# Multiple-Attribute Decision Making ELECTRE II Method under Bipolar Fuzzy Model
## 2 The BF-ELECTRE II Method
### Construction of Outranking Graphs
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Alternatives** Compared & \(\mathbb{C}_{\mathbf{\theta}}\) & \(\mathbb{D}_{\mathbf{\theta}}\) & \(\mathbf{c_{\mathbf{\theta}}}\) & \(\mathbf{d_{\mathbf{\theta}}}\) & \(\mathbf{g_{\mathbf{\theta}}}\) & \(\mathbf{h_{\mathbf{\theta}}}\) & \(\mathbf{q_{\mathbf{\theta}}}\) & **Outranking Relations** \\ \hline \((\mathcal{P}_{1},\mathcal{P}_{2})\) & \(\{2,3\}\) & \(\{1,4,5\}\) & 0.4202 & 1 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{1},\mathcal{P}_{3})\) & \(\{2,3\}\) & \(\{1,4,5\}\) & 0.4202 & 0.5893 & 0 & 1 & 0 & Incomparable \\ \((\mathcal{P}_{1},\mathcal{P}_{4})\) & \(\{2,3,5\}\) & \(\{1,4\}\) & 0.6380 & 0.5187 & 1 & 1 & 1 & \(\mathcal{P}_{1}\rightarrow\mathcal{P}_{4}\) \\ \((\mathcal{P}_{1},\mathcal{P}_{5})\) & \(\{2,5\}\) & \(\{1,3,4\}\) & 0.4401 & 1 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{2},\mathcal{P}_{1})\) & \(\{1,4,5\}\) & \(\{2,3\}\) & 0.5798 & 0.5514 & 1 & 1 & 1 & \(\mathcal{P}_{2}\rightarrow\mathcal{P}_{1}\) \\ \((\mathcal{P}_{2},\mathcal{P}_{3})\) & \(\{1,2,3,4,5\}\) & \(\{\}\) & 1 & 0 & 1 & 1 & 1 & \(\mathcal{P}_{2}\rightarrow\mathcal{P}_{3}\) \\ \((\mathcal{P}_{2},\mathcal{P}_{4})\) & \(\{1,3,4,5\}\) & \(\{2\}\) & 0.7777 & 0.0336 & 1 & 1 & 1 & \(\mathcal{P}_{2}\rightarrow\mathcal{P}_{4}\) \\ \((\mathcal{P}_{2},\mathcal{P}_{5})\) & \(\{4,5\}\) & \(\{1,2,3\}\) & 0.3834 & 1 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{3},\mathcal{P}_{1})\) & \(\{1,4,5\}\) & \(\{2,3\}\) & 0.5798 & 1 & 1 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{3},\mathcal{P}_{2})\) & \(\{\}\) & \(\{1,2,3,4\}\) & 0 & 1 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{3},\mathcal{P}_{4})\) & \(\{3,5\}\) & \(\{1,2,4\}\) & 0.4157 & 0.4312 & 0 & 1 & 0 & Incomparable \\ \((\mathcal{P}_{3},\mathcal{P}_{5})\) & \(\{5\}\) & \(\{1,2,3,4\}\) & 0.2178 & 1 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{4},\mathcal{P}_{1})\) & \(\{1,4\}\) & \(\{2,3,5\}\) & 0.3620 & 1 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{4},\mathcal{P}_{2})\) & \(\{2\}\) & \(\{1,3,4,5\}\) & 0.2223 & 1 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{4},\mathcal{P}_{3})\) & \(\{1,2,4\}\) & \(\{3,5\}\) & 0.5843 & 1 & 1 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{4},\mathcal{P}_{5})\) & \(\{2,5\}\) & \(\{1,3,4\}\) & 0.4401 & 0.8237 & 0 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{5},\mathcal{P}_{1})\) & \(\{1,3,4\}\) & \(\{2,5\}\) & 0.5599 & 0.1332 & 1 & 1 & 1 & \(\mathcal{P}_{5}\rightarrow\mathcal{P}_{1}\) \\ \((\mathcal{P}_{5},\mathcal{P}_{2})\) & \(\{1,2,3,\}\) & \(\{4,5\}\) & 0.6166 & 0.8744 & 1 & 0 & 0 & Incomparable \\ \((\mathcal{P}_{5},\mathcal{P}_{3})\) & \(\{1,2,3,4\}\) & \(\{5\}\) & 0.7822 & 0.2973 & 1 & 1 & 1 & \(\mathcal{P}_{5}\rightarrow\mathcal{P}_{3}\) \\ \((\mathcal{P}_{5},\mathcal{P}_{4})\) & \(\{1,3,4\}\) & \(\{2,5\}\) & 0.5599 & 1 & 1 & 0 & 0 & Incomparable \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Bipolar Fuzzy ELECTRE I results for selection of Business location. Figure 5: Graph representing the outranking relation of alternatives. criteria. Thus, we present BF-ELECTRE II method to rank the alternatives as an improvement of other successful versions of ELECTRE II method. ## 5 Insights and Limitations of Proposed Method

Some insights of proposed BF-ELECTRE II method are:

* The presented BF-ELECTRE II method is an extension of other existing versions of ELECTRE II method to rank alternatives, as it deals with bipolar fuzzy information. * An optimization technique based on maximizing deviation method is used to calculate the normalized weights of criteria to minimize the personal interest of decision makers towards the criteria. * Two types of outranking relations are used to obtain more appropriate results and an iterative procedure is followed to rank the alternatives. * Two numerical examples are explained by using this method. Beside of all above discussion, this method also has some limitations:
* This method is appropriate for dealing with two-sided information of human thinking, but it cannot appropriately handle the \(m\)-sided or \(m\)-polar information.