1 Introduction
The MULTIMOORA (Multi-Objective Optimization by a Ratio Analysis plus the Full Multiplicative Form) was proposed by Brauers and Zavadskas (2010).
The ordinary MULTIMOORA method has been proposed for usage with crisp numbers. In order to enable its use in solving a larger number of complex decision-making problems, several extensions have been proposed, out of which only the most prominent are mentioned: Brauers 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.
The MULTIMOORA method has been applied for the purpose of solving a wide range of problems.
As some of the most cited, the studies that consider different problems in economics (Brauers and Zavadskas, 2010, 2011; Brauers, 2010), personnel selection (Balezentis 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.
As some of the newest studies in which the MULTIMOORA method is used for solving various decision-making problems, the following ones can be mentioned: material selection (Hafezalkotob and Hafezalkotob, 2016; Hafezalkotob et al., 2016) and the CNC machine tool evaluation (Sahu et al., 2016).
A significant approach in solving complex decision-making problems was formed by adapting the multiple criteria decision-making methods for the purpose of using fuzzy numbers, proposed by Zadeh in the fuzzy set theory (Zadeh, 1965).
Based on the fuzzy set theory, some extensions are also proposed, such as: interval-valued fuzzy sets (Turksen, 1986), intuitionistic fuzzy sets (Atanassov, 1986) and interval-valued intuitionistic fuzzy sets (Atanassov and Gargov, 1989).
In addition to the membership function proposed in fuzzy sets, Atanassov (1986) introduced the non-membership function that expresses non-membership to a set, thus having created the basis for solving a much larger number of decision-making problems.
The intuitionistic fuzzy set is composed of membership (the so-called truth-membership) ${T_{A}}(x)$ and non-membership (the so-called falsity-membership) ${F_{A}}(x)$, which satisfies the conditions ${T_{A}}(x),{F_{A}}(x)\in [0,1]$ and $0\leqslant {T_{A}}(x)+{F_{A}}(x)\leqslant 1$. Therefore, intuitionistic fuzzy sets are capable of operating with incomplete pieces of information, but do not include intermediate and inconsistent information (Li et al., 2016).
In intuitionistic fuzzy sets, indeterminacy ${\pi _{A}}(x)$ is $1-{T_{A}}(x)-{F_{A}}(x)$ by default. Smarandache (1998, 1999) further extended intuitionistic fuzzy sets by proposing Neutrosophic, also introducing independent indeterminacy-membership.
Such a proposed neutrosophic set is composed of three independent membership functions named the truth-membership ${T_{A}}(x)$, the falsity-membership ${F_{A}}(x)$ and the indeterminacy-membership ${I_{A}}(x)$ functions.
Smarandache (1999) and Wang 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).
Compared with the fuzzy set and its extensions, the single valued neutrosophic set can be identified as more flexible, for which reason an extension of the MULTIMOORA method adapted for the purpose of using the single valued neutrosophic set is proposed in this approach.
Therefore, the rest of this paper is organized as follows: in Section 2, some basic definitions related to the single valued neutrosophic set are given. In Section 3, the ordinary MULTIMOORA method is presented, whereas in Section 4, the Single Valued Neutrosophic Extension of the MULTIMOORA method is proposed. In Section 5, an example is considered with the aim to explain in detail the proposed methodology. The conclusions are presented in the final section.
2 The Single Valued Neutrosophic Set
Definition 1 (See Smarandache, 1999).
Let 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:
where ${T_{A}}(x)$, ${I_{A}}(x)$ and ${F_{A}}(x)$ are the truth-membership function, the indeterminacy-membership function and the falsity-membership function, respectively,
(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).
Let X be the universe of discourse. The Single valued neutrosophic set (SVNS) A over X is an object having the following form:
where ${T_{A}}(x)$, ${I_{A}}(x)$ and ${F_{A}}(x)$ are the truth-membership function, the intermediacy-membership function and the falsity-membership function, respectively,
(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).
For an SVNS A in X, the triple $\langle {t_{A}},{i_{A}},{f_{A}}\rangle $ is called the single valued neutrosophic number (SVNN).
