## 4 Application in Circular Model Selection with Significant Challenges
### Stage 1: Z-Number Based Fuzzy DEMATEL for Global Weights  
Step 1: Develop a hierarchical framework shown in the Figure 5 The building of a hierarchy model necessitates the completion of a judgement matrix by experts about the assessment of all criteria and sub-criteria. Step 2: Construct a direct relation matrix (DRM) with confidence part. To determine the global weights of considered criteria and sub-criteria that influence of sustainable circular model. To adhere towards the Z-number-based fuzzy DEMATEL in Section 3.3 to construct DRMs with reliability among all criteria and sub-criteria in the degree of the hierarchy systems that depend on the preferences of the experts with confidence based on the Table 5 using triangular fuzzy number for restriction part and Table 4 using triangular fuzzy number for reliability part (i.e., DRM created using from the Expert 1 (E1) models given in Tables 7 and 8 utilizing crisp numbers instead of triangular fuzzy numbers for the impact element in order to ease comprehension.). Similarly, the remaining DRMs for the criteria and sub-criteria from the three experts can be tabulated. Step 3: Convert Z-numbers representing the preferences of experts with a reliability component into normal fuzzy numbers. Using Equation (3), convert the Z-number to crisp number of the reliability component. That crisp number employed as the reliability weight. For the investigation of criteria and sub-criteria by experts at this level, the greatest degree of reliability agreed upon for their judgements is 1. The fuzzy DRMs representing the preferences of experts are collected as illustrated in the example below. E1--Economic Objectives (O1) \(*\) Environmental Objectives (O2) from the Table 7 (i.e., VHI, S)  
\[=(0.75,\ 1,\ 1),\ \widetilde{m}=(0.9,\ 0.9,\ 1)\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**DM 1** & **C1** & **C2** & **C3** & **C4** & **C5** \\ \cline{2-6}
**(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** \\ \hline O1 & (NI, S) & (VHI, S) & (MI, SS) & (MI, SS) & (HI, S) & (VHI, S) \\ O2 & (VHI, S) & (NI, S) & (MI, S) & (HI, S) & (HI, S) & (HI, S) \\ O3 & (HI, SS) & (MI, S) & (NI, S) & (NI, S) & (MI, S) & (MI, SS) \\ O4 & (VHI, S) & (VHI, S) & (LI, S) & (NI, S) & (NI, S) & (VII, SS) \\ O5 & (VHI, S) & (VHI, SS) & (VHI, SS) & (MI, SS) & (HI, SS) & (HI, SS) & (VII, S) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Initial Relationship Matrix of sub-criteria with confidence part from E1 w.r.t goal. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**DM 1** & **C1** & **C2** & **C3** & **C4** & **C5** \\ \cline{2-6}
**(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** & **(\(a,m\))** \\ \hline C1 & (NI, S) & (VHI, S) & (HI, S) & (MI, SS) & (LI, S) & (HI, S) & (HI, S) \\ O2 & (VHI, S) & (NI, S) & (MI, S) & (MI, S) & (HIwhereby the \(\tilde{a}\), \(\widetilde{\widetilde{m}}\) values are taken from the Tables 4 and 5. The competence of the experts can be stated as the Z-number \(Z=(\tilde{a},\widetilde{\widetilde{m}})\): Initially, the part of confidence must be changed to crisp using Equation (3) to employed as the confidence weight. The confidence weight can be denoted as \(\alpha\). \[Z=[(0.75,\;1,\;1)\;(0.965)]\]  
Step 4: The next step is to convert weighted Z-numbers to regular fuzzy number. Augment the weight of the confidence part to the impact part using Equation (5). The regular fuzzy number representing the preferences of experts are collected as illustrated in the sample example used in the previous step is below. \[\widetilde{Z}^{\prime}=\left(\sqrt{0.965}\times 75,\sqrt{0.965}\times 1,\sqrt{0.9 65}\times 1\right)\]  
\[\widetilde{Z}^{\prime}=(0.7238,\;0.965,\;0.96