## 1 Introduction

*et al.*(2011) proposed a fuzzy extension of the MULTIMOORA method; Balezentis and Zeng (2013) proposed an interval-valued fuzzy extension; Balezentis

*et al.*(2014) proposed an intuitionistic fuzzy extension and Zavadskas

*et al.*(2015) proposed an interval-valued intuitionistic extension of the MULTIMOORA method.

*et al.*, 2012a, 2012b), construction (Kracka

*et al.*, 2015), risk management (Liu

*et al.*, 2014a) and waste treatment (Liu

*et al.*, 2014b) can be mentioned.

*et al.*, 2016) and the CNC machine tool evaluation (Sahu

*et al.*, 2016).

*et al.*, 2016).

*et al.*(2010) further proposed a single valued neutrosophic set, by modifying the conditions ${T_{A}}(x)$, ${I_{A}}(x)$ and ${F_{A}}(x)\in [0,1]$ and $0\leqslant {T_{A}}(x)+{I_{A}}(x)+{F_{A}}(x)\leqslant 3$, which are more suitable for solving scientific and engineering problems (Li

*et al.*, 2016).

## 2 The Single Valued Neutrosophic Set

##### Definition 1 (*See* Smarandache, 1999).

*X*be the universe of discourse, with a generic element in

*X*denoted by

*x*. Then, the Neutrosophic Set (NS)

*A*in

*X*is as follows:

##### (1)

\[ A=\big\{x\big\langle {T_{A}}(x),{I_{A}}(x),{F_{A}}(x)\big\rangle \hspace{0.1667em}\big|\hspace{0.1667em}x\in X\big\},\]##### Definition 2 (*See* Smarandache, 1999; Wang *et al.*, 2010).

*X*be the universe of discourse. The Single valued neutrosophic set (SVNS)

*A*over

*X*is an object having the following form:

##### (2)

\[ A=\big\{x\big\langle {T_{A}}(x),{I_{A}}(x),{F_{A}}(x)\big\rangle \hspace{0.1667em}\big|\hspace{0.1667em}x\in X\big\},\]##### Definition 3 (*See* Smarandache, 1999).

*A*in

*X*, the triple $\langle {t_{A}},{i_{A}},{f_{A}}\rangle $ is called the single valued neutrosophic number (SVNN).

##### Definition 4.

##### (3)

\[ {x_{1}}+{x_{2}}=\langle {t_{1}}+{t_{2}}-{t_{1}}{t_{2}},{i_{1}}{i_{2}},{f_{1}}{f_{2}}\rangle ,\]##### (4)

\[ {x_{1}}\cdot {x_{2}}=\langle {t_{1}}{t_{2}},{i_{1}}+{i_{2}}-{i_{1}}{i_{2,}}{f_{1}}+{f_{2}}-{f_{1}}{f_{2}}\rangle ,\]##### Definition 5 (*See* Sahin, 2014).

##### Definition 6.

##### Definition 7 (*See* Sahin, 2014).

##### (9)

\[\begin{array}{l}\displaystyle \mathit{SVNWA}({A_{1}},{A_{2}},\dots ,{A_{n}})\\ {} \displaystyle \hspace{1em}={\sum \limits_{j=1}^{n}}{w_{j}}{A_{j}}=\Bigg(1-{\prod \limits_{j=1}^{n}}{(1-{t_{j}})^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{({i_{j}})^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{({f_{j}})^{{w_{j}}}}\Bigg).\end{array}\]*j*of the weighting vector, ${w_{j}}\in [0,1]$ and ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$.

##### Definition 8 (*See* Sahin, 2014).

##### (10)

\[\begin{array}{l}\displaystyle \mathit{SVNWG}({A_{1}},{A_{2}},\dots ,{A_{n}})\\ {} \displaystyle \hspace{1em}={\prod \limits_{j=1}^{n}}{({A_{j}})^{{w_{j}}}}=\Bigg({\prod \limits_{j=1}^{n}}{({t_{j}})^{{w_{j}}}},1-{\prod \limits_{j=1}^{n}}{(1-{i_{j}})^{{w_{j}}}},1-{\prod \limits_{j=1}^{n}}{(1-{f_{j}})^{{w_{j}}}}\Bigg).\end{array}\]*j*of the weighting vector, ${w_{j}}\in [0,1]$ and ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$.

## 3 The MULTIMOORA Method

**The ratio system approach****.**In this approach, the overall importance of the alternative

*i*can be calculated as follows: with:

*i*, obtained on the basis of all the criteria; ${y_{i}^{+}}$ and ${y_{i}^{-}}$ denote the overall importance of the alternative

*i*, obtained on the basis of the benefit and cost criteria, respectively; ${r_{ij}}$ denotes the normalized performance of the alternative

*i*with respect to the criterion

*j*; ${x_{ij}}$ denotes the performance of the alternative

*i*to the criterion

*j*; ${\Omega _{\max }}$ and ${\Omega _{\min }}$ denote the sets of the benefit cost criteria, respectively; $i=1,2,\dots ,m$;

*m*is the number of the alternatives, $j=1,2,\dots ,n$;

*n*is the number of the criteria.

