The Cosine Measure of Single Valued Neutrosophic Multisets for Multiple Attribute Decision-Makin

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Article

The Cosine Measure of Single Valued Neutrosophic Multisets for Multiple Attribute Decision-Making Changxing Fan1*, En Fan 1, and Jun Ye2 1

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Department of Computer Science, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang 312000, China; fcxjszj@usx.edu.cn; efan@usx.edu.cn (E.F.) Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng

West Road, Shaoxing, Zhejiang 312000, P.R. China; yehjun@aliyun.com * Correspondence: fcxjszj@usx.edu.cn; Tel.: +86-575-8820-2669

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Received: ; Accepted:; Published:

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Abstract: Based on the multiplicity evaluation in some real situations, this paper firstly

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Keywords: single valued neutrosophic set (SVNS); neutrosophic multiset (NM); single

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introduces a single valued neutrosophic multiset (SVNM) as a subclass of neutrosophic multiset (NM) to express the multiplicity information and the operational relations of SVNMs. Then, a cosine measure between SVNMs and weighted cosine measure between SVNMs are presented to measure the cosine degree between SVNMs, and their properties are investigated. Based on the weighted cosine measure of SVNMs, a multiple attribute decision-making method under a SVNM environment is proposed, in which the evaluated values of alternatives are taken the form of SVNMs. The ranking order of all alternatives and the best one can be determined by the weighted cosine measure between every alternative and the ideal alternative. Finally, an actual application on the selecting problem illustrates the effectiveness and application of the proposed method.

valued neutrosophic multiset (SVNM); cosine measure; multiple attribute decision-making.

1. Introduction In 1965, Zadeh [1] proposed the theory of fuzzy sets (FS), in which every fuzzy element is expressed by the membership degree đ?‘‡(đ?‘Ľ) belonging to the scope of [0, 1]. While the fuzzy membership degree of đ?‘‡(đ?‘Ľ) is difficult to be determined, or can't be expressed by an exact real number, the practicability of FS is limited. In order to avoid the above situation, Turksen [2] extended a single value membership to an interval-valued membership. Generally, when the membership degree đ?‘‡(đ?‘Ľ) is determined, the non-membership degree can be calculated by 1 đ?‘‡(đ?‘Ľ). Considering the role of the non-membership degree, Atanassov [3] put forward the intuitionistic fuzzy sets (IFS) and introduced the related theory of IFS. Then IFS has been widely used for solving the decision-making problems. Although the FS theory and IFS theory have been constantly extended and completed, they are not applicable to all the fuzzy problems. In 1998, Smarandache [4] added the uncertain degree to the IFS and put forward the theory of neutrosophic set (NS), which is a general form of the FS and IFS. NS is composed of Symmetry 2018, 4, x; doi:

www.mdpi.com/journal/symmetry

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the neutrosophic components of truth, indeterminacy, and falsity denoted by T, I, F,

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2. Some Concepts of SVNS and NM

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Definition 1. [5] Let X be a space of points (objects), with a generic element x in X. A SVNS R in X

respectively. Since then, many forms of neutrosophic set were proposed as extension of the neutrosophic set. Wang and Smarandache [5-6] introduced a single valued neutrosophic set (SVNS) and an interval neutrosophic set (INS). Smarandache [7] and Smarandache and Ye [8] presented n-value and refined-single valued neutrosophic sets (R-SVNSs). Fan and Ye [9] presented a refined-interval neutrosophic set (R-INS). Ye [10] presented a dynamic single valued neutrosophic multiset (DSVM), and so on. Now, more researches have been done on the NS theory by experts and scholars. Ye [11-12] proposed the correlation coefficient and the weighted coefficient correlation of SVNS and proved that cosine similarity is a special case of the SVNS correlation coefficient. Broumi and Smarandache [13] proposed three vector similarity methods to simplify the similarity of SVNS, including Jaccard similarity, Dice similarity, and cosine similarity. Majumdar and Samanta [14] gave the similarity formula of SVNSs. Broumi and Smarandache [15] gave the correlation coefficient of INSs. Based on the Hamming and Euclidean distances, Ye [16] defined similarity of INSs. For the operation rules of NSs, Smarandache, Ye and Chi [4, 16, 17] gave different operation rules, respectively, they all have certain rationality and applicability. Recently, Smarandache [18] introduced the neutrosophic multiset and the neutrosophic multiset algebraic structures, in which one or more elements are repeated times keeping the same or different neutrosophic components. Its concept is different from the concept of single valued neutrosophic multiset in [10, 19]. Till now, there are few studies and applications of neutrosophic multisets (NM) in science and engineering fields, so we introduce a single valued neutrosophic multiset (SVNM) as a subclass of neutrosophic multiset (NM) to express the multiplicity information and propose a decision-making methods based on the weighted cosine measures of SVNMs, and then provide a decision-making example to show its application under SVNM environments. The rest sections of this article are organized as follows. Section 2 describes some basic concepts of SVNS, NM and the cosine measure of SVNSs. Section 3 presents a SVNM and its basic operational relations. Section 4 proposes a cosine measure between SVNMs and a weighted cosine measure between SVNMs and investigates their properties. Section 5 establishes a multiple attribute decision-making method using the weighted cosine measure of SVNMs under SVNM environment. Section 6 presents an actual example to demonstrate the application of the proposed methods under SVNM environment. Section 7 gives a conclusion and further research.

can be characterized by a truth-membership function đ?‘‡đ?‘… (đ?‘Ľ) , an indeterminacy-membership function đ??źđ?‘… (đ?‘Ľ), and a falsity-membership function đ??šđ?‘… (đ?‘Ľ), where đ?‘‡đ?‘… (đ?‘Ľ), đ??źđ?‘… (đ?‘Ľ), đ??šđ?‘… (đ?‘Ľ) âˆˆ [0,1] for each point x in X. Then, a SVNS R can be expressed by the following form: đ?‘… = {âŒŠđ?‘Ľ, đ?‘‡đ?‘… (đ?‘Ľ), đ??źđ?‘… (đ?‘Ľ), đ??šđ?‘… (đ?‘Ľ)âŒŞ|đ?‘Ľ âˆˆ đ?‘‹}. Thus, the SVNS R satisfies the condition 0 â‰¤ đ?‘‡đ?‘… (đ?‘Ľ) + đ??źđ?‘… (đ?‘Ľ) + đ??šđ?‘… (đ?‘Ľ) â‰¤ 3. For two SVNSs M and N, the relations of them are defined as follows [5]: (1) M âŠ† N if and only if đ?‘‡đ?‘€ (đ?‘Ľ) â‰¤ đ?‘‡đ?‘ (đ?‘Ľ), đ??źđ?‘€ (đ?‘Ľ) â‰Ľ đ??źđ?‘ (đ?‘Ľ), đ??šđ?‘€ (đ?‘Ľ) â‰Ľ đ??šđ?‘ (đ?‘Ľ) for any x in X; (2) M = N if and only if M âŠ† N and N âŠ† M; (3) đ?‘€đ?‘? = {âŒŠđ?‘Ľ, đ??šđ?‘€ (đ?‘Ľ), 1 âˆ’ đ??źđ?‘€ (đ?‘Ľ), đ?‘‡đ?‘€ (đ?‘Ľ)âŒŞ|đ?‘Ľ âˆˆ đ?‘‹}.

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In order to writing convenience, an element (called single-valued neutrosophic number (SVNN) in the SVNS R can be denoted by đ?‘… = âŒŠđ?‘‡đ?‘… (đ?‘Ľ), đ??źđ?‘… (đ?‘Ľ), đ??šđ?‘… (đ?‘Ľ)âŒŞ for any x in X. For two SVNNs M and N, the operational relations of them can be defined as follows [5]: (1) đ?‘€ âˆŞ đ?‘ = <max(đ?‘‡đ?‘€ (đ?‘Ľ), đ?‘‡đ?‘ (đ?‘Ľ)), min(đ??źđ?‘€ (đ?‘Ľ), đ??źđ?‘ (đ?‘Ľ)), min(đ??šđ?‘€ (đ?‘Ľ), đ??šđ?‘ (đ?‘Ľ)) > for any x in X; (2) M âˆŠ N = <min(đ?‘‡đ?‘€ (đ?‘Ľ), đ?‘‡đ?‘ (đ?‘Ľ)), max(đ??źđ?‘€ (đ?‘Ľ), đ??źđ?‘ (đ?‘Ľ)), max(đ??šđ?‘€ (đ?‘Ľ), đ??šđ?‘ (đ?‘Ľ)) > for any x in X. For two SVNNs M and N, the operational rules of them can be defined as follows [5]: M+N =< đ?‘‡đ?‘€ (đ?‘Ľ) + đ?‘‡đ?‘ (đ?‘Ľ) âˆ’ đ?‘‡đ?‘€ (đ?‘Ľ)đ?‘‡đ?‘ (đ?‘Ľ), đ??źđ?‘€ (đ?‘Ľ)đ??źđ?‘ (đ?‘Ľ), đ??šđ?‘€ (đ?‘Ľ)đ??šđ?‘ (đ?‘Ľ) > for any x in X;

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MĂ—N = < đ?‘‡đ?‘€ (đ?‘Ľ)đ?‘‡đ?‘ (đ?‘Ľ), đ??źđ?‘€ (đ?‘Ľ) + đ??źđ?‘ (đ?‘Ľ) âˆ’ đ??źđ?‘€ (đ?‘Ľ)đ??źđ?‘ (đ?‘Ľ), đ??šđ?‘€ (đ?‘Ľ) + đ??šđ?‘ (đ?‘Ľ) âˆ’ đ??šđ?‘€ (đ?‘Ľ)đ??šđ?‘ (đ?‘Ľ) > for any x in X; đ?œ‘

đ?œ‘

đ?œ‘đ?‘€ = < 1â€“ (1â€“ đ?‘‡đ?‘€ (đ?‘Ľ)) , (đ??źđ?‘€ (đ?‘Ľ)) , (đ??šđ?‘€ (đ?‘Ľ)) > for đ?œ‘ > 0 and any x in X;

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đ?œ‘

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đ?œ‘

đ?‘€đ?œ‘ = < (đ?‘‡đ?‘€ (đ?‘Ľ)) ďźŒ (1â€“ đ??źđ?‘€ (đ?‘Ľ)) ďźŒ (1â€“ đ??šđ?‘€ (đ?‘Ľ)) > , for đ?œ‘ > 0 and any x in X.

