symmetry 1 2 3 4 5 6 7 8 9 10
Article
The Cosine Measure of Single Valued Neutrosophic Multisets for Multiple Attribute Decision-Making Changxing Fan1*, En Fan 1, and Jun Ye2 1
2
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
11
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 Δ?‘‡(Δ?‘Δ½) belonging to the scope of [0, 1]. While the fuzzy membership degree of Δ?‘‡(Δ?‘Δ½) 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 Δ?‘‡(Δ?‘Δ½) is determined, the non-membership degree can be calculated by 1 Δ?‘‡(Δ?‘Δ½). 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 Δ?‘‡Δ?‘… (Δ?‘Δ½) , an indeterminacy-membership function Δ??ΕΊΔ?‘… (Δ?‘Δ½), and a falsity-membership function Δ??Ε‘Δ?‘… (Δ?‘Δ½), where Δ?‘‡Δ?‘… (Δ?‘Δ½), Δ??ΕΊΔ?‘… (Δ?‘Δ½), Δ??Ε‘Δ?‘… (Δ?‘Δ½) Γ’ˆˆ [0,1] for each point x in X. Then, a SVNS R can be expressed by the following form: Δ?‘… = {Γ’ŒΕ Δ?‘Δ½, Δ?‘‡Δ?‘… (Δ?‘Δ½), Δ??ΕΊΔ?‘… (Δ?‘Δ½), Δ??Ε‘Δ?‘… (Δ?‘Δ½)Γ’ŒΕ|Δ?‘Δ½ Γ’ˆˆ Δ?‘‹}. Thus, the SVNS R satisfies the condition 0 Γ’‰Β€ Δ?‘‡Δ?‘… (Δ?‘Δ½) + Δ??ΕΊΔ?‘… (Δ?‘Δ½) + Δ??Ε‘Δ?‘… (Δ?‘Δ½) Γ’‰Β€ 3. For two SVNSs M and N, the relations of them are defined as follows [5]: (1) M Γ’Š† N if and only if Δ?‘‡Δ?‘€ (Δ?‘Δ½) Γ’‰Β€ Δ?‘‡Δ?‘ (Δ?‘Δ½), Δ??ΕΊΔ?‘€ (Δ?‘Δ½) Γ’‰Δ½ Δ??ΕΊΔ?‘ (Δ?‘Δ½), Δ??Ε‘Δ?‘€ (Δ?‘Δ½) Γ’‰Δ½ Δ??Ε‘Δ?‘ (Δ?‘Δ½) for any x in X; (2) M = N if and only if M Γ’Š† N and N Γ’Š† M; (3) Δ?‘€Δ?‘? = {Γ’ŒΕ Δ?‘Δ½, Δ??Ε‘Δ?‘€ (Δ?‘Δ½), 1 Γ’ˆ’ Δ??ΕΊΔ?‘€ (Δ?‘Δ½), Δ?‘‡Δ?‘€ (Δ?‘Δ½)Γ’ŒΕ|Δ?‘Δ½ Γ’ˆˆ Δ?‘‹}.
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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 Δ?‘… = Γ’ŒΕ Δ?‘‡Δ?‘… (Δ?‘Δ½), Δ??ΕΊΔ?‘… (Δ?‘Δ½), Δ??Ε‘Δ?‘… (Δ?‘Δ½)Γ’ŒΕ for any x in X. For two SVNNs M and N, the operational relations of them can be defined as follows [5]: (1) Δ?‘€ Γ’ˆΕ Δ?‘ = <max(Δ?‘‡Δ?‘€ (Δ?‘Δ½), Δ?‘‡Δ?‘ (Δ?‘Δ½)), min(Δ??ΕΊΔ?‘€ (Δ?‘Δ½), Δ??ΕΊΔ?‘ (Δ?‘Δ½)), min(Δ??Ε‘Δ?‘€ (Δ?‘Δ½), Δ??Ε‘Δ?‘ (Δ?‘Δ½)) > for any x in X; (2) M Γ’ˆΕ N = <min(Δ?‘‡Δ?‘€ (Δ?‘Δ½), Δ?‘‡Δ?‘ (Δ?‘Δ½)), max(Δ??ΕΊΔ?‘€ (Δ?‘Δ½), Δ??ΕΊΔ?‘ (Δ?‘Δ½)), max(Δ??Ε‘Δ?‘€ (Δ?‘Δ½), Δ??Ε‘Δ?‘ (Δ?‘Δ½)) > for any x in X. For two SVNNs M and N, the operational rules of them can be defined as follows [5]: M+N =< Δ?‘‡Δ?‘€ (Δ?‘Δ½) + Δ?‘‡Δ?‘ (Δ?‘Δ½) Γ’ˆ’ Δ?‘‡Δ?‘€ (Δ?‘Δ½)Δ?‘‡Δ?‘ (Δ?‘Δ½), Δ??ΕΊΔ?‘€ (Δ?‘Δ½)Δ??ΕΊΔ?‘ (Δ?‘Δ½), Δ??Ε‘Δ?‘€ (Δ?‘Δ½)Δ??Ε‘Δ?‘ (Δ?‘Δ½) > for any x in X;
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(1)
MΔ—N = < Δ?‘‡Δ?‘€ (Δ?‘Δ½)Δ?‘‡Δ?‘ (Δ?‘Δ½), Δ??ΕΊΔ?‘€ (Δ?‘Δ½) + Δ??ΕΊΔ?‘ (Δ?‘Δ½) Γ’ˆ’ Δ??ΕΊΔ?‘€ (Δ?‘Δ½)Δ??ΕΊΔ?‘ (Δ?‘Δ½), Δ??Ε‘Δ?‘€ (Δ?‘Δ½) + Δ??Ε‘Δ?‘ (Δ?‘Δ½) Γ’ˆ’ Δ??Ε‘Δ?‘€ (Δ?‘Δ½)Δ??Ε‘Δ?‘ (Δ?‘Δ½) > for any x in X; Δ?œ‘
Δ?œ‘
Δ?œ‘
Δ?œ‘Δ?‘€ = < 1Γ’€“ (1Γ’€“ Δ?‘‡Δ?‘€ (Δ?‘Δ½)) , (Δ??ΕΊΔ?‘€ (Δ?‘Δ½)) , (Δ??Ε‘Δ?‘€ (Δ?‘Δ½)) > for Δ?œ‘ > 0 and any x in X;
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Δ?œ‘
Δ?œ‘
(2) (3)
Δ?œ‘
Δ?‘€Δ?œ‘ = < (Δ?‘‡Δ?‘€ (Δ?‘Δ½)) ΔΕΊŒ (1Γ’€“ Δ??ΕΊΔ?‘€ (Δ?‘Δ½)) ΔΕΊŒ (1Γ’€“ Δ??Ε‘Δ?‘€ (Δ?‘Δ½)) > , for Δ?œ‘ > 0 and any x in X.
