The cosine measure of refined-single valued neutrosophic sets and refined-interval neutrosophic sets for multiple attribute decision-making

Abstract

This paper introduces a Refined-Interval Neutrosophic Set (R-INS) as an extension of a Refined-Single Valued Neutrosophic Set (R-SVNS) and puts forward the decision-making models based on the Cosine Measures of R-SVNSs and Refined-Interval Neutrosophic Sets (R-INSs). By the cosine measure between every alternative and the ideal alternative, all the alternatives can be ranked with the measure values, and the best one of all alternatives can be selected. The proposed methods for multiple attribute decision making (MADM) are more clear (more refined) and the refined evaluation results are credible, so the proposed decision-making models are very suitable for handling with the decision-making problems with refined neutrosophic information.

Keywords

Refined-single valued neutrosophic set refined-interval neutrosophic sets cosine measure decision making multiple attribute

1 Introduction

Neutrosophic set was proposed by Smarandache [1], which is a very powerful tool to deal with indeterminate and incomplete information. It is composed of the neutrosophic components of truth, indeterminacy, and falsity denoted by T, I, F. Since then, other forms of neutrosophic set were proposed by some researchers. Wang et al. [2, 3] proposed a single valued neutrosophic set (SVNS) and an interval neutrosophic set (INS) as extensions of the neutrosophic set. Further, Ye and Smarandache [4] presented a refined-single valued neutrosophic set (R-SVNS), in which neutrosophic components T, I, F are refined into T_1R, T_2R, … T_kR and I_1R, I_2R, … I_kR, and F_1R, F_2R, … F_kR, and its concept is different from the concept of single valued neutrosophic multisets (neutrosophic refined sets) [6 –9]. In [6 –9], they proposed the neutrosophic refined sets just is the multiple-valued neutrosophic form and they are not the actual refined neutrosophic set, so literature [4] redefined the refined neutrosophic set. In this paper, we propose a refined-interval neutrosophic set (R-INS) as an extension of R-SVNS. Till now, there are few studies and applications of refined neutrosophic sets (R-SVNSs and R-INSs) in science and engineering fields, so we propose decision-making methods based on the cosine measures of R-SVNSs and R-INSs, and then provide a decision-making example to show its application under refined neutrosophic environments.

The other sections of this paper are listed as follows. Section 2 describes some basic concepts of SVNS, R-SVNS, INS, and R-INS. Section 3 introduces cosine measures of R-SVNS and R-INS. Section 4 establishes multiple attribute decision-making methods using the cosine measures of R-SVNSs and R-INSs. Section 5 presents an actual example with R-SVNSs and R-INSs to indicate the application of the proposed methods. Section 6 contains a conclusion.

2 Some concepts of SVNS and R-SVNS

Definition 1. [2] Let X be a universe of discourse, then a SVNS R in X can be characterized by a truth-membership function T_R (x), an indeterminacy-membership function I_R (x) and a falsity-membership function F_R (x), T_R (x) , I_R (x) , F_R (x) ∈ [0, 1] , while 0 ≤ T_R (x) + I_R (x) + F_R (x) ≤ 3.

Then a SVNS R can be expressed by the following form: $R = {〈 x, T_{R} (x), I_{R} (x), F_{R} (x) 〉 | x \in X} .$

Definition 2. [5] Let X ={ x₁, x₂, …, x_n } be a universe of discourse, and M and N be two SVNSs, M ={ 〈 x, T_M (x_i) , I_M (x_i) , F_M (x_i) 〉 |x_i ∈ X } and N ={ 〈 x, T_N (x_i) , I_N (x_i) , F_N (x_i) 〉 |x_i ∈ X }. The cosine measure method of two SVNSs M and N is:

$\begin{matrix} D (M, N) & = & \frac{1}{n} \sum_{k = 1}^{n} cos {\frac{π}{6} (| T_{M} (x_{k}) - T_{N} (x_{k}) | \\ + | I_{M} (x_{k}) - I_{N} (x_{k}) | \\ + | F_{M} (x_{k}) - F_{N} (x_{k}) |)} \end{matrix}$ (1)

Obviously, the cosine measure D (M, N) satisfies the following properties (1–4) [5]:

0 ≤ D (M, N) ≤ 1;

D (M, N) =1 if and only if M = N ;

D (M, N) = D (N, M);

If L is a SVNS in X and M ⊆ N ⊆ L, then D (M, L) ≤ D (M, N) D (M, L) ≤ D (N, L).

Definition 3. [4] Let X ={ x₁, x₂, …, x_n } be a universe of discourse, and then an R-SVNS R in X can be expressed by the following form:

R = {〈x, (T_1R (x) , T_2R (x) , …, T_kR (x)) , (I_1R (x) , I_2R (x) , …, I_kR (x)) , (F_1R (x) , F_2R (x) , …, F_kR (x)) 〉|x ∈ X}, in which k is a positive integer, T_1R (x) , T_2R (x) , …, T_kR (x) ∈ [0, 1] , I_1R (x) , I_2R (x) , …, I_kR (x) ∈ [0, 1] , F_1R (x) , F_2R (x) , …, F_kR (x) ∈ [0, 1] , and 0 ≤ T_iR (x) + I_iR (x) + F_iR (x) ≤ 3 for i = 1, 2, …, k.

