# Multiple-Attribute Decision Making ELECTRE II Method under Bipolar Fuzzy Model
## 2 The BF-ELECTRE II Method
### Construction of Outranking Graphs
Step 4.The weighted distances between any two alternatives with respect to each criteria are calculated by using the bipolar fuzzy Euclidean distance and are shown in Table 2.

For example, the Euclidean distance between \(\ell_{11}\) and \(\ell_{21}\) with respect to criteria \(K_{1}\) is calculated as

\[d(\omega_{1}\ell_{11},\omega_{1}\ell_{21}) = \sqrt{\frac{1}{2}\times\omega_{1}[(\mu_{11}-\mu_{21})^{2}+(\
u_{1 1}-\
u_{21})^{2}]}\] \[= \sqrt{\frac{1}{2}\times 0.1964[(0.35-0.6)^{2}+(-0.7-(-0.25))^{2}]}\] \[= 0.1613.\]

Similarly, \(d(\omega_{2}\ell_{12},\omega_{2}\ell_{22})=0.2047\), \(d(\omega_{3}\ell_{13},\omega_{3}\ell_{23})=0.0786\), \(d(\omega_{4}\ell_{14},\omega_{4}\ell_{24})=0.1493\), \(d(\omega_{5}\ell_{15},\omega_{5}\ell_{25})=0.1361\) and others. Step 5.The importance weights of bipolar fuzzy strong, median and weak discordance sets assign by decision maker are shown in Equations (16). The bipolar fuzzy discordance indices \(t_{\alpha\beta},\ \ \{\alpha,\beta=1,2,\cdots,5,\ \alpha\
eq\beta\}\) are calculated by applying Equations (10), which are used as entries to construct the bipolar fuzzy discordance matrix \(H=(t_{\alpha\beta})_{5\times 5}\). \[\left(\omega_{d},\omega_{d},\omega_{d^{\prime}}\right)=\left(1,\frac{3}{4}



For instance, the bipolar fuzzy discordance index \(t_{14}\) is computed as: \[t_{14} = \frac{\max\left\{\omega_{d}\times d(\omega_{5}\ell_{15},\omega_{5} \ell_{45}),\omega_{d^{\prime\prime}}\times d(\omega_{1}\ell_{11},\omega_{1}\ell _{41}),\omega_{d^{\prime\prime}}\times d(\omega_{4}\ell_{14},\omega_{4}\ell_{44 })\right\}}{\max\limits_{n\in I}\,d(\omega_{n}\ell_{an},\omega_{n}\ell_{gn})}\] \[= \frac{\max\left\{1\times 0.0962,\frac{2}{4}\times 0.0407,\frac{2}{4} \times 0.0983\right\}}{\max\left\{0.0407,0.1184,0.0847,0.0983,0.0962\right\}}\] \[= \frac{\max\{0.0962,0.0204,0.0492\}}{\max\{0.0407,0.1184,0.0847,0. 0983,0.0962\}}\] \[= \frac{0.0962}{0.1184}=0.8125.\]
* The outranking relationships between the alternatives, as strong outranking relation \(R^{s}\) and weak outranking relation \(R^{w}\), are computed by comparing the entries of bipolar fuzzy concordance and discordance matrices along with the concordance and discordance threshold values which are specified by decision maker as: \[(f^{*},f^{\circ},f^{-})=(0.5,0.3,0.2),\ \ (t^{*},t^{\circ})=(0.5,0.7).\] The strong and weak outranking relations are established by applying Equations (11) and (12), respectively, and are shown in Table 2.0
* The strongly outrank graph \(G^{s}\) and weakly outrank graph \(G^{w}\) are drawn in Figure 1 according the strong and weak outranking relations, respectively. The outranking graphs are used to find the average ordering of alternatives by following the iterative procedure explained in Section 2.5. By analyzing these outranking graphs through the iterative procedure mentioned in Section 2.5, the results of forward ranking \(\lambda^{{}^{\prime}}\), reverse ranking \(\lambda^{{}^{\prime\prime}}\) and average ranking \(\lambda\) are obtained and these rankings are summarized in Table 2.1 On the basis of these rankings, the final ranking of the five alternatives is: Table 2.2 That is, \(\mathcal{P}_{2}\) is best among all other locations. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(\mathcal{P}_{1}\) & \(\mathcal{P}_{2}\) & \(\mathcal{P}_{3}\) & \(\mathcal{P}_{4}\) & \(\mathcal{P}_{5}\) \\ \hline Forward ranking \(\lambda^{{}^{\prime}}\) & 2 & 1 & 4 & 2 & 3 \\ Reverse ranking \(\lambda^{{}^{\prime\prime}}\) & 2 & 1 & 5 & 1 & 4 \\ Average ranking \(\lambda\) & 2 & 1 & 4.5 & 1.5 & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ranking results. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(\mathcal{P}_{1}\) & \(\mathcal{P}_{2}\) & \(\mathcal{P}_{3}\) & \(\mathcal{P}_{4}\) & \(\mathcal{P}_{5}\) \\ \hline \(\mathcal{P}_{1}\) & \(-\) & 0 & \(R^{s},R^{w}\) & 0 & \(R^{s},R^{w}\) \\ \(\mathcal{P}_{2}\) & \(R^{s},R^{w}\) & \(-\)

### Selection of a Supplier

Supplier selection is a procedure by which companies identify, evaluate and contract with suppliers.