# Multiple-Attribute Decision Making ELECTRE II Method under Bipolar Fuzzy Model
## 2 The BF-ELECTRE II Method
### Construction of Outranking Graphs  
On the basis of these two types of pairwise outranking relations, the strongly outrank graph \(G^{s}=(\mathcal{P},E^{s})\) and the weakly outrank graph \(G^{w}=(\mathcal{P},E^{w})\) are drawn for strong outranking relationship \(R^{s}\) and weak outranking relationship \(R^{w}\), respectively, where \(\mathcal{P}\) is the set of alternatives and \(E^{s}\) and \(E^{w}\) denote the set of respective arcs of strong and weak outranking relations. After that, we use these graphs in an iterative procedure to construct different rankings, namely forward ranking \(\lambda^{{}^{\prime}}\) and reverse ranking \(\lambda^{{}^{\prime\prime}}\), and, finally, the average ranking \(\lambda\) provides the final ranking. **(a)**: **Forward ranking** \(\lambda^{{}^{\prime}}\)**:**:  
Let \(\mathcal{P}=\{\mathcal{P}_{1},\mathcal{P}_{2},\cdots,\mathcal{P}_{r}\}\) be a set of actions or alternatives and \(\mathcal{P}(x)\) is a subset of \(\mathcal{P}\). The steps for forward ranking are described as follows:  
1. Starting from the strong outranking graph \(G^{s}\), identify the vertices having no incoming or precedent arc. Denote this non-dominated set of alternatives by \(V(x)\). 2. Next, consider the weak outranking graph \(G^{w}\) and choose the arcs from \(E^{s}\) with both extremities in \(V(x)\), denoted by \(\overline{E}^{s}\). 3. Compute the set \(B(x)\) consisting of vertices having no precedent arc in the graph \((V(x),\overline{E}^{s})\). \(B(x)\) coincides to the set of non-dominated solutions at iteration \(x\). 4. Determine the forward ranking \(\lambda^{{}^{\prime}}\) by following the iterative scheme as: 1. Consider \(x=1\) and set \(\mathcal{P}(1)=\mathcal{P}\). 2. Construct the sets \(\mathcal{P}(x)\) and \(B(x)\) according to Steps 1, 2, 3 and 4(iv). 3. Rank the alternative \(\mathcal{P}_{m}\) by \(\lambda^{{}^{\prime}}(\mathcal{P}_{m})=x,\ \forall\ \mathcal{P}_{m}\in B(x)\). 4. Delete the alternatives from the system that have been forwardly ranked and remove all corresponding arcs from the graphs \(G^{s}\) and \(G^{w}\), also set \(\mathcal{P}(x+1)=\mathcal{P}(x)-B(x)\). If \(\mathcal{P}(x+1)=\emptyset\), then all the alternatives are ranked otherwise set \(x=x+1\) and go to the Step 4(ii). **(b)**: **Reverse ranking** \(\lambda^{{}^{\prime\prime}}\)**:**:  
The steps of reverse ranking are as follows:  
1. A mirror image of direct outranking relations is obtained by reversing the direction of the arcs of strongly outrank graph \(G^{s}\) and weakly outrank graph \(G^{w}\), whereas the set of alternatives remain same. 2. A ranking \(\gamma(\mathcal{P}_{m})\) is obtained from these reversed graphs by following the steps of forward ranking. 3. The correct order for reverse ranking is established by using the formula as: \[\lambda^{{}^{\prime\prime}}(\mathcal{P}_{m})=1+\underset{\mathcal{P}_{m}\in \mathcal{P}}{\max}\ \gamma\ (\mathcal{P}_{m})-\gamma\ (\mathcal{P}_{m}).\] (13)
**(c)**: **Average ranking \(\lambda\)**:**:  
The final ordering of all alternatives is computed by:  
\[\lambda(\mathcal{P}_{m})=\frac{\lambda^{{}^{\prime}}(\mathcal{P}_{m})+\lambda^ {{}^{\prime\prime}}(\mathcal{P}_{m})}{2}. \tag{14}\]  
According to all above description, the procedure of BF-ELECTRE II method is summarized in Algorithm 1. ```
1: Construct a bipolar fuzzy decision matrix containing the performance values of each alternatives with respect to criteria in the form of bipolar fuzzy sets.