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Definition 2 [20]. Let X= {đ?‘Ľ1 , đ?‘Ľ2 , â€Ś ďźŒđ?‘Ľđ?‘› } be a space of points (objects), L and M be two SVNSs. The cosine measure between L and M is defined as follows: 1

đ?œ‹

đ?‘›

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đ?œŒ(đ??ż, đ?‘€) = âˆ‘đ?‘›đ?‘–=1 đ?‘?đ?‘œđ?‘ { (| đ?‘‡đ??ż (đ?‘Ľđ?‘– ) âˆ’ đ?‘‡đ?‘€ (đ?‘Ľđ?‘– )| + |đ??źđ??ż (đ?‘Ľđ?‘– ) âˆ’ đ??źđ?‘€ (đ?‘Ľđ?‘– )| + |đ??šđ??ż (đ?‘Ľđ?‘– ) âˆ’ đ??šđ?‘€ (đ?‘Ľđ?‘– )|)}

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Obviously, the cosine measure between L and M satisfies the following properties [20]:

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Definition 3 [18]. Let X be a space of points (objects), and a neutrosophic multiset is repeated by one or

ď Ş0 â‰¤ đ?œŒ(đ??ż, đ?‘€) â‰¤ 1; ď Ť đ?œŒ(đ??ż, đ?‘€) = 1 if and only if đ??ż = đ?‘€; ď Ź đ?œŒ(đ??ż, đ?‘€) = đ?œŒ(đ?‘€, đ??ż).

more elements with the same or different neutrosophic components. For

example,

neutrosophic

set

M

=

{(đ?‘š1 , âŒŠ0.7,0.2,0.1âŒŞ), (đ?‘š2 , âŒŠ0.6,0.4,0.1âŒŞ), (đ?‘š3 , âŒŠ0.8,0.3,0.2âŒŞ)}

rather

than

a

neutrosophic

multiset;

{ (đ?‘˜1 , âŒŠ0.7,0.2,0.1âŒŞ), (đ?‘˜1 , âŒŠ0.7,0.2,0.1âŒŞ), (đ?‘˜1 , âŒŠ0.7,0.2,0.1âŒŞ), (đ?‘˜2 , âŒŠ0.6,0.4,0.1âŒŞ) }

is

while a

is

a K=

neutrosophic

multiset, where the element đ?‘˜1 is repeated. Then we can say that the element đ?‘˜1 has neutrosophic multiplicity 3 with the same neutrosophic components. While L= { (đ?‘™1 , âŒŠ0.7,0.2,0.1âŒŞ), (đ?‘™1 , âŒŠ0.6,0.3,0.1âŒŞ), (đ?‘™1 , âŒŠ0.8,0.1,0.1âŒŞ), (đ?‘™2 , âŒŠ0.6,0.4,0.1âŒŞ) } is also a neutrosophic multiset since the element đ?‘™1 is repeated, and then we can say that the element đ?‘™1 has neutrosophic multiplicity 3 with different neutrosophic components. If the element đ?‘™1 is repeated times with the same neutrosophic comonents, we say đ?‘™1 has multiplicity. If the element đ?‘™1 is repeated times with different neutrosophic comonents, we say đ?‘™1 has the neutrosophic multiplicity (đ?‘›đ?‘š).the nm function can be defined as follow: đ?‘›đ?‘š: Xďƒ N= {1, 2, 3,â€Ś, âˆž} for any đ?‘&#x; âˆˆ đ?‘… đ?‘›đ?‘š(đ?‘&#x;) = {( đ?‘?1 , âŒŠđ?‘‡1 , đ??ź1 , đ??š1 âŒŞ), ( đ?‘?2 , âŒŠđ?‘‡2 , đ??ź2 , đ??š2 âŒŞ), â€Ś , ( đ?‘?đ?‘– , âŒŠđ?‘‡đ?‘– , đ??źđ?‘– , đ??šđ?‘– âŒŞ), â€Ś}, which means that r is repeated by đ?‘?1 times with the neutrosophic components âŒŠđ?‘‡1 , đ??ź1 , đ??š1 âŒŞ; r is repeated by đ?‘?2 times with the neutrosophic components âŒŠđ?‘‡2 , đ??ź2 , đ??š2 âŒŞ; â€Ś; r is repeated by đ?‘?đ?‘– times with the neutrosophic components âŒŠđ?‘‡đ?‘– , đ??źđ?‘– , đ??šđ?‘– âŒŞ ; and so on. đ?‘?1 , đ?‘?2 , â€Ś , đ?‘?đ?‘– , â€Ś âˆˆ đ?‘ , and âŒŠđ?‘‡đ?‘— , đ??źđ?‘— , đ??šđ?‘— âŒŞ â‰ âŒŠđ?‘‡đ?‘˜ , đ??źđ?‘˜ , đ??šđ?‘˜ âŒŞ, for đ?‘— â‰ đ?‘˜ and đ?‘—, đ?‘˜ âˆˆ đ?‘ . Then a neutrosophic multiset R can be written as: (R, đ?‘›đ?‘š(đ?‘&#x;)) or {(r, đ?‘›đ?‘š(đ?‘&#x;) ďźŒfor đ?‘&#x; âˆˆ đ?‘…)}

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Now, with respect to the previous neutrosophic multisets K, L, we compute the neutrosophic multiplicity function:

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đ?‘›đ?‘šđ??ž : đ??ž â†’ đ?‘ ;

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đ?‘›đ?‘šđ??ž (đ?‘˜1 ) = {(3, âŒŠ0.7,0.2,0.1âŒŞ)}; đ?‘›đ?‘šđ??ž (đ?‘˜2 ) = {(1, âŒŠ0.6,0.4,0.1âŒŞ)}; đ?‘›đ?‘šđ??ż : đ??ż â†’ đ?‘ ; đ?‘›đ?‘šđ??ż (đ?‘™1 ) = {(1, âŒŠ0.7,0.2,0.1âŒŞ), (1, âŒŠ0.6,0.3,0.1âŒŞ), (1, âŒŠ0.8,0.1,0.1âŒŞ)}; đ?‘›đ?‘šđ??ż (đ?‘™2 ) = {(1, âŒŠ0.6,0.4,0.1âŒŞ)}.

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3. Single Valued Neutrosophic Multiset

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Definition 4. Let X be a space of points (objects) with a generic element x in X and N= {1, 2, 3 ,â€Ś, âˆž}.

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R={đ?‘Ľ, ((đ?‘?đ?‘…1 , âŒŠđ?‘‡đ?‘…1 (đ?‘Ľ), đ??źđ?‘…1 (đ?‘Ľ), đ??šđ?‘…1 (đ?‘Ľ)âŒŞ), (đ?‘?đ?‘…2 , âŒŠđ?‘‡đ?‘…2 (đ?‘Ľ), đ??źđ?‘…2 (đ?‘Ľ), đ??šđ?‘…2 (đ?‘Ľ)âŒŞ), â€Ś , (đ?‘?đ?‘…đ?‘— , âŒŠđ?‘‡đ?‘…đ?‘— (đ?‘Ľ), đ??źđ?‘…đ?‘— (đ?‘Ľ), đ??šđ?‘…đ?‘— (đ?‘Ľ)âŒŞ)) |đ?‘Ľ âˆˆ

A SVNM R in X can be defined as follows:

