# Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I

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## Abstract

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## 1. Introduction

## 2. Bipolar Neutrosophic TOPSIS Method

**Definition**

**1.**

- (i)
- Each value of alternative is estimated with respect to n criteria. The value of each alternative under each criterion is given in the form of BNSs and they can be expressed in the decision matrix as$$K={\left[{k}_{ij}\right]}_{m\times n}=\left[\begin{array}{cccc}{k}_{11}& {k}_{12}& \cdots & {k}_{1n}\\ {k}_{21}& {k}_{22}& \cdots & {k}_{2n}\\ \xb7& \xb7& \cdots & .\\ \xb7& \xb7& \cdots & \xb7\\ {k}_{m1}& {k}_{m2}& \cdots & {k}_{mn}\end{array}\right].$$Each entry ${k}_{ij}=<{T}_{ij}^{+},{I}_{ij}^{+},{F}_{ij}^{+},{T}_{ij}^{-},{I}_{ij}^{-},{F}_{ij}^{-}>$, where, ${T}_{ij}^{+},{I}_{ij}^{+}$ and ${F}_{ij}^{+}$ represent the degree of positive truth, indeterminacy and falsity membership, respectively, whereas, ${T}_{ij}^{-},{I}_{ij}^{-}$ and ${F}_{ij}^{-}$ represent the degree of negative truth, indeterminacy and falsity membership, respectively, such that ${T}_{ij}^{+},{I}_{ij}^{+},{F}_{ij}^{+}\in [0,1]$, ${T}_{ij}^{-},{I}_{ij}^{-},{F}_{ij}^{-}\in [-1,0]$ and $0\le {T}_{ij}^{+}+{I}_{ij}^{+}+{F}_{ij}^{+}-{T}_{ij}^{-}-{I}_{ij}^{-}-{F}_{ij}^{-}\le 6$, $i=1,2,3,\dots ,m$; $j=1,2,3,\dots ,n$.
- (ii)
- Suppose that the weights of the criteria are not equally assigned and they are totally unknown to the decision maker. We use the maximizing deviation method [30] to determine the unknown weights of the criteria. Therefore, the weight of the attribute ${T}_{j}$ is given as$$\begin{array}{c}\hfill {w}_{j}=\frac{{\displaystyle \sum _{i=1}^{m}}{\displaystyle \sum _{l=1}^{m}}|{k}_{ij}-{k}_{lj}|}{\sqrt{{\displaystyle \sum _{j=1}^{n}}{\left({\displaystyle \sum _{i=1}^{m}}{\displaystyle \sum _{l=1}^{m}}|{k}_{ij}-{k}_{lj}|\right)}^{2}}},\end{array}$$$$\begin{array}{c}\hfill {w}_{j}^{*}=\frac{{\displaystyle \sum _{i=1}^{m}}{\displaystyle \sum _{l=1}^{m}}|{k}_{ij}-{k}_{lj}|}{{\displaystyle \sum _{j=1}^{n}}\left({\displaystyle \sum _{i=1}^{m}}{\displaystyle \sum _{l=1}^{m}}|{k}_{ij}-{k}_{lj}|\right)}.\end{array}$$
- (iii)
- The accumulated weighted bipolar neutrosophic decision matrix is computed by multiplying the weights of the attributes to aggregated decision matrix as follows:$$K\otimes W={\left[{k}_{ij}^{{w}_{j}}\right]}_{m\times n}=\left[\begin{array}{cccc}{k}_{11}^{{w}_{1}}& {k}_{12}^{{w}_{2}}& \cdots & {k}_{1n}^{{w}_{n}}\\ {k}_{21}^{{w}_{1}}& {k}_{22}^{{w}_{2}}& \cdots & {k}_{2n}^{{w}_{n}}\\ \xb7& \xb7& \cdots & \xb7\\ \xb7& \xb7& \cdots & \xb7\\ {k}_{m1}^{{w}_{1}}& {k}_{m2}^{{w}_{2}}& \cdots & {k}_{mn}^{{w}_{n}}\end{array}\right].$$$$\begin{array}{cc}\hfill {k}_{ij}^{{w}_{j}}& =<{T}_{ij}^{{w}_{j}+},{I}_{ij}^{{w}_{j}+},{F}_{ij}^{{w}_{j}+},{T}_{ij}^{{w}_{j}-},{I}_{ij}^{{w}_{j}-},{F}_{ij}^{{w}_{j}-}>\hfill \\ & =<1-{(1-{T}_{ij}^{+})}^{{w}_{j}},{({I}_{ij}^{+})}^{{w}_{j}},{({F}_{ij}^{+})}^{{w}_{j}},-{(-{T}_{ij}^{-})}^{{w}_{j}},-{(-{I}_{ij}^{-})}^{{w}_{j}},-(1-{(1-(-{F}_{ij}^{-}))}^{{w}_{j}})>,\hfill \end{array}$$
- (iv)
- Two types of attributes, benefit type attributes and cost type attributes, are mostly applicable in real life decision making. The bipolar neutrosophic relative positive ideal solution (BNRPIS) and bipolar neutrosophic relative negative ideal solution (BNRNIS) for both type of attributes are defined as follows:$$\begin{array}{ccc}\hfill BNRPIS& =& (\u2329{}^{+}{T}_{1}^{{w}_{1}+}{,}^{+}{I}_{1}^{{w}_{1}+}{,}^{+}{F}_{1}^{{w}_{1}+}{,}^{+}{T}_{1}^{{w}_{1}-}{,}^{+}{I}_{1}^{{w}_{1}-}{,}^{+}{F}_{1}^{{w}_{1}-}\u232a,{\langle}^{+}{T}_{2}^{{w}_{2}+}{,}^{+}{I}_{2}^{{w}_{2}+}{,}^{+}{F}_{2}^{{w}_{2}+}{,}^{+}{T}_{2}^{{w}_{2}-},\hfill \\ & & \hfill {}^{+}{I}_{2}^{{w}_{2}-}{,}^{+}{F}_{2}^{{w}_{2}-}\rangle ,\dots ,\u2329{}^{+}{T}_{n}^{{w}_{n}+}{,}^{+}{I}_{n}^{{w}_{n}+}{,}^{+}{F}_{n}^{{w}_{n}+}{,}^{+}{T}_{n}^{{w}_{n}-}{,}^{+}{I}_{n}^{{w}_{n}-}{,}^{+}{F}_{n}^{{w}_{n}-}\u232a),\\ \hfill BNRNIS& =& (\u2329{}^{-}{T}_{1}^{{w}_{1}+}{,}^{-}{I}_{1}^{{w}_{1}+}{,}^{-}{F}_{1}^{{w}_{1}+}{,}^{-}{T}_{1}^{{w}_{1}-}{,}^{-}{I}_{1}^{{w}_{1}-}{,}^{-}{F}_{1}^{{w}_{1}-}\u232a,{\langle}^{-}{T}_{2}^{{w}_{2}+}{,}^{-}{I}_{2}^{{w}_{2}+}{,}^{-}{F}_{2}^{{w}_{2}+}{,}^{-}{T}_{2}^{{w}_{2}-},\hfill \\ & & \hfill {}^{-}{I}_{2}^{{w}_{2}-}{,}^{-}{F}_{2}^{{w}_{2}-}\rangle ,...,\u2329{}^{-}{T}_{n}^{{w}_{n}+}{,}^{-}{I}_{n}^{{w}_{n}+}{,}^{-}{F}_{n}^{{w}_{n}+}{,}^{-}{T}_{n}^{{w}_{n}-}{,}^{-}{I}_{n}^{{w}_{n}-}{,}^{-}{F}_{n}^{{w}_{n}-}\u232a),\end{array}$$$$\begin{array}{ccc}\hfill \u2329{}^{+}{T}_{j}^{{w}_{j}+}{,}^{+}{I}_{j}^{{w}_{j}+}{,}^{+}{F}_{j}^{{w}_{j}+}{,}^{+}{T}_{j}^{{w}_{j}-}{,}^{+}{I}_{j}^{{w}_{j}-}{,}^{+}{F}_{j}^{{w}_{j}-}\u232a& =& \langle max\left({T}_{ij}^{{w}_{j}+}\right),min\left({I}_{ij}^{{w}_{j}+}\right),min\left({F}_{ij}^{{w}_{j}+}\right),\hfill \\ & & min\left({T}_{ij}^{{w}_{j}-}\right),max\left({I}_{ij}^{{w}_{j}-}\right),max\left({F}_{ij}^{{w}_{j}-}\right)\rangle ,\hfill \\ \hfill \u2329{}^{-}{T}_{j}^{{w}_{j}+}{,}^{-}{I}_{j}^{{w}_{j}+}{,}^{-}{F}_{j}^{{w}_{j}+}{,}^{-}{T}_{j}^{{w}_{j}-}{,}^{-}{I}_{j}^{{w}_{j}-}{,}^{-}{F}_{j}^{{w}_{j}-}\u232a& =& \langle min\left({T}_{ij}^{{w}_{j}+}\right),max\left({I}_{ij}^{{w}_{j}+}\right),max\left({F}_{ij}^{{w}_{j}+}\right),\hfill \\ & & max\left({T}_{ij}^{{w}_{j}-}\right),min\left({I}_{ij}^{{w}_{j}-}\right),min\left({F}_{ij}^{{w}_{j}-}\right)\rangle .\hfill \end{array}$$Similarly, for cost type criteria, $j=1,2,\dots ,n$$$\begin{array}{ccc}\hfill \u2329{}^{+}{T}_{j}^{{w}_{j}+}{,}^{+}{I}_{j}^{{w}_{j}+}{,}^{+}{F}_{j}^{{w}_{j}+}{,}^{+}{T}_{j}^{{w}_{j}-}{,}^{+}{I}_{j}^{{w}_{j}-}{,}^{+}{F}_{j}^{{w}_{j}-}\u232a& =& \langle min\left({T}_{ij}^{{w}_{j}+}\right),max\left({I}_{ij}^{{w}_{j}+}\right),max\left({F}_{ij}^{{w}_{j}+}\right),\hfill \\ & & max\left({T}_{ij}^{{w}_{j}-}\right),min\left({I}_{ij}^{{w}_{j}-}\right),min\left({F}_{ij}^{{w}_{j}-}\right)\rangle ,\hfill \\ \hfill \u2329{}^{-}{T}_{j}^{{w}_{j}+}{,}^{-}{I}_{j}^{{w}_{j}+}{,}^{-}{F}_{j}^{{w}_{j}+}{,}^{-}{T}_{j}^{{w}_{j}-}{,}^{-}{I}_{j}^{{w}_{j}-}{,}^{-}{F}_{j}^{{w}_{j}-}\u232a& =& \langle max\left({T}_{ij}^{{w}_{j}+}\right),min\left({I}_{ij}^{{w}_{j}+}\right),min\left({F}_{ij}^{{w}_{j}+}\right),\hfill \\ & & min\left({T}_{ij}^{{w}_{j}-}\right),max\left({I}_{ij}^{{w}_{j}-}\right),max\left({F}_{ij}^{{w}_{j}-}\right)\rangle .\hfill \end{array}$$
- (v)
- The normalized Euclidean distance of each alternative $\u2329{T}_{ij}^{{w}_{j}+},{I}_{ij}^{{w}_{j}+},{F}_{ij}^{{w}_{j}+},{T}_{ij}^{{w}_{j}-},{I}_{ij}^{{w}_{j}-},{F}_{ij}^{{w}_{j}-}\u232a$ from the BNRPIS $\u2329{}^{+}{T}_{j}^{{w}_{j}+}{,}^{+}{I}_{j}^{{w}_{j}+}{,}^{+}{F}_{j}^{{w}_{j}+}{,}^{+}{T}_{j}^{{w}_{j}-}{,}^{+}{I}_{j}^{{w}_{j}-}{,}^{+}{F}_{j}^{{w}_{j}-}\u232a$ can be calculated as$${d}_{N}({S}_{i},BNRPIS)=\sqrt{\frac{1}{6n}{\displaystyle \sum _{j=1}^{n}}\left\{\begin{array}{c}{\left({T}_{ij}^{{w}_{j}+}{-}^{+}{T}_{j}^{{w}_{j}+}\right)}^{2}+{\left({I}_{ij}^{{w}_{j}+}{-}^{+}{I}_{j}^{{w}_{j}+}\right)}^{2}+{\left({F}_{ij}^{{w}_{j}+}{-}^{+}{F}_{j}^{{w}_{j}+}\right)}^{2}+\hfill \\ \hfill {\left({T}_{ij}^{{w}_{j}-}{-}^{+}{T}_{j}^{{w}_{j}-}\right)}^{2}+{\left({I}_{ij}^{{w}_{j}-}{-}^{+}{I}_{j}^{{w}_{j}-}\right)}^{2}+{\left({F}_{ij}^{{w}_{j}-}{-}^{+}{F}_{j}^{{w}_{j}-}\right)}^{2}\end{array}\right\}},$$$${d}_{N}({S}_{i},BNRNIS)=\sqrt{\frac{1}{6n}\underset{j=1}{\sum ^{n}}\left\{\begin{array}{c}{\left({T}_{ij}^{{w}_{j}+}{-}^{-}{T}_{j}^{{w}_{j}+}\right)}^{2}+{\left({I}_{ij}^{{w}_{j}+}{-}^{-}{I}_{j}^{{w}_{j}+}\right)}^{2}+{\left({F}_{ij}^{{w}_{j}+}{-}^{-}{F}_{j}^{{w}_{j}+}\right)}^{2}+\hfill \\ \hfill {\left({T}_{ij}^{{w}_{j}-}{-}^{-}{T}_{j}^{{w}_{j}-}\right)}^{2}+{\left({I}_{ij}^{{w}_{j}-}{-}^{-}{I}_{j}^{{w}_{j}-}\right)}^{2}+{\left({F}_{ij}^{{w}_{j}-}{-}^{-}{F}_{j}^{{w}_{j}-}\right)}^{2}\end{array}\right\}}.$$
- (vi)
- Revised closeness degree of each alternative to BNRPIS represented as ${\rho}_{i}$ and it is calculated using formula$$\begin{array}{c}\hfill \rho \left({S}_{i}\right)=\frac{{d}_{N}({S}_{i},BNRNIS)}{max\left\{{d}_{N}({S}_{i},BNRNIS)\right\}}-\frac{{d}_{N}({S}_{i},BNRPIS)}{min\left\{{d}_{N}({S}_{i},BNRPIS)\right\}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=1,2,\dots ,m.\end{array}$$
- (vii)
- By using the revised closeness degrees, the inferior ratio to each alternative is determined as follows:$$\begin{array}{c}\hfill IR\left(i\right)=\frac{\rho \left({S}_{i}\right)}{\underset{1\le i\le m}{min}\left(\rho \left({S}_{i}\right)\right)}.\end{array}$$It is clear that each value of $IR\left(i\right)$ lies in the closed unit interval [0,1].
- (viii)
- The alternatives are ranked according to the ascending order of inferior ratio values and the best alternative with minimum choice value is chosen.

