Page 299 - Materials Chemistry, Second Edition

298 14. LCS prioritization of energy storage under uncertainties

where w is the probability distribution.
Based on the multinomial distribution, the probability of the event j is proportionate to the
number of event occurrence to the total number of trials, namely:
a jW
n (14.7)
w j αX
a jW
j¼1
Then, it can be obtained:
w j
αa jW (14.8)
w W
Meanwhile, A B can be modeled using the multinomial distribution, but it is different from
A W because the operation orders of the pairwise comparisons for the best criterion and the
worst criterion are reverse. So, there exists:
A B α multinomial 1=wð Þ (14.9)
where/represents the element-wise division operator.
Similarly,
w B
αa Bj (14.10)
w j
Therefore, the criteria weights determination in the BWM is transferred to the estimation of
a probability distribution, and the statistical inference techniques can be used to find w in the
multinomial distribution.
The maximum likelihood estimation (MLE) is arguably the most popular inference tech-
nique that can find the optimal criteria weight vector, for which the Bayesian estimation
can be used. In the Bayesian inference, the Dirichlet distribution is employed to model the
criteria weights because of the non-negativity and sum-to-one properties of weight vector.
However, the MLE inference containing both A B and A W does not bear an analytical solution
due to the complexity of the corresponding optimization problem, and the simple Dirichlet-
multinomial conjugate cannot encompass A B and A W together. Thus, a Bayesian hierarchical
model is needed.
k
k
Assume that there are k decision-makers to assess n criteria using the vectors A B and A W ,
and w agg is the overall optimal weight. The w agg can be calculated based on the optimal weights
1:k
1:k
k
of k decision-makers shown by w . In the BBWM, A B and A W are given, and w 1:k and
w agg need to be estimated. Therefore, the following joint probability distribution can be
sought:
agg 1:k 1:k
Pw , w 1:k A , A (14.11)
B W
Then, the probability of each individual variable can be computed using the following
probability rule:
X
PxðÞ ¼ Px, yð Þ (14.12)
y
where x and y are two arbitrary random variables.