(4)
Definition 2 [20]. Let X= {Δ?‘Δ½1 , Δ?‘Δ½2 , Γ’€Ε ΔΕΊŒΔ?‘Δ½Δ?‘› } be a space of points (objects), L and M be two SVNSs. The cosine measure between L and M is defined as follows: 1
Δ?œ‹
Δ?‘›
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Δ?œŒ(Δ??ΕΌ, Δ?‘€) = Γ’ˆ‘Δ?‘›Δ?‘–=1 Δ?‘?Δ?‘œΔ?‘ { (| Δ?‘‡Δ??ΕΌ (Δ?‘Δ½Δ?‘– ) Γ’ˆ’ Δ?‘‡Δ?‘€ (Δ?‘Δ½Δ?‘– )| + |Δ??ΕΊΔ??ΕΌ (Δ?‘Δ½Δ?‘– ) Γ’ˆ’ Δ??ΕΊΔ?‘€ (Δ?‘Δ½Δ?‘– )| + |Δ??Ε‘Δ??ΕΌ (Δ?‘Δ½Δ?‘– ) Γ’ˆ’ Δ??Ε‘Δ?‘€ (Δ?‘Δ½Δ?‘– )|)}
(5)
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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
Δ Ε0 Γ’‰Β€ Δ?œŒ(Δ??ΕΌ, Δ?‘€) Γ’‰Β€ 1; Δ Ε€ Δ?œŒ(Δ??ΕΌ, Δ?‘€) = 1 if and only if Δ??ΕΌ = Δ?‘€; Δ ΕΉ Δ?œŒ(Δ??ΕΌ, Δ?‘€) = Δ?œŒ(Δ?‘€, Δ??ΕΌ).
more elements with the same or different neutrosophic components. For
example,
neutrosophic
set
M
=
{(Δ?‘š1 , Γ’ŒΕ 0.7,0.2,0.1Γ’ŒΕ), (Δ?‘š2 , Γ’ŒΕ 0.6,0.4,0.1Γ’ŒΕ), (Δ?‘š3 , Γ’ŒΕ 0.8,0.3,0.2Γ’ŒΕ)}
rather
than
a
neutrosophic
multiset;
{ (Δ?‘˜1 , Γ’ŒΕ 0.7,0.2,0.1Γ’ŒΕ), (Δ?‘˜1 , Γ’ŒΕ 0.7,0.2,0.1Γ’ŒΕ), (Δ?‘˜1 , Γ’ŒΕ 0.7,0.2,0.1Γ’ŒΕ), (Δ?‘˜2 , Γ’ŒΕ 0.6,0.4,0.1Γ’ŒΕ) }
is
while a
is
a K=
neutrosophic
multiset, where the element Δ?‘˜1 is repeated. Then we can say that the element Δ?‘˜1 has neutrosophic multiplicity 3 with the same neutrosophic components. While L= { (Δ?‘™1 , Γ’ŒΕ 0.7,0.2,0.1Γ’ŒΕ), (Δ?‘™1 , Γ’ŒΕ 0.6,0.3,0.1Γ’ŒΕ), (Δ?‘™1 , Γ’ŒΕ 0.8,0.1,0.1Γ’ŒΕ), (Δ?‘™2 , Γ’ŒΕ 0.6,0.4,0.1Γ’ŒΕ) } is also a neutrosophic multiset since the element Δ?‘™1 is repeated, and then we can say that the element Δ?‘™1 has neutrosophic multiplicity 3 with different neutrosophic components. If the element Δ?‘™1 is repeated times with the same neutrosophic comonents, we say Δ?‘™1 has multiplicity. If the element Δ?‘™1 is repeated times with different neutrosophic comonents, we say Δ?‘™1 has the neutrosophic multiplicity (Δ?‘›Δ?‘š).the nm function can be defined as follow: Δ?‘›Δ?‘š: XΔƒ N= {1, 2, 3,Γ’€Ε, Γ’ˆž} for any Δ?‘&#x; Γ’ˆˆ Δ?‘… Δ?‘›Δ?‘š(Δ?‘&#x;) = {( Δ?‘?1 , Γ’ŒΕ Δ?‘‡1 , Δ??ΕΊ1 , Δ??Ε‘1 Γ’ŒΕ), ( Δ?‘?2 , Γ’ŒΕ Δ?‘‡2 , Δ??ΕΊ2 , Δ??Ε‘2 Γ’ŒΕ), Γ’€Ε , ( Δ?‘?Δ?‘– , Γ’ŒΕ Δ?‘‡Δ?‘– , Δ??ΕΊΔ?‘– , Δ??Ε‘Δ?‘– Γ’ŒΕ), Γ’€Ε}, which means that r is repeated by Δ?‘?1 times with the neutrosophic components Γ’ŒΕ Δ?‘‡1 , Δ??ΕΊ1 , Δ??Ε‘1 Γ’ŒΕ; r is repeated by Δ?‘?2 times with the neutrosophic components Γ’ŒΕ Δ?‘‡2 , Δ??ΕΊ2 , Δ??Ε‘2 Γ’ŒΕ; Γ’€Ε; r is repeated by Δ?‘?Δ?‘– times with the neutrosophic components Γ’ŒΕ Δ?‘‡Δ?‘– , Δ??ΕΊΔ?‘– , Δ??Ε‘Δ?‘– Γ’ŒΕ ; and so on. Δ?‘?1 , Δ?‘?2 , Γ’€Ε , Δ?‘?Δ?‘– , Γ’€Ε Γ’ˆˆ Δ?‘ , and Γ’ŒΕ Δ?‘‡Δ?‘— , Δ??ΕΊΔ?‘— , Δ??Ε‘Δ?‘— Γ’ŒΕ Γ’‰ Γ’ŒΕ Δ?‘‡Δ?‘˜ , Δ??ΕΊΔ?‘˜ , Δ??Ε‘Δ?‘˜ Γ’ŒΕ, for Δ?‘— Γ’‰ Δ?‘˜ and Δ?‘—, Δ?‘˜ Γ’ˆˆ Δ?‘ . Then a neutrosophic multiset R can be written as: (R, Δ?‘›Δ?‘š(Δ?‘&#x;)) or {(r, Δ?‘›Δ?‘š(Δ?‘&#x;) ΔΕΊŒfor Δ?‘&#x; Γ’ˆˆ Δ?‘…)}
(6)
Now, with respect to the previous neutrosophic multisets K, L, we compute the neutrosophic multiplicity function:
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Δ?‘›Δ?‘šΔ??ΕΎ : Δ??ΕΎ Γ’†’ Δ?‘ ;
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Δ?‘›Δ?‘šΔ??ΕΎ (Δ?‘˜1 ) = {(3, Γ’ŒΕ 0.7,0.2,0.1Γ’ŒΕ)}; Δ?‘›Δ?‘šΔ??ΕΎ (Δ?‘˜2 ) = {(1, Γ’ŒΕ 0.6,0.4,0.1Γ’ŒΕ)}; Δ?‘›Δ?‘šΔ??ΕΌ : Δ??ΕΌ Γ’†’ Δ?‘ ; Δ?‘›Δ?‘šΔ??ΕΌ (Δ?‘™1 ) = {(1, Γ’ŒΕ 0.7,0.2,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.6,0.3,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.8,0.1,0.1Γ’ŒΕ)}; Δ?‘›Δ?‘šΔ??ΕΌ (Δ?‘™2 ) = {(1, Γ’ŒΕ 0.6,0.4,0.1Γ’ŒΕ)}.
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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 ,Γ’€Ε, Γ’ˆž}.
