## Appendix A  
\[\alpha_{k}=\prod_{i=1}^{T_{k}}(\alpha_{i}^{k})^{\tilde{\alpha}_{ki}} \tag{1}\]  
where \(\tilde{\delta}_{ki}=\frac{\delta_{ki}}{max_{i=1...T_{k}^{(k_{ki})}}}\) so that \(0\leq\tilde{\delta}_{ki}\leq 1\)  
Here, \(\alpha_{k}\) represents the degree to which an input matches with the antecedent attributes of the \(k^{\text{th}}\) rule. The total number of antecedent attributes in the \(k^{\text{th}}\) rule is denoted by \(T_{k}\). \(i^{th}\) antecedent attribute's weight is denoted by \(\delta_{ki}\). A rule is activated when the referential values of its antecedent attributes are assigned with matching degrees. Equation (1) calculates the activation weight of this rule, as shown below:  
\[w_{k}=\frac{\theta_{k}\alpha_{k}}{\sum_{i=1}^{L}(\theta_{i}\alpha_{i})} \tag{2}\]  
Here, \(w_{k}\) and \(\theta_{k}\) are the \(k^{th}\) rule's activation weight (\(0\leq w_{k}\leq 1\)) and rule weight (between 0 and 1), respectively. The total number of rules contained in the rule base of the BRBES is denoted by L.  
Uncertainty due to ignorance is triggered when input data to one or all of the antecedent attributes become unavailable. For example, energy consumption prediction requires input data for three antecedent attributes. However, if input data to any or all of these attributes become inaccessible, the initial belief degrees of the consequent referential values will be updated by Equation (2):  
\[\beta_{jk}=\tilde{\beta}_{jk}\frac{\sum_{t=1}^{T_{k}}(\lambda(t,k)\sum_{i=1}^ {L_{t}}(\alpha_{ti}))}{\sum_{t=1}^{T_{k}}\lambda(t,k)} \tag{3}\]  
\[\text{where }\lambda(t,k)=\begin{cases}1&\text{if the }t^{th}\text{ attribute is used in}\\ &\text{defining rule }R_{k}(k=1,...,T_{k})\\ 0&\text{otherwise}\end{cases}\]  
Here, \(\tilde{\beta_{ik}}\) and \(\beta_{ik}\) denote the \(k^{\text{th}}\) rule's original and updated belief degree, respectively. The degree to which an input value matches with an attribute is denoted by \(\alpha_{ti}\).  
In the inference procedure of the BRBES, the ER approach is employed to aggregate all the rules of its rule base. Either a recursive or an analytical ER approach is applied for this aggregation [47]. Because of lower computational complexity, we opt for the analytical ER approach in our framework [47]. Finally, Equation (A4) comes up with the ultimate decision C(Y), as shown below:  
\[C(Y)=\{(O_{j},\beta_{j}),j=1,...,N\} \tag{4}\]where \(O_{j}\) is the \(j^{th}\) referential value of the consequent attribute, \(\beta_{j}\) is the final aggregated belief degree related to the \(j^{th}\) referential value of the consequent attribute, and N is the total number of referential values of the consequent attribute. \(\beta_{j}\) is calculated by employing an analytical ER algorithm [47], as shown in Equation (A5):  
\[\beta_{j}=\frac{\mu\times[X-\prod_{k=1}^{L}(1-\omega_{k}\sum_{j=1}^{N}\beta_{jk })]}{1-\mu\times[\prod_{k=1}^{L}1-\omega_{k}]}\] (A5)  
\[\text{where }\mu=\bigg{[}\sum_{j=1}^{N}\prod_{k=1}^{L}(\omega_{k}\beta_{jk}+1- \omega_{k}\sum_{j=1}^{N}\beta_{jk})-(N-1)\times\prod_{k=1}^{L}(1-\omega_{k} \sum_{j=1}^{N}\beta_{jk})\bigg{]}^{-1}\]  
\[X=\prod_{k=1}^{L}(\omega_{k}\beta_{jk}+1-\omega_{k}\sum_{j=1}^{N}\beta_{jk})\]  
Here, L refers to the length of the rule base of the BRBES in terms of the total number of rules.