## 3. Applications

#### 3.1. Electronic Commerce Web Site

**Step****1.**- The decision matrix in the form of bipolar neutrosophic information is given as in Table 1:
**Step****2.**- The normalized weights of the criteria are calculated by using maximizing deviation method as given below:$${w}_{1}=0.2567,{w}_{2}=0.2776,{w}_{3}=0.2179,{w}_{4}=0.2478,\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{j=1}^{4}}{w}_{j}=1.$$
**Step****3.**- The weighted bipolar neutrosophic decision matrix is constructed by multiplying the weights to decision matrix as given in Table 2:
**Step****4.**- The BNRPIS and BNRNIS are given by$$\begin{array}{ccc}\hfill BNRPIS& =& \hfill <(0.21,0.662,0.554,-0.877,-0.554,-0.087),\\ & & \hfill (0.06,0.868,0.868,-0.528,-0.94,-0.284),\\ & & \hfill (0.395,0.704,0.86,-0.895,-0.769,-0.023),\\ & & \hfill (0.329,0.742,0.671,-0.842,-0.742,-0.062)>;\end{array}$$$$\begin{array}{ccc}\hfill BNRNIS& =& \hfill <(0.087,0.913,0.837,-0.662,-0.913,-0.266),\\ & & \hfill (0.36,0.716,0.528,-0.906,-0.64,-0.132),\\ & & \hfill (0.047,0.925,0.925,-0.605,-0.819,-0.075),\\ & & \hfill (0.054,0.915,0.881,-0.565,-0.915,-0.119)>.\end{array}$$
**Step****5.**- The normalized Euclidean distances of each alternative from the BNRPISs and the BNRNISs are given as follows:$$\begin{array}{cc}\hfill {d}_{N}({S}_{1},BNRPIS)& =0.1805,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{1},BNRNIS)=0.1125,\hfill \\ \hfill {d}_{N}({S}_{2},BNRPIS)& =0.1672,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{2},BNRNIS)=0.1485,\hfill \\ \hfill {d}_{N}({S}_{3},BNRPIS)& =0.135,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{3},BNRNIS)=0.1478,\hfill \\ \hfill {d}_{N}({S}_{4},BNRPIS)& =0.155,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{4},BNRNIS)=0.1678.\hfill \end{array}$$
**Step****6.**- The revised closeness degree of each alternative is given as$$\rho \left({S}_{1}\right)=-0.667,\rho \left({S}_{2}\right)=-0.334,\rho \left({S}_{3}\right)=-0.119,\rho \left({S}_{4}\right)=-0.148.$$
**Step****7.**- The inferior ratio to each alternative is given as$$IR\left(1\right)=1,IR\left(2\right)=0.52,IR\left(3\right)=0.18,IR\left(4\right)=0.22.$$
**Step****8.**- Ordering the web stores according to ascending order of alternatives, we obtain: ${S}_{3}<{S}_{4}<{S}_{2}<{S}_{1}$. Therefore, the person will choose the BigCommerce for opening a web store.

