Page 298 - Materials Chemistry, Second Edition

14.3 Methods 297

w B a Bj and

w B /w j ¼a Bj and w j /w W ¼a jW , it can minimize the maximum absolute differences
w j

w j
a jW . Therefore, the following optimization issue can be built.

w W

n
8
X
w j ¼ 1
>
>
>
<
w B w j
min max s:t: j¼1 (14.3)
,
j w j a Bj w W a jW w j 0
>
>
>
j ¼ 1,2,⋯,n
:
Then, the weight vector can also be calculated by the following equation:
min ξ

8
w B
> ξ

>
> a Bj

> w j
>
>

>
>
w j
>
> ξ

>
< a jW
w W (14.4)

s:t: n
X
w j ¼ 1
>
>
>
>
>
> j¼1
>
>
w j 0
>
>
>
:
j ¼ 1,2,⋯,n
To check the consistency degree of pairwise comparison for criteria weight determination,
the veracity between the pairwise comparisons and their associated weight ratios can be
checked using the following consistency ratio (CR):
ξ ∗
CR ¼ (14.5)
CI
∗
where ξ is the optimal value of ξ, and CI is the consistency index, which is listed in Table 14.1.
14.3.2.2 The basic theory BBWM
For BBWM, the inputs and outputs have probabilistic interpretations. The value of criteria
indicates the importance of the corresponding criteria. From a probabilistic perspective, the
decision criteria can be seen as the random events, and then the decision criteria weights are
their occurrence likelihoods. Therefore, all the inputs and outputs need to be modelled as the
probability distributions, and the multinomial distribution is employed (Mohammadi and
Rezaei, 2019). The probability mass function (PMF) of the multinomial distribution for A W is:

X n
j¼1 a jW ! Y n a jW
PA W jwð Þ ¼ Q n w j (14.6)
j¼1 jW ! j¼1
a
TABLE 14.1 CI Table.
1 2 3 4 5 6 7 8 9
a BW
CI 0.00 0.44 1.00 1.63 2.30 3.00 3.73 4.47 5.23