Page 278 - Materials Chemistry, Second Edition

276 13. Multi-criteria decision-making after life cycle sustainability assessment under hybrid information

Step 2: Normalizing the decision-making matrix. The decision-making matrix presented in
Eq. (13.17) should be normalized to avoid the influences caused by different dimensions or units.
As for the normalization of the data in the decision-making matrix, the data should be normal-
ized according to the types of the criteria. As for the data with respect to the benefit-type criteria:

" L U #
x x h i
ij ij ¼ y L y U (13.19)
ij
y ¼ ij ij
x j x j
where y ij , which is an interval number, represents the normalized data of the ith alternative

L
U
with respect to the jth criterion, and y ij and y ij represent the lower and upper bounds of the
interval numbery ij , respectively.

x j , which represents the average value of the upper bounds of the data with respect to the
jth criterion can be determined by Eq. (13.20).
M
X U
x
ij
i¼1
M
x j ¼ (13.20)
As for the data with respect to the cost-type criteria:
" U L #
1=x 1=x h i
ij ij L U
¼ y y (13.21)
ij
y ¼ ij ij
x j x j
where y ij , which is an interval number, represents the normalized data of the ith alternative

L
U
with respect to the jth criterion, and y ij and y ij represent the lower and upper bounds of the
interval numbery ij , respectively.

x j can be determined by Eq. (13.22).
M
X 1
x L
i¼1 ij
(13.22)
M
x j ¼
Then, the normalized decision-making matrix can be determined, as presented in Eq. (13.23).
C 1 C 2 ⋯ C N
A 1 y y ⋯ y
12
1N
11
Y ¼ A 2 y y ⋯ y (13.23)
2N
22
21
⋮ ⋮ ⋮ ⋱ ⋮
A M y y ⋯ y
M1 M2 MN
where Y represents the normalized decision-making matrix.
Step 3: Determining the weights of the N criteria for sustainability assessment. The interval
best-worst method developed by Ren (2018b) is based on the works of Rezaei (2015, 2016) and
Entani et al. (2001). It consists of three sub-steps:
Sub-step 1: Determining the most important (i.e., the best) criterion and the least important
(i.e., the worst) criterion according to the opinions of the stakeholders in the decision-making
process, denoted by C B and C W , respectively (Rezaei, 2015, 2016).