# Vector Similarity Measures between Refined Simplified Neutrosophic Sets and Their Multiple Attribute Decision-Making Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−}0, 1

^{+}[, Smarandache [4], Wang et al. [5,6], and Ye [7,8] constrained the three membership degrees in the neutrosophic set to the single-valued membership degrees and the interval membership degrees. These become a single-valued neutrosophic set (SVNS), an interval neutrosophic set (INS), and a simplified neutrosophic set (SNS) (including SVNS and INS), respectively. Obviously, they are subclasses of the neutrosophic set for convenient applications in science and engineering fields, such as decision-making [7,8,9,10,11,12,13] and fault diagnosis [14]. However, because there are both arguments and sub-arguments/refined arguments in the truth, indeterminacy, and falsity membership degrees of T, I, F in the neutrosophic set to express complex problems of the real world in detail, one needs to refine truth, indeterminacy, and falsity information. Hence, Smarandache [15] further extended the neutrosophic logic to n-valued refined neutrosophic logic, where he refined/split the truth, indeterminacy, and falsity functions T, I, F into T

_{1}, T

_{2}, ..., T

_{r}, I

_{1}, I

_{2}, ..., I

_{s}, and F

_{1}, F

_{2}, ..., F

_{t}, respectively, and constructed them as a n-valued refined neutrosophic set. Moreover, some researchers extended the neutrosophic set to multi-valued neutrosophic set/neutrosophic multiset/neutrosophic refined sets and applied them to medical diagnoses [16,17,18] and decision-making [19,20,21]. In fact, the multi-valued neutrosophic sets/neutrosophic refined sets are neutrosophic multisets in their expressed forms [22,23]. Hence, these multi-valued neutrosophic sets/neutrosophic refined sets, that is, neutrosophic multisets, and their decision-making methods cannot express and deal with decision-making problems with both attributes and sub-attributes. To solve the issue, Ye and Smarandache [22] proposed a refined single-valued neutrosophic set (RSVNS), where the neutrosophic set {T, I, F} was refined into the RSVNS {(T

_{1}, T

_{2}, ..., T

_{r}), (I

_{1}, I

_{2}, ..., I

_{r}), (F

_{1}, F

_{2}, ..., F

_{r})}, and proposed the similarity measures based on union and intersection operations of RSVNSs to solve decision-making problems with both attributes and sub-attributes. Then, Fan and Ye [23] further presented the cosine measures of RSVNSs and refined interval neutrosophic sets (RINSs) based the distance and cosine function and applied them to the decision-making problems with both attributes and sub-attributes under refined single-value/interval neutrosophic environments. However, these cosine measures cannot handle such a decision-making problem with the weights of both attributes and sub-attributes.

## 2. Basic Concepts of SNSs and Vector Similarity Measures of SNSs

^{−}0, 1

^{+}[. For convenient science and engineering applications, we need to constrain them in the real standard interval [0, 1] from a science and engineering point of view. Thus, Ye [7,8] introduced the concept of SNS as a simplified form/subclass of the neutrosophic set.

_{A}(x), I

_{A}(x), and F

_{A}(x), which is denoted as $A=\left\{\langle x,{T}_{A}(x),{I}_{A}(x),{F}_{A}(x)\rangle |x\in X\right\}$, where T

_{A}(x), I

_{A}(x) and F

_{A}(x) are singleton subintervals/subsets in the real standard [0, 1], such that T

_{A}(x): X → [0, 1], I

_{S}(x): X → [0, 1], and F

_{S}(x): X → [0, 1]. Then, the SNS A contains SVNS for T

_{A}(x), I

_{A}(x), F

_{A}(x) ∈ [0, 1] and INS for T

_{A}(x), I

_{A}(x), F

_{A}(x) ⊆ [0, 1].

_{a}, I

_{a}, F

_{a}>, where a contains a single-value neutrosophic number (SVNN) for T

_{a}, I

_{a}, F

_{a}∈ [0, 1] and an interval neutrosophic number (INN) for T

_{a}, I

_{a}, F

_{a}⊆ [0, 1].

_{1}, a

_{2}, …, a

_{n}} and B ={b

_{1}, b

_{2}, …, b

_{n}}, where a

_{j}= <T

_{aj}, I

_{aj}, F

_{aj}> and b

_{j}= <T

_{bj}, I

_{bj}, F

_{bj}> for j = 1, 2, …, n are two collections of SNNs. Based on the Jaccard, Dice, and cosine measures between two vectors, Ye [8] presented the their similarity measures between SNSs (SVNSs and INSs) A and B in vector space, respectively, as follows:

- (1)
- Three vector similarity measures between A and B for SVNSs:$${M}_{J}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{{T}_{aj}{T}_{bj}+{I}_{aj}{I}_{bj}+{F}_{aj}{F}_{bj}}{\left[\left({T}_{aj}^{2}+{I}_{aj}^{2}+{F}_{aj}^{2}\right)+\left({T}_{bj}^{2}+{I}_{bj}^{2}+{F}_{bj}^{2}\right)-\left({T}_{aj}{T}_{bj}+{I}_{aj}{I}_{bj}+{F}_{aj}{F}_{bj}\right)\right]}}$$$${M}_{D}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{2\left({T}_{aj}{T}_{bj}+{I}_{aj}{I}_{bj}+{F}_{aj}{F}_{aj}\right)}{\left({T}_{aj}^{2}+{I}_{aj}^{2}+{F}_{aj}^{2}\right)+\left({T}_{bj}^{2}+{I}_{bj}^{2}+{F}_{bj}^{2}\right)}}$$$${M}_{C}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{{T}_{aj}{T}_{bj}+{I}_{aj}{I}_{bj}+{F}_{aj}{F}_{bj}}{\sqrt{{T}_{aj}^{2}+{I}_{aj}^{2}+{F}_{aj}^{2}}\sqrt{{T}_{bj}^{2}+{I}_{bj}^{2}+{F}_{bj}^{2}}}}$$
- (2)
- Three vector similarity measures between A and B for INSs:$${M}_{J}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{\left(\mathrm{inf}{T}_{aj}\mathrm{inf}{T}_{bj}+\mathrm{sup}{T}_{aj}\mathrm{sup}{T}_{bj}+\mathrm{inf}{I}_{aj}\mathrm{inf}{I}_{bj}+\mathrm{sup}{I}_{aj}\mathrm{sup}{I}_{bj}+\mathrm{inf}{F}_{aj}\mathrm{inf}{F}_{bj}+\mathrm{sup}{F}_{aj}\mathrm{sup}{F}_{bj}\right)}{\left(\begin{array}{l}{(\mathrm{inf}{T}_{aj}^{})}^{2}+{(\mathrm{inf}{I}_{aj}^{})}^{2}+{(\mathrm{inf}{F}_{aj}^{})}^{2}+{(\mathrm{sup}{T}_{aj}^{})}^{2}+{(\mathrm{sup}{I}_{aj}^{})}^{2}+{(\mathrm{sup}{F}_{aj}^{})}^{2}\\ +{(\mathrm{inf}{T}_{bj}^{})}^{2}+{(\mathrm{inf}{I}_{bj}^{})}^{2}+{(\mathrm{inf}{F}_{bj}^{})}^{2}+{(\mathrm{sup}{T}_{bj}^{})}^{2}+{(\mathrm{sup}{I}_{bj}^{})}^{2}+{(\mathrm{sup}{F}_{bj}^{})}^{2}\\ -(\mathrm{inf}{T}_{aj}\mathrm{inf}{T}_{bj}+\mathrm{inf}{I}_{aj}\mathrm{inf}{I}_{bj}+\mathrm{inf}{F}_{aj}\mathrm{inf}{F}_{bj})\\ -(\mathrm{sup}{T}_{aj}\mathrm{sup}{T}_{bj}+\mathrm{sup}{I}_{aj}\mathrm{sup}{I}_{bj}+\mathrm{sup}{F}_{aj}\mathrm{sup}{F}_{bj})\end{array}\right)}}$$$${M}_{D}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{2\left(\mathrm{inf}{T}_{aj}\mathrm{inf}{T}_{bj}+\mathrm{inf}{I}_{aj}\mathrm{inf}{I}_{bj}+\mathrm{inf}{F}_{aj}\mathrm{inf}{F}_{bj}+\mathrm{sup}{T}_{aj}\mathrm{sup}{T}_{bj}+\mathrm{sup}{I}_{aj}\mathrm{sup}{I}_{bj}+\mathrm{sup}{F}_{aj}\mathrm{sup}{F}_{bj}\right)}{\left(\begin{array}{l}{(\mathrm{inf}{T}_{aj}^{})}^{2}+{(\mathrm{inf}{I}_{aj}^{})}^{2}+{(\mathrm{inf}{F}_{aj}^{})}^{2}+{(\mathrm{sup}{T}_{aj}^{})}^{2}+{(\mathrm{sup}{I}_{aj}^{})}^{2}+{(\mathrm{sup}{F}_{aj}^{})}^{2}\\ +{(\mathrm{inf}{T}_{bj}^{})}^{2}+{(\mathrm{inf}{I}_{bj}^{})}^{2}+{(\mathrm{inf}{F}_{bj}^{})}^{2}+{(\mathrm{sup}{T}_{bj}^{})}^{2}+{(\mathrm{sup}{I}_{bj}^{})}^{2}+{(\mathrm{sup}{F}_{bj}^{})}^{2}\end{array}\right)}}$$$${M}_{C}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{\left(\mathrm{inf}{T}_{aj}\mathrm{inf}{T}_{bj}+\mathrm{inf}{I}_{aj}\mathrm{inf}{I}_{bj}+\mathrm{inf}{F}_{aj}\mathrm{inf}{F}_{bj}+\mathrm{sup}{T}_{aj}\mathrm{sup}{T}_{bj}+\mathrm{sup}{I}_{aj}\mathrm{sup}{I}_{bj}+\mathrm{sup}{F}_{aj}\mathrm{sup}{F}_{bj}\right)}{\left(\begin{array}{l}\sqrt{{(\mathrm{inf}{T}_{aj}^{})}^{2}+{(\mathrm{inf}{I}_{aj}^{})}^{2}+{(\mathrm{inf}{F}_{aj}^{})}^{2}+{(\mathrm{sup}{T}_{aj}^{})}^{2}+{(\mathrm{sup}{I}_{aj}^{})}^{2}+{(\mathrm{sup}{F}_{aj}^{})}^{2}}\\ \sqrt{{(\mathrm{inf}{T}_{bj}^{})}^{2}+{(\mathrm{inf}{I}_{bj}^{})}^{2}+{(\mathrm{inf}{F}_{bj}^{})}^{2}+{(\mathrm{sup}{T}_{bj}^{})}^{2}+{(\mathrm{sup}{I}_{bj}^{})}^{2}+{(\mathrm{sup}{F}_{bj}^{})}^{2}}\end{array}\right)}}$$