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R={Δ?‘Δ½, ((Δ?‘?Δ?‘…1 , Γ’ŒΕ Δ?‘‡Δ?‘…1 (Δ?‘Δ½), Δ??ΕΊΔ?‘…1 (Δ?‘Δ½), Δ??Ε‘Δ?‘…1 (Δ?‘Δ½)Γ’ŒΕ), (Δ?‘?Δ?‘…2 , Γ’ŒΕ Δ?‘‡Δ?‘…2 (Δ?‘Δ½), Δ??ΕΊΔ?‘…2 (Δ?‘Δ½), Δ??Ε‘Δ?‘…2 (Δ?‘Δ½)Γ’ŒΕ), Γ’€Ε , (Δ?‘?Δ?‘…Δ?‘— , Γ’ŒΕ Δ?‘‡Δ?‘…Δ?‘— (Δ?‘Δ½), Δ??ΕΊΔ?‘…Δ?‘— (Δ?‘Δ½), Δ??Ε‘Δ?‘…Δ?‘— (Δ?‘Δ½)Γ’ŒΕ)) |Δ?‘Δ½ Γ’ˆˆ
A SVNM R in X can be defined as follows:
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Δ?‘‹},
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where
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Δ?‘‡Δ?‘…Δ?‘˜ (Δ?‘Δ½), Δ??ΕΊΔ?‘…Δ?‘˜ (Δ?‘Δ½), Δ??Ε‘Δ?‘…Δ?‘˜ (Δ?‘Δ½)
express
the
truth-membership
function,
the
indeterminacy-membership function and the falsity-membership function, respectively. Δ?‘‡Δ?‘…1 (Δ?‘Δ½), Δ?‘‡Δ?‘…2 (Δ?‘Δ½), Γ’€Ε , Δ?‘‡Δ?‘…Δ?‘˜ (Δ?‘Δ½) Γ’ˆˆ [0,1] , Δ??ΕΊΔ?‘…1 (Δ?‘Δ½), Δ??ΕΊΔ?‘…2 (Δ?‘Δ½), Γ’€Ε , Δ??ΕΊΔ?‘…Δ?‘˜ (Δ?‘Δ½) Γ’ˆˆ [0,1], Δ??Ε‘Δ?‘…1 (Δ?‘Δ½), Δ??Ε‘Δ?‘…2 (Δ?‘Δ½), Γ’€Ε , Δ??Ε‘Δ?‘…Δ?‘˜ (Δ?‘Δ½) Γ’ˆˆ [0,1] and 0 Γ’‰Β€ Δ?‘‡Δ?‘…Δ?‘˜ (Δ?‘Δ½) + Δ??ΕΊΔ?‘…Δ?‘˜ (Δ?‘Δ½) + Δ??Ε‘Δ?‘…Δ?‘˜ (Δ?‘Δ½) Γ’‰Β€ 3 , for k=1, 2, Γ’€Εj, Δ?‘— Γ’ˆˆ Δ?‘ , Δ?‘?Δ?‘…1 , Δ?‘?Δ?‘…2 , Γ’€Ε , Δ?‘?Δ?‘…Δ?‘— Γ’ˆˆ Δ?‘ and Δ?‘?Δ?‘…1 + Δ?‘?Δ?‘…2 + Γ’‹Ε» + Δ?‘?Δ?‘…Δ?‘— Γ’‰Δ½ 2. For convenience, a SVNM R can be denoted by the following simplified form: R={Δ?‘Δ½, (Δ?‘?Δ?‘…Δ?‘˜ , Γ’ŒΕ Δ?‘‡Δ?‘…Δ?‘˜ (Δ?‘Δ½), Δ??ΕΊΔ?‘…Δ?‘˜ (Δ?‘Δ½), Δ??Ε‘Δ?‘…Δ?‘˜ (Δ?‘Δ½)Γ’ŒΕ)|Δ?‘Δ½ Γ’ˆˆ Δ?‘‹}, for k=1, 2,Γ’€Ε,j. For example, with a universal set X= {Δ?‘Δ½1 , Δ?‘Δ½2 }, a SVNM R is given as: R={(Δ?‘Δ½1 , (2, Γ’ŒΕ 0.6,0.2,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.8,0.2,0.2Γ’ŒΕ)), (Δ?‘Δ½2 , (1, Γ’ŒΕ 0.7,0.3,0.1Γ’ŒΕ), (2, Γ’ŒΕ 0.7,0.2,0.3Γ’ŒΕ))}. Then Δ?‘›Δ?‘šΔ?‘… (Δ?‘Δ½1 ) = {(2, Γ’ŒΕ 0.6,0.2,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.8,0.2,0.2Γ’ŒΕ)}; Δ?‘›Δ?‘šΔ?‘… (Δ?‘Δ½2 ) = {(1, Γ’ŒΕ 0.7,0.3,0.1Γ’ŒΕ), (2, Γ’ŒΕ 0.7,0.2,0.3Γ’ŒΕ)}.
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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, Δ€ΕΊ Tr1 ( x), I r1 ( x), Fr1 ( x) Δ€ΕΎ),( pr 2 , Δ€ΕΊ Tr 2 ( x), I r 2 ( x), Fr 2 ( x) Δ€ΕΎ),Δ‚ΕΊ,( prj , Δ€ΕΊ Trj ( x), I rj ( x), Frj ( x) Δ€ΕΎ)) to express a basic element in a
SVNMs, Δ?‘€ = {Δ?‘Δ½, (Δ?‘?Δ?‘€Δ?‘˜ , Γ’ŒΕ Δ?‘‡Δ?‘€Δ?‘˜ (Δ?‘Δ½), Δ??ΕΊΔ?‘€Δ?‘˜ (Δ?‘Δ½), Δ??Ε‘Δ?‘€Δ?‘˜ (Δ?‘Δ½)Γ’ŒΕ)|Δ?‘Δ½ Γ’ˆˆ Δ?‘‹}, for k=1, 2, Γ’€Εj, Δ??ΕΌ = {Δ?‘Δ½, (Δ?‘?Δ??ΕΌΔ?‘˜ , Γ’ŒΕ Δ?‘‡Δ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??ΕΊΔ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??Ε‘Δ??ΕΌΔ?‘˜ (Δ?‘Δ½)Γ’ŒΕ)|Δ?‘Δ½ Γ’ˆˆ Δ?‘‹}, for k=1, 2, Γ’€Εj, then the relations of them are given as follows: Δ Ε Δ?‘€ = Δ??ΕΌ, if and only if Δ?‘?Δ?‘€Δ?‘˜ = Δ?‘?Δ??ΕΌΔ?‘˜ , Δ?‘‡Δ?‘€Δ?‘˜ (Δ?‘Δ½) = Δ?‘‡Δ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??ΕΊΔ?‘€Δ?‘˜ (Δ?‘Δ½) = Δ??ΕΊΔ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??Ε‘Δ?‘€Δ?‘˜ (Δ?‘Δ½) = Δ??Ε‘Δ??ΕΌΔ?‘˜ (Δ?‘Δ½), for k=1, 2,Γ’€Ε, j; Δ Ε€ Δ?‘€ Γ’ˆΕ Δ??ΕΌ={Δ?‘Δ½, ((Δ?‘?Δ?‘€Δ?‘˜ Γ’ˆΒ¨ Δ?‘?Δ??ΕΌΔ?‘˜ ), Γ’ŒΕ Δ?‘‡Δ?‘€Δ?‘˜ (Δ?‘Δ½) Γ’ˆΒ¨ Δ?‘‡Δ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??ΕΊΔ?‘€Δ?‘˜ (Δ?‘Δ½) Γ’ˆΒ§ Δ??ΕΊΔ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??Ε‘Δ?‘€Δ?‘˜ (Δ?‘Δ½) Γ’ˆΒ§ Δ??Ε‘Δ??ΕΌΔ?‘˜ (Δ?‘Δ½)Γ’ŒΕ)|Δ?‘Δ½ Γ’ˆˆ Δ?‘‹} Δ?‘“Δ?‘œΔ?‘&#x; Δ?‘˜ = 1, 2, Γ’€Ε , Δ?‘—; Δ ΕΉ
Δ?‘€ Γ’ˆΕ Δ??ΕΌ = {Δ?‘Δ½, ((Δ?‘?Δ?‘€Δ?‘˜ Γ’ˆΒ§ Δ?‘?Δ??ΕΌΔ?‘˜ ), Γ’ŒΕ Δ?‘‡Δ?‘€Δ?‘˜ (Δ?‘Δ½) Γ’ˆΒ§ Δ?‘‡Δ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??ΕΊΔ?‘€Δ?‘˜ (Δ?‘Δ½) Γ’ˆΒ¨ Δ??ΕΊΔ??ΕΌΔ?‘˜ (Δ?‘Δ½), Δ??Ε‘Δ?‘€Δ?‘˜ (Δ?‘Δ½) Γ’ˆΒ¨ Δ??Ε‘Δ??ΕΌΔ?‘˜ (Δ?‘Δ½)Γ’ŒΕ)|Δ?‘Δ½ Γ’ˆˆ
Δ?‘‹}, Δ?‘“Δ?‘œΔ?‘&#x; Δ?‘˜ = 1, 2, Γ’€Ε , Δ?‘—. 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= {Δ?‘Δ½1 , Δ?‘Δ½2 }, then two SVNMs M and L are given as: M={(Δ?‘Δ½1 , (2, Γ’ŒΕ 0.6,0.2,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.4,0.1,0.2Γ’ŒΕ)), (Δ?‘Δ½2 , (1, Γ’ŒΕ 0.7,0.3,0.1Γ’ŒΕ))}; L={(Δ?‘Δ½1 , (1, Γ’ŒΕ 0.6,0.2,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.8,0.2,0.1Γ’ŒΕ)), (Δ?‘Δ½2 , (1, Γ’ŒΕ 0.9,0.3,0.1Γ’ŒΕ))}; Δ?‘€Γ’ˆΕΔ??ΕΌ = {(Δ?‘Δ½1 , (2, Γ’ŒΕ 0.6,0.2,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.4,0.1,0.2Γ’ŒΕ), (1, Γ’ŒΕ 0.8,0.2,0.1Γ’ŒΕ)), (Δ?‘Δ½2 , (1, Γ’ŒΕ 0.7,0.3,0.1Γ’ŒΕ), (1, Γ’ŒΕ 0.9,0.3,0.1Γ’ŒΕ))}
Symmetry 2018, 4, x
5 of 13
π β© πΏ = {π₯1 , (1, β©0.6,0.2,0.1βͺ)}.
