INFORMATICA Informatica 1822-88440868-49520868-4952 Vilnius University INFO1131 10.15388/Informatica.2017.125 Research Article A Neutrosophic Extension of the MULTIMOORA Method StanujkicDragisadragisa.stanujkic@fmz.edu.rs1∗

D. Stanujkic is a full-time professor of Information Technology and Decision Sciences at the Faculty of Management in Zajecar, John Naisbitt University, Belgrade. He has received his MSc degree in Information Science and PhD in Organizational Sciences from the Faculty of Organizational Sciences, University of Belgrade. His current research is focused on decision-making theory, expert systems and intelligent decision support systems.

 ZavadskasEdmundas Kazimierasedmundas.zavadskas@vgtu.lt2

E.K. Zavadskas is a professor, head of the Department of Construction Technology and Management at Vilnius Gediminas Technical University, Vilnius, Lithuania, and senior researcher at Research Institute of Internet and Intelligent Technologies. He has a PhD in building structures (1973) and DrSc (1987) in building technology and management. He is a member of the Lithuanian and several foreign Academies of Sciences. He is doctor honoris causa at Poznan, Saint Petersburg, and Kiev universities. He is the editor in chief and a member of editorial boards of a number of research journals. He is an author and coauthor of more than 400 papers and a number of monographs. Research interests are: building technology and management, decision-making theory, automation in design and decision support systems.

 SmarandacheFlorentinfsmarandache@gmail.com3

F. Smarandache is a professor of mathematics at the University of New Mexico, USA. He has published many papers and books on neutrosophic set and logic and their applications and has presented to many international conferences. He got his MSc in Mathematics and Computer Science from the University of Craiova, Romania, PhD from the State University of Kishinev, and Post-Doctoral in Applied Mathematics from Okayama University of Sciences, Japan.

 BrauersWillem K.M.willem.brauers@uantwerpen.be4

W.K.M. Brauers, doctor honoris causa Vilnius Gediminas Technical University, was graduated as: PhD in economics (un. of Leuven), master of arts (in economics) of Columbia University (New York), master in economics, master in management and financial sciences, master in political and diplomatic sciences and bachelor in philosophy (all in the University of Leuven). He is professor ordinarius at the Faculty of Applied Economics of the University of Antwerp, honorary professor at the University of Leuven, the Belgian War College, the School of Military Administrators and the Antwerp Business School. His scientific publications consist of eighteen books and several hundreds of articles and reports in English, Dutch and French.

 KarabasevicDarjandarjankarabasevic@gmail.com5

D. Karabasevic is an assistant professor at the Faculty of Applied Management, Economics and Finance, Business Academy University. He obtained his degrees at all the levels of studies (BSc appl. in Economics, BSc in Economics, Academic Specialization in the Management of Business Information Systems and PhD in Management and Business) at the Faculty of Management in Zajecar, John Naisbitt University, Belgrade. His current research is focused on the human resource management, management and decision-making theory.

 Faculty of Management in Zajecar, John Naisbitt University, Belgrade, Serbia Research Institute of Smart Building Technologies, Civil Engineering Faculty, Vilnius Gediminas Technical University, Vilnius, Lithuania Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA Faculty of Applied Economics, University of Antwerp, Antwerp, Belgium Faculty of Applied Management, Economics and Finance, Business Academy University, Jevrejska 24, Belgrade, Serbia Corresponding author. 201711201728118119210201622017 © 2017 Vilnius University2017 Open access article under the CC BY license.

The aim of this manuscript is to propose a new extension of the MULTIMOORA method adapted for usage with a neutrosophic set. By using single valued neutrosophic sets, the MULTIMOORA method can be more efficient for solving complex problems whose solving requires assessment and prediction, i.e. those problems associated with inaccurate and unreliable data. The suitability of the proposed approach is presented through an example.

 neutrosophic set single valued neutrosophic set MULTIMOORA MCDM 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 ) ∈ [ 0 , 1 ] and 0 ⩽ T A ( x ) + F A ( x ) ⩽ 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 π 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 ) ∈ [ 0 , 1 ] and 0 ⩽ T A ( x ) + I A ( x ) + F A ( x ) ⩽ 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.