#### 3.2. Heart Surgeon

**Step****1.**- The decision matrix in the form of bipolar neutrosophic information is given as in Table 3:
**Step****2.**- The normalized weights of the criteria are calculated by using maximizing deviation method as given below:$${w}_{1}=0.2480,{w}_{2}=0.2424,{w}_{3}=0.2480,{w}_{4}=0.2616,\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{j=1}^{4}}{w}_{j}=1.$$
**Step****3.**- The weighted bipolar neutrosophic decision matrix is constructed by multiplying the weights to decision matrix as given in Table 4:
**Step****4.**- The BNRPIS and BNRNIS are given by$$\begin{array}{ccc}\hfill BNRPIS& =& \hfill <(0.435,0.742,0.671,-0.915,-0.797,-0.085),\\ & & \hfill (0.253,0.747,0.677,-0.917,-0.801,-0.116),\\ & & \hfill (0.085,0.915,0.915,-0.742,-0.915,-0.203),\\ & & \hfill (0.344,0.730,0.656,-0.875,-0.787,-0.089)>;\end{array}$$$$\begin{array}{ccc}\hfill BNRNIS& =& \hfill <(0.119,0.881,0.881,-0.742,-0.915,-0.158),\\ & & \hfill (0.116,0.917,0.884,-0.801,-0.917,-0.323),\\ & & \hfill (0.258,0.742,0.742,-0.915,-0.742,-0.054),\\ & & \hfill (0.125,0.911,0.911,-0.656,-0.911,-0.270)>.\end{array}$$
**Step****5.**- The normalized Euclidean distances of each alternative from the BNRPISs and the BNRNISs are given as follows:$$\begin{array}{cc}\hfill {d}_{N}({S}_{1},BNRPIS)& =0.1176,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{1},BNRNIS)=0.0945,\hfill \\ \hfill {d}_{N}({S}_{2},BNRPIS)& =0.0974,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{2},BNRNIS)=0.1402,\hfill \\ \hfill {d}_{N}({S}_{3},BNRPIS)& =0.1348,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{3},BNRNIS)=0.1043,\hfill \\ \hfill {d}_{N}({S}_{4},BNRPIS)& =0.1089,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{4},BNRNIS)=0.1093,\hfill \\ \hfill {d}_{N}({S}_{5},BNRPIS)& =0.1292,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{5},BNRNIS)=0.0837.\hfill \end{array}$$
**Step****6.**- The revised closeness degree of each alternative is given as$$\rho \left({S}_{1}\right)=-0.553,\rho \left({S}_{2}\right)=0,\rho \left({S}_{3}\right)=-0.64,\rho \left({S}_{4}\right)=-0.338,\rho \left({S}_{5}\right)=-0.729$$
**Step****7.**- The inferior ratio to each alternative is given as$$IR\left(1\right)=0.73,IR\left(2\right)=0,IR\left(3\right)=0.88,IR\left(4\right)=0.46,IR\left(5\right)=1.$$
**Step****8.**- Ordering the alternatives in ascending order, we obtain: ${S}_{2}<{S}_{4}<{S}_{1}<{S}_{3}<{S}_{5}$. Therefore, ${S}_{2}$ is best among all other alternatives.

#### 3.3. Employee (Marketing Manager)

**Step****1.**- The decision matrix in the form of bipolar neutrosophic information is given as in Table 5:
**Step****2.**- The normalized weights of the criteria are calculated by using maximizing deviation method as given below:$${w}_{1}=0.25,{w}_{2}=0.2361,{w}_{3}=0.2708,{w}_{4}=0.2431,\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{j=1}^{4}}{w}_{j}=1.$$
**Step****3.**- The weighted bipolar neutrosophic decision matrix is constructed by multiplying the weights to decision matrix as given in Table 6:
**Step****4.**- The BNRPIS and BNRNIS are given by$$\begin{array}{ccc}\hfill BNRPIS& =& \hfill <(0.3313,0.7401,0.7401,-0.9147,-0.6687,-0.1199),\\ & & \hfill (0.2474,0.7526,0.6839,-0.8864,-0.6839,-0.0808),\\ & & \hfill (0.2197,0.7803,0.7218,-0.9414,-0.7218,-0.0586),\\ & & \hfill (0.4287,0.7463,0.6762,-0.8832,-0.7463,-0.0528)>;\end{array}$$$$\begin{array}{ccc}\hfill BNRNIS& =& \hfill <(0.1199,0.9147,0.9457,-0.7401,-0.8801,-0.1591),\\ & & \hfill (0.1136,0.9192,0.8864,-0.6839,-0.8490,-0.1945),\\ & & \hfill (0.0921,0.9414,0.9079,-0.7218,-0.8289,-0.2782),\\ & & \hfill (0.1168,0.9169,0.9169,-0.7463,-0.8449,-0.1551)>.\end{array}$$
**Step****5.**- The normalized Euclidean distances of each alternative from the BNRPISs and the BNRNISs are given as follows:$$\begin{array}{cc}\hfill {d}_{N}({S}_{1},BNRPIS)& =0.0906,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{1},BNRNIS)=0.1393,\hfill \\ \hfill {d}_{N}({S}_{2},BNRPIS)& =0.1344,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{2},BNRNIS)=0.0953,\hfill \\ \hfill {d}_{N}({S}_{3},BNRPIS)& =0.1286,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{3},BNRNIS)=0.1011,\hfill \\ \hfill {d}_{N}({S}_{4},BNRPIS)& =0.1293,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{d}_{N}({S}_{4},BNRNIS)=0.0999.\hfill \end{array}$$
**Step****6.**- The revised closeness degree of each alternative is given as$$\rho \left({S}_{1}\right)=0,\rho \left({S}_{2}\right)=-0.799,\rho \left({S}_{3}\right)=-0.694,\rho \left({S}_{4}\right)=-0.780.$$
**Step****7.**- The inferior ratio to each alternative is given as$$IR\left(1\right)=0,IR\left(2\right)=1,IR\left(3\right)=0.87,IR\left(4\right)=0.98.$$
**Step****8.**- Ordering the alternatives in ascending order, we obtain: ${S}_{1}<{S}_{3}<{S}_{4}<{S}_{2}$. Therefore, the company will select the candidate ${S}_{1}$ for this post.