166 167 168 169 170
Definition 6. Let X be a space of points (objects) with a generic element x in X and M be a SVNM, we Μ by using the operational rules of SVNS. can change a SVNM M into a SVNS π
171
(7)
172 173 174 175 176 177
π = {π₯, (πππ , β©πππ (π₯), πΌππ (π₯), πΉππ (π₯)βͺ)|π₯ β π}, πππ π = 1, 2, β¦ , π. Then Μ = {β©π₯, 1 β βππ=1(1 β πππ (π₯))πππ , βππ=1(πΌππ (π₯))πππ , βππ=1(πΉππ (π₯))πππ βͺ|π₯ β π}. π Proof: Set π1 , π2 , β¦ ππ are basic elements in M. When k=1, we can get π1 = (β©ππ1 (π₯), πΌπ1 (π₯), πΉπ1 (π₯)βͺ, β©ππ1 (π₯), πΌπ1 (π₯), πΉπ1 (π₯)βͺ, β¦ , β©ππ1 (π₯), πΌπ1 (π₯), πΉπ1 (π₯)βͺ) , which has neutrosophic multiplicity ππ1 . According to the operational rules of SVNSs, we can get: ππ1
178 179
+π1 =1 β (1 β ππ1 (π₯))
180 181
+π2 =1 β (1 β ππ2 (π₯))
182
π1 +π2
183
ππ1
, (πΌπ1 (π₯))
, (πΉπ1 (π₯))
ππ2
ππ2
, (πΌπ1 (π₯))
, (πΉπ2 (π₯))
ππ2 (π₯))
ππ2
ππ1
1 β (1 β ππ1 (π₯))
ππ1
, (πΌπ1 (π₯))
(πΌπ1 (π₯))
ππ2
, (πΉπ2 (π₯)) πππ
=1 β β2π=1(1 β πππ (π₯))
ππ1
(1 β
ππ2
(πΉπ2 (π₯))
, β2π=1(πΌππ (π₯))
πππ
πππ
, β2π=1(πΉππ (π₯))
;
Suppose when k=i, the formula (7) is established, then we can get: πππ
188
π1 +π2 + β― + ππ + ππ+1 = 1 β βππ=1(1 β πππ (π₯))
, βππ=1(πΌππ (π₯))
πππ
πππ
, βππ=1(πΉππ (π₯))
;
Then
πππ
189
πππ (π₯))
190
ππ(π+1) (π₯))
πππ
+ 1 β (1 β ππ(π+1) (π₯))
ππ(π+1)
β (1 β βππ=1(1 β
)(1 β (1 β ππ(π+1)
), (βππ=1(πΌππ (π₯))
=1 β βπ+1 π=1(1 β πππ (π₯))
πππ
πππ
) (πΌπ(π+1) (π₯))
ππ(π+1)
πππ
, βπ+1 π=1(πΌππ (π₯))
πππ
, (βππ=1(πΉππ (π₯))
, βπ+1 π=1(πΉππ (π₯))
πππ
) (πΉπ(π+1) (π₯))
ππ(π+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 π = {π₯1 , π₯2 , β¦ οΌπ₯π } be a universe of discourse, and M and N be two SVNMs, and then the operational rules of SVNMs are defined as follows: π = {π₯, (πππ , β©πππ (π₯), πΌππ (π₯), πΉππ (π₯)βͺ)|π₯ β π}, for k=1, 2,β¦j; π = {π₯, (πππ , β©πππ (π₯), πΌππ (π₯), πΉππ (π₯)βͺ)|π₯ β π}, πππ π = 1, 2, β¦ π ; πβπ = {β©π₯, 1 π
π πππ
β β(1 β πππ (π₯)) π=1
β(1 π=1
π
201
.
=
π1 +π2 + β― + ππ =1 β βππ=1(1 β πππ (π₯))
200
ππ2
Then
186 187
194 195 196 197 198 199
.
As the same reason, when k=2, we can get
184 185
191 192 193
ππ1
πππ
β πππ (π₯))
π πππ
, β(πΌππ (π₯)) π=1
β(πΌππ (π₯)) π=1
π πππ
π πππ
, β(πΉππ (π₯)) π=1
β(πΉππ (π₯)) π=1
πππ
βͺ |π₯ β π}
Symmetry 2018, 4, x
6 of 13
π
202
π πππ
π β π = {β©π₯, (1 β β(1 β πππ (π₯))
) (1 β β(1 β πππ (π₯))
π=1
πππ
π=1
+ β(πΉππ (π₯))
π=1
π
β β(πΉππ (π₯))
πππ
β(πΉππ (π₯)) π
π πππ
) ) , (β(πΌππ (π₯))
π=1
πππ
ππ = {β©π₯, (1 β β(1 β πππ (π₯))
) βͺ |π₯ β π}
π=1
π πππ
π
π
) , (β(πΉππ (π₯))
π=1
π
βͺ |π₯ β π}
π=1
π πππ
πππ
, β(πΉππ (π₯))
π=1
π=1
ππ = {β©π₯, (1 β (β(1 β πππ (π₯))
π
π
) , 1 β (1 β β(πΌππ (π₯))
π=1
πππ
) ,1
π=1 π
π πππ
β (1 β β(πΉππ (π₯))
207
β(πΌππ (π₯))
πππ
π
π πππ
π πππ
π=1
206
πππ
π=1
π
205
π
β β(πΌππ (π₯))
πππ
π=1
π
+ β(πΌππ (π₯))
204
) , β(πΌππ (π₯))
π=1
π
203
π πππ
) βͺ |π₯ β π}
π=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 π = {π₯1 , π₯2 , β¦ οΌπ₯π } be a universe of discourse, M and N be two SVNMs,
214 215 216 217
π},
218 219
π= {π₯π , (ππ1 , β©ππ1 (π₯π ), πΌπ1 (π₯π ), πΉπ1 (π₯π )βͺ), (ππ2 , β©ππ2 (π₯π ), πΌπ2 (π₯π ), πΉπ2 (π₯π )βͺ), β¦ , (πππ , β©πππ (π₯π ), πΌππ (π₯π ), πΉππ (π₯π )βͺ)|π₯π β π = {β©π₯π , (ππ1 , β©ππ1 (π₯π ), πΌπ1 (π₯π ), πΉπ1 (π₯π )βͺ), (ππ2 , β©ππ2 (π₯π ), πΌπ2 (π₯π ), πΉπ2 (π₯π )βͺ), β¦ , (πππ , β©πππ (π₯π ), πΌππ (π₯π ), πΉππ (π₯π )βͺ)βͺ|π₯π β π}
Then, a cosine measure between two SVNMs M and N is defined as follows: π
π
π
π=1
π=1
π=1
1 π πππ πππ π(π, π) = β πππ { |β(1 β πππ (π₯π )) β β(1 β πππ (π₯π )) | π 6 π
+ |β(πΌππ (π₯π ))
220
π=1
π πππ
β β(πΌππ (π₯π )) π=1
π πππ
π πππ
| + |β(πΉππ (π₯π )) π=1
β β(πΉππ (π₯π ))
πππ
|}
π=1
221
(8)
222 223 224 225 226
Theorem 1. The cosine measure π(π, π) between two SVNMs M and N satisfies the following
227
Proof οͺ:
properties: οͺ π(π, π) = π(π, π); ο« 0 β€ π(π, π) β€ 1; ο¬ π(π, π) = 1, ππ πππ ππππ¦ ππ π = π.