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đ?‘‹},

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where

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đ?‘‡đ?‘…đ?‘˜ (đ?‘Ľ), đ??źđ?‘…đ?‘˜ (đ?‘Ľ), đ??šđ?‘…đ?‘˜ (đ?‘Ľ)

express

the

truth-membership

function,

the

indeterminacy-membership function and the falsity-membership function, respectively. đ?‘‡đ?‘…1 (đ?‘Ľ), đ?‘‡đ?‘…2 (đ?‘Ľ), â€Ś , đ?‘‡đ?‘…đ?‘˜ (đ?‘Ľ) âˆˆ [0,1] , đ??źđ?‘…1 (đ?‘Ľ), đ??źđ?‘…2 (đ?‘Ľ), â€Ś , đ??źđ?‘…đ?‘˜ (đ?‘Ľ) âˆˆ [0,1], đ??šđ?‘…1 (đ?‘Ľ), đ??šđ?‘…2 (đ?‘Ľ), â€Ś , đ??šđ?‘…đ?‘˜ (đ?‘Ľ) âˆˆ [0,1] and 0 â‰¤ đ?‘‡đ?‘…đ?‘˜ (đ?‘Ľ) + đ??źđ?‘…đ?‘˜ (đ?‘Ľ) + đ??šđ?‘…đ?‘˜ (đ?‘Ľ) â‰¤ 3 , for k=1, 2, â€Śj, đ?‘— âˆˆ đ?‘ , đ?‘?đ?‘…1 , đ?‘?đ?‘…2 , â€Ś , đ?‘?đ?‘…đ?‘— âˆˆ đ?‘ and đ?‘?đ?‘…1 + đ?‘?đ?‘…2 + â‹Ż + đ?‘?đ?‘…đ?‘— â‰Ľ 2. For convenience, a SVNM R can be denoted by the following simplified form: R={đ?‘Ľ, (đ?‘?đ?‘…đ?‘˜ , âŒŠđ?‘‡đ?‘…đ?‘˜ (đ?‘Ľ), đ??źđ?‘…đ?‘˜ (đ?‘Ľ), đ??šđ?‘…đ?‘˜ (đ?‘Ľ)âŒŞ)|đ?‘Ľ âˆˆ đ?‘‹}, for k=1, 2,â€Ś,j. For example, with a universal set X= {đ?‘Ľ1 , đ?‘Ľ2 }, a SVNM R is given as: R={(đ?‘Ľ1 , (2, âŒŠ0.6,0.2,0.1âŒŞ), (1, âŒŠ0.8,0.2,0.2âŒŞ)), (đ?‘Ľ2 , (1, âŒŠ0.7,0.3,0.1âŒŞ), (2, âŒŠ0.7,0.2,0.3âŒŞ))}. Then đ?‘›đ?‘šđ?‘… (đ?‘Ľ1 ) = {(2, âŒŠ0.6,0.2,0.1âŒŞ), (1, âŒŠ0.8,0.2,0.2âŒŞ)}; đ?‘›đ?‘šđ?‘… (đ?‘Ľ2 ) = {(1, âŒŠ0.7,0.3,0.1âŒŞ), (2, âŒŠ0.7,0.2,0.3âŒŞ)}.

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Definition 5. Let X be a space of points (objects) with a generic element x in X, M and L be two

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(( pr1, ď€ź Tr1 ( x), I r1 ( x), Fr1 ( x) ď€ž),( pr 2 , ď€ź Tr 2 ( x), I r 2 ( x), Fr 2 ( x) ď€ž),ď‚ź,( prj , ď€ź Trj ( x), I rj ( x), Frj ( x) ď€ž)) to express a basic element in a

SVNMs, đ?‘€ = {đ?‘Ľ, (đ?‘?đ?‘€đ?‘˜ , âŒŠđ?‘‡đ?‘€đ?‘˜ (đ?‘Ľ), đ??źđ?‘€đ?‘˜ (đ?‘Ľ), đ??šđ?‘€đ?‘˜ (đ?‘Ľ)âŒŞ)|đ?‘Ľ âˆˆ đ?‘‹}, for k=1, 2, â€Śj, đ??ż = {đ?‘Ľ, (đ?‘?đ??żđ?‘˜ , âŒŠđ?‘‡đ??żđ?‘˜ (đ?‘Ľ), đ??źđ??żđ?‘˜ (đ?‘Ľ), đ??šđ??żđ?‘˜ (đ?‘Ľ)âŒŞ)|đ?‘Ľ âˆˆ đ?‘‹}, for k=1, 2, â€Śj, then the relations of them are given as follows: ď Ş đ?‘€ = đ??ż, if and only if đ?‘?đ?‘€đ?‘˜ = đ?‘?đ??żđ?‘˜ , đ?‘‡đ?‘€đ?‘˜ (đ?‘Ľ) = đ?‘‡đ??żđ?‘˜ (đ?‘Ľ), đ??źđ?‘€đ?‘˜ (đ?‘Ľ) = đ??źđ??żđ?‘˜ (đ?‘Ľ), đ??šđ?‘€đ?‘˜ (đ?‘Ľ) = đ??šđ??żđ?‘˜ (đ?‘Ľ), for k=1, 2,â€Ś, j; ď Ť đ?‘€ âˆŞ đ??ż={đ?‘Ľ, ((đ?‘?đ?‘€đ?‘˜ âˆ¨ đ?‘?đ??żđ?‘˜ ), âŒŠđ?‘‡đ?‘€đ?‘˜ (đ?‘Ľ) âˆ¨ đ?‘‡đ??żđ?‘˜ (đ?‘Ľ), đ??źđ?‘€đ?‘˜ (đ?‘Ľ) âˆ§ đ??źđ??żđ?‘˜ (đ?‘Ľ), đ??šđ?‘€đ?‘˜ (đ?‘Ľ) âˆ§ đ??šđ??żđ?‘˜ (đ?‘Ľ)âŒŞ)|đ?‘Ľ âˆˆ đ?‘‹} đ?‘“đ?‘œđ?‘&#x; đ?‘˜ = 1, 2, â€Ś , đ?‘—; ď Ź

đ?‘€ âˆŠ đ??ż = {đ?‘Ľ, ((đ?‘?đ?‘€đ?‘˜ âˆ§ đ?‘?đ??żđ?‘˜ ), âŒŠđ?‘‡đ?‘€đ?‘˜ (đ?‘Ľ) âˆ§ đ?‘‡đ??żđ?‘˜ (đ?‘Ľ), đ??źđ?‘€đ?‘˜ (đ?‘Ľ) âˆ¨ đ??źđ??żđ?‘˜ (đ?‘Ľ), đ??šđ?‘€đ?‘˜ (đ?‘Ľ) âˆ¨ đ??šđ??żđ?‘˜ (đ?‘Ľ)âŒŞ)|đ?‘Ľ âˆˆ

đ?‘‹}, đ?‘“đ?‘œđ?‘&#x; đ?‘˜ = 1, 2, â€Ś , đ?‘—. For

convenience,

we

can

use

r

=

SVNM R and call r a single valued neutrosophic multiset element (SVNME). For example, with a universal set X= {đ?‘Ľ1 , đ?‘Ľ2 }, then two SVNMs M and L are given as: M={(đ?‘Ľ1 , (2, âŒŠ0.6,0.2,0.1âŒŞ), (1, âŒŠ0.4,0.1,0.2âŒŞ)), (đ?‘Ľ2 , (1, âŒŠ0.7,0.3,0.1âŒŞ))}; L={(đ?‘Ľ1 , (1, âŒŠ0.6,0.2,0.1âŒŞ), (1, âŒŠ0.8,0.2,0.1âŒŞ)), (đ?‘Ľ2 , (1, âŒŠ0.9,0.3,0.1âŒŞ))}; đ?‘€âˆŞđ??ż = {(đ?‘Ľ1 , (2, âŒŠ0.6,0.2,0.1âŒŞ), (1, âŒŠ0.4,0.1,0.2âŒŞ), (1, âŒŠ0.8,0.2,0.1âŒŞ)), (đ?‘Ľ2 , (1, âŒŠ0.7,0.3,0.1âŒŞ), (1, âŒŠ0.9,0.3,0.1âŒŞ))}

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𝑀 ∩ 𝐿 = {𝑥1 , (1, 〈0.6,0.2,0.1〉)}.

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Definition 6. Let X be a space of points (objects) with a generic element x in X and M be a SVNM, we ̃ by using the operational rules of SVNS. can change a SVNM M into a SVNS 𝑀

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(7)

172 173 174 175 176 177

𝑀 = {𝑥, (𝑝𝑀𝑘 , 〈𝑇𝑀𝑘 (𝑥), 𝐼𝑀𝑘 (𝑥), 𝐹𝑀𝑘 (𝑥)〉)|𝑥 ∈ 𝑋}, 𝑓𝑜𝑟 𝑘 = 1, 2, … , 𝑗. Then ̃ = {〈𝑥, 1 − ∏𝑗𝑘=1(1 − 𝑇𝑀𝑘 (𝑥))𝑝𝑀𝑘 , ∏𝑗𝑘=1(𝐼𝑀𝑘 (𝑥))𝑝𝑀𝑘 , ∏𝑗𝑘=1(𝐹𝑀𝑘 (𝑥))𝑝𝑀𝑘 〉|𝑥 ∈ 𝑋}. 𝑀 Proof: Set 𝑚1 , 𝑚2 , … 𝑚𝑗 are basic elements in M. When k=1, we can get 𝑚1 = (〈𝑇𝑚1 (𝑥), 𝐼𝑚1 (𝑥), 𝐹𝑚1 (𝑥)〉, 〈𝑇𝑚1 (𝑥), 𝐼𝑚1 (𝑥), 𝐹𝑚1 (𝑥)〉, … , 〈𝑇𝑚1 (𝑥), 𝐼𝑚1 (𝑥), 𝐹𝑚1 (𝑥)〉) , which has neutrosophic multiplicity 𝑝𝑚1 . According to the operational rules of SVNSs, we can get: 𝑝𝑚1