_{aj}= [inf T

_{aj}, sup T

_{aj}] I

_{aj}= [inf I

_{aj}, sup I

_{aj}], F

_{aj}= [inf F

_{aj}, sup F

_{aj}], T

_{bj}= [inf T

_{bj}, sup T

_{bj}], I

_{bj}= [inf I

_{bj}, sup I

_{bj}], and F

_{bj}= [inf F

_{bj}, sup F

_{bj}] are equal.

_{k}(A, B) (k = J, D, C) contains the following properties [8]:

- (P1) 0 ≤ M
_{k}(A, B) ≤ 1; - (P2) M
_{k}(A, B) = M_{k}(B, A); - (P3) M
_{k}(A, B) = 1 if A = B, i.e., T_{aj}, = T_{bj}, I_{a}_{j}= I_{bj}, and F_{aj}= F_{bj}for j = 1, 2, …, n.

## 3. Refined Simplified Neutrosophic Sets

_{A}(x), I

_{A}(x) and F

_{A}(x) for x∈ X are single-value and/or interval values in [0, 1]. Then, SNS contain INS and/or SVNS.

_{A}(x), I

_{A}(x), F

_{A}(x) in SNS are refined (split) into T

_{A}(x

_{1}), T

_{A}(x

_{2}), ..., T

_{A}(x

_{r}), I

_{A}(x

_{1}), I

_{A}(x

_{2}), ..., I

_{A}(x

_{r}), and F

_{A}(x

_{1}), F

_{A}(x

_{2}), ..., F

_{A}(x

_{r}), respectively, for x∈ X, x = {x

_{1}, x

_{2}, ..., x

_{r}}, and a positive integer r, then they can be constructed as RSNS by the refinement of SNS, which is defined below.

**Definition**

**1.**

_{A}(x

_{1}), T

_{A}(x

_{2}), ..., T

_{A}(x

_{r}), I

_{A}(x

_{1}), I

_{A}(x

_{2}), ..., I

_{A}(x

_{r}), F

_{A}(x

_{1}), F

_{A}(x

_{2}), ..., F

_{A}(x

_{r}) for x∈ X, x

_{j}∈ x = {x

_{1}, x

_{2}, ..., x

_{r}} (j = 1, 2, …, r), and a positive integer r are subintervals/subsets in the real standard interval [0, 1], such that T

_{A}(x

_{1}), T

_{A}(x

_{2}), ..., T

_{A}(x

_{r}): X → [0, 1], I

_{A}(x

_{1}), I

_{A}(x

_{2}), ..., I

_{A}(x

_{r}): X → [0, 1], and F

_{A}(x

_{1}), F

_{A}(x

_{2}), ..., F

_{A}(x

_{r}): X → [0, 1].

- (1)
- If T
_{A}(x_{1}), T_{A}(x_{2}), ..., T_{A}(x_{r}) ∈ [0, 1], I_{A}(x_{1}), I_{A}(x_{2}), ..., I_{A}(x_{r}) ∈ [0, 1], and F_{A}(x_{1}), F_{A}(x_{2}), ..., F_{A}(x_{r}) ∈ [0, 1] in A for x∈ X and x_{j}∈ x (j =1, 2, …, r) are considered as single/exact values in [0, 1], then A reduces to RSVNS [22], which satisfies the condition $0\le {T}_{A}({x}_{j})+{I}_{A}({x}_{j})+{F}_{A}({x}_{j})\le 3$ for j = 1, 2, …, r; - (2)
- If T
_{A}(x_{1}), T_{A}(x_{2}), ..., T_{A}(x_{r}) ⊆ [0, 1], I_{A}(x_{1}), I_{A}(x_{2}), ..., I_{A}(x_{r}) ⊆ [0, 1], and F_{A}(x_{1}), F_{A}(x_{2}), ..., F_{A}(x_{r}) ⊆ [0, 1] in A for x∈ X and x_{j}∈ x (j =1, 2, …, r) are considered as interval values in [0, 1], then A reduces to RINS [23], which satisfies the condition $0\le \mathrm{sup}{T}_{A}({x}_{j})+\mathrm{sup}{I}_{A}({x}_{j})+\mathrm{sup}{F}_{A}({x}_{j})\le 3$ for j = 1, 2, …, r.