Definition 4. [4] Let X = {x₁, x₂, …, x_n} be a universe of discourse, and M and N be two R-SVNSs $\begin{matrix} M & = & {〈 \begin{matrix} x, (T_{1 M} (x), T_{2 M} (x), \dots, T_{kM} (x)), \\ (I_{1 M} (x), I_{2 M} (x), \dots, I_{kM} (x)), \\ (F_{1 M} (x), F_{2 M} (x), \dots, F_{kM} (x)) \end{matrix} 〉 | x \in X} \\ N & = & {〈 \begin{matrix} x, (T_{1 N} (x), T_{2 N} (x), \dots, T_{kN} (x)), \\ (I_{1 N} (x), I_{2 N} (x), \dots, I_{kN} (x)), \\ (F_{1 N} (x), F_{2 N} (x), \dots, F_{kN} (x)) \end{matrix} 〉 | x \in X} \end{matrix}$

Then the relations of M and N are given as follows [4]:

M ⊆ N, if and only if T_iM (x) ≤ T_iN (x), I_iM (x) ≤ I_iN (x) , F_iM (x) ≤ F_iN (x) for i = 1, 2, …, k;

M = N, if and only if T_iM (x) = T_iN (x), I_iM (x) = I_iN (x) , F_iM (x) = F_iN (x) for i = 1, 2, …, k;

$M \cup N = {〈 \begin{matrix} x, (T_{1 M} (x) \lor T_{1 N} (x), T_{2 M} (x) \lor T_{2 N} (x), \\ \dots, T_{kM} (x) \lor T_{kN} (x)), \\ (I_{1 M} (x) \land I_{1 N} (x), I_{2 M} (x) \land I_{2 N} (x), \\ \dots, I_{kM} (x) \land I_{kN} (x)), \\ (F_{1 M} (x) \land F_{1 N} (x), F_{2 M} (x) \land F_{2 N} (x), \\ \dots, F_{kM} (x) \land F_{kN} (x)) \end{matrix} 〉 | x \in X};$

$M \cap N = {〈 \begin{matrix} x, (T_{1 M} (x) \land T_{1 N} (x), T_{2 M} (x) \land T_{2 N} (x), \\ \dots, T_{kM} (x) \land T_{kN} (x)), \\ (I_{1 M} (x) \lor I_{1 N} (x), I_{2 M} (x) \lor I_{2 N} (x), \\ \dots, I_{kM} (x) \lor I_{kN} (x)), \\ (F_{1 M} (x) \lor F_{1 N} (x), F_{2 M} (x) \lor F_{2 N} (x), \\ \dots, F_{kM} (x) \lor F_{kN} (x)) \end{matrix} 〉 | x \in X} .$

Definition 5. [3] Let X be a universe of discourse, then an INS R can be expressed by the following form: $R = {〈 \begin{matrix} x, [T_{R} (x)_\inf, T_{R} (x)_top], \\ [I_{R} (x)_\inf, I_{R} (x)_top], \\ [F_{R} (x)_\inf, F_{R} (x)_top] \end{matrix} 〉 | x \in X} .$

Then, the sum of T_R (x) _ top, I_R (x) _ top and F_R (x) _ top satisfies the condition 0 ≤ T_R (x) _ top + I_R (x) _ top + F_R (x) _ top ≤ 3.

As an extension of R-SVNS, we present the following definition of R-INS.

Definition 6. Let X ={ x₁, x₂, …, x_n } be a universe of discourse, and then an R-INS R in X can be expressed by the following form: $R = {〈 \begin{matrix} x, ([T_{1 R} (x)_\inf, T_{1 R} (x)_top], \\ [T_{2 R} (x)_\inf, T_{2 R} (x)_top], \\ \dots, \\ [T_{kR} (x)_\inf, T_{kR} (x)_top]), \\ ([I_{1 R} (x)_\inf, I_{1 R} (x)_top] \\ [I_{1 R} (x)_\inf, I_{2 R} (x)_top] \\ \dots, \\ [I_{kR} (x)_\inf, I_{kR} (x)_top]), \\ ([F_{1 R} (x)_\inf, F_{1 R} (x)_top], \\ [F_{2 R} (x)_\inf, F_{2 R} (x)_top], \\ \dots, \\ [F_{kR} (x)_\inf, F_{kR} (x)_top]) \end{matrix} 〉 | x \in X},$ in which k is a positive integer and T_1R (x) _ inf, T_1R (x) _ top, T_2R (x) _ inf, T_2R (x) _ top, …, T_kR (x) _ inf, T_kR (x) _ top ∈ [0, 1], I_1R (x) _ inf, I_1R (x) _ top, I_2R (x) _ inf, I_2R (x) _ top, …, I_kR (x) _ inf, I_kR (x) _ top ∈ [0, 1], F_1R (x) _ inf, F_1R (x) _ top, F_2R (x) _ inf, F_2R (x) _ top, …, F_kR (x) _ inf, F_kR (x) _ top ∈ [0, 1], and 0 ≤ T_iR (x) _ top + I_iR (x) _ top + F_iR (x) top ≤ 3 for i = 1, 2, …, k.

