## 6 A Numerical Example of the Supplier Assessment Process  
The supplier assessment process is critical for any business that wants to ensure its entire supply chain will be both efficient and reliable while also being cost-effective. The supplier assessment process involves evaluating potential and existing suppliers to determine their ability to meet the firm's requirements in terms of quality, price, delivery time, and other defining criteria. Supplier assessment data were collected in order to create long-term relationships with suppliers who can always provide products or services that meet company standards. This process also allows businesses to mitigate risks, like supply chain disruptions or compromised quality, and can lead to cost savings by making supplier partnerships more effective. In this part, three criteria are used by a firm to assess four suppliers (\(\tau_{1}\), \(\tau_{2}\), \(\tau_{3}\), \(\tau_{4}\)) for a long-term contract. The criteria are as follows: \(C_{1}:\) cost (minimization), where lower costs are preferred; \(C_{2}:\) quality (maximization): superior quality is appreciated; \(C_{3}:\) delivery time (minimization): It is preferable to have faster delivery times, where the weight vector for the criteria is defined as \(\epsilon=[0.4\ 0.4\ 0.2]\). To apply the first algorithm:  
1. We obtain the B\({}_{\text{pqr}}\)SF evaluation matrix \(\bar{D}=(\bar{s}_{ij})\) in Table 2.  
2. The weight vector for the criteria above is \(\epsilon=[0.4\ \ 0.4\ \ 0.2]\). 3. In this step, normalizing is needed as criteria are not on the same scale; in order to normalize them, we use step (3) in Definition 8.Criteria \(C_{1 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Alternatives** & \(\bar{D}(\tilde{\tau}_{i},B_{pqr}SFPIS)\) & \(\bar{D}(\tilde{\tau}_{i},B_{pqr}SFNIS)\) \\ \hline \(\tilde{\tau}_{1}\) & 0.121 & 0.1036 \\ \(\tilde{\tau}_{2}\) & 0.1043 & 0.0435 \\ \(\tilde{\tau}_{3}\) & 0.052 & 0.0967 \\ \(\tilde{\tau}_{4}\) & 0.0701 & 0.0841 \\ \hline \hline \end{tabular}  
7. The relative closeness degree of the corresponding alternative to the bipolar pqr-spherical fuzzy \(+\)ve ideal solution, denoted by \(\varpi_{i}\) for \(i=1,2,\ldots,n\), is computed by Formula (10), which is shown in Table 7 and Figure 2.  
\end{table} 
Table 6: The Euclidean distances of all alternatives with \(\mathbb{B}_{pqr}\)SFPIS and \(\mathbb{B}_{pqr}\)SFNIS, with the weighted normalized \(\mathbb{B}_{pqr}\)SF evaluation matrix. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\tilde{\tau}_{j}\) & **Cost \(C_{1}\)** & **Quality \(C_{2}\)** & **Delivery Time \(C_{3}\)** \\ \hline \multirow{3}{*}{\(\tau_{3}\)} & \(\left\{0.0018,0.1585,0.9456,\right.\) & \(\left\{0.3655,0.246,0.5544,\right.\) & \(\left\{0.0169,0.631,0.939,\right.\) \\  & \(-0.6178,-0.0007,-0.0141\}\) & \(-0.5357,-0.0003,-0.0101\}\) & \(-0.8152,-0.0006,-0.0351\}\) \\ \hline \multirow{3}{*}{\(\tau_{4}\)} & \(\left\{0.0012,0.3817,0.9352,\right.\) & \(\left\{0.5129,0.3245,0.2512,\right.\) & \(\left\{0.0058,0.6178,0.9587,\right.\) \\  & \(-0.634,-0.0013,-0.048\}\) & \(-0.4282,-0.0039,-0.2082\}\) & \(-0.7638,-0.0006,-0.0104\}\) \\ \hline \end{tabular}  
5. The \(\mathbb{B}_{pqr}\)SFPIS and \(\mathbb{B}_{pqr}\)SFNIS are computed in Table 5: \(\mathbb{B}_{pqr}\)SFPIS and \(\mathbb{B}_{pqr}\)SFNIS.  
\end{table} 
Table 4: \(Cont.\)  
Figure 2: Ranking scores. The alternatives are ranked based on their final scores (relative closeness values) as follows: \[\tau_{3}>\tau_{4}>\tau_{1}>\tau_{2}.\] Therefore, \(\tau_{3}\) is the best supply to be selected. To apply the second algorithm to the same numerical example used for the first algorithm: 1. We obtain the B\({}_{\text{pqr}}\)SF evaluation matrix \(\tilde{D}=(\tilde{\tau}_{ij})\), as in Table 2. 2. The weight vector for criteria is \(\epsilon=[0.4\ \ 0.4\ \ 0.2]\). 3. Normalizing, in this step, is needed, as criteria are not on the same scale; and in order to normalize them, we use step 3 in Definition 8. Criteria \(C_{1}\) and \(C_{3}\) need to be normalized by using Equation (6), as shown in Table 3. 4.