## 1 Introduction

*et al.*(2010) point out the fact that it is the study of the methods and procedures aimed at making a proposal for solutions in terms of multiple, often conflicting criteria. Hwang and Yoon (1981) states that MCDM is based on the two basic approaches, i.e. on multiple attribute decision-making (MADM), which implies a choice of courses in the presence of multiple, and often conflicting criteria, i.e. a selection of the best alternative from a finite set of possible alternatives. Unlike MADM, in multiple objective decision-making (MODM), the best alternative is that which is formed with multiple goals, based on the continuous variables of the decision with additional constraints.

*et al.*(2012); Zavadskas

*et al.*(2014). The increasing application of the MCDM method to solving various problems has caused an exceptional growth of multi-criteria decision-making as an important field of operational research, especially since 1980 Aouni

*et al.*(2018); Masri

*et al.*(2018); Wallenius

*et al.*(2008); Dyer

*et al.*(1992).

*et al.*(1994), Analytic Network Process (ANP) method Saaty (1996), Vlse Kriterijumska Optimizacija i kompromisno Resenje (VIKOR) Opricovic (1998), Multi-Objective Optimization on basis of Ratio Analysis (MOORA) method Brauers and Zavadskas (2006), Additive Ratio ASsessment (ARAS) method Zavadskas and Turskis (2010), Multi-Objective Optimization by Ratio Analysis plus the Full Multiplicative Form (MULTIMOORA) method Brauers and Zavadskas (2010a), and so on. While within the MODM methods that have been proposed can be stated: Data envelopment analysis (DEA) method Charnes

*et al.*(1978), Linear Programming (LP) and Nonlinear Programming (NP) Luenberger and Ye (1984), Multi-Objective Programming (MOP) technique Charnes

*et al.*(1989), Multi-Objective Linear Programming Ecker and Kouada (1978), and so on.

*et al.*(2015).

*et al.*(2018), project critical path selection Dorfeshan

*et al.*(2018), the selection of the optimal mining method Liang

*et al.*(2018), pharmacological therapy selection Eghbali-Zarch

*et al.*(2018), ICT hardware selection Adali and Işik (2017), industrial robot selection Karande

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

*et al.*(2016), personnel selection Karabasevic

*et al.*(2015); Baležentis

*et al.*(2012), the economy (Baležentis and Zeng (2013); Brauers and Zavadskas (2011a), 2010b; Brauers and Ginevičius (2010)), and so on.

*et al.*(2011) proposed a fuzzy extension of the MULTIMOORA method; Baležentis and Zeng (2013) proposed an IVFN extension of the MULTIMOORA method; Baležentis

*et al.*(2014) also proposed an IFN extension of the MULTIMOORA method; Stanujkic

*et al.*(2015) proposed an extension of the MULTIMOORA method based on the use of interval-valued triangular fuzzy numbers; Zavadskas

*et al.*(2015) proposed an IVIF-based extension of the MULTIMOORA method; Hafezalkotob

*et al.*(2016) proposed an extension of the MULTIMOORA method based on the use of interval numbers; Stanujkic

*et al.*(2017a) proposed a neutrosophic extension of the MULTIMOORA method, and so on.

*et al.*(2018) and Akram and Arshad (2018) proposed bipolar fuzzy extensions of TOPSIS and ELECTRE I methods; while Han

*et al.*(2018) provide a comprehensive mathematical approach based on the TOPSIS method for improving the accuracy of bipolar disorder clinical diagnosis.

*et al.*(2018), Pramanik

*et al.*(2018) and Tian

*et al.*(2016) can be cited as some of the current researches.

## 2 The Basic Elements of a Bipolar Fuzzy Set

##### Definition 1 (*Fuzzy set*, Zadeh 1965).

*X*be a nonempty set, with a generic element in

*X*denoted by

*x*. Then, a fuzzy set

*A*in

*X*is a set of ordered pairs: where the membership function ${\mu _{A}}(x)$ denotes the degree of the membership of the element

*x*to the set

*A*, and ${\mu _{A}}(x)\in [0,1]$.