178 179

+𝑚1 =1 − (1 − 𝑇𝑚1 (𝑥))

180 181

+𝑚2 =1 − (1 − 𝑇𝑚2 (𝑥))

182

𝑚1 +𝑚2

183

𝑝𝑚1

, (𝐼𝑚1 (𝑥))

, (𝐹𝑚1 (𝑥))

𝑝𝑚2

, (𝐼𝑚1 (𝑥))

, (𝐹𝑚2 (𝑥))

𝑇𝑚2 (𝑥))

𝑝𝑚2

𝑝𝑚1

1 − (1 − 𝑇𝑚1 (𝑥))

𝑝𝑚1

, (𝐼𝑚1 (𝑥))

(𝐼𝑚1 (𝑥))

𝑝𝑚2

, (𝐹𝑚2 (𝑥)) 𝑝𝑚𝑘

=1 − ∏2𝑘=1(1 − 𝑇𝑚𝑘 (𝑥))

𝑝𝑚1

(1 −

𝑝𝑚2

(𝐹𝑚2 (𝑥))

, ∏2𝑘=1(𝐼𝑚𝑘 (𝑥))

𝑝𝑚𝑘

, ∏2𝑘=1(𝐹𝑚𝑘 (𝑥))

;

Suppose when k=i, the formula (7) is established, then we can get: 𝑝𝑚𝑘

188

𝑚1 +𝑚2 + ⋯ + 𝑚𝑖 + 𝑚𝑖+1 = 1 − ∏𝑖𝑘=1(1 − 𝑇𝑚𝑘 (𝑥))

, ∏𝑖𝑘=1(𝐼𝑚𝑘 (𝑥))

𝑝𝑚𝑘

, ∏𝑖𝑘=1(𝐹𝑚𝑘 (𝑥))

;

Then

𝑝𝑚𝑘

189

𝑇𝑚𝑘 (𝑥))

190

𝑇𝑚(𝑖+1) (𝑥))

𝑝𝑚𝑘

+ 1 − (1 − 𝑇𝑚(𝑖+1) (𝑥))

𝑝𝑚(𝑖+1)

− (1 − ∏𝑖𝑘=1(1 −

)(1 − (1 − 𝑝𝑚(𝑖+1)

), (∏𝑖𝑘=1(𝐼𝑚𝑘 (𝑥))

=1 − ∏𝑖+1 𝑘=1(1 − 𝑇𝑚𝑘 (𝑥))

𝑝𝑚𝑘

) (𝐼𝑚(𝑖+1) (𝑥))

𝑝𝑚(𝑖+1)

𝑝𝑚𝑘

, ∏𝑖+1 𝑘=1(𝐼𝑚𝑘 (𝑥))

𝑝𝑚𝑘

, (∏𝑖𝑘=1(𝐹𝑚𝑘 (𝑥))

, ∏𝑖+1 𝑘=1(𝐹𝑚𝑘 (𝑥))

𝑝𝑚𝑘

) (𝐹𝑚(𝑖+1) (𝑥))

𝑝𝑚(𝑖+1)

.

To sum up, when k = i+1, the formula (7) is true, and then according to the mathematical induction, we can get that the aggregation result is also true. Definition 7. Let 𝑋 = {𝑥1 , 𝑥2 , … ，𝑥𝑛 } be a universe of discourse, and M and N be two SVNMs, and then the operational rules of SVNMs are defined as follows: 𝑀 = {𝑥, (𝑝𝑀𝑘 , 〈𝑇𝑀𝑘 (𝑥), 𝐼𝑀𝑘 (𝑥), 𝐹𝑀𝑘 (𝑥)〉)|𝑥 ∈ 𝑋}, for k=1, 2,…j; 𝑁 = {𝑥, (𝑝𝑁𝑘 , 〈𝑇𝑁𝑘 (𝑥), 𝐼𝑁𝑘 (𝑥), 𝐹𝑁𝑘 (𝑥)〉)|𝑥 ∈ 𝑋}, 𝑓𝑜𝑟 𝑘 = 1, 2, … 𝑗 ; 𝑀⊕𝑁 = {〈𝑥, 1 𝑗

𝑗 𝑝𝑀𝑘

− ∏(1 − 𝑇𝑀𝑘 (𝑥)) 𝑘=1

∏(1 𝑘=1

𝑗

201

.

=

𝑚1 +𝑚2 + ⋯ + 𝑚𝑖 =1 − ∏𝑖𝑘=1(1 − 𝑇𝑚𝑘 (𝑥))

200

𝑝𝑚2

Then

186 187

194 195 196 197 198 199

.

As the same reason, when k=2, we can get

184 185

191 192 193

𝑝𝑚1

𝑝𝑁𝑘

− 𝑇𝑁𝑘 (𝑥))

𝑗 𝑝𝑀𝑘

, ∏(𝐼𝑀𝑘 (𝑥)) 𝑘=1

∏(𝐼𝑁𝑘 (𝑥)) 𝑘=1

𝑗 𝑝𝑁𝑘

𝑗 𝑝𝑀𝑘

, ∏(𝐹𝑀𝑘 (𝑥)) 𝑘=1

∏(𝐹𝑁𝑘 (𝑥)) 𝑘=1

𝑝𝑁𝑘

〉 |𝑥 ∈ 𝑋}

Symmetry 2018, 4, x

6 of 13

𝑗

202

𝑗 𝑝𝑀𝑘

𝑀 ⊗ 𝑁 = {〈𝑥, (1 − ∏(1 − 𝑇𝑀𝑘 (𝑥))

) (1 − ∏(1 − 𝑇𝑁𝑘 (𝑥))

𝑘=1

𝑝𝑁𝑘

𝑘=1

+ ∏(𝐹𝑁𝑘 (𝑥))

𝑘=1

𝑗

− ∏(𝐹𝑀𝑘 (𝑥))

𝑝𝑁𝑘

∏(𝐹𝑁𝑘 (𝑥)) 𝜑

𝑗 𝑝𝑀𝑘

) ) , (∏(𝐼𝑀𝑘 (𝑥))

𝑘=1

𝑝𝑁𝑘

𝑀𝜑 = {〈𝑥, (1 − ∏(1 − 𝑇𝑀𝑘 (𝑥))

) 〉 |𝑥 ∈ 𝑋}

𝑘=1

𝜑 𝑝𝑀𝑘

𝜑

𝑗

) , (∏(𝐹𝑁𝑘 (𝑥))

𝑘=1

𝑗

〉 |𝑥 ∈ 𝑋}

𝑘=1

𝜑 𝑝𝑀𝑘

𝑝𝑀𝑘

, ∏(𝐹𝑀𝑘 (𝑥))

𝑘=1

𝜑𝑀 = {〈𝑥, (1 − (∏(1 − 𝑇𝑀𝑘 (𝑥))

𝜑

𝑗

) , 1 − (1 − ∏(𝐼𝑀𝑘 (𝑥))

𝑘=1

𝑝𝑀𝑘

) ,1

𝑘=1 𝜑

𝑗 𝑝𝑁𝑘

− (1 − ∏(𝐹𝑁𝑘 (𝑥))

207

∏(𝐼𝑁𝑘 (𝑥))

𝑝𝑀𝑘

𝑗

𝑗 𝑝𝑁𝑘

𝑘=1

206

𝑝𝑀𝑘

𝑘=1

𝑗

205

𝑗

− ∏(𝐼𝑀𝑘 (𝑥))

𝑝𝑀𝑘

𝑘=1

𝑗

+ ∏(𝐼𝑁𝑘 (𝑥))

204

) , ∏(𝐼𝑀𝑘 (𝑥))

𝑘=1

𝑗

203

𝑗 𝑝𝑁𝑘

) 〉 |𝑥 ∈ 𝑋}

𝑘=1

208 209 210

4. Cosine Measures of Single Value Neutrosophic Multisets Cosine measures are usually used in science and engineering applications. In this section, we propose a cosine measure of SVNMs and a weighted cosine measure of SVNMs.

