Page 279 - Materials Chemistry, Second Edition

13.2 Decision-making under multi-type data condition 277
Sub-step 2: Determine the BO and OW vectors by using the interval numbers (Ren, 2018a, b).
The comparison method is usually used in many weighting methods, and the numbers
from 1 to 9 (corresponding to different linguistic variables, as presented in Table 13.1) and
their reciprocals are used to describe the relative importance of one criterion over another.
The single number approach sometimes cannot describe the relative importance of one cri-
terion over another accurately because of the vagueness, hesitations, and ambiguity existing
in the minds of the stakeholders. The interval numbers such as [1 3] and [2 4] rather than the
single numbers are used to describe the relative importance. Then, the BO and the OW vectors
can be determined:

BO ¼ a a ⋯ a (13.24)
B1 B2 BT

OW ¼ a a ⋯ a (13.25)
1W 2W TW
where a Bj (j¼1,2,⋯,T) and a jW (j¼1,2,⋯,T) represent the relative preference of the most

important criterion comparing with the jth criterion and that of the jth criterion comparing
L
U
with the worst criterion; a Bj and a Bj are the upper and lower bounds of a Bj (j¼1,2,⋯,T), respec-

L
U
tively; and a jW and a jW are the upper and lower bounds of a jW (j¼1,2,⋯,T), respectively.

Sub-step 3: Determining the central weights of the criteria (Rezaei, 2015, 2016). The central
weights of the criteria can be determined by solving Eq. (13.26).
minξ
s:t:
L
ω C a + a U
Bj
B Bj ξ, j ¼ 1,2,⋯,n
ω 2
C
j

ω
C L
a + a U
j jW (13.26)
jW ξ, j ¼ 1,2,⋯,n
ω 2
C
W

n
X C
ω ¼ 1
j
j¼1
C
ω 0, j ¼ 1,2,⋯,T
j
C
C
C
where ω j ,ω B and ω W represent the central weight of the jth criterion, the central weights of the
best criterion and that of the worst criterion, respectively.
TABLE 13.1 Nine-scale in Saaty method (Saaty, 1978).
Scale Definition Scale Definition
1 Equally important 2 Between equally important and moderately important
3 Moderately important 4 Between moderately important and essentially important
5 Essentially important 6 Between essentially important and very strongly important
7 Very Strongly important 8 Between very strongly important and absolutely important
9 Absolutely important