##### Definition 2 (*Bipolar fuzzy set*, Lee 2000).

*X*be a nonempty set. Then, a bipolar fuzzy set (BFS) is defined as: where: the positive membership function ${\mu _{A}^{+}}(x)$ denotes the satisfaction degree of the element

*x*to the property corresponding to the bipolar-valued fuzzy set, and the negative membership function ${\nu _{A}^{-}}(x)$ denotes the degree of the satisfaction of the element

*x*to some implicit counter-property corresponding to the bipolar-valued fuzzy set, respectively; ${\mu _{A}^{+}}(x):X\to [0,1]$ and ${\nu _{A}^{-}}(x):X\to [-1,0]$.

##### Definition 3.

*R*, whose positive membership and negative membership function are as follows: respectively.

##### Definition 4.

##### (5)

\[ {a_{1}}+{a_{2}}=\big\langle {a_{1}^{+}}+{a_{2}^{+}}-{a_{1}^{+}}{a_{2}^{+}},-{a_{1}^{-}}{a_{2}^{-}}\big\rangle ,\]##### (6)

\[ {a_{1}}\cdot {a_{2}}=\big\langle {a_{1}^{+}}{a_{2}^{+}},-\big(-{a_{1}^{-}}-{a_{2}^{-}}-{a_{1}^{-}}{a_{2}^{-}}\big)\big\rangle ,\]##### Definition 6.

##### Definition 7.

##### Definition 8.

*n*dimensions is a mapping as follows:

##### (11)

\[\begin{aligned}{}{A_{w}}({a_{1}},{a_{2}},\dots ,{a_{n}})=& {\sum \limits_{j=1}^{n}}{w_{j}}{a_{j}}\\ {} =& \bigg(1-{\prod \limits_{j=1}^{n}}{\big(1-{a_{j}^{+}}\big)^{{w_{j}}}},-\bigg(1-{\prod \limits_{j=1}^{n}}\big(1-{\big(-{a_{j}^{-}}\big)\big)^{{w_{j}}}}\bigg)\bigg),\end{aligned}\]*j*of the weighting vector, ${w_{j}}\in [0,1]$ and ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$.

## 3 The MULTIMOORA Method

**Step 1.**

*Construct a decision matrix and determine the weights of criteria.*

**Step 2.**

*Calculate a normalized decision matrix*, as follows: where: ${r_{ij}}$ denotes the normalized performance of the alternative

*i*with respect to the criterion

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

*i*to the criterion

*j*.

**Step 3.**

*Calculate the overall significance of each alternative*, as follows:

##### (14)

\[ {y_{i}}=\sum \limits_{j\in {\Omega _{\max }}}{w_{j}}{r_{ij}}-\sum \limits_{j\in {\Omega _{\min }}}{w_{j}}{r_{ij}},\]*i*, ${\Omega _{\max }}$ and ${\Omega _{\min }}$ denote the sets of the benefit cost criteria, respectively.

**Step 4.**

*Determine the reference point*, as follows:

**Step 5.**

*Determine the maximal distance between each alternative and the reference point*, as follows: where: ${d_{i}^{\max }}$ denotes the maximal distance of the alternative

*i*to the reference point.

**Step 6.**

*Determine the overall utility of each alternative*, as follows:

##### (17)

\[ {u_{i}}=\frac{{\textstyle\prod _{j\in {\Omega _{\max }}}}{w_{j}}{r_{ij}}}{{\textstyle\prod _{j\in {\Omega _{\min }}}}{w_{j}}{r_{ij}}},\]*i*.

**Step 7.**

*Determine the final ranking order of the considered alternatives and select the most appropriate one*. In this step, the considered alternatives are ranked based on their:

## 4 An Extension of the MULTIMOORA Method Based on Single-Valued Bipolar Fuzzy Numbers

**Step 1.**

*Evaluate the alternatives in relation to the evaluation criteria*, and do that for each DM. In this step, each DM evaluates the alternatives and forms an evaluation matrix.

##### Table 1

Satisfaction level | Numerical value |

Neutral/without attitude | 0 |

Extremely low | 1 |

Very low | 2 |

Low | 3 |

Medium low | 4 |

Medium | 5 |

Medium high | 6 |

High | 7 |

Very high | 8 |

Extremely high | 9 |

Absolute | 10 |

**Step 2.**

*Determine the importance of the evaluation criteria*, and do that for each DM. In this step, each DM determines the weights of the criteria by using one of several existing methods for determining the weights of criteria.

