2. Measuring the Importance and the Weight of Decision Makers in the Criteria
Weighting Activities of Group Decision Making Process 7
making, is a vital issue in most areas of scientific activities that includes, best alternative finding from
the set of available options (Asgharpour, 2006). Criteria weights change in decision making process,
those have important affect on results of the decision making (Asgharpour, 2006). Methods such as,
LINMAP, SMART, Eigenvector and something like that, use for finding the criteria weights.
In This paper, the eigenvector method use for criteria weighting and then the process of finding
weights use for decision makers importunacy in group decision making. Often seen these decision
makers in all methods of group decision making (even in voting methods) have same weight of
importance for participate in the decision making process that has its drawbacks.
Generally multiple criteria decision making, including several in facts as follow:
1. Criteria Recognition and evaluation.
2. Allocate weight to each criterion
3. Best alternative selection or ranking with one of MADM methods
4. Make sensitivity analysis operations.
2. Eigenvector Method for Criteria Weighting
There are some methods for criteria weighting in decision making process (toloie, 2006). Eigenvector
method is one of these methods that use in circumstancing which decision making matrix is not
available. This method is based on pair wise comparison (Saati, 1980). In pair wises comparison
method, criteria preference find by using below table (Table 1). The preference measurement scales are
shown on table 2.
Table 1: Criteria pair wise comparison matrix
Criteria Criterion 1 Criterion 2 Criterion j
Criterion 1 Criterion 1 to criterion 1 Criterion 1 to criterion 2 Criterion 1 to criterion j
Criterion 2 Criterion 2 to criterion 1 Criterion 2 to criterion 2 Criterion 2 to criterion j
Criterion i Criterion i to criterion 1 Criterion i to criterion 2 Criterion i to criterion j
Table 2: Pair wise comparison scales (i to j)
1 Equal preference
3 Poor preference
5 Strong preference
7 Very strong preference
9 Absolute preference
2,4,6,8 Intermediate preference
Matrix that introduced in Table 1, always, is a square matrix and criteria that shown in rows
and column will be the same. As is clear, main diameter values of the matrix will equal one, because in
fact, it shows relative value and importance of each criterion to own. What happened for the rest matrix
members? Let's suppose that criterion1 has strong preference to criterion 2 then decision maker should
be settling 5, in cellule 12 that is calling f12. However, filling the matrix should be noticing two
important following principal:
• Reciprocal principal: If suppose that the value preference of ith criterion, to jth criterion is a
(means decision maker preference ith criterion to jth criterion, a times), logically, decision maker
have to prefer 1 / a, jth criterion to ith criterion.
1
ij 1 2 3
= i, j = , , ,....,n
(1)
f
f
ji
• Consistency principal: decision maker should be fully remembering that if:
Criterion 1 Criterion 2
3. 8 Abbas Toloie-Eshlaghy and Ebrahim Nazari Farokhi
And
Criterion 2 Criterion 3
Then:
Criterion 1 Criterion 3
In total consistency, have to:
f f f i, j ,k , , ,....,n ik kj ij = = 1 2 3 (2)
In addition, the decision maker should be sure that if:
Preference of 1th criterion to 2th criterion is equal 3 and also, preference of 2th criterion to 3th
criterion is equal 2, then the preference of 1th criterion to 3th criterion have to be 2*3 = 6.
The second principal, in fact, formed the basic and core concepts of this article. After establish
pair wise matrix, by using following formula, the matrix must be iterated multiple times, to finally be
close to convergence vector.
k
D .e
e .D .e
W Lim
t k
j
= (3)
k ®¥
k Î Integer
That:
Wj is jth weights vector
D is initial pair wise comparison matrix
e unit column vector that all elements are equal 1
et is transposing matrix of e
Number of iterations depends on the following two cases:
• If the number of criteria increases then the number of iterations of matrix for achieving to
convergence vector also increases (However, this relation is not linear).
• If the decision maker inconsistency increase then the number of matrix iterations also
increase.