**The reference point approach****.**The optimization based on this approach can be shown as follows: where: ${d_{i}^{\max }}$ denotes the maximum distance of the alternative

*i*to the reference point and ${r_{j}^{\ast }}$ denotes the coordinate

*j*of the reference point as follows:

##### (16)

\[ {r_{j}^{\ast }}=\left\{\begin{array}{l@{\hskip4.0pt}l}\underset{i}{\max }{r_{ij}},\hspace{1em}& j\in {\Omega _{\max }},\\ {} \underset{i}{\min }{r_{ij}},\hspace{1em}& j\in {\Omega _{\min }}.\end{array}\right.\]

**The full multiplicative form****.**In the FMF, the overall utility of the alternative

*i*can be determined in the following manner: with:

*i*, ${a_{i}}$ denotes the product of the weighted performance ratings of the benefit criteria and ${b_{i}}$ denotes the product of the weighted performance ratings of the cost criteria of the alternative

*i*.

**The final ranking of alternatives based on the MULTIMOORA method****.**As a result of evaluation by applying the MULTIMOORA method, three ranking lists of the considered alternatives are obtained. Based on Brauers and Zavadskas (2011), the final ranking order of the alternatives is determined based on the theory of dominance.

## 4 An Extension of the MULTIMOORA Method Based on Single Valued Neutrosophic Numbers

*m*alternatives and

*n*criteria, whereby the performances of the alternatives are expressed by using SVNS, the calculation procedure of the extended MULTIMOORA method can be expressed as follows:

**Step 1.**

**Determine the ranking order of the alternatives based on the RS approach.**The ranking of the alternatives and the selection of the best one based on this approach in the proposed extension of the MULTIMOORA method can be expressed through the following sub steps:

**Step 1.1.**

*Calculate*${Y_{i}^{+}}$ and ${Y_{i}^{-}}$ by using the SVNWA operator, as follows:

*i*obtained based on the benefit and cost criteria, respectively; ${Y_{i}^{+}}$ and ${Y_{i}^{-}}$ are SVNNs.

**Step 1.2.**

**Step 1.3.**

**Step 1.4.**

*Rank the alternatives and select the best one*. The ranking of the alternatives can be performed in the same way as in the RS approach of the ordinary MULTIMOORA method.

**Step 2.**

**Determine the ranking order of the alternatives based on the RP approach.**The ranking of the alternatives and the selection of the best one, based on the RP approach, can be expressed through the following substeps:

**Step 2.1.**

*Determine the reference point*. In this approach, each coordinate of the reference point ${r^{\ast }}=\{{r_{1}^{\ast }},{r_{2}^{\ast }},\dots ,{r_{n}^{\ast }}\}$ is an SVNN, ${r_{j}^{\ast }}=\langle {t_{j}^{\ast }},{i_{j}^{\ast }},{f_{j}^{\ast }}\rangle $, whose values are determined as follows:

##### (25)

\[ {r_{j}^{\ast }}=\left\{\begin{array}{l@{\hskip4.0pt}l}\big\langle \underset{i}{\max }{t_{ij}},\underset{i}{\min }{i_{ij}},\underset{i}{\min }{f_{ij}}\big\rangle ,\hspace{1em}& j\in {\Omega _{\max }},\\ {} \big\langle \underset{i}{\min }{t_{ij}},\underset{i}{\min }{i_{ij}},\underset{i}{\max }{f_{ij}}\big\rangle ,\hspace{1em}& j\in {\Omega _{\min }},\end{array}\right.\]*j*of the reference point.

**Step 2.2.**

*Determine the maximum distance from each alternative to all the coordinates of the reference point*as follows: where ${d_{ij}^{\max }}$ denotes the maximum distance of the alternative

*i*obtained based on the criterion

*j*determined by Eq. (8).

**Step 2.3.**

**Step 2.4.**

*Rank the alternatives and select the best one*. At this step, the ranking of the alternatives can be done in the same way as in the RPA of the ordinary MULTIMOORA method.

**Step 3.**

**Determine the ranking order of the alternatives and select the best one based on the FMF.**The ranking of the alternatives and the selection of the best one can be expressed through the following sub steps:

**Step 3.1.**

*Calculate*${A_{i}}$ and ${B_{i}}$ as follows:

**Step 3.2.**

**Step 3.3.**

**Sep 3.4.**

*Rank the alternatives and select the best one*. The ranking of the alternatives can be performed in the same way as in the FMF of the ordinary MULTIMOORA method.

**Step 4.**

**Determine the final ranking order of the alternatives**. The final ranking order of the alternatives can be determined as in the case of the ordinary MULTIMOORA method, i.e. based on the dominance theory.