Symmetry 2018, 4, x
228 229 230 231 232 233 234 235 236
7 of 13
π
For |βπ=1(1 β πππ (π₯π ))
πππ
π
β βπ=1(1 β πππ (π₯π ))
πππ
πππ
πππ
πππ
π
| + |βπ=1(πΌππ (π₯π ))
πππ π π βππ=1(πΌππ (π₯π )) | + |βππ=1(πΉππ (π₯π )) β βπ=1(πΉππ (π₯π )) | = |βπ=1(1 β πππ (π₯π )) β πππ πππ πππ πππ π π π π βπ=1(1 β πππ (π₯π )) | + |βπ=1(πΌππ (π₯π )) β βπ=1(πΌππ (π₯π )) | + |βπ=1(πΉππ (π₯π )) β πππ π βπ=1(πΉππ (π₯π )) | , so we can get π(π, π) = π(π, π) .
ο«: 0 β€ πππ (π₯π ) β€ 1 , 0 β€ πΌππ (π₯π ) β€ 1 , 0 β€ πΉππ (π₯π ) β€ 1,0 β€ πππ (π₯π ) β€ 1 , 0 β€ πΌππ (π₯π ) β€
For
1 , 0 β€ πΉππ (π₯π ) β€ 1 ; Then we can get 0 β€ 1 β πππ (π₯π ) β€ 1 πππ 0 β€ 1 β πππ (π₯π ) β€ 1; π
πππ
237 238
0 β€ βπ=1(1 β πππ (π₯π ))
239 240
0 β€ |βπ=1(1 β πππ (π₯π ))
241 242 243 244
0 β€ |βπ=1(πΌππ (π₯π ))
π
β€ 1 πππ 0 β€ βπ=1(1 β πππ (π₯π ))
πππ
β€ 1;
So, πππ
π
π
β βπ=1(1 β πππ (π₯π ))
πππ
| β€ 1.
As the same reason, we can get π
π βππ=1(πΉππ (π₯π )) ππ |
Above
πππ
π
β βπ=1(πΌππ (π₯π ))
|β€1
π
0 β€ |βπ=1(πΉππ (π₯π ))
and
we
β
can
π βππ=1(πΌππ (π₯π )) ππ |
π
πππ
πππ
π
+
πππ π |βπ=1(πΉππ (π₯π ))
π
πππ
β
πππ
βππ=1(πΌππ (π₯π ))
247
1 π πππ πππ π(π, π) = β πππ { |β(1 β πππ (π₯π )) β β(1 β πππ (π₯π )) | π 6
πππ
π
π
| + |βπ=1(πΉππ (π₯π ))
β βπ=1(πΉππ (π₯π ))
π
π
π
π=1
π=1
π=1
π
248
πππ
1
=
π
βππ=1(πΌππ (π₯1 ))
251
πππ (π₯2 ))
252
|βπ=1(πΉππ (π₯2 ))
253
πππ (π₯π ))
254
βππ=1(πΉππ (π₯π ))
πππ
π
πππ
πππ
π
πππ
π
| + |βπ=1(πΉππ (π₯1 ))
π
πππ π
πππ
π
| + |βπ=1(πΌππ (π₯1 ))
π
π
πππ
πππ
β
π
|)} + πππ { (|βπ=1(1 β π
β βπ=1(πΌππ (π₯2 )) π
πππ
|+
πππ
6
β βπ=1(πΌππ (π₯π ))
|}
6
π
πππ
π
6
256
βππ=1(πΌππ (π₯π ))
257
π(π, π) = {π1 + π2 + β― + ππ }.
258
πππ
π
| + |βπ=1(πΉππ (π₯π ))
πππ
π
β βπ=1(1 β
β
|)}). π
πππ
πππ
β
β β(πΉππ (π₯π ))
|)} + β― + πππ { (|βπ=1(1 β πππ (π₯π ))
πππ { (|βπ=1(1 β πππ (π₯π ))
Let
πππ
β βπ=1(πΉππ (π₯1 )) π
πππ
π
πππ
π
| + |βπ=1(πΌππ (π₯2 ))
β βπ=1(πΉππ (π₯2 ))
| + |βπ=1(πΌππ (π₯π )) πππ
πππ
β€ 3 and 0 β€
π=1
β βπ=1(1 β πππ (π₯1 ))
π
|+
π πππ
π=1
6
β βπ=1(1 β πππ (π₯2 ))
π
β
|} β€ 1;
| + |β(πΉππ (π₯π ))
π=1
(πππ { (|βπ=1(1 β πππ (π₯1 ))
πππ
250
πππ
π πππ
β β(πΌππ (π₯π ))
π=1
249
π
| + |βπ=1(πΌππ (π₯π ))
π
+ |β(πΌππ (π₯π ))
β βπ=1(1 β πππ (π₯π ))
π βππ=1(πΉππ (π₯π )) ππ |
246
β βπ=1(1 β πππ (π₯π ))
6
πππ
π
0 β€ |βπ=1(1 β πππ (π₯π ))
get
βππ=1 πππ { |βππ=1(1 β πππ (π₯π )) πππ
πππ
β€1
all,
πππ π |βπ=1(πΌππ (π₯π ))
πππ
245
255
β
πππ
π
πππ
| + |βπ=1(πΉππ (π₯π ))
πππ
π
β βπ=1(1 β πππ (π₯π ))
π
β βπ=1(πΉππ (π₯π ))
πππ
π
| + |βπ=1(πΌππ (π₯π ))
|)} = ππ (π = 1,2, β¦ ποΌ
,
πππ
β
then
1
π
According
to
π
π
πππ
0 β€ πππ { (|βπ=1(1 β πππ (π₯π )) 6
π
πππ
β βπ=1(1 β πππ (π₯π ))
|+
Symmetry 2018, 4, x
Δ?‘?Δ?‘€Δ?‘˜
Δ?‘—
8 of 13
Δ?‘—
Γ’ˆ’ Γ’ˆ?Δ?‘˜=1(Δ??ΕΊΔ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
Δ?‘?Δ?‘ Δ?‘˜
Δ?‘?Δ?‘€Δ?‘˜
Δ?‘—
| + |Γ’ˆ?Δ?‘˜=1(Δ??Ε‘Δ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
Δ?‘—
Γ’ˆ’ Γ’ˆ?Δ?‘˜=1(Δ??Ε‘Δ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
Δ?‘?Δ?‘ Δ?‘˜
259
|Γ’ˆ?Δ?‘˜=1(Δ??ΕΊΔ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
260
we can obtain 0 Γ’‰Β€ Δ?‘Ž1 + Δ?‘Ž2 + Γ’‹Ε» + Δ?‘ŽΔ?‘› Γ’‰Β€ Δ?‘› and 0 Γ’‰Β€ (Δ?‘Ž1 + Δ?‘Ž2 + Γ’‹Ε» + Δ?‘ŽΔ?‘› ) Γ’‰Β€ 1, so we can get 0 Γ’‰Β€
261 262 263 264 265 266 267
Δ?œŒ(Δ?‘€, Δ?‘ ) Γ’‰Β€ 1 .