 The Single Valued Neutrosophic Set (<italic>See</italic> Smarandache, <xref ref-type="bibr" rid="j_info1131_ref_020">1999</xref>).

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: A = { x ⟨ T A ( x ) , I A ( x ) , F A ( x ) ⟩ | x ∈ X } , 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, T A , I A , F A : X → ] − 0 , 1 + [ and − 0 ⩽ T A ( x ) + I A ( x ) + F A ( x ) ⩽ 3 + .

(<italic>See</italic> Smarandache, <xref ref-type="bibr" rid="j_info1131_ref_020">1999</xref>; Wang <italic>et al.</italic>, <xref ref-type="bibr" rid="j_info1131_ref_023">2010</xref>).

Let X be the universe of discourse. The Single valued neutrosophic set (SVNS) A over X is an object having the following form: A = { x ⟨ T A ( x ) , I A ( x ) , F A ( x ) ⟩ | x ∈ X } , 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, T A , I A , F A : X → [ 0 , 1 ] and 0 ⩽ T A ( x ) + I A ( x ) + F A ( x ) ⩽ 3 .

(<italic>See</italic> Smarandache, <xref ref-type="bibr" rid="j_info1131_ref_020">1999</xref>).

For an SVNS A in X, the triple ⟨ t A , i A , f A ⟩ is called the single valued neutrosophic number (SVNN).

Let x 1 = ⟨ t 1 , i 1 , f 1 ⟩ and x 2 = ⟨ t 2 , i 2 , f 2 ⟩ be two SVNNs and λ > 00$]]>; then the basic operations are defined as follows: x 1 + x 2 = ⟨ t 1 + t 2 − t 1 t 2 , i 1 i 2 , f 1 f 2 ⟩ , x 1 · x 2 = ⟨ t 1 t 2 , i 1 + i 2 − i 1 i 2 , f 1 + f 2 − f 1 f 2 ⟩ , λ x 1 = ⟨ 1 − ( 1 − t 1 ) λ , i 1 λ , f 1 λ ⟩ , x 1 λ = ⟨ t 1 λ , 1 − ( 1 − i 1 ) λ , 1 − ( 1 − f 1 ) λ ⟩ .

(<italic>See</italic> Sahin, <xref ref-type="bibr" rid="j_info1131_ref_017">2014</xref>).

Let x = ⟨ t x , i x , f x ⟩ be an SVNN; then the score function s x of x can be as follows: s x = ( 1 + t x − 2 i x − f x ) / 2 , where s x ∈ [ − 1 , 1 ].

Let x 1 = ⟨ t 1 , i 1 , f 1 ⟩ and x 2 = ⟨ t 2 , i 2 , f 2 ⟩ be two SVNNs. Then the maximum distance between x 1 and x 2 is as follows: d max ( x 1 , x 2 ) = | t 1 − t 2 | , x 1 , x 2 ∈ Ω max , | f 1 − f 2 | , x 1 , x 2 ∈ Ω min .

(<italic>See</italic> Sahin, <xref ref-type="bibr" rid="j_info1131_ref_017">2014</xref>).

Let A j = ⟨ t j , i j , f j ⟩ be a collection of SVNSs and W = ( w 1 , w 2 , … , w n ) T be an associated weighting vector. Then the Single Valued Neutrosophic Weighted Average (SVNWA) operator of A j is as follows: SVNWA ( A 1 , A 2 , … , A n ) = ∑ j = 1 n w j A j = ( 1 − ∏ j = 1 n ( 1 − t j ) w j , ∏ j = 1 n ( i j ) w j , ∏ j = 1 n ( f j ) w j ) . where: w j is the element j of the weighting vector, w j ∈ [ 0 , 1 ] and ∑ j = 1 n w j = 1.