_{A}(x

_{j}) = [inf T

_{A}(x

_{j}), sup T

_{A}(x

_{j})], I

_{A}(x

_{j}) = [inf I

_{A}(x

_{j}), sup I

_{A}(x

_{j})] and F

_{A}(x

_{j}) = [inf F

_{A}(x

_{j}), sup F

_{A}(x

_{j})] in A for x ∈ X and x

_{j}∈ x (j = 1, 2, …, r) are equal, the RINS A reduces to the RSVNS A. Clearly, RSVNS is a special case of RINS. If some lower and upper limits of T

_{A}(x

_{j}) = [inf T

_{A}(x

_{j}), sup T

_{A}(x

_{j})]/I

_{A}(x

_{j}) = [inf I

_{A}(x

_{j}), sup I

_{A}(x

_{j})]/F

_{A}(x

_{j}) = [inf F

_{A}(x

_{j}), sup F

_{A}(x

_{j})] in RINS are equal, then it can be denoted as a special interval (equal interval of the lower and upper limits) T

_{A}(x

_{j}) = [T

_{A}(x

_{j}), T

_{A}(x

_{j})]/I

_{A}(x

_{j}) = [I

_{A}(x

_{j}), I

_{A}(x

_{j})]/F

_{A}(x

_{j}) = [F

_{A}(x

_{j}), F

_{A}(x

_{j})]. Hence, RINS can contain RINS and/or SVNS information (hybrid information of both).

_{A}(x

_{1}), T

_{A}(x

_{2}), ..., T

_{A}(x

_{r})), (I

_{A}(x

_{1}), I

_{A}(x

_{2}), ..., I

_{A}(x

_{r})), (F

_{A}(x

_{1}), F

_{A}(x

_{2}), ..., F

_{A}(x

_{r}))> in A is simply denoted as a = <(T

_{a}

_{1}, T

_{a}

_{2}, …, T

_{ar}), (I

_{a}

_{1}, I

_{a}

_{2}, …, I

_{ar}), (F

_{a}

_{1}, F

_{a}

_{2}, …, F

_{ar})>, which is called a refined simplified neutrosophic number (RSNN).

_{a}

_{1}, T

_{a}

_{2}, …, T

_{ar}), (I

_{a}

_{1}, I

_{a}

_{2}, …, I

_{ar}), (F

_{a}

_{1}, F

_{a}

_{2}, …, F

_{ar})> and b = <(T

_{b}

_{1}, T

_{b}

_{2}, …, T

_{br}), (I

_{b}

_{1}, I

_{b}

_{2}, …, I

_{br}), (F

_{b}

_{1}, F

_{b}

_{2}, …, F

_{br})> for T

_{aj}, T

_{bj}, I

_{aj}, I

_{bj}, F

_{aj}, F

_{bj}∈ [0, 1] (j = 1, 2, …, r). Then, there are the following relations between a and b:

- (1)
- Containment: a ⊆ b, if and only if T
_{aj}≤ T_{bj}, I_{aj}≥ I_{bj}, F_{aj}≥ F_{bj}for j = 1, 2, …, r; - (2)
- Equality: a = b, if and only if a ⊆ b and b ⊆ a, i.e., T
_{aj}= T_{bj}, I_{aj}= I_{bj}, F_{aj}= F_{bj}for j = 1, 2, …, r; - (3)
- Union:$$a\cup b=\langle ({T}_{a1}\vee {T}_{b1},{T}_{a2}\vee {T}_{b2},\dots ,{T}_{ar}\vee {T}_{br}),({I}_{a1}\wedge {I}_{b1},{I}_{a2}\wedge {I}_{b2},\dots ,{I}_{ar}\wedge {I}_{br}),({F}_{a1}\wedge {F}_{b1},{F}_{a2}\wedge {F}_{b2},\dots ,{F}_{ar}\wedge {F}_{br})\rangle ;$$
- (4)
- Intersection:$$a\cap b=\langle ({T}_{a1}\wedge {T}_{b1},{T}_{a2}\wedge {T}_{b2},\dots ,{T}_{ar}\wedge {T}_{br}),({I}_{a1}\vee {I}_{b1},{I}_{a2}\vee {I}_{b2},\dots ,{I}_{ar}\vee {I}_{br}),({F}_{a1}\vee {F}_{b1},{F}_{a2}\vee {F}_{b2},\dots ,{F}_{ar}\vee {F}_{br})\rangle .$$

_{aj}, T

_{bj}, I

_{aj}, I

_{bj}, F

_{aj}, F

_{bj}⊆ [0, 1] (j = 1, 2, …, r). Then, there are the following relations of a and b:

- (1)
- Containment: a ⊆ b, if and only if inf T
_{aj}≤ inf T_{bj}, sup T_{aj}≤ sup T_{bj}, inf I_{aj}≥ inf I_{bj}, sup I_{aj}≥ sup I_{bj}, inf F_{aj}≥ inf F_{bj}, and sup F_{aj}≥ sup F_{bj}for j = 1, 2, …, r; - (2)
- Equality: a = b, if and only if a ⊆ b and b ⊆ a, i.e., inf T
_{aj}= inf T_{bj}, sup T_{aj}= sup T_{bj}, inf I_{aj}= inf I_{bj}, sup I_{aj}= sup I_{bj}, inf F_{aj}= inf F_{bj}, and sup F_{aj}= sup F_{bj}for j = 1, 2, …, r; - (3)
- Union:$$a\cup b=\langle \begin{array}{l}([\mathrm{inf}{T}_{a1}\vee \mathrm{inf}{T}_{b1},\mathrm{sup}{T}_{a1}\vee \mathrm{sup}{T}_{b1}],[\mathrm{inf}{T}_{a2}\vee \mathrm{inf}{T}_{b2},\mathrm{sup}{T}_{a2}\vee \mathrm{sup}{T}_{b2}],\dots ,[\mathrm{inf}{T}_{ar}\vee \mathrm{inf}{T}_{br},\mathrm{sup}{T}_{ar}\vee \mathrm{sup}{T}_{br}]),\\ ([\mathrm{inf}{I}_{a1}\wedge \mathrm{inf}{I}_{b1},\mathrm{sup}{I}_{a1}\wedge \mathrm{sup}{I}_{b1}],[\mathrm{inf}{I}_{a2}\wedge \mathrm{inf}{I}_{b2},\mathrm{sup}{I}_{a2}\wedge \mathrm{sup}{I}_{b2}],\dots ,[\mathrm{inf}{I}_{ar}\wedge \mathrm{inf}{I}_{br},\mathrm{sup}{I}_{ar}\wedge \mathrm{sup}{I}_{br}]),\\ ([\mathrm{inf}{F}_{a1}\wedge \mathrm{inf}{F}_{b1},\mathrm{sup}{F}_{a1}\wedge \mathrm{sup}{F}_{b1}],[\mathrm{inf}{F}_{a2}\wedge \mathrm{inf}{F}_{b2},\mathrm{sup}{F}_{a2}\wedge \mathrm{sup}{F}_{b2}],\dots ,[\mathrm{inf}{F}_{ar}\wedge \mathrm{inf}{F}_{br},\mathrm{sup}{F}_{ar}\wedge \mathrm{sup}{F}_{br}])\end{array}\rangle $$
- (4)
- Intersection:$$a\cap b=\langle \begin{array}{l}([\mathrm{inf}{T}_{a1}\wedge \mathrm{inf}{T}_{b1},\mathrm{sup}{T}_{a1}\wedge \mathrm{sup}{T}_{b1}],[\mathrm{inf}{T}_{a2}\wedge \mathrm{inf}{T}_{b2},\mathrm{sup}{T}_{a2}\wedge \mathrm{sup}{T}_{b2}],\dots ,[\mathrm{inf}{T}_{ar}\wedge \mathrm{inf}{T}_{br},\mathrm{sup}{T}_{ar}\wedge \mathrm{sup}{T}_{br}]),\\ ([\mathrm{inf}{I}_{a1}\vee \mathrm{inf}{I}_{b1},\mathrm{sup}{I}_{a1}\vee \mathrm{sup}{I}_{b1}],[\mathrm{inf}{I}_{a2}\vee \mathrm{inf}{I}_{b2},\mathrm{sup}{I}_{a2}\vee \mathrm{sup}{I}_{b2}],\dots ,[\mathrm{inf}{I}_{ar}\vee \mathrm{inf}{I}_{br},\mathrm{sup}{I}_{ar}\vee \mathrm{sup}{I}_{br}]),\\ ([\mathrm{inf}{F}_{a1}\vee \mathrm{inf}{F}_{b1},\mathrm{sup}{F}_{a1}\vee \mathrm{sup}{F}_{b1}],[\mathrm{inf}{F}_{a2}\vee \mathrm{inf}{F}_{b2},\mathrm{sup}{F}_{a2}\vee \mathrm{sup}{F}_{b2}],\dots ,[\mathrm{inf}{F}_{ar}\vee \mathrm{inf}{F}_{br},\mathrm{sup}{F}_{ar}\vee \mathrm{sup}{F}_{br}])\end{array}\rangle $$