211 212 213

Definition 8. Let 𝑋 = {𝑥1 , 𝑥2 , … ，𝑥𝑛 } be a universe of discourse, M and N be two SVNMs,

214 215 216 217

𝑋},

218 219

𝑀= {𝑥𝑖 , (𝑝𝑀1 , 〈𝑇𝑀1 (𝑥𝑖 ), 𝐼𝑀1 (𝑥𝑖 ), 𝐹𝑀1 (𝑥𝑖 )〉), (𝑝𝑀2 , 〈𝑇𝑀2 (𝑥𝑖 ), 𝐼𝑀2 (𝑥𝑖 ), 𝐹𝑀2 (𝑥𝑖 )〉), … , (𝑝𝑀𝑗 , 〈𝑇𝑀𝑗 (𝑥𝑖 ), 𝐼𝑀𝑗 (𝑥𝑖 ), 𝐹𝑀𝑗 (𝑥𝑖 )〉)|𝑥𝑖 ∈ 𝑁 = {〈𝑥𝑖 , (𝑝𝑁1 , 〈𝑇𝑁1 (𝑥𝑖 ), 𝐼𝑁1 (𝑥𝑖 ), 𝐹𝑁1 (𝑥𝑖 )〉), (𝑝𝑁2 , 〈𝑇𝑁2 (𝑥𝑖 ), 𝐼𝑁2 (𝑥𝑖 ), 𝐹𝑁2 (𝑥𝑖 )〉), … , (𝑝𝑁𝑗 , 〈𝑇𝑁𝑗 (𝑥𝑖 ), 𝐼𝑁𝑗 (𝑥𝑖 ), 𝐹𝑁𝑗 (𝑥𝑖 )〉)〉|𝑥𝑖 ∈ 𝑋}

Then, a cosine measure between two SVNMs M and N is defined as follows: 𝑛

𝑗

𝑖=1

𝑘=1

1 𝜋 𝑝𝑀𝑘 𝑝𝑁𝑘 𝜌(𝑀, 𝑁) = ∑ 𝑐𝑜𝑠 { |∏(1 − 𝑇𝑀𝑘 (𝑥𝑖 )) − ∏(1 − 𝑇𝑁𝑘 (𝑥𝑖 )) | 𝑛 6 𝑗

+ |∏(𝐼𝑀𝑘 (𝑥𝑖 ))

220

𝑘=1

𝑗 𝑝𝑀𝑘

− ∏(𝐼𝑁𝑘 (𝑥𝑖 )) 𝑘=1

𝑗 𝑝𝑁𝑘

𝑗 𝑝𝑀𝑘

| + |∏(𝐹𝑀𝑘 (𝑥𝑖 )) 𝑘=1

− ∏(𝐹𝑁𝑘 (𝑥𝑖 ))

𝑝𝑁𝑘

|}

𝑘=1

221

(8)

222 223 224 225 226

Theorem 1. The cosine measure 𝜌(𝑀, 𝑁) between two SVNMs M and N satisfies the following

227

Proof :

properties:  𝜌(𝑀, 𝑁) = 𝜌(𝑁, 𝑀);  0 ≤ 𝜌(𝑀, 𝑁) ≤ 1;  𝜌(𝑀, 𝑁) = 1, 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑀 = 𝑁.

Symmetry 2018, 4, x

228 229 230 231 232 233 234 235 236

7 of 13

𝑗

For |∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑖 ))

𝑝𝑀𝑘

𝑗

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥𝑖 ))

𝑝𝑁𝑘

𝑝𝑀𝑘

𝑗

| + |∏𝑘=1(𝐼𝑀𝑘 (𝑥𝑖 ))

: 0 ≤ 𝑇𝑀𝑘 (𝑥𝑖 ) ≤ 1 , 0 ≤ 𝐼𝑀𝑘 (𝑥𝑖 ) ≤ 1 , 0 ≤ 𝐹𝑀𝑘 (𝑥𝑖 ) ≤ 1,0 ≤ 𝑇𝑁𝑘 (𝑥𝑖 ) ≤ 1 , 0 ≤ 𝐼𝑁𝑘 (𝑥𝑖 ) ≤

For

1 , 0 ≤ 𝐹𝑁𝑘 (𝑥𝑖 ) ≤ 1 ; Then we can get 0 ≤ 1 − 𝑇𝑀𝑘 (𝑥𝑖 ) ≤ 1 𝑎𝑛𝑑 0 ≤ 1 − 𝑇𝑁𝑘 (𝑥𝑖 ) ≤ 1; 𝑗

𝑝𝑀𝑘

237 238

0 ≤ ∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑖 ))

239 240

0 ≤ |∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑖 ))

241 242 243 244

0 ≤ |∏𝑘=1(𝐼𝑀𝑘 (𝑥𝑖 ))

𝑗

≤ 1 𝑎𝑛𝑑 0 ≤ ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥𝑖 ))

𝑝𝑁𝑘

≤ 1;

So, 𝑝𝑀𝑘

𝑗

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥𝑖 ))

𝑝𝑁𝑘

| ≤ 1.

As the same reason, we can get 𝑗

𝑝 ∏𝑗𝑘=1(𝐹𝑁𝑘 (𝑥𝑖 )) 𝑁𝑘 |

Above

𝑝𝑁𝑘

𝑗

− ∏𝑘=1(𝐼𝑁𝑘 (𝑥𝑖 ))

|≤1

𝑗

0 ≤ |∏𝑘=1(𝐹𝑀𝑘 (𝑥𝑖 ))

and

we

−

can

𝑝 ∏𝑗𝑘=1(𝐼𝑁𝑘 (𝑥𝑖 )) 𝑁𝑘 |

𝜋

𝑝𝑀𝑘

𝑗

+

𝑝𝑀𝑘 𝑗 |∏𝑘=1(𝐹𝑀𝑘 (𝑥𝑖 ))

𝑗

𝑝𝑁𝑘

−

𝑝𝑀𝑘

∏𝑗𝑘=1(𝐼𝑁𝑘 (𝑥𝑖 ))

247

1 𝜋 𝑝𝑀𝑘 𝑝𝑁𝑘 𝜌(𝑀, 𝑁) = ∑ 𝑐𝑜𝑠 { |∏(1 − 𝑇𝑀𝑘 (𝑥𝑖 )) − ∏(1 − 𝑇𝑁𝑘 (𝑥𝑖 )) | 𝑛 6

𝑝𝑀𝑘

𝑗

| + |∏𝑘=1(𝐹𝑀𝑘 (𝑥𝑖 ))

− ∏𝑘=1(𝐹𝑁𝑘 (𝑥𝑖 ))

𝑛

𝑗

𝑖=1

𝑘=1

𝑗

248

𝑝𝑀𝑘

1

=

𝑛

∏𝑗𝑘=1(𝐼𝑁𝑘 (𝑥1 ))

251

𝑇𝑀𝑘 (𝑥2 ))

252

|∏𝑘=1(𝐹𝑀𝑘 (𝑥2 ))

253

𝑇𝑁𝑘 (𝑥𝑛 ))

254

∏𝑗𝑘=1(𝐹𝑁𝑘 (𝑥𝑛 ))

𝑝𝑀𝑘

𝜋

𝑝𝑁𝑘

𝑝𝑀𝑘

𝑗

𝑝𝑀𝑘

𝑗

| + |∏𝑘=1(𝐹𝑀𝑘 (𝑥1 ))

𝑗

𝑝𝑀𝑘 𝑗

𝑝𝑀𝑘

𝑗

| + |∏𝑘=1(𝐼𝑀𝑘 (𝑥1 ))

𝜋

𝑗

𝑝𝑁𝑘

𝑝𝑀𝑘

−

𝑗

|)} + 𝑐𝑜𝑠 { (|∏𝑘=1(1 − 𝑗

− ∏𝑘=1(𝐼𝑁𝑘 (𝑥2 )) 𝑗

𝑝𝑁𝑘

|+

𝑝𝑀𝑘

6

− ∏𝑘=1(𝐼𝑁𝑘 (𝑥𝑛 ))

|}

6

𝜋

𝑝𝑀𝑘

𝑗

6

256

∏𝑗𝑘=1(𝐼𝑁𝑘 (𝑥𝑖 ))

257

𝜌(𝑀, 𝑁) = {𝑎1 + 𝑎2 + ⋯ + 𝑎𝑛 }.

258

𝑝𝑁𝑘

𝑗

| + |∏𝑘=1(𝐹𝑀𝑘 (𝑥𝑛 ))

𝑝𝑀𝑘

𝑗

− ∏𝑘=1(1 −

−

|)}). 𝜋

𝑝𝑁𝑘

𝑝𝑀𝑘

−

− ∏(𝐹𝑁𝑘 (𝑥𝑖 ))

|)} + ⋯ + 𝑐𝑜𝑠 { (|∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑛 ))

𝑐𝑜𝑠 { (|∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑖 ))

Let

𝑝𝑁𝑘

− ∏𝑘=1(𝐹𝑁𝑘 (𝑥1 )) 𝑗

𝑝𝑁𝑘

𝑗

𝑝𝑁𝑘

𝑗

| + |∏𝑘=1(𝐼𝑀𝑘 (𝑥2 ))

− ∏𝑘=1(𝐹𝑁𝑘 (𝑥2 ))

| + |∏𝑘=1(𝐼𝑀𝑘 (𝑥𝑛 )) 𝑝𝑁𝑘

𝑝𝑁𝑘

≤ 3 and 0 ≤

𝑘=1

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥1 ))

𝑗

|+

𝑗 𝑝𝑀𝑘

𝑘=1

6

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥2 ))

𝑗

−

|} ≤ 1;

| + |∏(𝐹𝑀𝑘 (𝑥𝑖 ))

𝑘=1

(𝑐𝑜𝑠 { (|∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥1 ))

𝑝𝑁𝑘

250

𝑝𝑁𝑘

𝑗 𝑝𝑁𝑘

− ∏(𝐼𝑁𝑘 (𝑥𝑖 ))

𝑘=1

249

𝑗

| + |∏𝑘=1(𝐼𝑀𝑘 (𝑥𝑖 ))

𝑗

+ |∏(𝐼𝑀𝑘 (𝑥𝑖 ))

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥𝑖 ))

𝑝 ∏𝑗𝑘=1(𝐹𝑁𝑘 (𝑥𝑖 )) 𝑁𝑘 |

246

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥𝑖 ))