## 4. Bipolar Neutrosophic ELECTRE-I Method

- (i–iii)
- As in the section of bipolar neutrosophic TOPSIS, the rating values of alternatives with respect to the criteria are expressed in the form of matrix ${\left[{k}_{ij}\right]}_{m\times n}$. The weights ${w}_{j}$ of the criteria ${T}_{j}$ are evaluated by maximizing deviation method and the weighted bipolar neutrosophic decision matrix ${\left[{k}_{ij}^{{w}_{j}}\right]}_{m\times n}$ is constructed.
- (iv)
- The bipolar neutrosophic concordance sets ${E}_{xy}$ and bipolar neutrosophic discordance sets ${F}_{xy}$ are defined as follows:$$\begin{array}{c}\hfill {E}_{xy}=\{1\le j\le n\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\rho}_{xj}\ge {\rho}_{yj}\},\phantom{\rule{3.33333pt}{0ex}}x\ne y,\phantom{\rule{3.33333pt}{0ex}}x,y=1,2,\cdots ,m,\\ \hfill {F}_{xy}=\{1\le j\le n\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{\rho}_{xj}\le {\rho}_{yj}\},\phantom{\rule{3.33333pt}{0ex}}x\ne y,\phantom{\rule{3.33333pt}{0ex}}x,y=1,2,\cdots ,m,\end{array}$$
- (v)
- The bipolar neutrosophic concordance matrix E is constructed as follows:$$E=\left[\begin{array}{cccccc}-& {e}_{12}& \xb7& \xb7& \xb7& {e}_{1m}\\ {e}_{21}& -& \xb7& \xb7& \xb7& {e}_{2m}\\ \xb7\\ \xb7\\ \xb7\\ {e}_{m1}& {e}_{m2}& \xb7& \xb7& \xb7& -\end{array}\right],$$$$\begin{array}{c}\hfill {e}_{xy}=\sum _{j\in {E}_{xy}}{w}_{j}.\end{array}$$
- (vi)
- The bipolar neutrosophic discordance matrix F is constructed as follows:$$F=\left[\begin{array}{cccccc}-& {f}_{12}& \xb7& \xb7& \xb7& {f}_{1m}\\ {f}_{21}& -& \xb7& \xb7& \xb7& {f}_{2m}\\ \xb7\\ \xb7\\ \xb7\\ {f}_{m1}& {f}_{m2}& \xb7& \xb7& \xb7& -\end{array}\right],$$$$\begin{array}{c}\hfill {f}_{xy}=\frac{\underset{j\in {F}_{xy}}{max}\sqrt{\frac{1}{6n}\left\{\begin{array}{c}{({T}_{xj}^{{w}_{j}+}-{T}_{yj}^{{w}_{j}+})}^{2}+{({I}_{xj}^{{w}_{j}+}-{I}_{yj}^{{w}_{j}+})}^{2}+{({F}_{xj}^{{w}_{j}+}-{F}_{yj}^{{w}_{j}+})}^{2}+\hfill \\ \hfill {({T}_{xj}^{{w}_{j}-}-{T}_{yj}^{{w}_{j}-})}^{2}+{({I}_{xj}^{{w}_{j}-}-{I}_{yj}^{{w}_{j}-})}^{2}+{({F}_{xj}^{{w}_{j}-}-{F}_{yj}^{{w}_{j}-})}^{2}\end{array}\right\}}}{\underset{j}{max}\sqrt{\frac{1}{6n}\left\{\begin{array}{c}{({T}_{xj}^{{w}_{j}+}-{T}_{yj}^{{w}_{j}+})}^{2}+{({I}_{xj}^{{w}_{j}+}-{I}_{yj}^{{w}_{j}+})}^{2}+{({F}_{xj}^{{w}_{j}+}-{F}_{yj}^{{w}_{j}+})}^{2}+\hfill \\ \hfill {({T}_{xj}^{{w}_{j}-}-{T}_{yj}^{{w}_{j}-})}^{2}+{({I}_{xj}^{{w}_{j}-}-{I}_{yj}^{{w}_{j}-})}^{2}+{({F}_{xj}^{{w}_{j}-}-{F}_{yj}^{{w}_{j}-})}^{2}\end{array}\right\}}}.\end{array}$$
- (vii)
- Concordance and discordance levels are computed to rank the alternatives. The bipolar neutrosophic concordance level $\widehat{e}$ is defined as the average value of the bipolar neutrosophic concordance indices as$$\begin{array}{c}\hfill \widehat{e}=\frac{1}{m(m-1)}\sum _{\begin{array}{c}x=1,\\ x\ne y\end{array}}^{m}\sum _{\begin{array}{c}y=1,\\ y\ne x\end{array}}^{m}{e}_{xy},\end{array}$$$$\begin{array}{c}\hfill \widehat{f}=\frac{1}{m(m-1)}\sum _{\begin{array}{c}x=1,\\ x\ne y\end{array}}^{m}\sum _{\begin{array}{c}y=1,\\ y\ne x\end{array}}^{m}{f}_{xy}.\end{array}$$
- (viii)
- The bipolar neutrosophic concordance dominance matrix $\varphi $ on the basis of $\widehat{e}$ is determined as follows:$$\varphi =\left[\begin{array}{cccccc}-& {\varphi}_{12}& \xb7& \xb7& \xb7& {\varphi}_{1m}\\ {\varphi}_{21}& -& \xb7& \xb7& .& {\varphi}_{2m}\\ \xb7\\ \xb7\\ \xb7\\ {\varphi}_{m1}& {\varphi}_{m2}& \xb7& \xb7& \xb7& -\end{array}\right],$$$$\begin{array}{c}\hfill {\varphi}_{xy}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{e}_{xy}\ge \widehat{e},\hfill \\ 0,\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{e}_{xy}<\widehat{e}.\hfill \end{array}\right.\end{array}$$
- (ix)
- The bipolar neutrosophic discordance dominance matrix $\psi $ on the basis of $\widehat{f}$ is determined as follows:$$\psi =\left[\begin{array}{cccccc}-& {\psi}_{12}& \xb7& \xb7& \xb7& {\psi}_{1m}\\ {\psi}_{21}& -& \xb7& \xb7& \xb7& {\psi}_{2m}\\ \xb7\\ \xb7\\ \xb7\\ {\psi}_{m1}& {\psi}_{m2}& \xb7& \xb7& \xb7& -\end{array}\right],$$$$\begin{array}{c}\hfill {\psi}_{xy}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{f}_{xy}\le \widehat{f},\hfill \\ 0,\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{f}_{xy}>\widehat{f}.\hfill \end{array}\right.\end{array}$$
- (x)
- Consequently, the bipolar neutrosophic aggregated dominance matrix $\pi $ is evaluated by multiplying the corresponding entries of $\varphi $ and $\psi $, that is$$\pi =\left[\begin{array}{cccccc}-& {\pi}_{12}& \xb7& \xb7& \xb7& {\pi}_{1m}\\ {\pi}_{21}& -& \xb7& \xb7& \xb7& {\pi}_{2m}\\ \xb7\\ \xb7\\ \xb7\\ {\pi}_{m1}& {\pi}_{m2}& \xb7& \xb7& \xb7& -\end{array}\right],$$$$\begin{array}{c}\hfill {\pi}_{xy}={\varphi}_{xy}{\psi}_{xy}.\end{array}$$
- (xi)
- Finally, the alternatives are ranked according to the outranking values ${\pi}_{{xy}^{\prime}}$. That is, for each pair of alternatives ${S}_{x}$ and ${S}_{y}$, an arrow from ${S}_{x}$ to ${S}_{y}$ exists if and only if ${\pi}_{xy}=1$. As a result, we have three possible cases:
- (a)
- There exits a unique arrow from ${S}_{x}$ into ${S}_{y}$.
- (b)
- There exist two possible arrows between ${S}_{x}$ and ${S}_{y}$.
- (c)
- There is no arrow between ${S}_{x}$ and ${S}_{y}$.

