## 4 Application in Circular Model Selection with Significant Challenges
### Stage 2: Z-Number Based Fuzzy TOPSIS for Ranking  
Step 1: Construct the Z-Fuzzy Initial Decision Matrix (ZFIDM), the ZFIDM is constructed, and the linguistic variables from Tables 4 and 6 are used to assess the parameters respect to the criteria and sub-criteria. The assessments of the circular models are reported in Table 16 (i.e., ZFIDM created using from the Experts (E1, E2, E3)). Step 2: Converting the reliability part and then combine with the restriction part are computed in the above Section 4.1. similarly follow for those steps to convert the Z-numbers to normal fuzzy numbers. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Challenges** & **CMs** & **E1** & **E2** & **E3** & **Challenges** & **CMs** & **E1** & **E2** & **E3** \\ \hline \multirow{3}{*}{C1 (Min)} & CM1 & (VHL S) & (VHL S) & (VHL S) & (VHL S) & & CM1 & (HL S) & (HL S) \\  & CM2 & (VHL S) & (VHL S) & (HL MB) & C9 (Max) & CM2 & (HL S) & (HL S) & (HL S) \\  & CM3 & (MHL S) & (MHL S) & (MHL S) & & CM3 & (VHL S) & (VHL S) & (VHL S) \\ \hline \multirow{3}{*}{C2 (Min)} & CM1 & (HIL S) & (HIL S) & (HIL S) & & CM1 & (HIL S) & (HL S) & (HIL S) \\  & CM2 & (VHL S) & (VHL S) & (VHL S) & (VHL S) & & CM3 & (HIL S) & (HL S) \\  & CM3 & (MHL S) & (VHL S) & (VHL S) & & & CM3 & (HIL S) & (HIL S) \\ \hline \multirow{3}{*}{C3 (Min)} & CM1 & (VHL S) & (VHL S) & (HIL S) & & CM1 & (VHL S) & (VHL S) & (VHL S) \\  & CM2 & (VHL S) & (VHL S) & (VHL S) & & & CM1 & (VHL S) & (VHL S) \\  & CM3 & (VHL S) & (VHL S) & (VHL S) & & & CM3 & (HIL S) & (MHL S) \\ \hline \multirow{3}{*}{C4 (Min)} & CM1 & (VHL S) & (VHL S) & (VHL S) & & CM1 & (HIL S) & (HH L S) & (HIL S) \\  & CM3 & (MHL S) & (VHL S) & (VHL S) & & & CM3 & (HIL S) & (HL S) \\ \hline \multirow{3}{*}{C5 (Max)} & CM1 & (VHL S) & (VHL S) & (VHL S) & & CM1 & (MHL S) & (MHL S) & (VHL S) \\  & CM2 & (VHL S) & (VHL S) & (VHL S) & & & CM2 & (VHL S) & (VHL S) \\  & CM3 & (VHL S) & (VHL S) & (VHL S) & & & CM3 & (MHL S) & (VHL S) \\ \hline \multirow{3}{*}{C6 (Max)} & CM1 & (VHL S) & (VHL S) & (VHL S) & & & CM1 & (VHL S) & (VHL S) \\  & CM2 & (VHL S) & (VHL S) & (VHL S) & & & CM2 & (VHL S) & (VHL S) \\ \hline \multirow{3}{*}{C7 (Max)} & CM1 & (VHL S) & (VHL S) & (VHL S) & & & & \\  & CM2 & (HIL S) & (HIL S) & (HIL S) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 16: ZIFDM given by the experts w.r.t Transport criteria. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Objectives** & **Local Weights** & **Challenges** & **Local Weights** & **Global Weights** & **Rank** \\ \hline \multirow{3}{*}{O1} & \multirow{3}{*}{0.219} & C1 & 0.316 & 0.069 & 9 \\  & & C2 & 0.353 & 0.078 & 1 \\  & & C3 & 0.331 & 0.073 & 4 \\ \hline \multirow{3}{*}{O2} & \multirow{3}{*}{0.209} & C4 & 0.328 & 0.069 & 10 \\  & & C5 & 0.326 & 0.068 & 11 \\  & & C6 & 0.346 & 0.072 & 6 \\ \hline \multirow{3}{*}{O3} & \multirow{3}{*}{0.140} & C7 & 0.330 & 0.046 & 14 \\  & & C8 & 0.311 & 0.044 & 15 \\  & & C9 & 0.358 & 0.050 & 13 \\ \hline \multirow{3}{*}{O4} & \multirow{3}{*}{0.216} & C10 & 0.352 & 0.076 & 2 \\  & & C11 & 0.313 & 0.068 & 12 \\  & & C12 & 0.335 & 0.073 & 5 \\ \hline \multirow{3}{*}{O5} & \multirow{3}{*}{0.215} & C13 & 0.335 & 0.072 & 7 \\  & & C14 & 0.340 & 0.073 & 3 \\ \cline{1-1}  & & C15 & 0.325 & 0.070 & 8 \\ \hline \end{tabular}
\end{table} 
Table 15: Computed global weights from criteria and sub-criteria. Step 3: Defuzzify the fuzzy numbers of the decision matrix of the experts accumulated preferences using Equation (2). the fuzzy initial decision matrix is defuzzified for converting into crisp decision matrix. Step 4: Compute the defuzzified weighted normalized matrix to proceed for the further steps of the computations. Step 5: In the next step, High Impact Ideal Solution (HIIS, H\({}^{+}\)) and Low Impact Ideal Solution (LIIS L\({}^{-}\)) are calculated using Equations (12) and (13). Step 6: Calculate the distance of each circular model from HIIS and LIIS.