|)} = Δ?‘ŽΔ?‘– Γ’‰Β€ 1
,
1
Δ?‘›
Δ ΕΉ If M=N then Δ?‘‡Δ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– ) = Δ?‘‡Δ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– ), Δ??ΕΊΔ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– ) = Δ??ΕΊΔ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– ), Δ?‘ŽΔ?‘›Δ?‘‘ Δ??Ε‘Δ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– ) = Δ??Ε‘Δ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– ) for any Δ?‘Δ½Δ?‘– Γ’ˆˆ Δ?‘‹ Δ?‘ŽΔ?‘›Δ?‘‘ Δ?‘– = 1,2, Γ’€Ε Δ?‘›, so we can get Δ?œŒ(Δ?‘€, Δ?‘ ) = 1, if and only if Δ?‘€ = Δ?‘ . Now, we consider different weights for each element Δ?‘Δ½Δ?‘– (Δ?‘– = 1,2, Γ’€Ε , Δ?‘›) in X. Then, let Δ?‘Β€ = (Δ?‘Β€1 , Δ?‘Β€2 , Γ’€Ε , Δ?‘Β€Δ?‘› )Δ?‘‡ be the weight vector of each element Δ?‘Δ½Δ?‘– (Δ?‘– = 1,2, Γ’€Ε , Δ?‘›) with Δ?‘Β€Δ?‘– Γ’ˆˆ [0,1], and Γ’ˆ‘Δ?‘›Δ?‘–=1 Δ?‘Β€Δ?‘– = 1. Hence, we further extend the cosine measure of Equation (8) to the following weighted cosine measure of SVNM: Δ?‘›
Δ?‘—
Δ?‘—
Δ?‘–=1
Δ?‘˜=1
Δ?‘˜=1
Δ?œ‹ Δ?‘?Δ?‘€Δ?‘˜ Δ?‘?Δ?‘ Δ?‘˜ Δ?œŒΔ?‘Β€ (Δ?‘€, Δ?‘ ) = Γ’ˆ‘ Δ?‘Β€Δ?‘– Δ?‘?Δ?‘œΔ?‘ { (|Γ’ˆ?(1 Γ’ˆ’ Δ?‘‡Δ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– )) Γ’ˆ’ Γ’ˆ?(1 Γ’ˆ’ Δ?‘‡Δ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– )) | 6
268
Δ?‘—
Δ?‘— Δ?‘?Δ?‘€Δ?‘˜
+ |Γ’ˆ?(Δ??ΕΊΔ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
269
Γ’ˆ’ Γ’ˆ?(Δ??ΕΊΔ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
Δ?‘˜=1
|
Δ?‘˜=1
Δ?‘—
Δ?‘— Δ?‘?Δ?‘€Δ?‘˜
+ |Γ’ˆ?(Δ??Ε‘Δ?‘€Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
270
Δ?‘?Δ?‘ Δ?‘˜
Δ?‘˜=1
Γ’ˆ’ Γ’ˆ?(Δ??Ε‘Δ?‘ Δ?‘˜ (Δ?‘Δ½Δ?‘– ))
Δ?‘?Δ?‘ Δ?‘˜
|)}
Δ?‘˜=1
271
(9)
272 273 274 275 276 277
Theorem 2. The cosine measure Δ?œŒΔ?‘Β€ (Δ?‘€, Δ?‘ ) 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: Δ Ε Δ?œŒΔ?‘Β€ (Δ?‘€, Δ?‘ ) = Δ?œŒΔ?‘Β€ (Δ?‘ , Δ?‘€); Δ Ε€ 0 Γ’‰Β€ Δ?œŒΔ?‘Β€ (Δ?‘€, Δ?‘ ) Γ’‰Β€ 1; Δ ΕΉ Δ?œŒΔ?‘Β€ (Δ?‘€, Δ?‘ ) = 1, Δ?‘–Δ?‘“ Δ?‘ŽΔ?‘›Δ?‘‘ Δ?‘œΔ?‘›Δ?‘™Δ?‘Ε Δ?‘–Δ?‘“ Δ?‘€ = Δ?‘ . 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 Δ??Ε = {Δ?‘”1 , Δ?‘”2 , Γ’€Ε ΔΕΊŒΔ?‘”Δ?‘š } as a set of alternatives and Δ?‘‹ = {Δ?‘Δ½1 , Δ?‘Δ½2 , Γ’€Ε ΔΕΊŒΔ?‘Δ½Δ?‘› } as a set of attributes, then they can be established in a decision-making problem. But sometimes Δ?‘Δ½Δ?‘– (Δ?‘– = 1,2, Γ’€Ε , Δ?‘›) maybe have multiplicity, and then we can use the form of a SVNM to represent the evaluation value. Let Δ?‘”Δ?‘&#x; = {Δ?‘Δ½Δ?‘– , (Δ?‘?Δ?‘” 1 , Γ’ŒΕ Δ?‘‡Δ?‘” 1 (Δ?‘Δ½Δ?‘– ), Δ??ΕΊΔ?‘” 1 (Δ?‘Δ½Δ?‘– ), Δ??Ε‘Δ?‘” 1 (Δ?‘Δ½Δ?‘– )Γ’ŒΕ), (Δ?‘?Δ?‘” 2 , Γ’ŒΕ Δ?‘‡Δ?‘” 2 (Δ?‘Δ½Δ?‘– ), Δ??ΕΊΔ?‘” 2 (Δ?‘Δ½Δ?‘– ), Δ??Ε‘Δ?‘” 2 (Δ?‘Δ½Δ?‘– )Γ’ŒΕ), Γ’€Ε , (Δ?‘?Δ?‘” Δ?‘— , Γ’ŒΕ Δ?‘‡Δ?‘” Δ?‘— (Δ?‘Δ½Δ?‘– ), Δ??ΕΊΔ?‘” Δ?‘— (Δ?‘Δ½Δ?‘– ), Δ??Ε‘Δ?‘” Δ?‘— (Δ?‘Δ½Δ?‘– )Γ’ŒΕ)|Δ?‘Δ½Δ?‘– Γ’ˆˆ Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘&#x; Δ?‘‹},
for r=1,2,Γ’€Ε,m and i=1,2,Γ’€Ε,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 , Γ’ŒΕ Tg11 (x1 ), Ig1 1 (x1 ), Fg1 1 (x1 )Γ’ŒΕ), Γ’€Ε , (pg1j , Γ’ŒΕ Tg1 j (x1 ), Ig1 j (x1 ), Fg1 j (x1 )Γ’ŒΕ)
Γ’€Ε Γ’€Ε
Symmetry 2018, 4, x
g2
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x1 , (pg2 1 , Γ’ŒΕ Tg21 (x1 ), Ig2 1 (x1 ), Fg2 1 (x1 )Γ’ŒΕ), Γ’€Ε , (pg2j , Γ’ŒΕ Tg2 j (x1 ), Ig2 j (x1 ), Fg2 j (x1 )Γ’ŒΕ)
Γ’€Ε gm
Γ’€Ε
Γ’€Ε
Γ’€Ε
x1 , (pgm 1 , Γ’ŒΕ Tgm1 (x1 ), Igm 1 (x1 ), Fgm1 (x1 )Γ’ŒΕ), Γ’€Ε , (pgmj , Γ’ŒΕ Tgmj (x1 ), Igmj (x1 ), Fgm j (x1 )Γ’ŒΕ)
Γ’€Ε