3 Cosine measures of refined neutrosophic sets

3.1 Cosine measure method of R-SVNSs

Definition 7. Let X ={ x₁, x₂, …, x_n } be a universe of discourse, and M and N be two R-SVNSs

$\begin{matrix} M & = & {〈 \begin{matrix} x_{i}, (T_{1 M} (x_{i}), T_{2 M} (x_{i}), \dots, T_{k_{i} M} (x_{i})), \\ (I_{1 M} (x_{i}), I_{2 M} (x_{i}), \dots, I_{k_{i} M} (x_{i})), \\ (F_{1 M} (x_{i}), F_{2 M} (x_{i}), \dots, F_{k_{i} M} (x_{i})) \end{matrix} 〉 | x_{i} \in X}, \\ N & = & {〈 \begin{matrix} x_{i}, (T_{1 N} (x_{i}), T_{2 N} (x_{i}), \dots, T_{k_{i} N} (x_{i})), \\ (I_{1 N} (x_{i}), I_{2 N} (x_{i}), \dots, I_{k_{i} N} (x_{i})), \\ (F_{1 N} (x_{i}), F_{2 N} (x_{i}), \dots, F_{k_{i} N} (x_{i})) \end{matrix} 〉 | x_{i} \in X} . \end{matrix}$

Where k_i is a positive integer, and all T_jM (x_i), I_jM (x_i) , F_jM (x_i) ∈ [0, 1] and all T_jN (x_i) , I_jN (x_i) , F_jN (x_i) ∈ [0, 1] (i = 1, 2, … n ; j = 1, 2, …, k_i).

As an extension of Definition 2, we present a cosine measure between two R-SVNSs M and N as follows:

$\begin{matrix} D (M, N) & = & \frac{1}{n} \sum_{i = 1}^{n} \sum_{j = 1}^{k_{i}} cos {\frac{π}{6} (| T_{jM} (x_{i}) - T_{jN} (x_{i}) | \\ + | I_{jM} (x_{i}) - I_{jN} (x_{i}) | \\ + | F_{jM} (x_{i}) - F_{jN} (x_{i}) |)} / k_{i} \end{matrix}$ (2)

Theorem 1.The cosine measureD (M, N) between two R-SVNSsMandNsatisfies the following properties:

D (M, N) = D (N, M);

0 ≤ D (M, N) ≤1;

D (M, N) = 1, if and only if M = N .

➁ For 0 ≤ T_jM (x_i) ≤ 1 then 0 ≤ |T_jM (x_i) - T_jN (x_i) |≤1, so we can get 0 ≤ |T_jM (x_i) - T_jN (x_i) | + |I_jM (x_i) - I_jN (x_i) | + |F_jM (x_i) - F_jN (x_i) |≤3 and $0 \leq \cos {\frac{π}{6} (| T_{jM} (x_{i}) - T_{jN} (x_{i}) | + | I_{jM} (x_{i}) - I_{jN} (x_{i}) | + | F_{jM} (x_{i}) - F_{jN} (x_{i}) |)} \leq 1$ ; $\begin{matrix} D (M, N) & = & \frac{1}{n} \sum_{i = 1}^{n} \sum_{j = 1}^{k_{i}} cos {\frac{π}{6} (| T_{jM} (x_{i}) - T_{jN} (x_{i}) | + | I_{jM} (x_{i}) - I_{jN} (x_{i}) | + | F_{jM} (x_{i}) - F_{jN} (x_{i}) |)} / k_{i} \\ = & \frac{1}{n} {\frac{1}{k_{1}} {cos {\frac{π}{6} (| T_{1 M} (x_{1}) - T_{1 N} (x_{1}) | + | I_{1 M} (x_{1}) - I_{1 N} (x_{1}) | + | F_{1 M} (x_{1}) - F_{1 N} (x_{1}) |)} + \dots \\ + cos {\frac{π}{6} (| T_{k_{1} M} (x_{1}) - T_{k_{1} N} (x_{1}) | + | I_{k_{1} M} (x_{1}) - I_{k_{1} N} (x_{1}) | + | F_{k_{1} M} (x_{1}) - F_{k_{1} N} (x_{1}) |)}} + \dots \\ + \frac{1}{k_{n}} {cos {\frac{π}{6} (| T_{1 M} (x_{n}) - T_{1 N} (x_{n}) | + | I_{1 M} (x_{n}) - I_{1 N} (x_{n}) | + | F_{1 M} (x_{n}) - F_{1 N} (x_{n}) |)} + \dots \\ + cos {\frac{π}{6} (| T_{k_{n} M} (x_{n}) - T_{k_{n} N} (x_{n}) | + | I_{k_{n} M} (x_{n}) - I_{k_{n} N} (x_{n}) | + | F_{k_{n} M} (x_{n}) - F_{k_{n} N} (x_{n}) |)}}} \end{matrix}$

Let $cos {\frac{π}{6} (| T_{jM} (x_{i}) - T_{jN} (x_{i}) | + | I_{jM} (x_{i}) - I_{jN} (x_{i}) | + | F_{jM} (x_{i}) - F_{jN} (x_{i}) |)} = α_{j}^{i} (i = 1, 2, \dots n; j = 1 2 \dots k_{i})$ then $D (M, N) = \frac{1}{n} {\frac{1}{k_{1}} {α_{1}^{1} + \dots + α_{k_{1}}^{1}} + \dots + \frac{1}{k_{n}} {α_{1}^{n} + \dots + α_{k_{n}}^{n}}}$ .