## 4. Vector Similarity Measures of RSNSs

**Definition**

**2.**

_{1}, a

_{2}, …, a

_{n}} and B ={b

_{1}, b

_{2}, …, b

_{n}}, where ${a}_{j}=\langle ({T}_{{a}_{j}1},{T}_{{a}_{j}2},\dots ,{T}_{{a}_{j}{r}_{j}}),({I}_{{a}_{j}1},{I}_{{a}_{j}2},\dots ,{I}_{{a}_{j}{r}_{j}}),({F}_{{a}_{j}1},{F}_{{a}_{j}2},\dots ,{F}_{{a}_{j}{r}_{j}})\rangle $ and ${b}_{j}=\langle ({T}_{{b}_{j}1},{T}_{{b}_{j}2},\dots ,{T}_{{b}_{j}{r}_{j}}),({I}_{{b}_{j}1},{I}_{{b}_{j}2},\dots ,{I}_{{b}_{j}{r}_{j}}),({F}_{{b}_{j}1},{F}_{{b}_{j}2},\dots ,{F}_{{b}_{j}{r}_{j}})\rangle $ for j = 1, 2, …, n are two collections of RSNNs for ${T}_{{a}_{j}k},{I}_{{a}_{j}k},{F}_{{a}_{j}k}$, ${T}_{{b}_{j}k},{I}_{{b}_{j}k},{F}_{{b}_{j}k}$∈ [0, 1] or ${T}_{{a}_{j}k},{I}_{{a}_{j}k},{F}_{{a}_{j}k}$, ${T}_{{b}_{j}k},{I}_{{b}_{j}k},{F}_{{b}_{j}k}$⊆ [0, 1] (j = 1, 2, …, n; k = 1, 2, …, r

_{j}). Then, the Jaccard, Dice, and cosine measures between A and B are defined, respectively, as follows:

- (1)
- Three vector similarity measures between A and B for RSVNSs:$${R}_{J}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{1}{{r}_{j}}{\displaystyle \sum _{k=1}^{{r}_{j}}\frac{{T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}}{\left[\left({T}_{{a}_{j}k}^{2}+{I}_{{a}_{j}k}^{2}+{F}_{{a}_{j}k}^{2}\right)+\left({T}_{{b}_{j}k}^{2}+{I}_{{b}_{j}k}^{2}+{F}_{{b}_{j}k}^{2}\right)-\left({T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}\right)\right]}}}$$$${R}_{D}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{1}{{r}_{j}}{\displaystyle \sum _{k=1}^{{r}_{j}}\frac{2\left({T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}\right)}{\left({T}_{{a}_{j}k}^{2}+{I}_{{a}_{j}k}^{2}+{F}_{{a}_{j}k}^{2}\right)+\left({T}_{{b}_{j}k}^{2}+{I}_{{b}_{j}k}^{2}+{F}_{{b}_{j}k}^{2}\right)}}}$$$${R}_{C}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{1}{{r}_{j}}{\displaystyle \sum _{k=1}^{{r}_{j}}\frac{{T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}}{\sqrt{{T}_{{a}_{j}k}^{2}+{I}_{{a}_{j}k}^{2}+{F}_{{a}_{j}k}^{2}}\sqrt{{T}_{{b}_{j}k}^{2}+{I}_{{b}_{j}k}^{2}+{F}_{{b}_{j}k}^{2}}}}}$$
- (2)
- Three vector similarity measures between A and B for RINSs:$${R}_{J}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{1}{{r}_{j}}{\displaystyle \sum _{k=1}^{{r}_{j}}\frac{\left(\begin{array}{l}\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}\\ +\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k}\end{array}\right)}{\left(\begin{array}{l}{(\mathrm{inf}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{a}_{j}k}^{})}^{2}\\ +{(\mathrm{inf}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{b}_{j}k}^{})}^{2}\\ -(\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k})\\ -(\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k})\end{array}\right)}}},$$$${R}_{D}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{1}{{r}_{j}}{\displaystyle \sum _{k=1}^{{r}_{j}}\frac{2\left(\begin{array}{l}\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k}\\ +\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k}\end{array}\right)}{\left(\begin{array}{l}{(\mathrm{inf}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{a}_{j}k}^{})}^{2}\\ +{(\mathrm{inf}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{b}_{j}k}^{})}^{2}\end{array}\right)}}},$$$${R}_{C}(A,B)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}\frac{1}{{r}_{j}}{\displaystyle \sum _{k=1}^{{r}_{j}}\frac{\left(\begin{array}{l}\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k}\\ +\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k}\end{array}\right)}{\left(\begin{array}{l}\sqrt{{(\mathrm{inf}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{a}_{j}k}^{})}^{2}}\\ \sqrt{{(\mathrm{inf}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{b}_{j}k}^{})}^{2}}\end{array}\right)}}}$$

_{j}) are equal. Especially when k = 1, Equations (7)–(12) are reduced to Equations (1)–(6).

_{s}(A, B) (s = J, D, C) also contain the following properties (P1)–(P3):

- (P1) 0 ≤ R
_{s}(A, B) ≤ 1; - (P2) R
_{s}(A, B) = R_{s}(B, A); - (P3) R
_{s}(A, B) = 1 if A = B, i.e., ${T}_{{a}_{j}k}={T}_{{b}_{j}k},{I}_{{a}_{j}k}={I}_{{b}_{j}k},{F}_{{a}_{j}k}={F}_{{b}_{j}k}$ for j = 1, 2, …, n and k = 1, 2, ..., r_{j}.

_{j}and b

_{j}(j = 1, 2, …, n) in the RSNSs A and B is given as w

_{j}∈ [0, 1] with ${\sum}_{j=1}^{n}{w}_{j}}=1$ and the weight of the refined components (sub-elements) ${T}_{{a}_{j}k},{I}_{{a}_{j}k},{F}_{{a}_{j}k}$ and ${T}_{{b}_{j}k},{I}_{{b}_{j}k},{F}_{{b}_{j}k}$ (k = 1, 2, …, r

_{j}) in ${a}_{j}=\langle ({T}_{{a}_{j}1},{T}_{{a}_{j}2},\dots ,{T}_{{a}_{j}{r}_{j}}),({I}_{{a}_{j}1},{I}_{{a}_{j}2},\dots ,{I}_{{a}_{j}{r}_{j}}),({F}_{{a}_{j}1},{F}_{{a}_{j}2},\dots ,{F}_{{a}_{j}{r}_{j}})\rangle $ and ${b}_{j}=\langle ({T}_{{b}_{j}1},{T}_{{b}_{j}2},\dots ,{T}_{{b}_{j}{r}_{j}}),({I}_{{b}_{j}1},{I}_{{b}_{j}2},\dots ,{I}_{{b}_{j}{r}_{j}}),({F}_{{b}_{j}1},{F}_{{b}_{j}2},\dots ,{F}_{{b}_{j}{r}_{j}})\rangle $ (j = 1, 2, …, n) is considered as ω

_{k}∈ [0, 1] with ${\sum}_{k=1}^{{r}_{j}}{\omega}_{k}}=1$, the weighted Jaccard, Dice, and cosine measures between A and B are presented, respectively, as follows:

