## Appendix B  
The detailed derivation of Equation (18) is presented as follow: We use the arithmetic operation rules of interval numbers in the proof progress. Therefore, the basic operation rules between interval numbers are introduced first.  
For arbitrary \(a^{down},a^{up},b^{down},b^{up}\in R\) which subject to \(a^{down}<a^{up},b^{down}<b^{up}\), \(A^{I}=[a^{down},a^{up}]\), \(B^{I}=[b^{down},b^{up}]\), the rules for interval addition, subtraction, multiplication and division are:  
\[A^{I}+B^{I}=\left[a^{down}+b^{down},a^{up}+b^{up}\right]\] (B1)  
\[A^{I}-B^{I}=\left[a^{down}-b^{up},a^{up}-b^{down}\right]\] (B2)  
\[A^{I}\times B^{I}=\left[\begin{array}{c}\min\Bigl{(}a^{down}b^{down},a^{down }b^{up},a^{up}b^{down},a^{up}b^{up}\Bigr{)},\\ \max\Bigl{(}a^{down}b^{down},a^{down}b^{up},a^{up}b^{down},a^{up}b^{up}\Bigr{)} \end{array}\right]\] (B3)  
\[A^{I}/B^{I}=\left[\begin{array}{c}\min\Bigl{(}a^{down}/b^{down},a^{down}/b^{ up},a^{up}/b^{down},a^{up}/b^{up}\Bigr{)},\\ \max\Bigl{(}a^{down}/b^{down},a^{down}/b^{up},a^{up}/b^{down},a^{up}/b^{up} \Bigr{)}\end{array}\right]\] (B4)  
Also, in order to make the expression simpler, we set  
\[d_{n}^{U}=g(a_{d},\Gamma_{d})=d_{n}^{\min}+a_{d}\Gamma_{d}\Delta d_{n}=[g(a_{d }^{\min},\Gamma_{d}),g(a_{d}^{\max},\Gamma_{d})]\] (B5)  
\[\theta_{e,n}^{U}=h(a_{\theta},\Gamma_{\theta})=\theta_{e,n}^{\max}+a_{\theta} \Gamma_{\theta}\Delta\theta_{e,n}=[h(a_{\theta}^{\max},\Gamma_{\theta}),h(a_{ \theta}^{\min},\Gamma_{\theta})]\] (B6)  
Therefore, after bringing the above formulation into the Equation (A3), we can find that the uncertain hot water temperature is transformed into intervals as follows:  
\[\begin{array}{l}\theta_{n+1}^{U}=[(1-\partial)(h(a_{\theta}^{\max},\Gamma_ {\theta})+U_{heat,n})+\partial\theta_{n}^{U},(1-\partial)(h(a_{\theta}^{\min },\Gamma_{\theta})+U_{heat,n})+\partial\theta_{n}^{U}]-\\ \bigl{[}((1-\partial)U_{heat,n}+\partial\bigl{(}\theta_{n}^{U}-h(a_{\theta}^ {\min},\Gamma_{\theta})\bigr{)}\bigr{)},\bigl{(}(1-\partial)U_{heat,n}+ \partial\bigl{(}\theta_{n}^{U}-h(a_{\theta}^{\max},\Gamma_{\theta})\bigr{)} \bigr{)}\bigr{]}\\ \times[g(a_{d}^{\min},\Gamma_{d}),g(a_{d}^{\max},\Gamma_{d})]/M\end{array}\] (B7)  
Using the arithmetic operation rules of interval numbers above, equation can be further derived as follows:  
\[\begin{array}{l}\theta_{n+1}^{U}=[(1-\partial)(h(a_{\theta}^{\max},\Gamma_ {\theta})+U_{heat,n})+\partial\theta_{n}^{U},(1-\partial)(h(a_{\theta}^{\min },\Gamma_{\theta})+U_{heat,n})+\partial\theta_{n}^{U}]-[p,q]\\ =[(1-\partial)(h(a_{\theta}^{\max},\Gamma_{\theta})+U_{heat,n})+\partial \theta_{n}^{U}-p,(1-\partial)(h(a_{\theta}^{\min},\Gamma_{\theta})+U_{heat,n} )+\partial\theta_{n}^{U}-q]\\ p=\bigl{(}(1-\partial)U_{heat,n}+\partial\bigl{(}\theta_{n}^{U}-h(a_{\theta}^ {\min},\Gamma_{\theta})\bigr{)}\bigr{)}\cdot g(a_{d}^{\min},\Gamma_{d})/M\\ q=\bigl{(}(1-\partial)U_{heat,n}+\partial\bigl{(}\theta_{n}^{U}-h(a_{\theta}^ {\max},\Gamma_{\theta})\bigr{)}\bigr{)}\cdot g(a_{d}^{\max},\Gamma_{d})/M \end{array}\] (B8)  
\[\theta_{n+1}^{U}=\left[\theta_{n+1}^{U}(X_{n},a_{\theta}^{\max},a_{d}^{\max}, \Gamma_{d},\Gamma_{\theta}),\theta_{n+1}^{U}\Bigl{(}X_{n},a_{\theta}^{\min},a _{d}^{\min},\Gamma_{d},\Gamma_{\theta}\Bigr{)}\right]\] (B9)  
Because \(a_{d}^{\min}=a_{\theta}^{\min}=0\), \(a_{d}^{\max}=a_{\theta}^{\max}=1\), therefore Equation (18) holds as follow:  
\[\left\{\begin{array}{l}\theta_{n+1}^{\min}(X_{n},a_{d},a_{\theta},\Gamma_{d },\Gamma_{\theta})=\theta_{n+1}^{U}(X_{n},1,1,\Gamma_{d},\Gamma_{\theta})\\ \theta_{n+1}^{\max}(X_{n},a_{d},a_{\theta},\Gamma_{d},\Gamma_{\theta})=\theta_{ n+1}^{U}(X_{n},0,0,\Gamma_{d},\Gamma_{\theta})\end{array}\right.\] (B10)