According to $0 \leq cos {\frac{π}{6} (| T_{jM} (x_{i}) - T_{jN} (x_{i}) | + | I_{jM} (x_{i}) - I_{jN} (x_{i}) | + | F_{jM} (x_{i}) - F_{jN} (x_{i}) |)} = α_{j}^{i} \leq 1$ , we can obtain $0 \leq \frac{1}{k_{1}} {α_{1}^{1} + \dots + α_{k_{1}}^{1}} \leq 1$ and $0 \leq \frac{1}{n} {\frac{1}{k_{1}} {α_{1}^{1} + \dots + α_{k_{1}}^{1}} + \dots + \frac{1}{k_{n}} {α_{1}^{n} + \dots + α_{k_{n}}^{n}}} \leq 1$ , so we can get 0 ≤ D (M, N) ≤1.

➂ If M = N then T_jM (x_i) = T_jN (x_i) , I_jM (x_i) = I_jN (x_i) , andF_jM (x_i) = F_jN (x_i) for any x_i ∈ X andi = 1, 2, … n, so we can get D (M, N) = 1, if and only if M = N.

Considering the weights of attributes, we assume that the weight of each attribute G_i (i = 1, 2, …, n) with w_i ∈ [0, 1] , and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the weighted cosine measure formula of R-SVNS can be introduced as follows:

$\begin{matrix} W (M, N) \\ = \sum_{i = 1}^{n} w_{i} \sum_{j = 1}^{k_{i}} cos {\frac{π}{6} (| T_{jM} (x_{i}) - T_{jN} (x_{i}) | \\ + | I_{jM} (x_{i}) - I_{jN} (x_{i}) | \\ + | F_{jM} (x_{i}) - F_{jN} (x_{i}) |)} / k_{i} \end{matrix}$ (3)

3.2 Cosine measure method of R-INSs

Definition 8. Let X ={ x₁, x₂, …, x_n } be a universe of discourse, and M and N be two R-INSs, $M = {〈 \begin{matrix} x_{i}, ([T_{1 M} (x_{i})_\inf, T_{1 M} (x_{i})_top], \\ [T_{2 M} (x_{i})_\inf, T_{2 M} (x_{i})_top], \\ \dots, \\ [T_{k_{i} M} (x_{i})_\inf, T_{k_{i} M} (x_{i})_top]), \\ ([I_{1 M} (x_{i})_\inf, I_{1 M} (x_{i})_top] \\ [I_{2 M} (x_{i})_\inf, I_{2 M} (x_{i})_top] \\ \dots, \\ [I_{k_{i} M} (x_{i})_\inf, I_{k_{i} M} (x_{i})_top]), \\ ([F_{1 M} (x_{i})_\inf, F_{1 M} (x_{i})_top], \\ [F_{2 M} (x_{i})_\inf, F_{2 M} (x_{i})_top], \\ \dots, \\ [F_{k_{i} M} (x_{i})_\inf, F_{k_{i} M} (x_{i})_top]) \end{matrix} 〉 | x \in X},$

$N = {〈 \begin{matrix} x_{i}, ([T_{1 N} (x_{i})_\inf, T_{1 N} (x_{i})_top], \\ [T_{2 N} (x_{i})_\inf, T_{2 N} (x_{i})_top], \\ \dots, \\ [T_{k_{i} N} (x_{i})_\inf, T_{k_{i} N} (x_{i})_top]), \\ ([I_{1 N} (x_{i})_\inf, I_{1 N} (x_{i})_top] \\ [I_{2 N} (x_{i})_\inf, I_{2 N} (x_{i})_top] \\ \dots, \\ [I_{k_{i} N} (x_{i})_\inf, I_{k_{i} N} (x_{i})_top]), \\ ([F_{1 N} (x_{i})_\inf, F_{1 N} (x_{i})_top], \\ [F_{2 N} (x_{i})_\inf, F_{2 N} (x_{i})_top], \\ \dots, \\ [F_{k_{i} N} (x_{i})_\inf, F_{k_{i} N} (x_{i})_top]) \end{matrix} 〉 | x \in X},$

where k_i is a positive integer, and all T_jM (x_i) _ inf, T_jM (x_i) _ top, I_jM (x_i) _ inf, I_jM (x_i) _ top, F_jM (x_i) _ inf, F_jM (x_i) _ top ∈ [0, 1], and all T_jN (x_i) _ inf, T_jN (x_i) _ top, I_jN (x_i) _ inf, I_jN (x_i) _ top, F_jN (x_i) _ inf, F_jN (x_i) _ top ∈ [0, 1] (i = 1, 2, … n ; j = 1, 2, …, k_i).

We present a cosine measure between two R-INSs M and N as follows:

$\begin{matrix} D (M, N) \\ = \frac{1}{n} \sum_{i = 1}^{n} \sum_{j = 1}^{k_{i}} cos {\frac{π}{12} (| T_{jM} (x_{i})_\inf \\ - T_{jN} (x_{i})_\inf | + | T_{jM} (x_{i})_top - T_{jN} (x_{i})_top | \\ + | I_{jM} (x_{i})_\inf - I_{jN} (x_{i})_\inf | + | I_{jM} (x_{i})_top \\ - I_{jN} (x_{i})_top | + | F_{jM} (x_{i})_\inf - F_{jN} (x_{i})_\inf | \\ + | F_{jM} (x_{i})_top - F_{jN} (x_{i})_top |)} / k_{i} \end{matrix}$ (4)

Theorem 2.The cosine measureD (M, N) between two R-INSsMandNalso satisfies the following properties:

D (M, N) = D (N, M);

0 ≤ D (M, N) ≤1;

D (M, N) = 1, if and only if M = N .