- (1)
- Three weighted vector similarity measures between A and B for RSVNSs:$${R}_{WJ}(A,B)={\displaystyle \sum _{j=1}^{n}{w}_{j}{\displaystyle \sum _{k=1}^{{r}_{j}}{\omega}_{k}\frac{{T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}}{\left[\left({T}_{{a}_{j}k}^{2}+{I}_{{a}_{j}k}^{2}+{F}_{{a}_{j}k}^{2}\right)+\left({T}_{{b}_{j}k}^{2}+{I}_{{b}_{j}k}^{2}+{F}_{{b}_{j}k}^{2}\right)-\left({T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}\right)\right]}}}$$$${R}_{WD}(A,B)={\displaystyle \sum _{j=1}^{n}{w}_{j}{\displaystyle \sum _{k=1}^{{r}_{j}}{\omega}_{k}\frac{2\left({T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}\right)}{\left({T}_{{a}_{j}k}^{2}+{I}_{{a}_{j}k}^{2}+{F}_{{a}_{j}k}^{2}\right)+\left({T}_{{b}_{j}k}^{2}+{I}_{{b}_{j}k}^{2}+{F}_{{b}_{j}k}^{2}\right)}}}$$$${R}_{WC}(A,B)={\displaystyle \sum _{j=1}^{n}{w}_{j}{\displaystyle \sum _{k=1}^{{r}_{j}}{\omega}_{k}\frac{{T}_{{a}_{j}k}{T}_{{b}_{j}k}+{I}_{{a}_{j}k}{I}_{{b}_{j}k}+{F}_{{a}_{j}k}{F}_{{b}_{j}k}}{\sqrt{{T}_{{a}_{j}k}^{2}+{I}_{{a}_{j}k}^{2}+{F}_{{a}_{j}k}^{2}}\sqrt{{T}_{{b}_{j}k}^{2}+{I}_{{b}_{j}k}^{2}+{F}_{{b}_{j}k}^{2}}}}}$$
- (2)
- Three weighted vector similarity measures between A and B for RINSs:$${R}_{WJ}(A,B)={\displaystyle \sum _{j=1}^{n}{w}_{j}{\displaystyle \sum _{k=1}^{{r}_{j}}{\omega}_{k}\frac{\left(\begin{array}{l}\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}\\ +\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k}\end{array}\right)}{\left(\begin{array}{l}{(\mathrm{inf}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{a}_{j}k}^{})}^{2}\\ +{(\mathrm{inf}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{b}_{j}k}^{})}^{2}\\ -(\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k})\\ -(\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k})\end{array}\right)}}},$$$${R}_{WD}(A,B)={\displaystyle \sum _{j=1}^{n}{w}_{j}{\displaystyle \sum _{k=1}^{{r}_{j}}{\omega}_{k}\frac{2\left(\begin{array}{l}\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k}\\ +\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k}\end{array}\right)}{\left(\begin{array}{l}{(\mathrm{inf}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{a}_{j}k}^{})}^{2}\\ +{(\mathrm{inf}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{b}_{j}k}^{})}^{2}\end{array}\right)}}},$$$${R}_{WC}(A,B)={\displaystyle \sum _{j=1}^{n}{w}_{j}{\displaystyle \sum _{k=1}^{{r}_{j}}{\omega}_{k}\frac{\left(\begin{array}{l}\mathrm{inf}{T}_{{a}_{j}k}\mathrm{inf}{T}_{{b}_{j}k}+\mathrm{inf}{I}_{{a}_{j}k}\mathrm{inf}{I}_{{b}_{j}k}+\mathrm{inf}{F}_{{a}_{j}k}\mathrm{inf}{F}_{{b}_{j}k}\\ +\mathrm{sup}{T}_{{a}_{j}k}\mathrm{sup}{T}_{{b}_{j}k}+\mathrm{sup}{I}_{{a}_{j}k}\mathrm{sup}{I}_{{b}_{j}k}+\mathrm{sup}{F}_{{a}_{j}k}\mathrm{sup}{F}_{{b}_{j}k}\end{array}\right)}{\left(\begin{array}{l}\sqrt{{(\mathrm{inf}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{a}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{a}_{j}k}^{})}^{2}}\\ \sqrt{{(\mathrm{inf}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{inf}{F}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{T}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{I}_{{b}_{j}k}^{})}^{2}+{(\mathrm{sup}{F}_{{b}_{j}k}^{})}^{2}}\end{array}\right)}}}.$$

_{j}) are equal. Especially when each w

_{j}= 1/n and ω

_{k}= 1/r

_{j}(j = 1, 2, …, n; k = 1, 2, …, r

_{j}), Equations (13)–(18) are reduced to Equations (7)–(12).

_{Ws}(A, B) (s = J, D, C) also satisfies the following properties (P1)–(P3):

- (P1) 0 ≤ R
_{Ws}(A, B) ≤ 1; - (P2) R
_{Ws}(A, B) = R_{Ws}(B, A); - (P3) R
_{Ws}(A, B) = 1 if A = B, i.e., ${T}_{{a}_{j}k}={T}_{{b}_{j}k},{I}_{{a}_{j}k}={I}_{{b}_{j}k},{F}_{{a}_{j}k}={F}_{{b}_{j}k}$ for j = 1, 2, …, n and k = 1, 2, ..., r_{j}.

## 5. Decision-Making Method Using the Vector Similarity Measures

_{1}, A

_{2}, …, A

_{m}} is a set of m alternatives, which needs to satisfies a set of n attributes B = {b

_{1}, b

_{2}, …, b

_{n}}, where b

_{j}(j = 1, 2, …, n) may be refined/split into a set of r

_{j}sub-attributes ${b}_{j}=\{{b}_{j1},{b}_{j2},\dots ,{b}_{j{r}_{j}}\}$ (j = 1, 2, …, n). If the decision-maker provides the suitability evaluation values of attributes ${b}_{j}=\{{b}_{j1},{b}_{j2},\dots ,{b}_{j{r}_{j}}\}$ (j =1, 2, ..., n) on the alternative A

_{i}(i = 1, 2,…, m) by using RSNS: ${A}_{i}=\left\{\langle {b}_{j},({T}_{{A}_{i}}({b}_{j1}),{T}_{{A}_{i}}({b}_{j2}),\dots ,{T}_{{A}_{i}}({b}_{j{r}_{j}})),({I}_{{A}_{i}}({b}_{j1}),{I}_{{A}_{i}}({b}_{j2}),\dots ,{I}_{{A}_{i}}({b}_{j{r}_{j}})),({F}_{{A}_{i}}({b}_{j1}),{F}_{{A}_{i}}({b}_{j2}),\dots ,{F}_{{A}_{i}}({b}_{j{r}_{j}}))\rangle |{b}_{j}\in B,{b}_{jk}\in {b}_{j}\right\}$.

_{i}is represented by RSNN: ${a}_{ij}=\langle ({T}_{{a}_{ij}1},{T}_{{a}_{ij}2},\dots ,{T}_{{a}_{ij}{r}_{j}}),({I}_{{a}_{ij}1},{I}_{{a}_{ij}2},\dots ,{I}_{{a}_{ij}{r}_{j}}),({F}_{{a}_{ij}1},{F}_{{a}_{ij}2},\dots ,{F}_{{a}_{ij}{r}_{j}})\rangle $ for i = 1, 2, …, m and j = 1, 2, …, n.

_{ij})

_{m}

_{×n}, as shown in Table 1.