For case a, we decide that ${S}_{x}$ is preferred to ${S}_{y}$. For the second case, ${S}_{x}$ and ${S}_{y}$ are indifferent, whereas, ${S}_{x}$ and ${S}_{y}$ are incomparable in case c.

#### Numerical Example

**Step****4.**- The bipolar neutrosophic concordance sets ${E}_{{xy}^{\prime}}$ are given as in Table 7:
**Step****5.**- The bipolar neutrosophic discordance sets ${F}_{{xy}^{\prime}}$ are given as in Table 8.
**Step****6.**- The bipolar neutrosophic concordance matrix E is computed as follows$$E=\left[\begin{array}{cccc}-& 0.7522& 0.5343& 0\\ 0.2478& -& 0.2478& 0\\ 0.4657& 0.7522& -& 0.2179\\ 1& 1& 0.7821& -\end{array}\right]$$
**Step****7.**- The bipolar neutrosophic discordance matrix F is computed as follows$$F=\left[\begin{array}{cccc}-& 0.5826& 0.9464& 1\\ 1& -& 1& 1\\ 1& 0.3534& -& 1\\ 0& 0& 0.6009& -\end{array}\right]$$
**Step****8.**- The bipolar neutrosophic concordance level is $\widehat{e}=0.5003$ and bipolar neutrosophic discordance level is $\widehat{f}=0.7069$. The bipolar neutrosophic concordance dominance matrix $\varphi $ and bipolar neutrosophic discordance dominance matrix $\psi $ are as follows$$\varphi =\left[\begin{array}{cccc}-& 1& 1& 0\\ 0& -& 0& 0\\ 0& 1& -& 0\\ 1& 1& 1& -\end{array}\right],\psi =\left[\begin{array}{cccc}-& 1& 0& 0\\ 0& -& 0& 0\\ 0& 1& -& 0\\ 0& 0& 0& -\end{array}\right].$$
**Step****9.**- The bipolar neutrosophic aggregated dominance matrix $\pi $ is computed as$$\pi =\left[\begin{array}{cccc}-& 1& 0& 0\\ 0& -& 0& 0\\ 0& 1& -& 0\\ 0& 0& 0& -\end{array}\right].$$

## 5. Comparison of Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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$\mathit{S}\setminus \mathit{T}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | ($0.4,0.2,0.5,$ | ($0.5,0.3,0.3,$ | ($0.2,0.7,0.5,$ | ($0.4,0.6,0.5,$ |

$-0.6,-0.4,-0.4$) | $-0.7,-0.2,-0.4$) | $-0.4,-0.4,-0.3$) | $-0.3,-0.7,-0.4$) | |

