## 3 An Expert System to Estimate RIF
### Bayesian Best-Worst Method  
The BWM was proposed by [PERSON] and applied in multi-criteria decision-making techniques to determine the weight of criteria [18, 20, 37]; this method has attracted many scholars' attention due to its efficiency and good performance in maintaining consistency between judgments [17]. Nevertheless, the BWM can obtain optimal weights of a set of criteria from only one expert. Different experts will give different opinions based on their knowledge background and understanding of the product, and a group of experts' judgments must therefore be aggregated into a single opinion. Normally, equal weights are given for all experts and the opinions of all people are arithmetically averaged, which results in ignoring the experts' knowledge, expertise and experience. On the other hand, if each expert gives different weights based on their experience, the opinions of the higher-scoring experts will dominate the results [38]. There are indeed some studies that have been conducted on multiple BWMs as listed below, which were used to fuse the opinions of a group of experts [39, 40]. However, these methods did not determine the final weight of the group by probability. While the Bayesian BWM computes the aggregated distribution and all the individual preferences at once using probabilistic modeling, at the same time, we can check the ranking of the weights from a probabilistic point of view [23, 41]. If the relationship between the two criteria is associated with a high confidence level, the panel will be more certain, while a low-confidence-level relationship should be interpreted more carefully. In this paper, we introduced Bayesian BWM to obtain the aggregated criterion without needing to consider expert weights; the steps of this are as follows:* Determination of criteria A set of criteria \(C=\{c_{1},c_{2},\cdots,c_{m}\}\) is determined that impact the reliability of a product. * Determination of the best and worst criteria The best and worst criteria represent the criteria that have the most influential and least significant impact on the reliability of the set of criteria \(C\). Each expert should select the best and the worst criteria from the set of criteria. * Determination of vectors of the best-to-others and others-to-worst vectors Based on step 2, the relative importance of the best criteria to others, and of the worst to others, can be obtained. The best-to-others and others-to-worst vectors are defined as follows: \[A_{B}^{i}=\left(d_{b1}^{i},a_{b2}^{j},\cdots,a_{bm}^{i}\right)\] (3) where \(a_{bj}^{i}\) indicates the relative importance of the best of the criteria \(j(1\leq j\leq m)\); similarly, \[A_{W}^{i}=\left(a_{w1}^{i},a_{w2}^{i},\cdots,a_{wm}^{i}\right)\] (4) where \(A_{W}^{i}\) indicates the relative importance of the worst to other criteria \(j(1\leq j\leq m)\). * Determination of the aggregated weight The aggregated weight \(w^{\text{agg}}\) is calculated by each expert's optimal weight \(w_{k}(k=1,2,\cdots,n)\), and the Bayesian BWM can simultaneously calculate \(w^{\text{agg}}\) and \(w_{k}\). The best-to-others and others-to-worst vectors make up the priors that could be considered to be posterior. Thus, the following joint probability distribution is sought \(P\left(w^{\text{agg}},w^{1:K}\mid A_{B}^{1:K},A_{W}^{1:K}\right)\), which is the posterior of Bayesian inference. The distribution of the aggregated weight prior \(P\left(w^{\text{agg}},w^{1:K}\right)\) is assumed to follow the Dirichlet distribution, and \(P\left(A_{B}^{1:K},A_{W}^{1:K}\mid w^{\text{agg}},w^{1:K}\right)\) is the likelihood function; thus, by the Bayes rule, the posterior distribution could be written as \[ \begin{split} P\left(w^{\text{agg}},w^{1:n}\mid A_{B}^{1:n},A_{W }^{1:n}\right)\propto& P\left(A_{B}^{1:n}\mid A_{W}^{1:n}\mid w^{ \text{agg}}\,,w^{1:n}\right)\,P\left(w^{\text{agg}}\,,w^{1:n}\right)\\ &=P\left(w^{\text{agg}}\,\right)\prod_{i=1}^{n}P\left(A_{W}^{i} \mid w^{i}\right)P\left(A_{B}^{i}\mid w^{i}\right)P\left(w^{i}\mid w^{\text{agg }}\,\right)\end{split}