**Step 3.**

*Determine the group decision matrix*. In order to transform individual into group preferences, individual evaluation matrices are transformed into group one by applying Eq. (11).

**Step 4.**

*Determine the group weights of the criteria*. In order to transform individual into group preferences with respect to the weights of criteria, the group weights of criteria can be determined as follows: where: ${w_{j}}$ denotes the weight of the criterion

*j*, and ${w_{j}^{k}}$ denotes the weight of the criterion

*j*obtained from the DM

*k*.

*et al.*(2017b), the remainder steps of the proposed approach are as follows:

**Step 5.**

*Determine the significance of the evaluated alternatives based on the RS approach*. This step can be explained through the following sub-steps:

**Step 5.1.**

*Determine the impact of the benefit and cost criteria to the importance of each alternative*, as follows:

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

**Step. 5.2.**

*Transform*${Y_{i}^{+}}$

*and*${Y_{i}^{-}}$

*into crisp values by using the Score Function*, as follows:

**Step 5.3.**

*Calculate the overall importance for each alternative*, as follows:

**Step 6.**

*Determine the significance of the evaluated alternatives based on the RP approach*. This step can be explained through the following sub-steps:

**Step 6.1.**

*Determine the reference point*. The coordinates on the bipolar fuzzy reference point ${r^{\ast }}=\{{r_{1}^{\ast }},{r_{2}^{\ast }},\dots ,{r_{n}^{\ast }}\}$ can be determined as follows:

##### (25)

\[ {r^{\ast }}=\Big\{\Big(\Big\langle \underset{i}{\max }{r_{ij}},\underset{i}{\min }{r_{ij}}\Big\rangle \Big|j\in {\Omega _{\max }}\Big),\Big(\Big\langle \underset{i}{\min }{r_{ij}},\underset{i}{\max }{r_{ij}}\Big\rangle \Big|j\in {\Omega _{\min }}\Big)\Big\}\]*j*of the reference point.

**Step 6.2.**

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

*i*to the criterion

*j*determined by Eq. (10).

**Step 6.3.**

*Determine the maximum distance of each alternative*, as follows: where ${d_{i}^{\max }}$ denotes the maximum distance of the alternative

*i*.

**Step 7.**

*Determine the significance of the evaluated alternatives based on*the FMF.

**Step 7.1.**

*Calculate the utility obtained based on the benefit*${U_{i}^{+}}$

*and cost*${U_{i}^{-}}$

*criteria*, for each alternative, as follows:

**Step 7.2.**

*Transform*${U_{i}^{+}}$

*and*${U_{i}^{-}}$

*into crisp values by using the Score Function*, as follows:

**Step 7.3.**

*Determine the overall utility for each alternative*, as follows:

**Step 8.**

*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 Brauers and Zavadskas (2011b).

## 5 A Numerical Example

##### Table 2

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

${a^{+}}$ | ${a^{-}}$ | ${a^{+}}$ | ${a^{-}}$ | ${a^{+}}$ | ${a^{-}}$ | ${a^{+}}$ | ${a^{-}}$ | ${a^{+}}$ | ${a^{-}}$ | |

$\hspace{2.5pt}{A_{1}}$ | 7 | −2 | 7 | −3 | 5 | −1 | 7 | −5 | 8 | −1 |

$\hspace{2.5pt}{A_{2}}$ | 4 | −1 | 5 | −2 | 4 | −2 | 4 | −6 | 7 | −1 |

$\hspace{2.5pt}{A_{3}}$ | 7 | −1 | 3 | −1 | 2 | 0 | 2 | −1 | 7 | −2 |

$\hspace{2.5pt}{A_{4}}$ | 9 | −1 | 4 | −1 | 3 | 0 | 3 | −1 | 6 | −1 |

##### Table 3

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

${A_{1}}$ | $\langle 0.70,-0.20\rangle $ | $\langle 0.70,-0.30\rangle $ | $\langle 0.50,-0.10\rangle $ | $\langle 0.70,-0.50\rangle $ | $\langle 0.80,-0.10\rangle $ |

${A_{2}}$ | $\langle 0.40,-0.10\rangle $ | $\langle 0.50,-0.20\rangle $ | $\langle 0.40,-0.20\rangle $ | $\langle 0.40,-0.60\rangle $ | $\langle 0.70,-0.10\rangle $ |