6

𝑝𝑁𝑘

𝑗

0 ≤ |∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑖 ))

get

∑𝑛𝑖=1 𝑐𝑜𝑠 { |∏𝑗𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑖 )) 𝑝𝑁𝑘

𝑝𝑀𝑘

≤1

all,

𝑝𝑀𝑘 𝑗 |∏𝑘=1(𝐼𝑀𝑘 (𝑥𝑖 ))

𝑝𝑀𝑘

245

255

−

𝑝𝑁𝑘

𝑗

𝑝𝑀𝑘

| + |∏𝑘=1(𝐹𝑀𝑘 (𝑥𝑖 ))

𝑝𝑁𝑘

𝑗

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥𝑖 ))

𝑗

− ∏𝑘=1(𝐹𝑁𝑘 (𝑥𝑖 ))

𝑝𝑁𝑘

𝑗

| + |∏𝑘=1(𝐼𝑀𝑘 (𝑥𝑖 ))

|)} = 𝑎𝑖 (𝑖 = 1,2, … 𝑛）

,

𝑝𝑀𝑘

−

then

1

𝑛

According

to

𝜋

𝑗

𝑝𝑀𝑘

0 ≤ 𝑐𝑜𝑠 { (|∏𝑘=1(1 − 𝑇𝑀𝑘 (𝑥𝑖 )) 6

𝑗

𝑝𝑁𝑘

− ∏𝑘=1(1 − 𝑇𝑁𝑘 (𝑥𝑖 ))

|+

Symmetry 2018, 4, x

đ?‘?đ?‘€đ?‘˜

đ?‘—

8 of 13

đ?‘—

âˆ’ âˆ?đ?‘˜=1(đ??źđ?‘ đ?‘˜ (đ?‘Ľđ?‘– ))

đ?‘?đ?‘ đ?‘˜

đ?‘?đ?‘€đ?‘˜

đ?‘—

| + |âˆ?đ?‘˜=1(đ??šđ?‘€đ?‘˜ (đ?‘Ľđ?‘– ))

đ?‘—

âˆ’ âˆ?đ?‘˜=1(đ??šđ?‘ đ?‘˜ (đ?‘Ľđ?‘– ))

đ?‘?đ?‘ đ?‘˜

259

|âˆ?đ?‘˜=1(đ??źđ?‘€đ?‘˜ (đ?‘Ľđ?‘– ))

260

we can obtain 0 â‰¤ đ?‘Ž1 + đ?‘Ž2 + â‹Ż + đ?‘Žđ?‘› â‰¤ đ?‘› and 0 â‰¤ (đ?‘Ž1 + đ?‘Ž2 + â‹Ż + đ?‘Žđ?‘› ) â‰¤ 1, so we can get 0 â‰¤

261 262 263 264 265 266 267

đ?œŒ(đ?‘€, đ?‘ ) â‰¤ 1 .

|)} = đ?‘Žđ?‘– â‰¤ 1

,

1

đ?‘›

ď Ź If M=N then đ?‘‡đ?‘€đ?‘˜ (đ?‘Ľđ?‘– ) = đ?‘‡đ?‘ đ?‘˜ (đ?‘Ľđ?‘– ), đ??źđ?‘€đ?‘˜ (đ?‘Ľđ?‘– ) = đ??źđ?‘ đ?‘˜ (đ?‘Ľđ?‘– ), đ?‘Žđ?‘›đ?‘‘ đ??šđ?‘€đ?‘˜ (đ?‘Ľđ?‘– ) = đ??šđ?‘ đ?‘˜ (đ?‘Ľđ?‘– ) for any đ?‘Ľđ?‘– âˆˆ đ?‘‹ đ?‘Žđ?‘›đ?‘‘ đ?‘– = 1,2, â€Ś đ?‘›, so we can get đ?œŒ(đ?‘€, đ?‘ ) = 1, if and only if đ?‘€ = đ?‘ . Now, we consider different weights for each element đ?‘Ľđ?‘– (đ?‘– = 1,2, â€Ś , đ?‘›) in X. Then, let đ?‘¤ = (đ?‘¤1 , đ?‘¤2 , â€Ś , đ?‘¤đ?‘› )đ?‘‡ be the weight vector of each element đ?‘Ľđ?‘– (đ?‘– = 1,2, â€Ś , đ?‘›) with đ?‘¤đ?‘– âˆˆ [0,1], and âˆ‘đ?‘›đ?‘–=1 đ?‘¤đ?‘– = 1. Hence, we further extend the cosine measure of Equation (8) to the following weighted cosine measure of SVNM: đ?‘›

đ?‘—

đ?‘–=1

đ?‘˜=1

đ?œ‹ đ?‘?đ?‘€đ?‘˜ đ?‘?đ?‘ đ?‘˜ đ?œŒđ?‘¤ (đ?‘€, đ?‘ ) = âˆ‘ đ?‘¤đ?‘– đ?‘?đ?‘œđ?‘ { (|âˆ?(1 âˆ’ đ?‘‡đ?‘€đ?‘˜ (đ?‘Ľđ?‘– )) âˆ’ âˆ?(1 âˆ’ đ?‘‡đ?‘ đ?‘˜ (đ?‘Ľđ?‘– )) | 6

268

đ?‘—

đ?‘— đ?‘?đ?‘€đ?‘˜

+ |âˆ?(đ??źđ?‘€đ?‘˜ (đ?‘Ľđ?‘– ))

269

âˆ’ âˆ?(đ??źđ?‘ đ?‘˜ (đ?‘Ľđ?‘– ))

đ?‘˜=1

|

đ?‘˜=1

đ?‘—

đ?‘— đ?‘?đ?‘€đ?‘˜

+ |âˆ?(đ??šđ?‘€đ?‘˜ (đ?‘Ľđ?‘– ))

270

đ?‘?đ?‘ đ?‘˜

đ?‘˜=1

âˆ’ âˆ?(đ??šđ?‘ đ?‘˜ (đ?‘Ľđ?‘– ))

đ?‘?đ?‘ đ?‘˜

|)}

đ?‘˜=1

271

(9)

272 273 274 275 276 277

Theorem 2. The cosine measure đ?œŒđ?‘¤ (đ?‘€, đ?‘ ) between two SVNMs M and N satisfies the following

278

5. Cosine Measure of SVNM for Multiple Attribute Decision-making

279 280 281 282 283 284 285 286 287 288

properties: ď Ş đ?œŒđ?‘¤ (đ?‘€, đ?‘ ) = đ?œŒđ?‘¤ (đ?‘ , đ?‘€); ď Ť 0 â‰¤ đ?œŒđ?‘¤ (đ?‘€, đ?‘ ) â‰¤ 1; ď Ź đ?œŒđ?‘¤ (đ?‘€, đ?‘ ) = 1, đ?‘–đ?‘“ đ?‘Žđ?‘›đ?‘‘ đ?‘œđ?‘›đ?‘™đ?‘Ś đ?‘–đ?‘“ đ?‘€ = đ?‘ . The proof of Theorem 2 is similar to that of the Theorem 1, so we omitted it here.

In this section, we use the weighted cosine measure of SVNM to deal with the multiple attribute decision-making problems with SVNM information. Let đ??ş = {đ?‘”1 , đ?‘”2 , â€Ś ďźŒđ?‘”đ?‘š } as a set of alternatives and đ?‘‹ = {đ?‘Ľ1 , đ?‘Ľ2 , â€Ś ďźŒđ?‘Ľđ?‘› } as a set of attributes, then they can be established in a decision-making problem. But sometimes đ?‘Ľđ?‘– (đ?‘– = 1,2, â€Ś , đ?‘›) maybe have multiplicity, and then we can use the form of a SVNM to represent the evaluation value. Let đ?‘”đ?‘&#x; = {đ?‘Ľđ?‘– , (đ?‘?đ?‘” 1 , âŒŠđ?‘‡đ?‘” 1 (đ?‘Ľđ?‘– ), đ??źđ?‘” 1 (đ?‘Ľđ?‘– ), đ??šđ?‘” 1 (đ?‘Ľđ?‘– )âŒŞ), (đ?‘?đ?‘” 2 , âŒŠđ?‘‡đ?‘” 2 (đ?‘Ľđ?‘– ), đ??źđ?‘” 2 (đ?‘Ľđ?‘– ), đ??šđ?‘” 2 (đ?‘Ľđ?‘– )âŒŞ), â€Ś , (đ?‘?đ?‘” đ?‘— , âŒŠđ?‘‡đ?‘” đ?‘— (đ?‘Ľđ?‘– ), đ??źđ?‘” đ?‘— (đ?‘Ľđ?‘– ), đ??šđ?‘” đ?‘— (đ?‘Ľđ?‘– )âŒŞ)|đ?‘Ľđ?‘– âˆˆ đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘&#x; đ?‘‹},

for r=1,2,â€Ś,m and i=1,2,â€Ś,n. Then we can establish the SVNM decision matrix D, which is

shown in Table 1.