_{j}(j = 1, 2, …, n) and its sub-attributes are considered as having different importance, the weight vector of the attributes is given by

**W**= (w

_{1}, w

_{2}, …, w

_{n}) with w

_{j}∈ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}=1$ and the weight vector for each sub-attribute set $\{{b}_{j1},{b}_{j2},\dots ,{b}_{j{r}_{j}}\}$ is given as ${\omega}_{j}=\{{\omega}_{j1},{\omega}_{j2},\dots ,{\omega}_{j{r}_{j}}\}$ (j = 1, 2, …, n) with ω

_{jk}∈ [0, 1] and ${\sum}_{k=1}^{{r}_{j}}{\omega}_{jk}=1$. Thus, the decision steps are described as follows:

**Step****1:**- We determine the ideal solution (ideal RSNN) from the refined simplified neutrosophic decision matrix M(a
_{ij})_{m}_{×n}as follows:$${a}_{j}^{*}=\langle \begin{array}{l}(\underset{i}{\mathrm{max}}({T}_{j1}),\underset{i}{\mathrm{max}}({T}_{j2}),\dots ,\underset{i}{\mathrm{max}}({T}_{j{r}_{j}})),(\underset{i}{\mathrm{min}}({I}_{j1}),\underset{i}{\mathrm{min}}({I}_{j2}),\\ \dots ,\underset{i}{\mathrm{min}}({I}_{j{r}_{j}})),(\underset{i}{\mathrm{min}}({F}_{j1}),\underset{i}{\mathrm{min}}({F}_{j2}),\dots ,\underset{i}{\mathrm{min}}({F}_{j{r}_{j}}))\end{array}\rangle \text{}\mathrm{for}\text{}\mathrm{RSVNN}$$$$\mathrm{or}\text{}{a}_{j}^{*}=\langle \begin{array}{l}([\underset{i}{\mathrm{max}}(\mathrm{inf}{T}_{j1}),\underset{i}{\mathrm{max}}(\mathrm{sup}{T}_{j1})],[\underset{i}{\mathrm{max}}(\mathrm{inf}{T}_{j2}),\underset{i}{\mathrm{max}}(\mathrm{sup}{T}_{j2})],\dots ,[\underset{i}{\mathrm{max}}(\mathrm{inf}{T}_{j{r}_{j}}),\underset{i}{\mathrm{max}}(\mathrm{sup}{T}_{j{r}_{j}})]),\\ ([\underset{i}{\mathrm{min}}(\mathrm{inf}{I}_{j1}),\underset{i}{\mathrm{min}}(\mathrm{sup}{I}_{j1})],[\underset{i}{\mathrm{min}}(\mathrm{inf}{I}_{j2}),\underset{i}{\mathrm{min}}(\mathrm{sup}{I}_{j2})],\dots ,[\underset{i}{\mathrm{min}}(\mathrm{inf}{I}_{j{r}_{j}}),\underset{i}{\mathrm{min}}(\mathrm{sup}{I}_{j{r}_{j}})]),\\ ([\underset{i}{\mathrm{min}}(\mathrm{inf}{F}_{j1}),\underset{i}{\mathrm{min}}(\mathrm{sup}{F}_{j1})],[\underset{i}{\mathrm{min}}(\mathrm{inf}{F}_{j2}),\underset{i}{\mathrm{min}}(\mathrm{sup}{F}_{j2})],\dots ,[\underset{i}{\mathrm{min}}(\mathrm{inf}{F}_{j{r}_{j}}),\underset{i}{\mathrm{min}}(\mathrm{sup}{F}_{j{r}_{j}})])\end{array}\rangle \text{}\mathrm{for}\text{}\mathrm{RINN},$$ **Step****2:**- The similarity measure between each alternative A
_{i}(i = 1, 2, …, m) and the ideal alternative A^{*}can be calculated by using one of Equations (13)–(15) or Equations (16)–(18), and obtained as the values of R_{Ws}(A_{i}, A^{*}) for i = 1, 2, …, m and s = J or D or C. **Step****3:**- According to the values of R
_{Ws}(A_{i}, A^{*}) for i = 1, 2, …, m and s = J or D or C, the alternatives are ranked in a descending order. The greater value of R_{Ws}(A_{i}, A^{*}) means the best alternative. **Step****4:**- End.

## 6. Illustrative Example on the Selection of Construction Projects

_{1}, A

_{2}, A

_{3}, A

_{4}}. To select the best one of them, experts or decision-makers need to make a decision of these construction projects corresponding to three attributes and their seven sub-attributes, which are described as follows:

- (1)
- Financial state (b
_{1}) contains two sub-attributes: budget control (b_{11}) and risk/return ratio (b_{12}); - (2)
- Environmental protection (b
_{2}) contains three sub-attributes: public relation (b_{21}), geographical location (b_{22}), and health and safety (b_{23}); - (3)
- Technology (b
_{3}) contains tow sub-attributes: technical know-how (b_{31}), technological capability (b_{32}).

**W**= (0.4, 0.3, 0.3) and the weight vectors of the three sub-attribute sets {b

_{11}, b

_{12}}, {b

_{21}, b

_{22}, b

_{23}}, and {b

_{31}, b

_{32}} are given, respectively, by

**ω**

_{1}= (0.6, 0.4),

**ω**

_{2}= (0.25, 0.4, 0.35), and

**ω**

_{3}= (0.45, 0.55).

_{1}= 2, r

_{2}= 3, r

_{3}= 2). Thus, we can construct the following refined simplified neutrosophic decision matrix M(a

_{ij})

_{4}

_{×3}, which is shown in Table 2.

**Step****1:**- By Equation (19), the ideal solution (ideal RSVNS) can be determined as the following ideal alternative: A
^{*}= {<(0.8, 0.8), (0.1, 0.1), (0.2, 0.2)>, <(0.9, 0.8, 0.8), (0.1, 0.1, 0.1), (0.1, 0.1, 0.1)>, <(0.8, 0.8), (0.1, 0.2), (0.1, 0.1)>}. **Step****2:**- According to one of Equations (13)–(15), the weighted similarity measure values between each alternative A
_{i}(i = 1, 2, 3, 4) and the ideal alternative A^{*}can be obtained and all the results are shown in Table 3. **Step****3:**- In Table 3, since all the measure values are R
_{Ws}(A_{2}, A^{*}) > R_{Ws}(A_{4}, A^{*}) > R_{Ws}(A_{3}, A^{*}) > R_{Ws}(A_{1}, A^{*}) for s = J, D, C, all the ranking orders of the four alternatives are A_{2}≻A_{4}≻A_{3}≻A_{1}. Hence, the alternative A_{2}is the best choice among all the construction projects.

_{1}= 2, r

_{2}= 3, r

_{3}= 2). Thus, we can construct the following refined simplified neutrosophic decision matrix M(a

_{ij})

_{4}

_{×3}, which is shown in Table 4.

**Step****1:**- By Equation (20), the ideal solution (ideal RINS) can be determined as the following ideal alternative:A
^{*}= {<([0.8, 0.9], [0.8, 0.9]), ([0.1, 0.2], [0.1, 0.2]), ([0.2, 0.3], [0.2, 0.3])>, <([0.8, 0.9], [0.8, 0.9], [0.8, 0.9]), ([0.1, 0.2], [0.1, 0.2], [0.1, 0.2]), ([0.1, 0.2], [0.1, 0.2], [0.1, 0.2])>, <([0.8, 0.9], [0.8, 0.9]), ([0.1, 0.2], [0.2, 0.3]), ([0.1, 0.2], [0.1, 0.2])>}. **Step****2:**- By using one of Equations (16)–(18), the weighted similarity measure values between each alternative A
_{i}(i = 1, 2, 3, 4) and the ideal alternative A^{*}can be calculated, and then all the results are shown in Table 5. **Step****3:**- In Table 5, since all the measure values are R
_{Ws}(A_{2}, A^{*}) > R_{Ws}(A_{4}, A^{*}) > R_{Ws}(A_{3}, A^{*}) > R_{Ws}(A_{1}, A^{*}) for s = J, D, C, all the ranking orders of the four alternatives are A_{2}≻A_{4}≻A_{3}≻A_{1}. Hence, the alternative A_{2}is the best choice among all the construction projects.