${A_{3}}$ | $\langle 0.70,-0.10\rangle $ | $\langle 0.30,-0.10\rangle $ | $\langle 0.20,0.00\rangle $ | $\langle 0.20,-0.10\rangle $ | $\langle 0.70,-0.20\rangle $ |

${A_{4}}$ | $\langle 0.90,-0.10\rangle $ | $\langle 0.40,-0.10\rangle $ | $\langle 0.30,0.00\rangle $ | $\langle 0.30,-0.10\rangle $ | $\langle 0.60,-0.10\rangle $ |

##### Table 4

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

${A_{1}}$ | $\langle 0.70,-0.20\rangle $ | $\langle 0.70,-0.50\rangle $ | $\langle 0.40,-0.20\rangle $ | $\langle 0.70,-0.50\rangle $ | $\langle 0.80,-0.10\rangle $ |

${A_{2}}$ | $\langle 0.60,-0.10\rangle $ | $\langle 0.40,-0.60\rangle $ | $\langle 0.40,-0.20\rangle $ | $\langle 0.40,-0.60\rangle $ | $\langle 0.80,-0.10\rangle $ |

${A_{3}}$ | $\langle 0.80,-0.10\rangle $ | $\langle 0.20,-0.10\rangle $ | $\langle 0.20,-0.10\rangle $ | $\langle 0.20,-0.10\rangle $ | $\langle 0.70,-0.10\rangle $ |

${A_{4}}$ | $\langle 0.90,-0.10\rangle $ | $\langle 0.30,-0.10\rangle $ | $\langle 0.30,-0.10\rangle $ | $\langle 0.30,-0.10\rangle $ | $\langle 0.60,-0.10\rangle $ |

##### Table 5

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

${A_{1}}$ | $\langle 0.60,-0.10\rangle $ | $\langle 0.90,-0.20\rangle $ | $\langle 1.00,0.00\rangle $ | $\langle 1.00,0.00\rangle $ | $\langle 0.80,-0.10\rangle $ |

${A_{2}}$ | $\langle 0.40,-0.60\rangle $ | $\langle 0.40,-0.60\rangle $ | $\langle 1.00,-0.40\rangle $ | $\langle 1.00,0.00\rangle $ | $\langle 0.80,-0.10\rangle $ |

${A_{3}}$ | $\langle 0.20,-0.10\rangle $ | $\langle 0.90,-0.40\rangle $ | $\langle 0.80,-0.30\rangle $ | $\langle 0.70,-0.10\rangle $ | $\langle 0.70,-0.10\rangle $ |

${A_{4}}$ | $\langle 0.30,-0.10\rangle $ | $\langle 1.00,-0.30\rangle $ | $\langle 0.80,-0.20\rangle $ | $\langle 0.80,-0.10\rangle $ | $\langle 0.60,-0.10\rangle $ |

##### Table 6

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

${A_{1}}$ | $\langle 0.67,-0.16\rangle $ | $\langle 0.79,-0.32\rangle $ | $\langle 1.00,0.00\rangle $ | $\langle 1.00,0.00\rangle $ | $\langle 0.80,-0.10\rangle $ |

${A_{2}}$ | $\langle 0.47,-0.18\rangle $ | $\langle 0.43,-0.42\rangle $ | $\langle 1.00,-0.26\rangle $ | $\langle 1.00,0.00\rangle $ | $\langle 0.77,-0.10\rangle $ |

${A_{3}}$ | $\langle 0.64,-0.10\rangle $ | $\langle 0.61,-0.16\rangle $ | $\langle 0.49,0.00\rangle $ | $\langle 0.41,-0.10\rangle $ | $\langle 0.70,-0.13\rangle $ |

${A_{4}}$ | $\langle 0.81,-0.10\rangle $ | $\langle 1.00,-0.15\rangle $ | $\langle 0.53,0.00\rangle $ | $\langle 0.53,-0.10\rangle $ | $\langle 0.60,-0.10\rangle $ |

*et al.*(2017b) are accounted for in Table 7, while the group weights of the criteria, calculated by applying Eq. (19), are shown in Table 8.