Definition 4.
Let ${x_{1}}=\langle {t_{1}},{i_{1}},{f_{1}}\rangle $ and ${x_{2}}=\langle {t_{2}},{i_{2}},{f_{2}}\rangle $ be two SVNNs and $\lambda >0$; then the basic operations are defined as follows:
(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.
Let ${x_{1}}=\langle {t_{1}},{i_{1}},{f_{1}}\rangle $ and ${x_{2}}=\langle {t_{2}},{i_{2}},{f_{2}}\rangle $ be two SVNNs. Then the maximum distance between ${x_{1}}$ and ${x_{2}}$ is as follows:
Definition 7 (See Sahin, 2014).
Let ${A_{j}}=\langle {t_{j}},{i_{j}},{f_{j}}\rangle $ be a collection of SVNSs and $W={({w_{1}},{w_{2}},\dots ,{w_{n}})^{T}}$ be an associated weighting vector. Then the Single Valued Neutrosophic Weighted Average (SVNWA) operator of ${A_{j}}$ is as follows:
where: ${w_{j}}$ is the element j of the weighting vector, ${w_{j}}\in [0,1]$ and ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$.
(9)
\[\begin{array}{l}\displaystyle \mathit{SVNWA}({A_{1}},{A_{2}},\dots ,{A_{n}})\\ {} \displaystyle \hspace{1em}={\sum \limits_{j=1}^{n}}{w_{j}}{A_{j}}=\Bigg(1-{\prod \limits_{j=1}^{n}}{(1-{t_{j}})^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{({i_{j}})^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{({f_{j}})^{{w_{j}}}}\Bigg).\end{array}\]Definition 8 (See Sahin, 2014).
Let ${A_{j}}=\langle {t_{j}},{i_{j}},{f_{j}}\rangle $ be a collection of SVNSs and $W={({w_{1}},{w_{2}},\dots ,{w_{n}})^{T}}$ be an associated weighting vector. Then the Single Valued Neutrosophic Weighted Geometric (SVNWG) operator of ${A_{j}}$ is as follows:
where: ${w_{j}}$ is the element j of the weighting vector, ${w_{j}}\in [0,1]$ and ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$.
(10)
\[\begin{array}{l}\displaystyle \mathit{SVNWG}({A_{1}},{A_{2}},\dots ,{A_{n}})\\ {} \displaystyle \hspace{1em}={\prod \limits_{j=1}^{n}}{({A_{j}})^{{w_{j}}}}=\Bigg({\prod \limits_{j=1}^{n}}{({t_{j}})^{{w_{j}}}},1-{\prod \limits_{j=1}^{n}}{(1-{i_{j}})^{{w_{j}}}},1-{\prod \limits_{j=1}^{n}}{(1-{f_{j}})^{{w_{j}}}}\Bigg).\end{array}\]3 The MULTIMOORA Method
The MULTIMOORA method consists of three approaches named as follows: the Ratio System (RS) Approach, the Reference Point (RP) Approach and the Full Multiplicative Form (FMF).
The considered alternatives are ranked based on all three approaches and the final ranking order and the final decision is made based on the theory of dominance. In other words, the alternative with the highest number of appearances in the first positions on all ranking lists is the best-ranked alternative.
The ratio system approach. In this approach, the overall importance of the alternative i can be calculated as follows:
with:
where: ${y_{i}}$ denotes the overall importance of the alternative 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.
In this approach, the compared alternatives are ranked based on ${y_{i}}$ in descending order and the alternative with the highest value of ${y_{i}}$ is considered to be the best-ranked.
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:
In this approach, the compared alternatives are ranked based on ${d_{i}^{\max }}$ in ascending order and the alternative with the lowest value of ${d_{i}^{\max }}$ is considered the best-ranked.
(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:
where: ${u_{i}}$ denotes the overall utility of the alternative 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.
As in the RSA, the compared alternatives are ranked based on their ${u_{i}}$ in descending order and the alternative with the highest value of ${u_{i}}$ is considered the best-ranked.