${S}_{2}$ | ($0.3,0.6,0.1,$ | ($0.2,0.6,0.1,$ | ($0.4,0.2,0.5,$ | ($0.2,0.7,0.5,$ |

$-0.5,-0.7,-0.5$) | $-0.5,-0.3,-0.7$) | $-0.6,-0.3,-0.1$) | $-0.5,-0.3,-0.2$) | |

${S}_{3}$ | ($0.3,0.5,0.2,$ | ($0.4,0.5,0.2,$ | ($0.9,0.5,0.7,$ | ($0.3,0.7,0.6,$ |

$-0.4,-0.3,-0.7$) | $-0.3,-0.8,-0.5$) | $-0.3,-0.4,-0.3$) | $-0.5,-0.5,-0.4$) | |

${S}_{4}$ | ($0.6,0.7,0.5,$ | ($0.8,0.4,0.6,$ | ($0.6,0.3,0.6,$ | ($0.8,0.3,0.2,$ |

$-0.2,-0.1,-0.3$) | $-0.1,-0.3,-0.4$) | $-0.1,-0.4,-0.2$) | $-0.1,-0.3,-0.1$) |

$\mathit{S}\setminus \mathit{T}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | $(0.123,0.662,0.837,$ | $(0.175,0.716,0.716,$ | $(0.047,0.925,0.86,$ | $(0.119,0.881,0.842,$ |

$-0.877,-0.79,-0.123)$ | $-0.906,-0.64,-0.132)$ | $-0.819,-0.819,-0.075)$ | $-0.742,-0.915,-0.119)$ | |

${S}_{2}$ | $(0.087,0.877,0.554,$ | $(0.06,0.868,0.528,$ | $(0.105,0.704,0.86,$ | $(0.054,0.915,0.842,$ |

$-0.837,-0.913,-0.163)$ | $-0.825,-0.716,-0.284)$ | $-0.895,-0.769,-0.023)$ | $-0.842,-0.742,-0.054)$ | |

${S}_{3}$ | $(0.087,0.837,0.662,$ | $(0.132,0.825,0.64,$ | $(0.395,0.86,0.925,$ | $(0.085,0.915,0.881,$ |

$-0.79,-0.734,-0.226)$ | $-0.716,-0.94,-0.175)$ | $-0.769,-0.819,-0.75)$ | $-0.842,-0.842,-0.119)$ | |

${S}_{4}$ | $(0.21,0.913,0.837,$ | $(0.36,0.775,0.868,$ | $(0.181,0.769,0.895,$ | $(0.329,0.742,0.671,$ |

$-0.662,-0.554,-0.087)$ | $-0.528,-0.716,-0.132)$ | $-0.605,-0.819,-0.047)$ | $-0.565,-0.742,-0.026)$ |

$\mathit{S}\setminus \mathit{T}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | $(0.6,0.5,0.3,$ | $(0.5,0.7,0.4,$ | $(0.3,0.5,0.5,$ | $(0.5,0.3,0.6,$ |

$-0.5,-0.7,-0.4)$ | $-0.6,-0.4,-0.5)$ | $-0.7,-0.3,-0.4)$ | $-0.4,-0.7,-0.5)$ | |

${S}_{2}$ | $(0.9,0.3,0.2,$ | $(0.7,0.4,0.2,$ | $(0.4,0.7,0.6,$ | $(0.8,0.3,0.2,$ |

$-0.3,-0.6,-0.5)$ | $-0.4,-0.5,-0.7)$ | $-0.6,-0.3,-0.3)$ | $-0.5,-0.5,-0.7)$ | |

${S}_{3}$ | $(0.4,0.6,0.6,$ | $(0.5,0.3,0.6,$ | $(0.7,0.5,0.3,$ | $(0.4,0.6,0.7,$ |

$-0.7,-0.4,-0.3)$ | $-0.6,-0.4,-0.4)$ | $-0.4,-0.4,-0.6)$ | $-0.5,-0.4,-0.4)$ | |

${S}_{4}$ | $(0.8,0.5,0.3,$ | $(0.6,0.4,0.3,$ | $(0.4,0.5,0.7,$ | $(0.5,0.4,0.6,$ |

$-0.3,-0.4,-0.5)$ | $-0.5,-0.7,-0.8)$ | $-0.5,-0.4,-0.2)$ | $-0.6,-0.7,-0.3)$ | |

${S}_{5}$ | $(0.6,0.4,0.6,$ | $(0.4,0.7,0.6,$ | $(0.6,0.3,0.5,$ | $(0.5,0.7,0.4$, |

$-0.4,-0.7,-0.3)$ | $-0.7,-0.5,-0.6)$ | $-0.3,-0.7,-0.4)$ | $-0.3,-0.6,-0.5)$ |

$\mathit{S}\setminus \mathit{T}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | $(0.203,0.842,0.742,$ | $(0.155,0.917,0.801,$ | $(0.085,0.842,0.842,$ | $(0.166,0.730,0.875,$ |

$-0.842,-0.915,-0.119)$ | $-0.884,-0.801,-0.677)$ | $-0.915,-0.742,-0.119)$ | $-0.787,-0.911,-0.166)$ | |

${S}_{2}$ | $(0.435,0.742,0.671,$ | $(0.253,0.801,0.677,$ | $(0.119,0.915,0.881,$ | $(0.344,0.730,0.656,$ |

$-0.742,-0.881,-0.158)$ | $-0.801,-0.845,-0.253)$ | $-0.881,-0.742,-0.085)$ | $-0.656,-0.834,-0.270)$ | |

${S}_{3}$ | $(0.119,0.881,0.881,$ | $(0.155,0.747,0.884,$ | $(0.258,0.842,0.742,$ | $(0.125,0.875,0.911,$ |

$-0.915,-0.797,-0.085)$ | $-0.884,-0.801,-0.116)$ | $-0.797,-0.797,-0.203)$ | $-0.834,-0.787,-0.125)$ | |