Proves of these properties are similar to those in Theorem 1, so we don’t repeat it here.

$\begin{matrix} W (M, N) \\ = \sum_{i = 1}^{n} w_{i} \sum_{j = 1}^{k_{i}} cos {\frac{π}{12} (| T_{jM} (x_{i})_\inf \\ - T_{jN} (x_{i})_\inf | + | T_{jM} (x_{i})_top - T_{jN} (x_{i})_top | \\ + | I_{jM} (x_{i})_\inf - I_{jN} (x_{i})_\inf | + | I_{jM} (x_{i})_top \\ - I_{jN} (x_{i})_top | + | F_{jM} (x_{i})_\inf - F_{jN} (x_{i})_\inf | \\ + | F_{jM} (x_{i})_top - F_{jN} (x_{i})_top |)} / k_{i} \end{matrix}$ (5)

4 Decision-making models

Usually, a set of alternatives S = {S₁, S₂, …, S_m} and a set of attributes G = {G₁, G₂, …, G_n} can be established in a decision-making problem. But G_i (i = 1, 2, …, n) maybe have some sub-attribute G_ij (i = 1, 2, …, n, j = 1, 2, … k_i) , for this phenomenon. We can use R-SVNS or R-INS to express it.

4.1 Decision-making method using the cosine of R-SVNSs

Let $S_{r} = {〈 \begin{matrix} G_{i}, (T_{1 S_{r}} (G_{i}), T_{2 S_{r}} (G_{i}), \dots, T_{k_{i} S_{r}} (G_{i})), \\ (I_{1 S_{r}} (G_{i}), I_{2 S_{r}} (G_{i}), \dots, I_{k_{i} S_{r}} (G_{i})), \\ (F_{1 S_{r}} (G_{i}), F_{2 S_{r}} (G_{i}), \dots, F_{k_{i} S_{r}} (G_{i})) \end{matrix} 〉 | G_{i} \in G}$ , r = 1, 2, …, m and i = 1, 2, …, n.

The single values of the three functions T_{k
_i
S
_r} (G_i) , I_{k
_i
S
_r} (G_i) , F_{k
_i
S
_r} (G_i) , can be denoted by R-SVNSs, so we establish the R-SVNS decision matrix D, which is shown in Table 1.

Table 1
The R-SVNS decision matrix D

G₁ (G₁₁, G₁₂, …, G_1k₁) … G_n (G_n1, G_n2, …, G_{nk
_n})

S ₁ 〈 (T_1S₁, T_2S₁, … , T_{k
₁
S
₁}) , (I_1S₁, I_2S₁, …, I_{k
₁
S
₁}) , (F_1S₁, F_2S₁, …, F_{k
₁
S
₁}) 〉 … 〈 (T_1S₁, T_2S₁, … , T_{k
_n
S
₁}) , (I_1S₁, I_2S₁, …, I_{k
_n
S
₁}) , (F_1S₁, F_2S₁, …, F_{k
_n
S
₁}) 〉

S ₂ 〈 (T_1S₂, T_2S₂, … , T_{k
₁
S
₂}) , (I_1S₂, I_2S₂, …, I_{k
₁
S
₂}) , (F_1S₂, F_2S₂, …, F_{k
₁
S
₂}) 〉 … 〈 (T_1S₂, T_2S₂, … , T_{k
_n
S
₂}) , (I_1S₂, I_2S₂, …, I_{k
_n
S
₂}) , (F_1S₂, F_2S₂, …, F_{k
_n
S
₂}) 〉

… … … …

S _m 〈 (T_{1S_m}, T_{2S_m}, … , T_{k
₁
S
_m}) , (I_{1S_m}, I_{2S_m}, …, I_{k
₁
S
_m}) , (F_{1S_m}, F_{2S_m}, …, F_{k
₁
S
_n}) 〉 … 〈 (T_1S₁, T_2S₁, … , T_{k
_n
S
_m}) , (I_1S₁, I_2S₁, …, I_{k
_n
S
_m}) , (F_1S₁, F_{2S_m}, …, F_{k
_n
S
_m}) 〉

	G₁ (G₁₁, G₁₂, …, G_1k₁)	…	G_n (G_n1, G_n2, …, G_{nk _n})
S ₁	〈 (T_1S₁, T_2S₁, … , T_{k ₁ S ₁}) , (I_1S₁, I_2S₁, …, I_{k ₁ S ₁}) , (F_1S₁, F_2S₁, …, F_{k ₁ S ₁}) 〉	…	〈 (T_1S₁, T_2S₁, … , T_{k _n S ₁}) , (I_1S₁, I_2S₁, …, I_{k _n S ₁}) , (F_1S₁, F_2S₁, …, F_{k _n S ₁}) 〉
S ₂	〈 (T_1S₂, T_2S₂, … , T_{k ₁ S ₂}) , (I_1S₂, I_2S₂, …, I_{k ₁ S ₂}) , (F_1S₂, F_2S₂, …, F_{k ₁ S ₂}) 〉	…	〈 (T_1S₂, T_2S₂, … , T_{k _n S ₂}) , (I_1S₂, I_2S₂, …, I_{k _n S ₂}) , (F_1S₂, F_2S₂, …, F_{k _n S ₂}) 〉
…	…	…	…
S _m	〈 (T_{1S_m}, T_{2S_m}, … , T_{k ₁ S _m}) , (I_{1S_m}, I_{2S_m}, …, I_{k ₁ S _m}) , (F_{1S_m}, F_{2S_m}, …, F_{k ₁ S _n}) 〉	…	〈 (T_1S₁, T_2S₁, … , T_{k _n S _m}) , (I_1S₁, I_2S₁, …, I_{k _n S _m}) , (F_1S₁, F_{2S_m}, …, F_{k _n S _m}) 〉