 (<italic>See</italic> Sahin, <xref ref-type="bibr" rid="j_info1131_ref_017">2014</xref>).

Let A j = ⟨ t j , i j , f j ⟩ be a collection of SVNSs and W = ( w 1 , w 2 , … , w n ) T be an associated weighting vector. Then the Single Valued Neutrosophic Weighted Geometric (SVNWG) operator of A j is as follows: SVNWG ( A 1 , A 2 , … , A n ) = ∏ j = 1 n ( A j ) w j = ( ∏ j = 1 n ( t j ) w j , 1 − ∏ j = 1 n ( 1 − i j ) w j , 1 − ∏ j = 1 n ( 1 − f j ) w j ) . where: w j is the element j of the weighting vector, w j ∈ [ 0 , 1 ] and ∑ j = 1 n w j = 1.

 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: y i = y i + − y i − , with: y i + = ∑ j ∈ Ω max w j r i j , and y i − = ∑ j ∈ Ω min w j r i j , r i j = x i j ∑ i = 1 n x i j 2 , 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 i j denotes the normalized performance of the alternative i with respect to the criterion j; x i j denotes the performance of the alternative i to the criterion j; Ω max and Ω min denote the sets of the benefit cost criteria, respectively; i = 1 , 2 , … , m; m is the number of the alternatives, j = 1 , 2 , … , 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: d i max = max j ( w j | r j ∗ − r i j | ) , where: d i max denotes the maximum distance of the alternative i to the reference point and r j ∗ denotes the coordinate j of the reference point as follows: r j ∗ = max i r i j , j ∈ Ω max , min i r i j , j ∈ Ω min . 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.

The full multiplicative form. In the FMF, the overall utility of the alternative i can be determined in the following manner: u i = a i b i , with: a i = ∏ j ∈ Ω max w j r i j , b i = ∏ j ∈ Ω min w j r i j , 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.

 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: Y i + = ( 1 − ∏ j ∈ Ω max ( 1 − t j ) w j , ∏ j ∈ Ω max ( i j ) w j , ∏ j ∈ Ω max ( f j ) w j ) , Y i − = ( 1 − ∏ j ∈ Ω min ( 1 − t j ) w j , ∏ j ∈ Ω min ( i j ) w j , ∏ j ∈ Ω min ( f j ) w j ) , 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.

Calculate y i + and y i − by using the Score Function, as follows: y i + = s ( Y i + ) , y i − = s ( Y i − ) .

Step 1.3.

Calculate the overall importance for each alternative, as follows: y i = y i + − y i − .

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 ∗ = { r 1 ∗ , r 2 ∗ , … , r n ∗ } is an SVNN, r j ∗ = ⟨ t j ∗ , i j ∗ , f j ∗ ⟩, whose values are determined as follows: r j ∗ = ⟨ max i t i j , min i i i j , min i f i j ⟩ , j ∈ Ω max , ⟨ min i t i j , min i i i j , max i f i j ⟩ , j ∈ Ω min , where: r j ∗ denotes the coordinate j of the reference point.

For the sake of simplicity, r j ∗ could be determined as follows: r j ∗ = ⟨ 1 , 0 , 0 ⟩ , j ∈ Ω max , ⟨ 0 , 0 , 1 ⟩ , j ∈ Ω min .

Step 2.2.

Determine the maximum distance from each alternative to all the coordinates of the reference point as follows: d i j max = d max ( r i j , r j ∗ ) w j , where d i j max denotes the maximum distance of the alternative i obtained based on the criterion j determined by Eq. (8).

Step 2.3.

Determine the maximum distance of each alternative, as follows: d i max = max j d i j max .

