
and other values are calculated at the them.
However, if values are obtained by comparing
the pair of elements as AHP, we adopt the Saaty’s
method. Saaty proved that for consistent reciprocal
matrix, the largest eigen value is equal to the size of
comparison matrix, or λ
max
= n. Then a measure of
consistency, called Consistency Index is given as de-
viation or degree of consistency using the following
formula
CI =
λmax− n
n− 1
. (5)
3.3.2 Consistency in Duality
We propose how to judge the consistency in duality
between P and N. The consistency in duality means
that positive and negative have same order of priority
in criteria and alternatives. It is necessary to examine
the consistency of duality to discover implicit contra-
diction of the decision-maker.
We apply the concept of reachability matrix to
our procedure. If proposed consistency is satisfied
completely, the order relation between P and N is
duality. We defined, the relation of priorities in
positive and negative matrices is not reversed and it
should be considered within the acceptable limits.
Step1) creation of adjacency matrix
Both matrix P and N can be represented by a ma-
trix M, called the adjacency matrix, as shown below.
There is a row and column for each node; M[i, j] = 1
if (i, j) element of matrix is more than 1, if (i, j) ele-
ment is otherwise M[i, j] = 0.
M
P
= [m
P
ij
]
M
N
= [m
N
ij
] (6)
m
ij
=
1 :ω
ij
≥ 1
0 :ω
ij
< 1orunknown
where M
P
is the adjacency matrix of P and M
N
is the
adjacency matrix of N.
Step2) creation of reachability matrix
The next step derives the reachability matrices R
P
and R
N
. It is calculated on gotten the adjacency ma-
trices with Boolean OR operation as follows:
R
P
=
m
∑
k=1
M
k
P
(7)
R
N
=
m
∑
k=1
M
k
N
(8)
When the number of k(≤ m) is added up, the ele-
ments of adjacency matrix M converge in specific k.
Then, the convergent matrix is reachability matrix. If
R(i, j) = 1 then it means i element is more important
than j element. if R(i, j) = 0 then i is less important
than j, respectively. Thus, R represents the relation
of important degree in n-tuple pairwise comparison.
Step3) judgement of the consistency of duality
In the last step, we judge the consistency of dual-
ity on P and N. When obtaining R
P
and R
N
in step2
have same elements, it is defined by ω = 1/ω
′
that
both matrices are consistent in duality. To judge the
consistency, matrix C is defined as
R
P
⊕ R
T
N
= C (9)
where operator ⊕ represents exclusive OR. As C =
O(where O is zero matrix)Cwe define the relation be-
tween P and N as being consistent. On the other hand,
as C 6= O, it suggests that there is an inconsistency of
order about i and j where 1 in C.
DECISION MAKING BASED ON DUALITY BETWEEN POSITIVE AND NEGATIVE EVALUATIONS
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