In actual conditions, decision makers have different levels of accessible information, thinking
capabilities and experience. It is impossible that in decision making process, two individual decision
makers have same judgment. However, it happened by different reasons, subject to the talents and
capabilities of different people, cannot achieve to the same access of resources of information and so
on (Asgharpour, 2006). Therefore, the decision makers’ pair wise comparison matrixes, always, are
inconsistence. It seems that if the decision makers be inconsistence in decision making process, then
number of iterations to reach a convergence vector increase. So, the number of iterations maybe a good
basis, for measuring accuracy and consistency of decision makers. Calculated weights and importance
can be used in group decision making process method such as, BORDA technique, DEMATEL
technique, and something like that.
When the number of iterations, for each person by using the eigenvector method, achieved and
since the sum of weights of participants in the group decision making process, should be equal to 1
(because the relative importance of decision makers should consider) , then by using the following
relation, weight and importance of each decision maker could be calculate:
Absolute weight of each decision maker = 1 – (number of iterations for each decision makers
/total number of iterations for all decision makers) (4)
And, finally by using probability scale less method:
Relative weight of each decision maker = absolute weight of each decision maker / sum of all
decision maker absolute weights (5)
3. Case Study
In same conditions of space, location and time (for controlling the circumstances), following decision
matrix is completed by decision makers. Since this section of paper, takes a case study to identify the
4. Measuring the Importance and the Weight of Decision Makers in the Criteria
Weighting Activities of Group Decision Making Process 9
level of matrix inconsistency, so, the type of criteria are not important. Also for achieving more
effective visual perception, some forms considered as criteria. Therefore decision making process for
criteria weighting followed with 4 criteria as below:
And then, pair wise comparison matrix must be as follows:
Table 3: Pair wise comparison matrix for form selection
criteria
In this case study, three decision makers, play his role for measuring weights of them. The three
people have shown with capital letters A, B and C. The completed matrix for each decision maker,
shown as follows:
Table 4: Completed pair wise comparison matrix for decision maker A
criteria
1 5 7 4
1/5 1 1/3 8
1/7 3 1 2
1/4 1/8 1/2 1
Table 5: Completed pair wise comparison matrix for decision maker B
criteria
1 6 9 4
1/6 1 3 7
1/9 1/3 1 8
1/4 1/7 1/8 1
5. 10 Abbas Toloie-Eshlaghy and Ebrahim Nazari Farokhi
Table 6: Completed pair wise comparison matrix for decision maker C
criteria
1 2 3 4
1/2 1 5 6
1/3 1/5 1 4
1/4 1/6 1/4 1
After obtaining the pair wise comparison matrix for each decision maker, regards to formula
(2), iterations must be calculated to achieving convergence vector. In this article MATLAB software
used for to this purpose. For decision maker A, for example:
Input data:
D = [1 5 7 4; 1/5 1 1/3 8; 1/7 3 1 2; 1/4 1/8 1/2 1];
e = [1; 1; 1; 1];
et =[1 1 1 1];
Then, in:
First iteration:
W1= (D^1*e)/ ( et *D^1*e)
W1 =
0.4920
0.2759
0.1778
0.0543
And finally after 8 iterations:
W8= (W^8*e)/ (et *W^8*e)
W8 =
0.5860
0.1708
0.1792
0.0639
Weights that obtained in the seventh iteration, identically repeated in eighth iteration. Therefore
for decision maker A, number of iterations to achieve convergence vector is equal 8.
Just like the above steps, for decision maker B, number of iterations is equal 7, and for decision
maker C, number of iterations is equal 5.
Now according formula (4) and (5) will be:
Weight of decision maker A:
.
1 .
0 3
0 6
. =
0 6 0 65 0 75
0 6
8
20
. + . +
.
= − =
Weight of decision maker B:
.
1 .
0 325
0 65
. =
0 6 0 65 0 75
0 65
7
20
. + . +
.
= − =
Weight of decision maker C:
.
1 .
0 375
0 75
. =
0 6 0 65 0 75
0 75
5
20
. + . +
.
= − =
Therefore, decision maker C has the highest weight and decision maker A has with the lowest
weight and so, these weights show decision maker importance in decision making group.
6. Measuring the Importance and the Weight of Decision Makers in the Criteria
Weighting Activities of Group Decision Making Process 11
4. Conclusion
Since the criteria weighting in decision making process, often, have done by humans and decision
makers, so, inconsistent decision makers have less weight and also the consistent decision makers have
more weight. In this article, by using eigenvector weighting method based on pair wise comparison, a
new method introduced to demonstrate the importance of decision makers.
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