## 5 A Numerical Example

*et al.*(2015). In order to briefly demonstrate the advantages of the proposed methodology, this example has been slightly modified.

*The ranking results and the ranking order of the alternatives obtained based on the RS approach, i.e. by applying Eqs. (19) to (23), are accounted for in Table 2.*

**The ranking based on the RS approach.**##### Table 1

${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ | ${C_{5}}$ | |

${A_{1}}$ | $\langle 0.9,0.1,0.2\rangle $ | $\langle 0.7,0.2,0.3\rangle $ | $\langle 0.9,0.1,0.2\rangle $ | $\langle 0.9,0.1,0.2\rangle $ | $\langle 0.9,0.1,0.2\rangle $ |

${A_{2}}$ | $\langle 0.8,0.1,0.3\rangle $ | $\langle 0.8,0.1,0.3\rangle $ | $\langle 0.8,0.1,0.3\rangle $ | $\langle 0.9,0.1,0.2\rangle $ | $\langle 0.8,0.1,0.3\rangle $ |

${A_{3}}$ | $\langle 1.0,0.1,0.3\rangle $ | $\langle 0.9,0.1,0.2\rangle $ | $\langle 0.9,0.1,0.2\rangle $ | $\langle 0.7,0.2,0.5\rangle $ | $\langle 0.7,0.2,0.3\rangle $ |

**The ranking based on the RPA****.**The ranking of the alternatives based on the RP approach begins by determining the reference point, as it is shown in Table 3.

##### Table 2

${Y_{i}^{+}}$ | ${Y_{i}^{-}}$ | ${y_{i}^{+}}$ | ${y_{i}^{-}}$ | ${y_{i}}$ | Rank | |

${A_{1}}$ | $\langle 0.73,0.25,0.38\rangle $ | $\langle 0.55,0.45,0.57\rangle $ | 0.425 | 0.045 | 0.380 | 2 |

${A_{2}}$ | $\langle 0.65,0.22,0.46\rangle $ | $\langle 0.51,0.45,0.60\rangle $ | 0.372 | 0.006 | 0.366 | 3 |

${A_{3}}$ | $\langle 1.0,0.22,0.39\rangle $ | $\langle 0.34,0.57,0.73\rangle $ | 0.583 | −0.263 | 0.845 | 1 |

##### Table 3

${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ | ${C_{5}}$ | |

${r_{j}^{\ast }}$ | $\langle 1.0,0.1,0.3\rangle $ | $\langle 0.9,0.2,0.3\rangle $ | $\langle 0.9,0.1,0.3\rangle $ | $\langle 0.7,0.1,0.2\rangle $ | $\langle 0.7,0.1,0.2\rangle $ |

*j*of the reference point obtained by using Eq. (25) and the maximum distance of each alternative obtained by using Eq. (26) are presented in Table 4. The ranking order of the alternatives is also presented in Table 4.

**The ranking based on the FMF****.**The ranking results and the ranking order of the alternatives obtained on the basis of the FMF approach, i.e. by applying Eqs. (27) to (31), are demonstrated in Table 5.

##### Table 4

I | II | III | IV | V | VI | VII | VI |

${r_{1}^{\ast }}$ | ${r_{2}^{\ast }}$ | ${r_{3}^{\ast }}$ | ${r_{4}^{\ast }}$ | ${r_{5}^{\ast }}$ | ${d_{i}^{\max }}$ | Rank | |

${A_{1}}$ | 0.02 | 0.03 | 0.00 | 0.00 | 0.00 | 0.034 | 1 |

${A_{2}}$ | 0.05 | 0.02 | 0.02 | 0.00 | 0.01 | 0.048 | 2 |

${A_{3}}$ | 0.00 | 0.00 | 0.00 | 0.06 | 0.01 | 0.063 | 3 |

##### Table 5

${A_{i}}$ | ${B_{i}}$ | ${a_{i}}$ | ${b_{i}}$ | ${u_{i}}$ | Rank | |

${A_{1}}$ | $\langle 0.89,0.25,0.15\rangle $ | $\langle 0.96,0.45,0.08\rangle $ | 0.618 | 0.498 | 1.242 | 3 |

${A_{2}}$ | $\langle 0.86,0.22,0.21\rangle $ | $\langle 0.95,0.45,0.09\rangle $ | 0.605 | 0.481 | 1.258 | 2 |

${A_{3}}$ | $\langle 0.96,0.22,0.16\rangle $ | $\langle 0.88,0.57,0.18\rangle $ | 0.674 | 0.283 | 2.379 | 1 |

##### Table 6

RS | RP | FMF | Rank | |

${A_{1}}$ | 2 | 1 | 3 | 3 |

${A_{2}}$ | 3 | 2 | 2 | 2 |

${A_{3}}$ | 1 | 3 | 1 | 1 |