${S}_{4}$ | $(0.329,0.842,0.742,$ | $(0.199,0.801,0.747,$ | $(0.119,0.842,0.915,$ | $(0.166,0.787,0.875,$ |

$-0.742,-0.797,-0.158)$ | $-0.845,-0.917,-0.323)$ | $-0.842,-0.797,-0.054)$ | $-0.875,-0.911,-0.089)$ | |

${S}_{5}$ | $(0.203,0.797,0.881,$ | $(0.116,0.917,0.884,$ | $(0.203,0.742,0.842,$ | $(0.166,0.911,0.787,$ |

$-0.797,-0.915,-0.085)$ | $-0.917,-0.845,-0.199)$ | $-0.742,-0.915,-0.119)$ | $-0.730,-0.875,-0.166)$ |

$\mathit{S}\setminus \mathit{T}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | $(0.8,0.5,0.3,$ | $(0.7,0.3,0.2,$ | $(0.5,0.4,0.6,$ | $(0.9,0.3,0.2,$ |

$-0.3,-0.6,-0.5)$ | $-0.3,-0.5,-0.4)$ | $-0.5,-0.3,-0.4)$ | $-0.3,-0.4,-0.2)$ | |

${S}_{2}$ | $(0.5,0.7,0.6$ | $(0.4,0.7,0.5,$ | $(0.6,0.8,0.5,$ | $(0.5,0.3,0.6,$ |

$-0.4,-0.2,-0.4)$ | $-0.6,-0.2,-0.3)$ | $-0.3,-0.5,-0.7)$ | $-0.6,-0.4,-0.3)$ | |

${S}_{3}$ | $(0.4,0.6,0.8,$ | $(0.6,0.3,0.5,$ | $(0.3,0.5,0.7,$ | $(0.5,0.7,0.4,$ |

$-0.7,-0.3,-0.4)$ | $-0.2,-0.4,-0.6)$ | $-0.8,-0.4,-0.2)$ | $-0.6,-0.3,-0.5)$ | |

${S}_{4}$ | $(0.7,0.3,0.5,$ | $(0.5,0.4,0.6,$ | $(0.6,0.4,0.3,$ | $(0.4,0.5,0.7,$ |

$-0.4,-0.2,-0.5)$ | $-0.4,-0.5,-0.3)$ | $-0.3,-0.5,-0.7)$ | $-0.6,-0.5,-0.3)$ |

$\mathit{S}\setminus \mathit{T}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | $(0.3313,0.8409,0.7401,$ | $(0.2474,0.7526,0.6839,$ | $(0.1711,0.7803,0.8708,$ | $(0.4287,0.7463,0.6762,$ |

$-0.7401,-0.8801,-0.1591)$ | $-0.7256,-0.8490,-0.1136)$ | $-0.8289,-0.7218,-0.1292)$ | $-0.7463,-0.8003,-0.0528)$ | |

${S}_{2}$ | $(0.1591,0.9147,0.8801,$ | $(0.1136,0.9192,0.8490,$ | $(0.2197,0.9414,0.8289,$ | $(0.1551,0.7463,0.8832,$ |

$-0.7953,-0.6687,-0.1199)$ | $-0.8864,-0.6839,-0.0808)$ | $-0.7218,-0.8289,-0.2782)$ | $-0.8832,-0.8003,-0.0831)$ | |

${S}_{3}$ | $(0.1199,0.8801,0.9457,$ | $(0.1945,0.7526,0.8490,$ | $(0.0921,0.8289,0.9079,$ | $(0.1551,0.9169,0.8003,$ |

$-0.9147,-0.7401,-0.1199)$ | $-0.6839,-0.8055,-0.1945)$ | $-0.9414,-0.7803,-0.0586)$ | $-0.8832,-0.7463,-0.1551)$ | |

${S}_{4}$ | $(0.2599,0.7401,0.8409,$ | $(0.1510,0.8055,0.8864,$ | $(0.2197,0.7803,0.7218,$ | $(0.1168,0.8449,0.9169,$ |

$-0.7953,-0.6687,-0.1591)$ | $-0.8055,-0.8490,-0.0808)$ | $-0.7218,-0.8289,-0.2782)$ | $-0.8832,-0.8449,-0.0831)$ |

${\mathit{E}}_{\mathit{xy}}\setminus \mathit{y}$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|

${E}_{1y}$ | - | {1, 2, 3} | {1, 2} | { } |

${E}_{2y}$ | {4} | - | {4} | { } |

${E}_{3y}$ | {3, 4} | {1, 2, 3} | - | {3} |

${E}_{4y}$ | {1, 2, 3, 4} | {1, 2, 3, 4} | {1, 2, 4} | - |

${\mathit{F}}_{\mathit{xy}}\setminus \mathit{y}$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|

${F}_{1y}$ | - | {4} | {3, 4} | {1, 2, 3, 4} |

${F}_{2y}$ | {1, 2, 3} | - | {1, 2, 3} | {1, 2, 3, 4} |

${F}_{3y}$ | {1, 2} | {4} | - | {1, 2, 4} |

${F}_{4y}$ | { } | { } | {3} | - |

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**MDPI and ACS Style**

Akram, M.; Shumaiza; Smarandache, F.
Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I. *Axioms* **2018**, *7*, 33.
https://doi.org/10.3390/axioms7020033

**AMA Style**

Akram M, Shumaiza, Smarandache F.
Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I. *Axioms*. 2018; 7(2):33.
https://doi.org/10.3390/axioms7020033

**Chicago/Turabian Style**

Akram, Muhammad, Shumaiza, and Florentin Smarandache.
2018. "Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I" *Axioms* 7, no. 2: 33.
https://doi.org/10.3390/axioms7020033