_{Ws}for s = J, D, C. However, the existing related decision-making methods with RSVNSs and RINSs [22,23] cannot deal with such a decision-making problem with both attribute weights and sub-attribute weights in this paper. Although the same computational complexity in decision-making algorithms is shown by comparison of the method of this study with the related methods introduced in [22,23], the developed method in this study extends the methods in [22,23] and is more feasible and more general than the existing related decision-making methods [22,23]. It is obvious that the new developed decision-making method in a RSNS (RINS and/or SVNS) setting is superior to the existing related methods in a RINS or SVNS setting [22,23].

_{4}into the RSNS A

_{4’}= {<([0.7,0.7], [0.6,0.6]), ([0.2,0.2], [0.2,0.2]), ([0.3,0.3], [0.3,0.3])>, <([0.7,0.8], [0.8,0.9], [0.7,0.8]), ([0.2,0.3], [0.2,0.3], [0.1,0.2]), ([0.1,0.2], [0.2,0.3], [0.2,0.3])>, <([0.7,0.8], [0.7,0.8]), ([0.2,0.3], [0.3,0.4]), ([0.2,0.3], [0.3,0.4])> with hybrid information of both RSVNNs and RINNs. Then, by above similar computation steps, we can obtain all the measure values, which are shown in Table 6.

_{W}

_{J}(A

_{i}, A

^{*}) and R

_{W}

_{D}(A

_{i}, A

^{*}) are the same, but their decision-making method can change the previous ranking orders and show some difference between two alternatives A

_{3}and A

_{4}; while the best one is still A

_{2}. Clearly, the decision-making approach based on the Jaccard and Dice measures shows some sensitivity in this case. However, the ranking order based on R

_{W}

_{C}(A

_{i}, A

^{*}) still keeps the previous ranking order, and then the decision-making approach based on the cosine measure shows some robustness/insensitivity in this case.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Zadeh, L.A. Fuzzy sets. Inf. Control.
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K.; Gargov, G. Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 31, 343–349. [Google Scholar] [CrossRef] - Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic; American Research Press: Rehoboth, IL, USA, 1998. [Google Scholar]
- Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005. [Google Scholar]
- Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistruct.
**2010**, 4, 410–413. [Google Scholar] - Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst.
**2014**, 26, 2459–2466. [Google Scholar] - Ye, J. Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making. Int. J. Fuzzy Syst.
**2014**, 16, 204–211. [Google Scholar] - Zavadskas, E.K.; Bausys, R.; Lazauskas, M. Sustainable assessment of alternative sites for the construction of a waste incineration plant by applying WASPAS method with single-valued neutrosophic set. Sustainability
**2015**, 7, 15923–15936. [Google Scholar] [CrossRef] - Zavadskas, E.K.; Bausys, R.; Kaklauskas, A.; Ubartė, I.; Kuzminskė, A.; Gudienė, N. Sustainable market valuation of buildings by the single-valued neutrosophic MAMVA method. Appl. Soft Comput.
**2017**, 57, 74–87. [Google Scholar] [CrossRef] - Lu, Z.K.; Ye, J. Cosine measures of neutrosophic cubic sets for multiple attribute decision-making. Symmetry
**2017**, 9, 121. [Google Scholar] [CrossRef] - Lu, Z.K.; Ye, J. Single-valued neutrosophic hybrid arithmetic and geometric aggregation operators and their decision-making method. Information
**2017**, 8, 84. [Google Scholar] [CrossRef] - Chen, J.Q.; Ye, J. Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision-making. Symmetry
**2017**, 9, 82. [Google Scholar] [CrossRef] - Ye, J. Single valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Soft Comput.
**2017**, 21, 817–825. [Google Scholar] [CrossRef] - Smarandache, F. n-Valued refined neutrosophic logic and its applications in physics. Prog. Phys.
**2013**, 4, 143–146. [Google Scholar] - Ye, S.; Ye, J. Dice similarity measure between single valued neutrosophic multisets and its application in medical diagnosis. Neutrosophic Sets Syst.
**2014**, 6, 49–54. [Google Scholar] - Broumi, S.; Smarandache, F. Neutrosophic refined similarity measure based on cosine function. Neutrosophic Sets Syst.
**2014**, 6, 42–48. [Google Scholar] - Broumi, S.; Deli, I. Correlation measure for neutrosophic refined sets and its application in medical diagnosis. Palest. J. Math.
**2014**, 3, 11–19. [Google Scholar] - Mondal, K.; Pramanik, S. Neutrosophic refined similarity measure based on cotangent function and its application to multi-attribute decision making. Glob. J. Adv. Res.
**2015**, 2, 486–496. [Google Scholar] - Ji, P.; Zhang, H.Y.; Wang, J.Q. A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput. Appl.
**2016**. [Google Scholar] [CrossRef] - Peng, J.J.; Wang, J.; Wu, X.H. An extension of the ELECTRE approach with multi-valued neutrosophic information. Neural Comput. Appl.
**2016**. [Google Scholar] [CrossRef] - Ye, J.; Smarandache, F. Similarity measure of refined single-valued neutrosophic sets and its multicriteria decision making method. Neutrosophic Sets Syst.
**2016**, 12, 41–44. [Google Scholar] - Fan, C.X.; Ye, J. The cosine measure of refined-single valued neutrosophic sets and refined-interval neutrosophic sets for multiple attribute decision-making. J. Intell. Fuzzy Syst.
**2017**, accepted (in press). [Google Scholar]

b_{1} | b_{2} | … | b_{n} | |
---|---|---|---|---|

$\{{\mathit{b}}_{11},{\mathit{b}}_{12},\dots ,{\mathit{b}}_{1{\mathit{r}}_{1}}\}$ | $\{{\mathit{b}}_{21},{\mathit{b}}_{22},\dots ,{\mathit{b}}_{2{\mathit{r}}_{2}}\}$ | $\{{\mathit{b}}_{\mathit{n}1},{\mathit{b}}_{\mathit{n}2},\dots ,{\mathit{b}}_{\mathit{n}{\mathit{r}}_{\mathit{n}}}\}$ | ||

A_{1} | a_{11} | a_{12} | … | a_{1n} |

A_{2} | a_{21} | a_{22} | ... | a_{2n} |

… | … | … | ... | … |

A_{m} | a_{m}_{1} | a_{m}_{2} | ... | a_{mn} |

**Table 2.**Defined simplified neutrosophic decision matrix M(a

_{ij})

_{4}

_{×3}under refined single-valued neutrosophic set (RSVNS) environment.

b_{1} | b_{2} | b_{3} | |
---|---|---|---|

{b_{11}, b_{12}} | {b_{21}, b_{22}, b_{23}} | {b_{31}, b_{32}} | |

A_{1} | <(0.6, 0.7), (0.2, 0.1), (0.2, 0.3)> | <(0.9, 0.7, 0.8), (0.1, 0.3, 0.2), (0.2, 0.2, 0.1)> | <(0.6, 0.8), (0.3, 0.2), (0.3, 0.4)> |

A_{2} | <(0.8, 0.7), (0.1, 0.2), (0.3, 0.2)> | <(0.7, 0.8, 0.7), (0.2, 0.4, 0.3), (0.1, 0.2, 0.1)> | <(0.8, 0.8), (0.1, 0.2), (0.1, 0.2)> |

A_{3} | <(0.6, 0.8), (0.1, 0.3), (0.3, 0.4)> | <(0.8, 0.6, 0.7), (0.3, 0.1, 0.1), (0.2, 0.1, 0.2)> | <(0.8, 0.7), (0.4, 0.3), (0.2, 0.1)> |

A_{4} | <(0.7, 0.6), (0.1, 0.2), (0.2, 0.3)> | <(0.7, 0.8, 0.7), (0.2, 0.2, 0.1), (0.1, 0.2, 0.2)> | <(0.7, 0.7), (0.2, 0.3), (0.2, 0.3)> |

**Table 3.**All the measure values between A

_{i}(i = 1, 2, 3, 4) and A

^{*}for RSVNSs and ranking orders of the four alternatives.