xn xn , (pg1 1 , Γ’ŒΕ Tg1 1 (xn ), Ig11 (xn ), Fg11 (xn )Γ’ŒΕ), Γ’€Ε , (pg1j , Γ’ŒΕ Tg1 j (xn ), Ig1 j (xn ), Fg1j (xn )Γ’ŒΕ) xn , (pg2 1 , Γ’ŒΕ Tg2 1 (xn ), Ig21 (xn ), Fg21 (xn )Γ’ŒΕ), Γ’€Ε , (pg2j , Γ’ŒΕ Tg2 j (xn ), Ig2 j (xn ), Fg2j (xn )Γ’ŒΕ) Γ’€Ε xn , (pgm 1 , Γ’ŒΕ Tgm 1 (xn ), Igm1 (xn ), Fgm1 (xn )Γ’ŒΕ), Γ’€Ε , (pgm j , Γ’ŒΕ Tgmj (xn ), Igm j (xn ), Fgmj (xn )Γ’ŒΕ)
290 291
Step 1: By using the Equation (7), we change the SVNM decision matrix D into SVNS Δƒ , which is shown in Table 2. decision matrix Δ??Λ
292
Δƒ Table 2. The SVNS decision matrix Δ??Λ x1 gΔƒ1
j
Γ’ŒΕ x1 , 1 Γ’ˆ’ Γ’ˆ? (1 Γ’ˆ’ Tg1 k (x1 ))
pg1 k
j
, Γ’ˆ? (Ig1k (x1 ))
k=1
gă 2
Γ’€Ε pg1 k
j
, Γ’ˆ? (Fg1 k (x1 ))
k=1
j
Γ’ŒΕ x1 , 1 Γ’ˆ’ Γ’ˆ? (1 Γ’ˆ’ Tg2 k (x1 ))
pg2 k
Γ’€Ε Γ’ŒΕ
k=1
j
, Γ’ˆ? (Ig2k (x1 ))
k=1
pg1 k
pg2 k
j
, Γ’ˆ? (Fg2 k (x1 ))
k=1
pg2 k
Γ’€Ε Γ’ŒΕ
k=1
Γ’€Ε
Γ’€Ε
gă m
j
Γ’ŒΕ x1 , 1 Γ’ˆ’ Γ’ˆ? (1 Γ’ˆ’ Tgm k (x1 ))
pgm k
k=1
Γ’€Ε
j
, Γ’ˆ? (Igm k (x1 ))
pgm k
j
, Γ’ˆ? (Fgmk (x1 ))
k=1
pgm k
Γ’€Ε Γ’ŒΕ
k=1
xn j
Γ’ŒΕ xn , 1 Γ’ˆ’ Γ’ˆ? (1 Γ’ˆ’ Tg1 k (xn ))
pg1 k
j
, Γ’ˆ? (Ig1 k (xn ))
k=1
pg1 k
k=1
j
Γ’ŒΕ xn , 1 Γ’ˆ’ Γ’ˆ? (1 Γ’ˆ’ Tg2 k (xn ))
pg2 k
, Γ’ˆ? (Fg1k (xn ))
pg1 k
Γ’ŒΕ
k=1
j
, Γ’ˆ? (Ig2 k (xn ))
k=1
j
pg2 k
k=1
j
, Γ’ˆ? (Fg2k (xn ))
pg2 k
Γ’ŒΕ
k=1
Γ’€Ε j
Γ’ŒΕ xn , 1 Γ’ˆ’ Γ’ˆ? (1 Γ’ˆ’ Tgm k (xn )) k=1
293 294 295 296 297
pgm k
j
, Γ’ˆ? (Igm k (xn ))
pgm k
k=1
j
, Γ’ˆ? (Fgmk (xn ))
pgm k
Γ’ŒΕ
k=1
Δ?‘‡ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– ) is the maximum truth value in each Δƒ column Δ?‘Δ½Δ?‘– of the decision matrix Δ??Λ , Δ??ΕΊ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– ) and Δ??Ε‘ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– ) are the minimum indeterminate and Δƒ , respectively, the ideal solution can be falsity values in each column Δ?‘Δ½Δ?‘– of the decision matrix Δ??Λ Step
2:
Setting
determined as Δ?‘Δ½Δ?‘–Γ’ˆ— . Δ?‘Δ½Δ?‘–Γ’ˆ— = Γ’ŒΕ Δ?‘‡ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– ), Δ??ΕΊ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– ), Δ??Ε‘ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– )Γ’ŒΕ, for i=1,2,Γ’€Εn.
Symmetry 2018, 4, x
298 299
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So we can get the ideal alternative Δ?‘”Γ’ˆ— = {Δ?‘Δ½1,Γ’ˆ— , Δ?‘Δ½2,Γ’ˆ— Γ’€Ε , Δ?‘Δ½Δ?‘›Γ’ˆ— }. Step 3: When the weight vector of attributes for the different importance of each attribute Δ?‘‡
300
Δ?‘Δ½Δ?‘– (Δ?‘– = 1,2, Γ’€Ε , Δ?‘›) is given by Δ?‘Β€ = (Δ?‘Β€1, Δ?‘Β€2, Γ’€Ε , Δ?‘Β€Δ?‘› ) with Δ?‘Β€Δ?‘– Γ’‰Δ½ 0 and Γ’ˆ‘Δ?‘›Δ?‘–=1 Δ?‘Β€Δ?‘– = 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 Δ?‘”ΔƒΔ?‘&#x; (Δ?‘&#x; = 1,2, Γ’€Ε ΔΕΊŒΔ?‘š)
304
Δ?œŒΔ?‘Β€ (Δ?‘”Δ?‘&#x; , Δ?‘”Γ’ˆ— ) = Δ?œŒΔ?‘Β€ (Δ?‘”ΔƒΔ?‘&#x; , Δ?‘”Γ’ˆ— ) = Γ’ˆ‘Δ?‘›Δ?‘–=1 Δ?‘Β€Δ?‘– Δ?‘?Δ?‘œΔ?‘ { (|Δ?‘‡Δ?‘”ΔƒΔ?‘&#x; (Δ?‘Δ½Δ?‘– ) Γ’ˆ’ Δ?‘‡ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– )| + |Δ??ΕΊΔ?‘”ΔƒΔ?‘&#x; (Δ?‘Δ½Δ?‘– ) Γ’ˆ’ Δ??ΕΊ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– )| + |Δ??Ε‘Δ?‘”ΔƒΔ?‘&#x; (Δ?‘Δ½Δ?‘– ) Γ’ˆ’
305
Δ??Ε‘ Δ?‘”Γ’ˆ— (Δ?‘Δ½Δ?‘– )|)}.
306 307 308 309 310
and the ideal alternative Δ?‘”Γ’ˆ— can be calculated by using the following formula: Δ?œ‹ 6
(10) Step 4: According to the values of Δ?œŒΔ?‘Β€ (Δ?‘”ΔƒΔ?‘&#x; , Δ?‘”
Γ’ˆ—)
for r=1, 2, Γ’€Ε, 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 (Δ?‘”1 , Δ?‘”2 , Δ?‘”3 , Δ?‘”4 ) to be considered. Decision should be taken according to four attributes: fuel economy (Δ?‘Δ½1 ), price (Δ?‘Δ½2 ), comfort (Δ?‘Δ½3 ), safety (Δ?‘Δ½4 ).The weight vector of these four attributes is given by Δ?‘Β€ = (0.5,0.25,0.125,0.125)Δ?‘‡ . 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.
