# Multiple-Attribute Decision Making ELECTRE II Method under Bipolar Fuzzy Model
## 2 The BF-ELECTRE II Method
### Construction of Outranking Graphs
\[\left(\omega_{d},\omega_{d}^{\prime},\omega_{d}^{\prime\prime}\right)=\left(1, \frac{



For instance, the bipolar fuzzy discordance index \(t_{12}\) is determined as:

\[t_{12} = \frac{\max\left\{\omega_{d^{{}^{\prime}}}\times d(\omega_{1}\ell_{11},\omega_{1}\ell_{21}),\omega_{d^{{}^{\prime\prime}}}\times d(\omega_{2}\ell_{12},\omega_{1}\ell_{22}),\omega_{d^{{}^{\prime\prime}}}\times d(\omega_{5}\ell_{15 },\omega_{4}\ell_{25})\right\}}{\max\limits_{n\in I}\,d(\omega_{n}\ell_{an}, \omega_{n}\ell_{gn})}\] \[= \frac{\max\left\{\frac{3}{4}\times 0.0344,\frac{2}{4}\times 0.1209,\frac{2}{4}\times 0.2233\right\}}{\max\left\{0.0344,0.1209,0,0.1483,0.2233 \right\}}\] \[= \frac{\max\{0.0258,0.0605,0.1117\}}{\max\{0.0344,0.1209,0,0.1483,0.2233\}}\] \[= \frac{0.1117}{0.2233}=0.5002.\]

Step 6.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 levels which are specified by decision maker as:

\[(f^{*},f^{\circ},f^{-})=(0.4,0.3,0.2),\ \ \ (t^{*},t^{\circ})=(0.3,0.5).\]

The strong and weak outranking relations are established by applying Equations (11) and (12), respectively, and are shown in Figure 1:0

Step 7.The strongly outrank graph \(G^{s}\) and weakly outrank graph \(G^{w}\) are drawn in Figure 2 according the strong and weak outranking relations, respectively. These outranking graphs are used to find the average ordering of alternatives by following the iterative procedure explained in Section 2.5. \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 \(\mathcal{P}_{1}\) & \(-\) & \(0\) & \(0\) & \(0\) & \(R^{s},R^{w}\) \\ \(\mathcal{P}_{2}\) & \(R^{s}\) & \(-\) & \(R^{w}\) & \(R^{s}\) & \(R^{s},R^{w}\) \\ \(\mathcal{P}_{3}\) & \(R^{s},R^{w}\) & \(0\) & \(-\) & \(0\) & \(R^{s},R^{w}\) \\ \(\mathcal{P}_{4}\) & \(0\) & \(0\) & \(0\) & \(-\) & \(R^{s},R^{w}\) \\ \(\mathcal{P}_{5}\) & \(0\) & \(0\) & \(0\) & \(0\) & \(-\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Outranking relation. Figure 2: Graphical representation of outranking relations between alternatives. 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 the average ranking \(\lambda\) are obtained and these rankings are summarized in Table 8. Finally, on the basis of average ranking, these five alternative are ranked as:

\[\mathcal{P}_{2}\succ\mathcal{P}_{3}\succ\mathcal{P}_{4}\succ\mathcal{P}_{1} \succ\mathcal{P}_{5}.\]

## 4 Discussion and Comparative Study

This section provides a comparative analysis of presented bipolar fuzzy ELECTRE II method with already existing MADM methods such as TOPSIS and ELECTRE I models under bipolar fuzzy environment, which were presented by [PERSON] et al. (2019). We applied these methods to the numerical problem presented in Section 3.1, as \"selection of business location\" to compare the different MCDM methods. A discussion and theoretical comparison of BF-ELECTRE II method is also presented with fuzzy ELECTRE II method, which was proposed by [PERSON] et al. (2019). ### Bipolar Fuzzy TOPSIS Method

A flow chart of general steps of bipolar fuzzy TOPSIS method, presented by [PERSON] et al. (2019), is given in Figure 3. In this subsection, the problem to select the business location is solved by using bipolar fuzzy TOPSIS method. The steps for the construction of bipolar fuzzy decision matrix and the calculation of weights are the same as described in bipolar fuzzy ELECTRE II method.