289

Table 1. The SVNM decision matrix D x1 g1

x1 , (pg1 1 , âŒŠTg11 (x1 ), Ig1 1 (x1 ), Fg1 1 (x1 )âŒŞ), â€Ś , (pg1j , âŒŠTg1 j (x1 ), Ig1 j (x1 ), Fg1 j (x1 )âŒŞ)

â€Ś â€Ś

Symmetry 2018, 4, x

g2

9 of 13

x1 , (pg2 1 , âŒŠTg21 (x1 ), Ig2 1 (x1 ), Fg2 1 (x1 )âŒŞ), â€Ś , (pg2j , âŒŠTg2 j (x1 ), Ig2 j (x1 ), Fg2 j (x1 )âŒŞ)

â€Ś gm

â€Ś

x1 , (pgm 1 , âŒŠTgm1 (x1 ), Igm 1 (x1 ), Fgm1 (x1 )âŒŞ), â€Ś , (pgmj , âŒŠTgmj (x1 ), Igmj (x1 ), Fgm j (x1 )âŒŞ)

â€Ś

xn xn , (pg1 1 , âŒŠTg1 1 (xn ), Ig11 (xn ), Fg11 (xn )âŒŞ), â€Ś , (pg1j , âŒŠTg1 j (xn ), Ig1 j (xn ), Fg1j (xn )âŒŞ) xn , (pg2 1 , âŒŠTg2 1 (xn ), Ig21 (xn ), Fg21 (xn )âŒŞ), â€Ś , (pg2j , âŒŠTg2 j (xn ), Ig2 j (xn ), Fg2j (xn )âŒŞ) â€Ś xn , (pgm 1 , âŒŠTgm 1 (xn ), Igm1 (xn ), Fgm1 (xn )âŒŞ), â€Ś , (pgm j , âŒŠTgmj (xn ), Igm j (xn ), Fgmj (xn )âŒŞ)

290 291

Step 1: By using the Equation (7), we change the SVNM decision matrix D into SVNS Ěƒ , which is shown in Table 2. decision matrix đ??ˇ

292

Ěƒ Table 2. The SVNS decision matrix đ??ˇ x1 gĚƒ1

j

âŒŠx1 , 1 âˆ’ âˆ? (1 âˆ’ Tg1 k (x1 ))

pg1 k

j

, âˆ? (Ig1k (x1 ))

k=1

gĚƒ 2

â€Ś pg1 k

j

, âˆ? (Fg1 k (x1 ))

k=1

j

âŒŠx1 , 1 âˆ’ âˆ? (1 âˆ’ Tg2 k (x1 ))

pg2 k

â€Ś âŒŞ

k=1

j

, âˆ? (Ig2k (x1 ))

k=1

pg1 k

pg2 k

j

, âˆ? (Fg2 k (x1 ))

k=1

pg2 k

â€Ś âŒŞ

k=1

â€Ś

gĚƒ m

j

âŒŠx1 , 1 âˆ’ âˆ? (1 âˆ’ Tgm k (x1 ))

pgm k

k=1

â€Ś

j

, âˆ? (Igm k (x1 ))

pgm k

j

, âˆ? (Fgmk (x1 ))

k=1

pgm k

â€Ś âŒŞ

k=1

xn j

âŒŠxn , 1 âˆ’ âˆ? (1 âˆ’ Tg1 k (xn ))

pg1 k

j

, âˆ? (Ig1 k (xn ))

k=1

pg1 k

k=1

j

âŒŠxn , 1 âˆ’ âˆ? (1 âˆ’ Tg2 k (xn ))

pg2 k

, âˆ? (Fg1k (xn ))

pg1 k

âŒŞ

k=1

j

, âˆ? (Ig2 k (xn ))

k=1

j

pg2 k

k=1

j

, âˆ? (Fg2k (xn ))

pg2 k

âŒŞ

k=1

â€Ś j

âŒŠxn , 1 âˆ’ âˆ? (1 âˆ’ Tgm k (xn )) k=1

293 294 295 296 297

pgm k

j

, âˆ? (Igm k (xn ))

pgm k

k=1

j

, âˆ? (Fgmk (xn ))

pgm k

âŒŞ

k=1

đ?‘‡ đ?‘”âˆ— (đ?‘Ľđ?‘– ) is the maximum truth value in each Ěƒ column đ?‘Ľđ?‘– of the decision matrix đ??ˇ , đ??ź đ?‘”âˆ— (đ?‘Ľđ?‘– ) and đ??š đ?‘”âˆ— (đ?‘Ľđ?‘– ) are the minimum indeterminate and Ěƒ , respectively, the ideal solution can be falsity values in each column đ?‘Ľđ?‘– of the decision matrix đ??ˇ Step

2:

Setting

determined as đ?‘Ľđ?‘–âˆ— . đ?‘Ľđ?‘–âˆ— = âŒŠđ?‘‡ đ?‘”âˆ— (đ?‘Ľđ?‘– ), đ??ź đ?‘”âˆ— (đ?‘Ľđ?‘– ), đ??š đ?‘”âˆ— (đ?‘Ľđ?‘– )âŒŞ, for i=1,2,â€Śn.

Symmetry 2018, 4, x

298 299

10 of 13

So we can get the ideal alternative đ?‘”âˆ— = {đ?‘Ľ1,âˆ— , đ?‘Ľ2,âˆ— â€Ś , đ?‘Ľđ?‘›âˆ— }. Step 3: When the weight vector of attributes for the different importance of each attribute đ?‘‡

300

đ?‘Ľđ?‘– (đ?‘– = 1,2, â€Ś , đ?‘›) is given by đ?‘¤ = (đ?‘¤1, đ?‘¤2, â€Ś , đ?‘¤đ?‘› ) with đ?‘¤đ?‘– â‰Ľ 0 and âˆ‘đ?‘›đ?‘–=1 đ?‘¤đ?‘– = 1. Then, we utilize

301

the weighted cosine measure to deal with multiple attribute decision-making problems with

302 303

SVNM information. The weighted cosine measure between an alternative đ?‘”Ěƒđ?‘&#x; (đ?‘&#x; = 1,2, â€Ś ďźŒđ?‘š)

304

đ?œŒđ?‘¤ (đ?‘”đ?‘&#x; , đ?‘”âˆ— ) = đ?œŒđ?‘¤ (đ?‘”Ěƒđ?‘&#x; , đ?‘”âˆ— ) = âˆ‘đ?‘›đ?‘–=1 đ?‘¤đ?‘– đ?‘?đ?‘œđ?‘ { (|đ?‘‡đ?‘”Ěƒđ?‘&#x; (đ?‘Ľđ?‘– ) âˆ’ đ?‘‡ đ?‘”âˆ— (đ?‘Ľđ?‘– )| + |đ??źđ?‘”Ěƒđ?‘&#x; (đ?‘Ľđ?‘– ) âˆ’ đ??ź đ?‘”âˆ— (đ?‘Ľđ?‘– )| + |đ??šđ?‘”Ěƒđ?‘&#x; (đ?‘Ľđ?‘– ) âˆ’

305

đ??š đ?‘”âˆ— (đ?‘Ľđ?‘– )|)}.

306 307 308 309 310

and the ideal alternative đ?‘”âˆ— can be calculated by using the following formula: đ?œ‹ 6

(10) Step 4: According to the values of đ?œŒđ?‘¤ (đ?‘”Ěƒđ?‘&#x; , đ?‘”

âˆ—)

for r=1, 2, â€Ś, m, we rank the alternatives

and select the best one. Step 5: End. The formalization of the steps is illustrated in Fig. 1. Establish the SVNM decision matrix

Chang SVNM decision matrix into SVNS decision matrix

Get the ideal alternative

Calculate the weighted cosine measure between every alternative and the ideal alternative Rank the alternatives and get the best one

311

Fig. 1. Flowchart of the decision steps

312

6. Numerical Example and Comparative Analysis

313

6.1 Numerical Example

314 315 316 317 318 319 320 321 322 323 324

Now, we utilize a practical example for the decision-making problem adapted from the literature [21] to demonstrate the applications of the proposed method under a SVNM environment. Now, one customer wants to buy a car, he selects four types of cars and evaluates them according to four attributes. Then, we build a decision model. There are four possible alternatives (đ?‘”1 , đ?‘”2 , đ?‘”3 , đ?‘”4 ) to be considered. Decision should be taken according to four attributes: fuel economy (đ?‘Ľ1 ), price (đ?‘Ľ2 ), comfort (đ?‘Ľ3 ), safety (đ?‘Ľ4 ).The weight vector of these four attributes is given by đ?‘¤ = (0.5,0.25,0.125,0.125)đ?‘‡ . Then, the customer testes the four cars on the road with less obstacles and on the road with more obstacles, respectively, and after testing, some attributes may have two different evaluated values or the same. So the customer evaluates the four cars (alternatives) under the four attributes by the form of SVNMs.

Symmetry 2018, 4, x

325 326

11 of 13

Step 1: Establish the SVNM decision matrix D provided by the customer, which is given as the following SVNM decision matrix D in Tables 3.