Step 1. Based on the R-SVNS decision matrix D, setting $T_{k_{i} S_{m}}^{\max}$ is the maximum truth value in each column k_iS_mof the decision matrix D, $I_{k_{i} S_{m}}^{\min} and F_{k_{i} S_{m}}^{\min}$ are the minimum indeterminate and falsity values in each column k_iS_m of the decision matrix D, the ideal solution (ideal R-SVNS) can be determined as $S_{i}^{*}$ .

$\begin{matrix} S_{i}^{*} & = & 〈 \begin{matrix} (T_{1 S_{m}}^{\max}, T_{2 S_{m}}^{\max}, \dots, T_{k_{i} S_{m}}^{\max}), \\ (I_{1 S_{m}}^{\min}, I_{2 S_{m}}^{\min}, \dots, I_{k_{i} S_{m}}^{\min}), \\ (F_{1 S_{m}}^{\min}, F_{2 S_{m}}^{\min}, \dots, F_{k_{i} S_{m}}^{\min}) \end{matrix} 〉, \\ for i = 1, 2, \dots n . \end{matrix}$ (6)

So we can get the ideal alternative $S^{*} = {S_{1}^{*}, S_{2}^{*}, \dots, S_{n}^{*}}$ .

Step 2. When the weights of attributes are given by w = (w₁, w₂, …, w_n) with w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . The cosine measure between each alternative S_r (r = 1, 2, …, m) and the ideal alternative S^* can be calculated according to the formula (3). Then we can obtain the values of W (S_r, S^*) for r = 1, 2, …, m.

Step 3. According the values of W (S_r, S^*) for r = 1, 2, …, m, we can rank the alternatives in a descending order, then the alternative of biggest W (S_r, S^*) value is the best choice.

Step 4: End.

4.2 Decision-making method using the cosine of R-INSs

Let $\begin{matrix} S_{r} & = & {〈 \begin{matrix} G_{i}, ([T_{1 S_{r}} (G_{i})_\inf, T_{1 S_{r}} (G_{i})_top], \\ [T_{2 S_{r}} (G_{i})_\inf, T_{2 S_{r}} (G_{i})_top], \\ \dots, \\ [T_{k_{i} S_{r}} (G_{i})_\inf, T_{k_{i} S_{r}} (G_{i})_top]), \\ ([I_{1 S_{r}} (G_{i})_\inf, I_{1 S_{r}} (G_{i})_top], \\ [I_{2 S_{r}} (G_{i})_\inf, I_{2 S_{r}} (G_{i})_top], \\ \dots, \\ [I_{k_{i} S_{G}} (x_{i})_\inf, I_{k_{i} S_{r}} (G_{i})_top]), \\ ([F_{1 S_{r}} (G_{i})_\inf, F_{1 N} (G_{i})_top], \\ [F_{2 S_{r}} (G_{i})_\inf, F_{2 N} (G_{i})_top], \\ \dots, \\ [F_{k_{i} S_{r}} (G_{i})_\inf, F_{k_{i} S_{r}} (G_{i})_top]) \end{matrix} 〉 | G_{i} \in G}, \\ r = 1, 2 \dots, m and i = 1, 2 \dots, n . \end{matrix}$

The interval values of the three functions T_{k
_i
S
_r} (G_i) , I_{k
_i
S
_r} (G_i) , F_{k
_i
S
_r} (G_i) are denoted by R-INSs, we use $T_{k_{i} S_{r}}^{L}$ to express T_{k
_i
S
_r} (G_i) _ inf and $T_{k_{i} S_{r}}^{U}$ to express T_{k
_i
S
_r} (G_i) _ top, $I_{k_{i} S_{r}}^{L}$ to express I_{k
_i
S
_r} (G_i) _ inf and $I_{k_{i} S_{r}}^{U}$ to express I_{k
_i
S
_r} (G_i) _ top, $F_{k_{i} S_{r}}^{L}$ to express F_{k
_i
S
_r} (G_i) _ inf and $F_{k_{i} S_{r}}^{U}$ to express F_{k
_i
S
_r} (G_i) _ top, so we establish the R-INS decision matrix D, which is shown in Table 2.

Table 2
The R-INS decision matrix D

G₁ (G₁₁, G₁₂, …, G_1k₁) …

S ₁ $〈 ([T_{1 S_{1}}^{L}, T_{1 S_{1}}^{U}], [T_{2 S_{1}}^{L}, T_{2 S_{1}}^{U}], \dots, [T_{k_{1} S_{1}}^{L}, T_{k_{1} S_{1}}^{U}]), ([I_{1 S_{1}}^{L}, I_{1 S_{1}}^{U}], [I_{2 S_{1}}^{L}, I_{2 S_{1}}^{U}], \dots, [I_{k_{1} S_{1}}^{L}, I_{k_{1} S_{1}}^{U}]), ([F_{1 S_{1}}^{L}, F_{1 S_{1}}^{U}], [F_{2 S_{1}}^{L}, F_{2 S_{1}}^{U}], \dots, [F_{k_{1} S_{1}}^{L}, F_{k_{1} S_{1}}^{U}]) 〉$ …

