Page 280 - Materials Chemistry, Second Edition

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

Sub-step 4: Consistency check (Rezaei, 2015). The consistency ratio can be determined by
Eq. (13.27) (Ren, 2018a, b).
ξ ∗
(13.27)
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
CR ¼ L U
a + a 4a L +4a U +1
BW BW +1 BW BW
2
∗
where ξ is the minimum value of the objective function in programming (13.26), and CR rep-
resents the consistency ratio.
The value of CR represents the consistency level of the decision-makers’ judgments in de-
termining the BO and OW vectors, and the closer to zero, the more consistent the
judgments are.
Sub-step 5: Calculating the interval weights of the criteria. The radius of the weight of each
criterion can be determined by solving the programming (13.28).
min λ
C
ω d B L
B
s:t: a
C Bj
ω + d j
j
C
ω + d B U
B a
C Bj
ω d j
j
C
ω d
j L
a (13.28)
C jW
ω + d W
W
C
ω + d j
j U
C a jW
ω d W
W
d j λ
C
ω d j 0
j
j ¼ 1,2,⋯,T
where d j , d B , and d W represent the radius of the weights of the jth criterion, the best criterion,
and the worst criterion, respectively.
After determining the radius of each weight, the interval weight of each criterion can be
determined by Eq. (13.29).
h i h i
ω ¼ ω L ω U C C (13.29)

j
j
j j j ¼ ω d j ω + d j
U
L
where ω j represents the interval weight of the jth criterion, and ω j and ω j represent the lower

and upper bounds of the interval weight of the jth criterion, respectively.
Based on the above-mentioned five sub-steps in Step 3, the weights of the three dimensions
of sustainability and the local of the criteria in each dimension can be determined. Then, the
global weight of each criterion can be determined by using the local weight of each criterion
multiplied with the weight of the corresponding dimension to which it belongs.
Step 4: Determining the weighted normalized decision-making matrix. The weighted nor-
malized decision-making matrix can be determined by Eqs. (13.30), (13.31) after determining
the normalized decision-making matrix and the global weights of the criteria.