##### Table 7

${s_{j}}$ | ${k_{j}}$ | ${q_{j}}$ | ${w_{j}}$ | |

${C_{1}}$ | 1 | 1 | 0.19 | |

${C_{2}}$ | 1.2 | 0.80 | 1.25 | 0.23 |

${C_{3}}$ | 0.9 | 1.10 | 1.14 | 0.21 |

${C_{4}}$ | 0.7 | 1.30 | 0.87 | 0.16 |

${C_{5}}$ | 1.2 | 0.80 | 1.09 | 0.20 |

5.00 | 5.35 | 1.00 |

##### Table 8

${w_{j}^{1}}$ | ${w_{j}^{2}}$ | ${w_{j}^{3}}$ | ${w_{j}}$ | |

${C_{1}}$ | 0.19 | 0.17 | 0.19 | 0.18 |

${C_{2}}$ | 0.23 | 0.24 | 0.23 | 0.24 |

${C_{3}}$ | 0.21 | 0.22 | 0.21 | 0.21 |

${C_{4}}$ | 0.16 | 0.17 | 0.16 | 0.16 |

${C_{5}}$ | 0.20 | 0.21 | 0.20 | 0.21 |

1.00 |

##### Table 9

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

${A_{1}}$ | $\langle 1.00,-0.11\rangle $ | $\langle 1.00,-0.02\rangle $ | 0.94 | 0.99 | -0.05 | 3 |

${A_{2}}$ | $\langle 1.00,-0.20\rangle $ | $\langle 1.00,-0.02\rangle $ | 0.90 | 0.99 | -0.09 | 4 |

${A_{3}}$ | $\langle 0.42,-0.06\rangle $ | $\langle 0.30,-0.05\rangle $ | 0.68 | 0.63 | 0.05 | 2 |

${A_{4}}$ | $\langle 1.00,-0.06\rangle $ | $\langle 0.29,-0.04\rangle $ | 0.97 | 0.62 | 0.35 | 1 |

##### Table 10

${\Omega _{\max }}$ | ${\Omega _{\min }}$ | |||

${r^{+}}$ | ${r^{-}}$ | ${r^{+}}$ | ${r^{-}}$ | |

${r^{\ast }}$ | 1.00 | −0.20 | 0.29 | −0.02 |

##### Table 11

${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ | ${C_{5}}$ | ${d_{i}^{\max }}$ | $\text{Rank}$ | |

${A_{1}}$ | 0.08 | 0.16 | 0.13 | 0.29 | 0.10 | 0.08 | 4 |

${A_{2}}$ | 0.17 | 0.28 | 0.00 | 0.29 | 0.09 | 0.00 | 1 |

${A_{3}}$ | 0.13 | 0.33 | 0.38 | 0.05 | 0.06 | 0.05 | 3 |

${A_{4}}$ | 0.04 | 0.14 | 0.36 | 0.11 | 0.00 | 0.00 | 1 |

##### Table 12

$\hspace{2.5pt}\hspace{2.5pt}{U_{i}^{+}}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}{U_{i}^{-}}$ | $\hspace{2.5pt}\hspace{2.5pt}{u_{i}^{+}}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}{u_{i}^{-}}$ | ${u_{i}}$ | $\text{Rank}$ | |||

${A_{1}}$ | $\langle 1.00,-0.11\rangle $ | $\langle 1.00,-0.02\rangle $ | 0.94 | 0.99 | −0.05 | 3 |

${A_{2}}$ | $\langle 1.00,-0.20\rangle $ | $\langle 1.00,-0.02\rangle $ | 0.90 | 0.99 | −0.09 | 4 |

${A_{3}}$ | $\langle 0.42,-0.06\rangle $ | $\langle 0.30,-0.05\rangle $ | 0.68 | 0.63 | 0.05 | 2 |

${A_{4}}$ | $\langle 1.00,-0.06\rangle $ | $\langle 0.29,-0.04\rangle $ | 0.97 | 0.62 | 0.35 | 1 |

##### Table 13

$\mathit{RS}$ | $\mathit{RP}$ | $\mathit{FMP}$ | $\text{Final rank}$ | |

${A_{1}}$ | 3 | 4 | 3 | 3 |

${A_{2}}$ | 4 | 1 | 4 | 4 |

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

${A_{4}}$ | 1 | 1 | 1 | 1 |