S ₂ $〈 ([T_{1 S_{2}}^{L}, T_{1 S_{2}}^{U}], [T_{2 S_{2}}^{L}, T_{2 S_{2}}^{U}], \dots, [T_{k_{1} S_{2}}^{L}, T_{k_{1} S_{2}}^{U}]), ([I_{1 S_{2}}^{L}, I_{1 S_{2}}^{U}], [I_{2 S_{2}}^{L}, I_{2 S_{2}}^{U}], \dots, [I_{k_{1} S_{2}}^{L}, I_{k_{1} S_{2}}^{U}]), ([F_{1 S_{2}}^{L}, F_{1 S_{2}}^{U}], [F_{2 S_{2}}^{L}, F_{2 S_{2}}^{U}], \dots, [F_{k_{1} S_{2}}^{L}, F_{k_{1} S_{2}}^{U}]) 〉$ …

… … …

S _m $〈 ([T_{1 S_{m}}^{L}, T_{1 S_{m}}^{U}], [T_{2 S_{m}}^{L}, T_{2 S_{m}}^{U}], \dots, [T_{k_{1} S_{m}}^{L}, T_{k_{1} S_{m}}^{U}]), ([I_{1 S_{m}}^{L}, I_{1 S_{m}}^{U}], [I_{2 S_{m}}^{L}, I_{2 S_{m}}^{U}], \dots, [I_{k_{1} S_{m}}^{L}, I_{k_{1} S_{m}}^{U}]), ([F_{1 S_{m}}^{L}, F_{1 S_{m}}^{U}], [F_{2 S_{m}}^{L}, F_{2 S_{m}}^{U}], \dots, [F_{k_{1} S_{m}}^{L}, F_{k_{1} S_{m}}^{U}]) 〉$ …

… G_n (G_n1, G_n2, …, G_{nk
_n})

S ₁ … $〈 ([T_{1 S_{1}}^{L}, T_{1 S_{1}}^{U}], [T_{2 S_{1}}^{L}, T_{2 S_{1}}^{U}], \dots, [T_{k_{n} S_{1}}^{L}, T_{k_{n} S_{1}}^{U}]), ([I_{1 S_{1}}^{L}, I_{1 S_{1}}^{U}], [I_{2 S_{1}}^{L}, I_{2 S_{1}}^{U}], \dots, [I_{k_{n} S_{1}}^{L}, I_{k_{n} S_{1}}^{U}]), ([F_{1 S_{1}}^{L}, F_{1 S_{1}}^{U}], [F_{2 S_{1}}^{L}, F_{2 S_{1}}^{U}], \dots, [F_{k_{n} S_{1}}^{L}, F_{k_{n} S_{1}}^{U}]) 〉$

S ₂ … $〈 ([T_{1 S_{2}}^{L}, T_{1 S_{2}}^{U}], [T_{2 S_{2}}^{L}, T_{2 S_{2}}^{U}], \dots, [T_{k_{n} S_{2}}^{L}, T_{k_{n} S_{2}}^{U}]), ([I_{1 S_{2}}^{L}, I_{1 S_{2}}^{U}], [I_{2 S_{2}}^{L}, I_{2 S_{2}}^{U}], \dots, [I_{k_{n} S_{2}}^{L}, I_{k_{n} S_{2}}^{U}]), ([F_{1 S_{2}}^{L}, F_{1 S_{2}}^{U}], [F_{2 S_{2}}^{L}, F_{2 S_{2}}^{U}], \dots, [F_{k_{n} S_{2}}^{L}, F_{k_{n} S_{2}}^{U}]) 〉$

… … …

S _m … $〈 ([T_{1 S_{m}}^{L}, T_{1 S_{m}}^{U}], [T_{2 S_{m}}^{L}, T_{2 S_{m}}^{U}], \dots, [T_{k_{n} S_{m}}^{L}, T_{k_{n} S_{m}}^{U}]), ([I_{1 S_{m}}^{L}, I_{1 S_{m}}^{U}], [I_{2 S_{m}}^{L}, I_{2 S_{m}}^{U}], \dots, [I_{k_{n} S_{m}}^{L}, I_{k_{n} S_{m}}^{U}]), ([F_{1 S_{m}}^{L}, F_{1 S_{m}}^{U}], [F_{2 S_{m}}^{L}, F_{2 S_{m}}^{U}], \dots, [F_{k_{n} S_{m}}^{L}, F_{k_{n} S_{m}}^{U}]) 〉$