327

Table 3. The SVNM decision matrix D đ?‘Ľ1

đ?‘Ľ2

đ?‘Ľ3

đ?‘Ľ4

đ?‘”1

(1, âŒŠ0.5,0.7,0.2âŒŞ), (1, âŒŠ0.7,0.3,0.6âŒŞ) 1, âŒŠ0.4,0.4,0.5âŒŞ

(1, âŒŠ0.7,0.7.0.5âŒŞ), (1, âŒŠ0.8,0.7,0.6âŒŞ)

(1, âŒŠ0.1,0.5,0.7âŒŞ), (1, âŒŠ0.5,0.2,0.8âŒŞ)

đ?‘”2

(1, âŒŠ0.9,0.7,0.5âŒŞ), (1, âŒŠ0.7,0.7,0.1âŒŞ) 1, âŒŠ0.7,0.6,0.8âŒŞ

2, âŒŠ0.9,0.4,0.6âŒŞ

(1, âŒŠ0.5,0.2,0.7âŒŞ, (1, âŒŠ0.5,0.1,0.9âŒŞ)

đ?‘”3

(1, âŒŠ0.3,0.4,0.2âŒŞ, (1, âŒŠ0.6,0.3,0.7âŒŞ) 1, âŒŠ0.2,0.2,0.2âŒŞ

(1, âŒŠ0.9,0.5,0.5âŒŞ, (1, âŒŠ0.6,0.5,0.2âŒŞ)

(1, âŒŠ0.7,0.5,0.3âŒŞ, (1, âŒŠ0.4,0.2,0.2âŒŞ)

đ?‘”4

(1, âŒŠ0.9,0.7,0.2âŒŞ, (1, âŒŠ0.8,0.6,0.1âŒŞ) 1, âŒŠ0.3,0.5,0.2âŒŞ

(1, âŒŠ0.5,0.4,0.5âŒŞ, (1, âŒŠ0.1,0.7,0.2âŒŞ)

2, âŒŠ0.4,0.2,0.8âŒŞ

328 329

Step 2: By using the Equation (7), we change the SVNM decision matrix D into SVNS Ěƒ , which is shown in Table 4. decision matrix đ??ˇ

330

Ěƒ Table 4. The SVNS decision matrix đ??ˇ

331 332 333 334 335 336 337 338 339 340 341 342 343

đ?‘Ľ1

đ?‘Ľ2

đ?‘Ľ3

đ?‘Ľ4

đ?‘”Ěƒ1

âŒŠ0.85,0.21,0.12âŒŞ

âŒŠ0.4,0.4,0.5âŒŞ

âŒŠ0.94,0.49,0.3âŒŞ

âŒŠ0.55,0.1,0.56âŒŞ

đ?‘”Ěƒ2

âŒŠ0.97,0.49,0.05âŒŞ

âŒŠ0.7,0.6,0.8âŒŞ

âŒŠ0.99,0.16,0.36âŒŞ

âŒŠ0.75,0.02,0.63âŒŞ

đ?‘”Ěƒ3

âŒŠ0.72,0.12,0.14âŒŞ

âŒŠ0.2,0.2,0.2âŒŞ

âŒŠ0.96,0.25,0.1âŒŞ

âŒŠ0.82,0.1,0.06âŒŞ

đ?‘”Ěƒ4

âŒŠ0.98,0.42,0.02âŒŞ

âŒŠ0.3,0.5,0.2âŒŞ

âŒŠ0.55,0.28,0.1âŒŞ

âŒŠ0.64,0.04,0.64âŒŞ

Step 3: According to the decision matrix Ěƒ đ??ˇ , we can get the ideal alternative đ?‘”âˆ— : đ?‘”âˆ— = {âŒŠ0.98,0.12,0.02âŒŞâŒŠ0.7,0.2,0.2âŒŞâŒŠ0.99,0.16,0.1âŒŞâŒŠ0.82,0.02,0.06âŒŞ}. Step 4: By applying the Equation (10), we can obtain the values of the weighted cosine measure between each alternative and the ideal alternative đ?‘”âˆ— as follows: đ?œŒđ?‘¤ (đ?‘”1 , đ?‘”âˆ— ) = 0.9535, đ?œŒđ?‘¤ (đ?‘”2 , đ?‘”âˆ— ) = 0.9511, đ?œŒđ?‘¤ (đ?‘”3 , đ?‘”âˆ— ) = 0.9813 and đ?œŒđ?‘¤ (đ?‘”4 , đ?‘”âˆ— ) = 9616. Step 5: According to the above values of weighted cosine measure, we can rank the four alternatives: đ?‘”3 â‰ť đ?‘”4 â‰ť đ?‘”1 â‰ť đ?‘”2 . Therefore, the alternative đ?‘”3 is the best choice. This example clearly indicates that the proposed decision-making method based on the weighted cosine measure of SVNMs is relatively simple and easy for dealing with multiple attribute decision-making problems under SVNM environment. 6.2 Comparative Analysis In what follows, we compare the proposed method for SVNM with other existing related methods for SVNM, all the results are shown in Table 5.

344

Table 5. The ranking orders by utilizing four different methods Method Method

1

based

Result on

correlation coefficient in [11]

đ?œŒđ?‘¤ (đ?‘”1 , đ?‘”

âˆ— )=0.9053,

đ?œŒđ?‘¤ (đ?‘”2 , đ?‘”

âˆ—)

=0.9017,

đ?œŒđ?‘¤ (đ?‘”3 , đ?‘”

âˆ—)

=0.9516,

đ?œŒđ?‘¤ (đ?‘”4 , đ?‘”

âˆ—)

=0.8816.

Ranking order đ?‘”3 â‰ť đ?‘”1 â‰ť đ?‘”2 â‰ť đ?‘”4

The best alternative đ?‘”3

Symmetry 2018, 4, x

Method

2

12 of 13

based

on

similarity in [16]

Method

3

based

similarity in [16]

The proposed method

on

đ?œŒđ?‘¤ (đ?‘”1 , đ?‘”âˆ— )=0.8204, đ?œŒđ?‘¤ (đ?‘”2 , đ?‘”

âˆ—)

=0.8108,

đ?œŒđ?‘¤ (đ?‘”3 , đ?‘”

âˆ—)

=0.8867,

đ?œŒđ?‘¤ (đ?‘”4 , đ?‘”

âˆ—)

=0.8358.

đ?œŒđ?‘¤ (đ?‘”1 , đ?‘”

âˆ— )=0.7898,

đ?œŒđ?‘¤ (đ?‘”2 , đ?‘”

âˆ—)

=0.7121,

đ?œŒđ?‘¤ (đ?‘”3 , đ?‘”

âˆ—)

=0.8125,

đ?œŒđ?‘¤ (đ?‘”4 , đ?‘”

âˆ—)

=0.7553.

đ?œŒđ?‘¤ (đ?‘”1 , đ?‘”

âˆ— )=0.9535,

đ?‘”3 â‰ť đ?‘”4 â‰ť đ?‘”2 â‰ť đ?‘”1

đ?‘”3

đ?‘”3 â‰ť đ?‘”1 â‰ť đ?‘”4 â‰ť đ?‘”2

đ?‘”3

đ?‘”3 â‰ť đ?‘”4 â‰ť đ?‘”1 â‰ť đ?‘”2

đ?‘”3

đ?œŒđ?‘¤ (đ?‘”2 , đ?‘”âˆ— ) =0.9511, đ?œŒđ?‘¤ (đ?‘”3 , đ?‘”âˆ— ) =0.9813, đ?œŒđ?‘¤ (đ?‘”4 , đ?‘”âˆ— ) =9616.

345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374

From Table 5, these four methods have the same best alternative đ?‘”3 . Many methods such as similarity measure, correlation coefficient, and cosine measure can all be used in SVNM to handle the multiple attribute decision-making problems and can get the similar results. The proposed decision-making method can express and handle the multiplicity evaluated data

given

by

decision makers

or

experts,

while

various

existing

neutrosophic

decision-making methods cannot deal with these problems. 7. Conclusions Based on the multiplicity evaluation in some real situations, this paper introduced a SVNM as a subclass of NM to express the multiplicity information and the operational relations of SVNMs. The SVNM is expressed by its one or more elements, wich may have multiplicity. Therefore, SVNM has the desirable advantages and characteristics of expressing and handling the multiplicity problems, while existing neutrosophic sets cannot deal with them. Then, we proposed the cosine measure of SVNMs and weighted cosine measure of SVNMs and investigated their properties. Based on the weighted cosine measure of SVNMs, the multiple attribute decision-making methods under SVNM environments was proposed, in which the evaluated values were taken the form of SVNMEs. Through the weighed cosine measure between each alternative and the ideal alternative, one can determine the ranking order of all alternatives and can select the best one. Finally, a practical example adapted from the literature [21] about buying cars was presented to demonstrate the effectiveness and practicality of the proposed method in this paper. According to the ranking orders, we can find that the ranking result with weighted cosine measures is agreement with the ranking results in literature [21]. Then the proposed method is suitable for actual applications in multiple attribute decision-making problems with single value neutrosophic multiplicity information. In the future, we shall extend SVNMs to interval neutrosophic multisets and develop the application of interval neutrosophic multisets for handling the decision-making methods or other domains. Acknowledgments: This work was supported by the National Natural Science Foundation of China under grant Nos.61703280, and Science and Technology Planning Project of Shaoxing City of China (no. 2017B70056).

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Author Contributions: Changxing Fan originally proposed the LNNNWBM and LNNNWGBM operators and investigated their properties; Jun Ye and En Fan provided the calculation and comparative analysis; and we wrote the paper together.

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Conflicts of Interest: The author declares no conflict of interest.

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