Page 185 - Materials Chemistry, Second Edition

9.3 Methodology 183
9.3.3 Step 3. Determine the criteria weights based on opinions of different
decision makers by group ZBWM

The criteria weights in this study were determined by multiple stakeholders by using the
group ZBWM. The BWM (Rezaei, 2015) is a weighting method to weight criteria by pairwise
comparison. However, unlike AHP, BWM simplifies the process by comparing all criteria
with the best criterion and comparing all criteria with the worst criterion, instead of compar-
ing every pair of criteria. In this case, the BWM has been widely adapted in ranking and se-
lection in industries such as marketing (Cohen, 2009), energy generation (van de Kaa et al.,
2017), and supply chain (Badri Ahmadi et al., 2017).
The first step is to list out the level of priority by pairwise comparison. The number or lin-
guistic term collected for comparison description in BWM is used for describing the pairwise
comparison between every criterion and the best criterion, and the pairwise comparison be-
tween every criterion and the worst criterion. In order to consider the hesitations of decision
makers, fuzzy numbers are adapted to better illustrate the linguistic preferences provided by
decision makers. The Z-number, denoted as e xl, m, uð Þ, is a fuzzy number raised by Zadeh
(2011) that can show both constraints and reliability. In a Z-number linguistic variable for
the ZBWM, there are two parts describing pairwise importance and reliabilities of this com-
parison, respectively. The linguistic variables of constraints include equally important (EI),
weakly important (WI), fairly important (FI), very important (VI), and absolutely important
(AI). As for linguistic variables of reliabilities, the linguistic terms can be denoted as very low
(VL), low (L), medium (M), high (H), and very high (VH). Either linguistic variables of con-
straints or of reliabilities have their own membership functions. The membership functions
for Z-number linguistic variables were calculated as shown in Table 9.3. The detailed calcu-
lation process can be referred to the work of Aboutorab et al. (2018).
Assume that p stakeholders participating in the decision-making process, the linguistic
opinions need to be collected in the same process as mentioned above.
The optimal fuzzy weights of multiple stakeholders can be solved using Eq. (9.3).
p
X
Min λ k ξ k
k¼1
8

W
W
> l , m , u W
> B B B
> ξ
> k
> l kBj , m kBj , u kBj
> W W W
> l , m , u
> j j j
>
>
>
>
>
> W W W
> l , m , u
> j j j (9.3)
>
< ξ

W
W
l , m , u W l kjW , m kjW , u kjW k
s:t: W W W
>
>
>
>
> W W W
n
> X l +4 m + u
> j j j
>
>
> ¼ 1
>
> 6
>
> j¼1
>
>
>
: W W W
0 l m u
j j j
where, λ k indicates the importance of the opinion of the kth stakeholder participating in de-
P p
λ
cision making and k¼1 k ¼ 1; ξ ¼ ξ , ξ , ξð k k k Þ represents the upper limits of differences
e
k