# Multiple-Attribute Decision Making ELECTRE II Method under Bipolar Fuzzy Model
## 2 The BF-ELECTRE II Method
### Construction of Outranking Graphs
In addition, determine the normalized weights of criteria by using Equations (1)
2: Compute the bipolar fuzzy concordance (strong, median and weak), indifference and discordance (strong, median and weak) sets by using Equations (2)-(8), respectively
3: Calculate the bipolar fuzzy concordance indices which are further used to construct the bipolar fuzzy concordance matrix
4: Determine the weighted distances between any two alternatives by using the bipolar fuzzy Euclidean distance formula
5: Calculate the bipolar fuzzy discordance indices which are further used to construct the bipolar fuzzy discordance matrix
6: Establish the strong and weak outranking relations by applying Equations (11) and (12), respectively
7: Draw the strongly outrank and weakly outrank graphs according the strong and weak outranking relations. Finally, rank the alternatives by analyzing these graphs through the iterative procedure given in Section 2.5

**Algorithm 1** BF-ELECTRE II method

## 3 Practical Examples

In this section, we solve some numerical problems to explain the above presented bipolar fuzzy ELECTRE-II method through a step by step procedure. ### Selection of a Business Location

Choosing a location for a new business is one of the most important decisions. The location of a business can affect many aspects such as total sales and how costly it is to run. Suppose that a businessman is confused in choosing a location for his business. After initial screening, five alternative locations \(\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3},\mathcal{P}_{4}\) and \(\mathcal{P}_{5}\) are chosen for further evaluation. Furthermore, these alternatives are assessed by a set of five criteria \(K=\{K_{1},K_{2},K_{3},K_{4},K_{5}\}\) representing the availability of labor \((K_{1})\), price \((K_{2})\), safety \((K_{3})\), government economic incentives \((K_{4})\) and transport costs \((K_{5})\). **Step 1. **: Performance values for each alternative according to the conflicting criteria in the form of bipolar fuzzy decision matrix given by decision maker are displayed in Table 1.

Where each entry \(\ell_{mn}\) in the matrix represents the positivity and negativity of an alternative \(\mathcal{P}_{m}\), \(m=1,2,...,5\) for criteria \(K_{n}\), \(n=1,2,...,5\). Further, the weights of the criteria are given in vector form as \(\omega\)=[ 0.1964 0.2223 0.1979 0.1656 0.2178 ]\({}^{T}\), which is determined by \begin{table}
\begin{tabular}{l l l l l l} \hline \hline \(\mathcal{L}\) & \(\mathbf{K_{1}}\) & \(\mathbf{K_{2}}\) & \(\mathbf{K_{3}}\) & \(\mathbf{K_{4}}\) & \(\mathbf{K_{5}}\) \\ \hline \(\mathcal{P}_{1}\) & \((0.35,-0.7)\) & \((0.9,-0.33)\) & \((0.5,-0.6)\) & \((0.43,-0.72)\) & \((0.6,-0.35)\) \\ \(\mathcal{P}_{2}\) & \((0.6,-0.25)\) & \((0.4,-0.7)\) & \((0.65,-0.8)\) & \((0.9,-0.5)\) & \((1.0,-0.25)\) \\ \(\mathcal{P}_{3}\) & \((0.2,-0.5)\) & \((0.28,-0.6)\) & \((0.37,-0.6)\) & \((0.33,-0.6)\) & \((0.8,-0.4)\) \\ \(\mathcal{P}_{4}\) & \((0.47,-0.65)\) & \((0.55,-0.27)\) & \((0.25,-0.5)\) & \((0.55,-0.4)\) & \((0.75,-0.6)\) \\ \(\mathcal{P}_{5}\) & \((0.8,-0.3)\) & \((0.3,-0.15)\) & \((1.0,-0.4)\) & \((0.7,-0.32)\) & \((0.3,-0.8)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bipolar fuzzy decision matrix. Equations (1). Clearly, each \(0<\omega_{n}<1\) and also satisfy the normalized condition, that is, \(\sum\limits_{n=1}^{s}\omega_{n}=1\). * The bipolar fuzzy concordance sets \(K_{\alpha\beta}\), \(K_{\alpha\beta}\), \(K_{\alpha\beta^{\prime\prime}}\) are enumerated by using Equations (2)-(4), respectively. 1. The bipolar fuzzy strong concordance sets \(K_{\alpha\beta}\) are given as: 2. The bipolar fuzzy median concordance sets \(K_{\alpha\beta}\) are as: 3. The bipolar fuzzy weak concordance sets \(K_{\alpha\beta^{\prime\prime}}\) are given as: 4. The indifferent sets \(I_{\alpha\beta}\) are computed by employing Equation (5) as: 5. The bipolar fuzzy discordance sets \(D_{\alpha\beta}\), \(D_{\alpha\beta^{\prime}}\), \(D_{\alpha\beta^{\prime\prime}}\) are established by applying Equations (6)-(8).