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325 326
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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 Δ?‘Δ½1
Δ?‘Δ½2
Δ?‘Δ½3
Δ?‘Δ½4
Δ?‘”1
(1, Γ’ŒΕ 0.5,0.7,0.2Γ’ŒΕ), (1, Γ’ŒΕ 0.7,0.3,0.6Γ’ŒΕ) 1, Γ’ŒΕ 0.4,0.4,0.5Γ’ŒΕ
(1, Γ’ŒΕ 0.7,0.7.0.5Γ’ŒΕ), (1, Γ’ŒΕ 0.8,0.7,0.6Γ’ŒΕ)
(1, Γ’ŒΕ 0.1,0.5,0.7Γ’ŒΕ), (1, Γ’ŒΕ 0.5,0.2,0.8Γ’ŒΕ)
Δ?‘”2
(1, Γ’ŒΕ 0.9,0.7,0.5Γ’ŒΕ), (1, Γ’ŒΕ 0.7,0.7,0.1Γ’ŒΕ) 1, Γ’ŒΕ 0.7,0.6,0.8Γ’ŒΕ
2, Γ’ŒΕ 0.9,0.4,0.6Γ’ŒΕ
(1, Γ’ŒΕ 0.5,0.2,0.7Γ’ŒΕ, (1, Γ’ŒΕ 0.5,0.1,0.9Γ’ŒΕ)
Δ?‘”3
(1, Γ’ŒΕ 0.3,0.4,0.2Γ’ŒΕ, (1, Γ’ŒΕ 0.6,0.3,0.7Γ’ŒΕ) 1, Γ’ŒΕ 0.2,0.2,0.2Γ’ŒΕ
(1, Γ’ŒΕ 0.9,0.5,0.5Γ’ŒΕ, (1, Γ’ŒΕ 0.6,0.5,0.2Γ’ŒΕ)
(1, Γ’ŒΕ 0.7,0.5,0.3Γ’ŒΕ, (1, Γ’ŒΕ 0.4,0.2,0.2Γ’ŒΕ)
Δ?‘”4
(1, Γ’ŒΕ 0.9,0.7,0.2Γ’ŒΕ, (1, Γ’ŒΕ 0.8,0.6,0.1Γ’ŒΕ) 1, Γ’ŒΕ 0.3,0.5,0.2Γ’ŒΕ
(1, Γ’ŒΕ 0.5,0.4,0.5Γ’ŒΕ, (1, Γ’ŒΕ 0.1,0.7,0.2Γ’ŒΕ)
2, Γ’ŒΕ 0.4,0.2,0.8Γ’ŒΕ
328 329
Step 2: By using the Equation (7), we change the SVNM decision matrix D into SVNS Δƒ , which is shown in Table 4. decision matrix Δ??Λ
330
Δƒ Table 4. The SVNS decision matrix Δ??Λ
331 332 333 334 335 336 337 338 339 340 341 342 343
Δ?‘Δ½1
Δ?‘Δ½2
Δ?‘Δ½3
Δ?‘Δ½4
Δ?‘”Δƒ1
Γ’ŒΕ 0.85,0.21,0.12Γ’ŒΕ
Γ’ŒΕ 0.4,0.4,0.5Γ’ŒΕ
Γ’ŒΕ 0.94,0.49,0.3Γ’ŒΕ
Γ’ŒΕ 0.55,0.1,0.56Γ’ŒΕ
Δ?‘”Δƒ2
Γ’ŒΕ 0.97,0.49,0.05Γ’ŒΕ
Γ’ŒΕ 0.7,0.6,0.8Γ’ŒΕ
Γ’ŒΕ 0.99,0.16,0.36Γ’ŒΕ
Γ’ŒΕ 0.75,0.02,0.63Γ’ŒΕ
Δ?‘”Δƒ3
Γ’ŒΕ 0.72,0.12,0.14Γ’ŒΕ
Γ’ŒΕ 0.2,0.2,0.2Γ’ŒΕ
Γ’ŒΕ 0.96,0.25,0.1Γ’ŒΕ
Γ’ŒΕ 0.82,0.1,0.06Γ’ŒΕ
Δ?‘”Δƒ4
Γ’ŒΕ 0.98,0.42,0.02Γ’ŒΕ
Γ’ŒΕ 0.3,0.5,0.2Γ’ŒΕ
Γ’ŒΕ 0.55,0.28,0.1Γ’ŒΕ
Γ’ŒΕ 0.64,0.04,0.64Γ’ŒΕ
Step 3: According to the decision matrix Δƒ Δ??Λ , we can get the ideal alternative Δ?‘”Γ’ˆ— : Δ?‘”Γ’ˆ— = {Γ’ŒΕ 0.98,0.12,0.02Γ’ŒΕΓ’ŒΕ 0.7,0.2,0.2Γ’ŒΕΓ’ŒΕ 0.99,0.16,0.1Γ’ŒΕΓ’ŒΕ 0.82,0.02,0.06Γ’ŒΕ}. Step 4: By applying the Equation (10), we can obtain the values of the weighted cosine measure between each alternative and the ideal alternative Δ?‘”Γ’ˆ— as follows: Δ?œŒΔ?‘Β€ (Δ?‘”1 , Δ?‘”Γ’ˆ— ) = 0.9535, Δ?œŒΔ?‘Β€ (Δ?‘”2 , Δ?‘”Γ’ˆ— ) = 0.9511, Δ?œŒΔ?‘Β€ (Δ?‘”3 , Δ?‘”Γ’ˆ— ) = 0.9813 and Δ?œŒΔ?‘Β€ (Δ?‘”4 , Δ?‘”Γ’ˆ— ) = 9616. Step 5: According to the above values of weighted cosine measure, we can rank the four alternatives: Δ?‘”3 Γ’‰Ε₯ Δ?‘”4 Γ’‰Ε₯ Δ?‘”1 Γ’‰Ε₯ Δ?‘”2 . Therefore, the alternative Δ?‘”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]
Δ?œŒΔ?‘Β€ (Δ?‘”1 , Δ?‘”
Γ’ˆ— )=0.9053,
Δ?œŒΔ?‘Β€ (Δ?‘”2 , Δ?‘”
Γ’ˆ—)
=0.9017,
Δ?œŒΔ?‘Β€ (Δ?‘”3 , Δ?‘”
Γ’ˆ—)
=0.9516,
Δ?œŒΔ?‘Β€ (Δ?‘”4 , Δ?‘”
Γ’ˆ—)
=0.8816.
Ranking order Δ?‘”3 Γ’‰Ε₯ Δ?‘”1 Γ’‰Ε₯ Δ?‘”2 Γ’‰Ε₯ Δ?‘”4
The best alternative Δ?‘”3
Symmetry 2018, 4, x
Method
2
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based
on
similarity in [16]
Method
3
based
similarity in [16]
The proposed method
on
Δ?œŒΔ?‘Β€ (Δ?‘”1 , Δ?‘”Γ’ˆ— )=0.8204, Δ?œŒΔ?‘Β€ (Δ?‘”2 , Δ?‘”
Γ’ˆ—)
=0.8108,
Δ?œŒΔ?‘Β€ (Δ?‘”3 , Δ?‘”
Γ’ˆ—)
=0.8867,
Δ?œŒΔ?‘Β€ (Δ?‘”4 , Δ?‘”
Γ’ˆ—)
=0.8358.
Δ?œŒΔ?‘Β€ (Δ?‘”1 , Δ?‘”
Γ’ˆ— )=0.7898,
Δ?œŒΔ?‘Β€ (Δ?‘”2 , Δ?‘”
Γ’ˆ—)
=0.7121,
Δ?œŒΔ?‘Β€ (Δ?‘”3 , Δ?‘”
Γ’ˆ—)
=0.8125,
Δ?œŒΔ?‘Β€ (Δ?‘”4 , Δ?‘”
Γ’ˆ—)
=0.7553.
Δ?œŒΔ?‘Β€ (Δ?‘”1 , Δ?‘”
Γ’ˆ— )=0.9535,
Δ?‘”3 Γ’‰Ε₯ Δ?‘”4 Γ’‰Ε₯ Δ?‘”2 Γ’‰Ε₯ Δ?‘”1
Δ?‘”3
Δ?‘”3 Γ’‰Ε₯ Δ?‘”1 Γ’‰Ε₯ Δ?‘”4 Γ’‰Ε₯ Δ?‘”2
Δ?‘”3
Δ?‘”3 Γ’‰Ε₯ Δ?‘”4 Γ’‰Ε₯ Δ?‘”1 Γ’‰Ε₯ Δ?‘”2
Δ?‘”3
Δ?œŒΔ?‘Β€ (Δ?‘”2 , Δ?‘”Γ’ˆ— ) =0.9511, Δ?œŒΔ?‘Β€ (Δ?‘”3 , Δ?‘”Γ’ˆ— ) =0.9813, Δ?œŒΔ?‘Β€ (Δ?‘”4 , Δ?‘”Γ’ˆ— ) =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 Δ?‘”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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375 376 377
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.
378
Conflicts of Interest: The author declares no conflict of interest.
379
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