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
For an MCDM problem involving 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:
where: ${Y_{i}^{+}}$ and ${Y_{i}^{-}}$ denote the importance of the alternative 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:
where: ${r_{j}^{\ast }}$ denotes the coordinate j of the reference point.
(25)
\[ {r_{j}^{\ast }}=\left\{\begin{array}{l@{\hskip4.0pt}l}\big\langle \underset{i}{\max }{t_{ij}},\underset{i}{\min }{i_{ij}},\underset{i}{\min }{f_{ij}}\big\rangle ,\hspace{1em}& j\in {\Omega _{\max }},\\ {} \big\langle \underset{i}{\min }{t_{ij}},\underset{i}{\min }{i_{ij}},\underset{i}{\max }{f_{ij}}\big\rangle ,\hspace{1em}& j\in {\Omega _{\min }},\end{array}\right.\]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:
where: ${A_{i}}=\langle {t_{Ai}},{i_{Ai}},{f_{Ai}}\rangle $ and ${B_{i}}=\langle {t_{Bi}},{i_{Bi}},{f_{Bi}}\rangle $ are SVNNs.
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
In order to demonstrate the applicability and efficiency of the proposed approach, an example has been adopted from Stanujkic et al. (2015). In order to briefly demonstrate the advantages of the proposed methodology, this example has been slightly modified.
Suppose that a mining and smelting company has to build a new flotation plant, for which reason an expert has been engaged to evaluate the three Comminution Circuit Designs (CCDs) listed below:
For the purpose of conducting an evaluation, the following criteria have been chosen:
The ratings obtained from the expert are shown in Table 1.
The ranking based on the RS approach. 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.
Table 1
The ratings of the three generic CCDs obtained from an expert.
| ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ | ${C_{5}}$ | |
| ${A_{1}}$ | $\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
The ranking orders of the alternatives obtained on the basis of the RS approach.
| ${Y_{i}^{+}}$ | ${Y_{i}^{-}}$ | ${y_{i}^{+}}$ | ${y_{i}^{-}}$ | ${y_{i}}$ | 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
The reference point.
| ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ | ${C_{5}}$ | |
| ${r_{j}^{\ast }}$ | $\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 $ |
The maximum distances from each alternative to the coordinate 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
The ranking order of the alternatives obtained based on the RP approach.
| 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
The ranking order of the alternatives obtained based on the FMF.
| ${A_{i}}$ | ${B_{i}}$ | ${a_{i}}$ | ${b_{i}}$ | ${u_{i}}$ | 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 |
The final ranking order of the alternatives which summarizes the three different ranks provided by the respective parts of the MULTIMOORA method is shown in Table 6.
Table 6
The final ranking order of the alternatives according to the MULTIMOORA method.
| RS | RP | FMF | Rank | |
| ${A_{1}}$ | 2 | 1 | 3 | 3 |
| ${A_{2}}$ | 3 | 2 | 2 | 2 |
| ${A_{3}}$ | 1 | 3 | 1 | 1 |
As it can be seen from Table 6, all three approaches, integrated in the MULTIMOORA, have resulted in different ranking orders, for which reason the final ranking order is determined based on the dominance theory.
6 Conclusion
The MULTIMOORA method has been proven in solving different decision-making problems. In order to enable its application in the solving of a larger number of complex decision-making problems, numerous extensions have been proposed for the MULTIMOORA method.
Compared to crisp, fuzzy, interval-valued and intuitionistic fuzzy numbers, the neutrosophic set provides significantly greater flexibility, which can be conducive to solving decision-making problems associated with uncertainty, estimations and predictions.
Therefore, an extension of the MULTIMOORA method enabling the use of single valued neutrosophic numbers is proposed in this paper.
The usability and efficiency of the proposed extension is presented in the example of the comminution circuit design selection.
Finally, it should be noted that the proposed extension of the MULTIMOORA method can be used for solving a much larger number of complex decision-making problems. A number of real-world decision making problems which have to be solved is based on the data acquired from respondents can be identified as one of the areas where the proposed extension of the MULTIMOORA method can reach its advantages.
