# Multiple-Attribute Decision Making ELECTRE II Method under Bipolar Fuzzy Model
## 2 The BF-ELECTRE II Method
### Construction of Outranking Graphs
Consider the bipolar fuzzy decision matrix given in Table 1, and construct a weighted bipolar fuzzy decision matrix by multiplying the weight vector to decision matrix, which is given in Table 9.

The bipolar fuzzy positive ideal solution (BFPIS) and bipolar fuzzy negative ideal solution (BFNIS) for each criteria are computed as follows:

\[BFPIS =[(0.1571,-0.0491),(0.2001,-0.0333),(0.1979,-0.0792),(0.1490,-0. 0530),(0.2178,-0.0545)],\] \[BFNIS =[(0.0393,-0.1375),(0.0622,-0.1556),(0.0495,-0.1583),(0.0546,-0. 1192),(0.0653,-0.1742)].\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathbf{K_{1}}\) & \(\mathbf{K_{2}}\) & \(\mathbf{K_{3}}\) & \(\mathbf{K_{4}}\) & \(\mathbf{K_{5}}\) \\ \hline \(\mathcal{P}_{1}\) & \((0.0687,-0.1375)\) & \((0.2001,-0.0734)\) & \((0.0990,-0.1187)\) & \((0.0712,-0.1192)\) & \((0.1307,-0.0762)\) \\ \(\mathcal{P}_{2}\) & \((0.1178,-0.0491)\) & \((0.0889,-0.1556)\) & \((0.1286,-0.1583)\) & \((0.1490,-0.0083)\) & \((0.2178,-0.0545)\) \\ \(\mathcal{P}_{3}\) & \((0.0393,-0.0982)\) & \((0.0622,-0.1334)\) & \((0.0732,-0.1187)\) & \((0.0546,-0.0994)\) & \((0.1742,-0.0871)\) \\ \(\mathcal{P}_{4}\) & \((0.0923,-0.1277)\) & \((0.1223,-0.0600)\) & \((0.0495,-0.0990)\) & \((0.0911,-0.0662)\) & \((0.1634,-0.1307)\) \\ \(\mathcal{P}_{5}\) & \((0.1571,-0.0589)\) & \((0.0667,-0.0333)\) & \((0.1979,-0.0792)\) & \((0.1159,-0.0530)\) & \((0.0653,-0.1742)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Weighted bipolar fuzzy decision matrix. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{\mathcal{P}_{1}}\) & \(\mathbf{\mathcal{P}_{2}}\) & \(\mathbf{\mathcal{P}_{3}}\) & \(\mathcal{P}_{4}\) & \(\mathbf{\mathcal{P}_{5}}\) \\ \hline Forward ranking \(\lambda^{{}^{\prime\prime}}\) & 3 & 1 & 2 & 2 & 4 \\ Reverse ranking \(\lambda^{{}^{\prime\prime}}\) & 3 & 1 & 2 & 3 & 4 \\ Average ranking \(\lambdaFurthermore, the Euclidean distance of each alternative from BFPIS and BFNIS is enumerated as:

\[D(\mathcal{P}_{1},BFPIS) =0.153, D(\mathcal{P}_{1},BFNIS) =0.150,\] \[D(\mathcal{P}_{2},BFPIS) =0.143, D(\mathcal{P}_{2},BFNIS) =0.185,\] \[D(\mathcal{P}_{3},BFPIS) =0.196, D(\mathcal{P}_{3},BFNIS) =0.110,\] \[D(\mathcal{P}_{4},BFPIS) =0.161, D(\mathcal{P}_{4},BFNIS) =0.132,\] \[D(\mathcal{P}_{5},BFPIS) =0.168, D(\mathcal{P}_{5},BFNIS) =0.189.\]

The relative closeness degree of each alternative to BFPIS is calculated as follows:

\[C(\mathcal{P}_{1})=0.495,\ C(\mathcal{P}_{2})=0.564,\ C(\mathcal{P}_{3})=0.359, \ C(\mathcal{P}_{4})=0.451,\ C(\mathcal{P}_{5})=0.529.\]

According to these closeness coefficients, the alternatives are ranked in descending order as \(\mathcal{P}_{2}\succ\mathcal{P}_{5}\succ\mathcal{P}_{1}\succ\mathcal{P}_{4} \succ\mathcal{P}_{3}\), and thus location \(\mathcal{P}_{2}\) is the best choice with maximum closeness degree. ### Bipolar Fuzzy ELECTRE I Method

A flow chart of the general steps of bipolar fuzzy ELECTRE I method is given in Figure 4.

Figure 3: The steps of bipolar fuzzy TOPSIS model. To solve the same problem of business location by employing the bipolar fuzzy ELECTRE I method, consider the weighted bipolar fuzzy decision matrix given in Table 9, and follow the next steps of bipolar fuzzy ELECTRE I method to determine an outranking relation of alternatives in order to compare these multi-attribute decision making methods. The evaluation of bipolar fuzzy concordance sets \(\mathbb{C}_{\alpha\beta}\), bipolar fuzzy discordance sets \(\mathbb{D}_{\alpha\beta}\), bipolar fuzzy concordance indices \(c_{\alpha\beta}\), bipolar fuzzy discordance indices \(d_{\alpha\beta}\), concordance dominance \(g_{\alpha\beta}\), discordance dominance \(h_{\alpha\beta}\), aggregated dominance \(q_{\alpha\beta}\) and outranking relations for this problem is briefly summarized in Table 10. The graph sketch by outranking relations is given in Figure 5 and the set of most favorable alternatives is \(\{\mathcal{P}_{2},\mathcal{P}_{5}\}\). \[D(\mathcal{P}_{1},BFPIS) =0.153, D(\mathcal{P}_{1},BFNIS) =0.150,\]0 The steps of bipolar fuzzy ELECTRE I method.