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: A i = ( ∏ j ∈ Ω max ( t j ) w j , 1 − ∏ j ∈ Ω max ( 1 − i j ) w j , 1 − ∏ j ∈ Ω max ( 1 − f j ) w j ) , B i = ( ∏ j ∈ Ω min ( t j ) w j , 1 − ∏ j ∈ Ω min ( 1 − i j ) w j , 1 − ∏ j ∈ Ω min ( 1 − f j ) w j ) , where: A i = ⟨ t A i , i A i , f A i ⟩ and B i = ⟨ t B i , i B i , f B i ⟩ are SVNNs.

Step 3.2.

Calculate a i and b i by using the Score Function as follows: a i = s ( A i ) , b i = s ( B i ) .

Step 3.3.

Determine the overall utility for each alternative as follows: u i = a i b i .

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.

 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:

A 1 , the CCDs based on the combined use of rod mills and ball mills;

A 2 , the CCDs based on the use of ball mills; and

A 3 , the CCDs based on the use of semi-autogenous mills.

 For the purpose of conducting an evaluation, the following criteria have been chosen:

C 1 , Grinding efficiency;

C 2 , Economic efficiency;

C 3 , Technological reliability;

C 4 , Capital investment costs; and

C 5 , Environmental impact.

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.

The ratings of the three generic CCDs obtained from an expert.

C 1 C 2 C 3 C 4 C 5
A 1 ⟨ 0.9 , 0.1 , 0.2 ⟩ ⟨ 0.7 , 0.2 , 0.3 ⟩ ⟨ 0.9 , 0.1 , 0.2 ⟩ ⟨ 0.9 , 0.1 , 0.2 ⟩ ⟨ 0.9 , 0.1 , 0.2 ⟩
A 2 ⟨ 0.8 , 0.1 , 0.3 ⟩ ⟨ 0.8 , 0.1 , 0.3 ⟩ ⟨ 0.8 , 0.1 , 0.3 ⟩ ⟨ 0.9 , 0.1 , 0.2 ⟩ ⟨ 0.8 , 0.1 , 0.3 ⟩
A 3 ⟨ 1.0 , 0.1 , 0.3 ⟩ ⟨ 0.9 , 0.1 , 0.2 ⟩ ⟨ 0.9 , 0.1 , 0.2 ⟩ ⟨ 0.7 , 0.2 , 0.5 ⟩ ⟨ 0.7 , 0.2 , 0.3 ⟩

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.

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 ⟨ 0.73 , 0.25 , 0.38 ⟩ ⟨ 0.55 , 0.45 , 0.57 ⟩ 0.425 0.045 0.380 2
A 2 ⟨ 0.65 , 0.22 , 0.46 ⟩ ⟨ 0.51 , 0.45 , 0.60 ⟩ 0.372 0.006 0.366 3
A 3 ⟨ 1.0 , 0.22 , 0.39 ⟩ ⟨ 0.34 , 0.57 , 0.73 ⟩ 0.583 −0.263 0.845 1

The reference point.

C 1 C 2 C 3 C 4 C 5
r j ∗ ⟨ 1.0 , 0.1 , 0.3 ⟩ ⟨ 0.9 , 0.2 , 0.3 ⟩ ⟨ 0.9 , 0.1 , 0.3 ⟩ ⟨ 0.7 , 0.1 , 0.2 ⟩ ⟨ 0.7 , 0.1 , 0.2 ⟩

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.

The ranking order of the alternatives obtained based on the RP approach.

I II III IV V VI VII VI
r 1 ∗ r 2 ∗ r 3 ∗ r 4 ∗ r 5 ∗ 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

The ranking order of the alternatives obtained based on the FMF.

A i B i a i b i u i Rank
A 1 ⟨ 0.89 , 0.25 , 0.15 ⟩ ⟨ 0.96 , 0.45 , 0.08 ⟩ 0.618 0.498 1.242 3
A 2 ⟨ 0.86 , 0.22 , 0.21 ⟩ ⟨ 0.95 , 0.45 , 0.09 ⟩ 0.605 0.481 1.258 2
A 3 ⟨ 0.96 , 0.22 , 0.16 ⟩ ⟨ 0.88 , 0.57 , 0.18 ⟩ 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.

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.

 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.

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