Measure Method | Measure Value | Ranking Order | The Best Choice |
---|---|---|---|

W(A_{i}, A^{*}) [23] | W(A_{1}, A^{*}) = 0.9848, W(A_{2}, A^{*}) = 0.9938,W(A _{3}, A^{*}) = 0.9858, W(A_{4}, A^{*}) = 0.9879 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

R_{W}_{J}(A_{i}, A^{*}) | R_{W}_{J}(A_{1}, A^{*}) = 0.9187, R_{W}_{J}(A_{2}, A^{*}) = 0.9610,R _{W}_{J}(A_{3}, A^{*}) = 0.9249, R_{W}_{J}(A_{4}, A^{*}) = 0.9320 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

R_{W}_{D}(A_{i}, A^{*}) | R_{W}_{D}(A_{1}, A^{*}) = 0.9568, R_{W}_{D}(A_{2}, A^{*}) = 0.9797,R _{W}_{D}(A_{3}, A^{*}) = 0.9607, R_{W}_{D}(A_{4}, A^{*}) = 0.9646 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

R_{W}_{C}(A_{i}, A^{*’}) | R_{W}_{C}(A_{1}, A^{*}) = 0.9646, R_{W}_{C}(A_{2}, A^{*}) = 0.9832,R _{W}_{C}(A_{3}, A^{*}) = 0.9731, R_{W}_{C}(A_{4}, A^{*}) = 0.9780 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

**Table 4.**Defined simplified neutrosophic decision matrix M(a

_{ij})

_{4}

_{×3}under refined interval neutrosophic set (RINS) environment.

b_{1} | b_{2} | b_{3} | |
---|---|---|---|

{b_{11}, b_{12}} | {b_{21}, b_{22}, b_{23}} | {b_{31}, b_{32}} | |

A_{1} | <([0.6,0.7],[ 0.7,0.8]),([0.2,0.3], [ 0.1,0.2]),([0.2,0.3],[ 0.3,0.4])> | <([0.8,0.9],[0.7,0.8],[0.8,0.9]), ([0.1,0.2],[0.3,0.4],[0.2,0.3]), ([0.2,0.3],[0.2,0.3],[0.1,0.2])> | <([0.6,0.7],[0.8,0.9]),([0.3,0.4], [0.2,0.3]),([0.3,0.4],[0.4,0.5])> |

A_{2} | <([0.8,0.9],[0.7,0.8]),([0.1,0.2], [ 0.2,0.3]),([0.3,0.4],[ 0.2,0.3])> | <([0.7,0.8],[0.8,0.9],[0.7,0.8]), ([0.2,0.3],[0.4,0.5],[0.3,0.4]), ([0.1,0.2],[0.2,0.3],[0.1,0.2])> | <([0.8,0.9],[0.8,0.9]),([0.1,0.2], [0.2,0.3]),([0.1,0.2],[0.2,0.3])> |

A_{3} | <([0.6,0.7],[0.8,0.9]),([0.1,0.2], [ 0.3,0.4]),([0.3,0.4],[0.4,0.5])> | <([0.8,0.9],[0.6,0.7],[0.7,0.8]), ([0.3,0.4],[0.1,0.2],[0.1,0.2]), ([0.2,0.3],[0.1,0.2],[0.2,0.3])> | <([0.8,0.9],[0.7,0.8]),([0.4,0.5], [0.3,0.4]),([0.2,0.3],[0.1,0.2])> |

A_{4} | <([0.7,0.8],[0.6,0.7]),([0.1,0.2], [0.2,0.3]),([0.2,0.3],[0.3,0.4])> | <([0.7,0.8],[0.8,0.9],[0.7,0.8]), ([0.2,0.3],[0.2,0.3],[0.1,0.2]), ([0.1,0.2],[0.2,0.3],[0.2,0.3])> | <([0.7,0.8],[0.7,0.8]),([0.2,0.3], [0.3,0.4]),([0.2,0.3],[0.3,0.4])> |

**Table 5.**All the measure values between A

_{i}(i = 1, 2, 3, 4) and A

^{*}for RINSs and ranking orders of the four alternatives.

Measure Method | Measure Value | Ranking Order | The Best Choice |
---|---|---|---|

W(A_{i}, A^{*}) [23] | W(A_{1}, A^{*}) = 0.9848, W(A_{2}, A^{*}) = 0.9932,W(A _{3}, A^{*}) = 0.9868, W(A_{4}, A^{*}) = 0.9886 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

R_{WJ}(A_{i}, A^{*}) | R_{WJ}(A_{1}, A^{*}) = 0.9314, R_{WJ}(A_{2}, A^{*}) = 0.9693,R _{WJ}(A_{3}, A^{*}) = 0.9369, R_{WJ}(A_{4}, A^{*}) = 0.9430 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

R_{WD}(A_{i}, A^{*}) | R_{WD}(A_{1}, A^{*}) = 0.9639, R_{WD}(A_{2}, A^{*}) = 0.9841,R _{WD}(A_{3}, A^{*}) = 0.9672, R_{WD}(A_{4}, A^{*}) = 0.9705 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

R_{WC}(A_{i}, A^{*}) | R_{WC}(A_{1}, A^{*}) = 0.9697, R_{WC}(A_{2}, A^{*}) = 0.9860,R _{WC}(A_{3}, A^{*}) = 0.9775, R_{WC}(A_{4}, A^{*}) = 0.9805 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

**Table 6.**All the measure values between A

_{i}(i = 1, 2, 3, 4’) and A

^{*}for RSNSs and ranking orders of the four alternatives.

Measure Method | Measure Value | Ranking Order | The Best Choice |
---|---|---|---|

R_{WJ}(A_{i}, A_{*}) | R_{WJ}(A_{1}, A_{*}) = 0.9314, R_{WJ}(A_{2}, A_{*}) = 0.9693,R _{WJ}(A_{3}, A_{*}) = 0.9369, R_{WJ}(A_{4}, A_{*}) = 0.9356 | A_{2}≻A_{3}≻A_{4}≻A_{1} | A_{2} |

R_{WD}(A_{i}, A_{*}) | R_{WD}(A_{1}, A_{*}) = 0.9639, R_{WD}(A_{2}, A_{*}) = 0.9841,R _{WD}(A_{3}, A_{*}) = 0.9672, R_{WD}(A_{4}, A_{*}) = 0.9665 | A_{2}≻A_{3}≻A_{4}≻A_{1} | A_{2} |

R_{WC}(A_{i}, A_{*}) | R_{WC}(A_{1}, A_{*}) = 0.9697, R_{WC}(A_{2}, A_{*}) = 0.9860,R _{WC}(A_{3}, A_{*}) = 0.9775, R_{WC}(A_{4}, A_{*}) = 0.9780 | A_{2}≻A_{4}≻A_{3}≻A_{1} | A_{2} |

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**MDPI and ACS Style**

Chen, J.; Ye, J.; Du, S.
Vector Similarity Measures between Refined Simplified Neutrosophic Sets and Their Multiple Attribute Decision-Making Method. *Symmetry* **2017**, *9*, 153.
https://doi.org/10.3390/sym9080153

**AMA Style**

Chen J, Ye J, Du S.
Vector Similarity Measures between Refined Simplified Neutrosophic Sets and Their Multiple Attribute Decision-Making Method. *Symmetry*. 2017; 9(8):153.
https://doi.org/10.3390/sym9080153

**Chicago/Turabian Style**

Chen, Jiqian, Jun Ye, and Shigui Du.
2017. "Vector Similarity Measures between Refined Simplified Neutrosophic Sets and Their Multiple Attribute Decision-Making Method" *Symmetry* 9, no. 8: 153.
https://doi.org/10.3390/sym9080153