	G₁ (G₁₁, G₁₂, …, G_1k₁)	…
S ₁		$〈 ([T_{1 S_{1}}^{L}, T_{1 S_{1}}^{U}], [T_{2 S_{1}}^{L}, T_{2 S_{1}}^{U}], \dots, [T_{k_{1} S_{1}}^{L}, T_{k_{1} S_{1}}^{U}]), ([I_{1 S_{1}}^{L}, I_{1 S_{1}}^{U}], [I_{2 S_{1}}^{L}, I_{2 S_{1}}^{U}], \dots, [I_{k_{1} S_{1}}^{L}, I_{k_{1} S_{1}}^{U}]), ([F_{1 S_{1}}^{L}, F_{1 S_{1}}^{U}], [F_{2 S_{1}}^{L}, F_{2 S_{1}}^{U}], \dots, [F_{k_{1} S_{1}}^{L}, F_{k_{1} S_{1}}^{U}]) 〉$	…
S ₂		$〈 ([T_{1 S_{2}}^{L}, T_{1 S_{2}}^{U}], [T_{2 S_{2}}^{L}, T_{2 S_{2}}^{U}], \dots, [T_{k_{1} S_{2}}^{L}, T_{k_{1} S_{2}}^{U}]), ([I_{1 S_{2}}^{L}, I_{1 S_{2}}^{U}], [I_{2 S_{2}}^{L}, I_{2 S_{2}}^{U}], \dots, [I_{k_{1} S_{2}}^{L}, I_{k_{1} S_{2}}^{U}]), ([F_{1 S_{2}}^{L}, F_{1 S_{2}}^{U}], [F_{2 S_{2}}^{L}, F_{2 S_{2}}^{U}], \dots, [F_{k_{1} S_{2}}^{L}, F_{k_{1} S_{2}}^{U}]) 〉$	…
…	…	…
S _m		$〈 ([T_{1 S_{m}}^{L}, T_{1 S_{m}}^{U}], [T_{2 S_{m}}^{L}, T_{2 S_{m}}^{U}], \dots, [T_{k_{1} S_{m}}^{L}, T_{k_{1} S_{m}}^{U}]), ([I_{1 S_{m}}^{L}, I_{1 S_{m}}^{U}], [I_{2 S_{m}}^{L}, I_{2 S_{m}}^{U}], \dots, [I_{k_{1} S_{m}}^{L}, I_{k_{1} S_{m}}^{U}]), ([F_{1 S_{m}}^{L}, F_{1 S_{m}}^{U}], [F_{2 S_{m}}^{L}, F_{2 S_{m}}^{U}], \dots, [F_{k_{1} S_{m}}^{L}, F_{k_{1} S_{m}}^{U}]) 〉$	…
	…	G_n (G_n1, G_n2, …, G_{nk _n})
S ₁	…	$〈 ([T_{1 S_{1}}^{L}, T_{1 S_{1}}^{U}], [T_{2 S_{1}}^{L}, T_{2 S_{1}}^{U}], \dots, [T_{k_{n} S_{1}}^{L}, T_{k_{n} S_{1}}^{U}]), ([I_{1 S_{1}}^{L}, I_{1 S_{1}}^{U}], [I_{2 S_{1}}^{L}, I_{2 S_{1}}^{U}], \dots, [I_{k_{n} S_{1}}^{L}, I_{k_{n} S_{1}}^{U}]), ([F_{1 S_{1}}^{L}, F_{1 S_{1}}^{U}], [F_{2 S_{1}}^{L}, F_{2 S_{1}}^{U}], \dots, [F_{k_{n} S_{1}}^{L}, F_{k_{n} S_{1}}^{U}]) 〉$
S ₂	…	$〈 ([T_{1 S_{2}}^{L}, T_{1 S_{2}}^{U}], [T_{2 S_{2}}^{L}, T_{2 S_{2}}^{U}], \dots, [T_{k_{n} S_{2}}^{L}, T_{k_{n} S_{2}}^{U}]), ([I_{1 S_{2}}^{L}, I_{1 S_{2}}^{U}], [I_{2 S_{2}}^{L}, I_{2 S_{2}}^{U}], \dots, [I_{k_{n} S_{2}}^{L}, I_{k_{n} S_{2}}^{U}]), ([F_{1 S_{2}}^{L}, F_{1 S_{2}}^{U}], [F_{2 S_{2}}^{L}, F_{2 S_{2}}^{U}], \dots, [F_{k_{n} S_{2}}^{L}, F_{k_{n} S_{2}}^{U}]) 〉$
…	…	…
S _m	…	$〈 ([T_{1 S_{m}}^{L}, T_{1 S_{m}}^{U}], [T_{2 S_{m}}^{L}, T_{2 S_{m}}^{U}], \dots, [T_{k_{n} S_{m}}^{L}, T_{k_{n} S_{m}}^{U}]), ([I_{1 S_{m}}^{L}, I_{1 S_{m}}^{U}], [I_{2 S_{m}}^{L}, I_{2 S_{m}}^{U}], \dots, [I_{k_{n} S_{m}}^{L}, I_{k_{n} S_{m}}^{U}]), ([F_{1 S_{m}}^{L}, F_{1 S_{m}}^{U}], [F_{2 S_{m}}^{L}, F_{2 S_{m}}^{U}], \dots, [F_{k_{n} S_{m}}^{L}, F_{k_{n} S_{m}}^{U}]) 〉$

Step 1: Based on the R-INS decision matrix D, setting ${T_{k_{i} S_{m}}^{L}}^{\max}$ and ${T_{k_{i} S_{m}}^{U}}^{\max}$ are the maximum truth values of all $T_{k_{i} S_{m}}^{L}$ and $T_{k_{i} S_{m}}^{U}$ in each column k_iS_m of thedecision matrix D, ${I_{k_{i} S_{m}}^{L}}^{\min}$ and ${I_{k_{i} S_{m}}^{U}}^{\min}, {F_{k_{i} S_{m}}^{L}}^{\min}$ and ${F_{k_{i} S_{m}}^{U}}^{\min}$ are the minimum indeterminate and falsity values in each column k_iS_m of the decision matrix D, the ideal solution (ideal R-INS) can be determined as $S_{i}^{*}$ .

$S_{i}^{*} = 〈 \begin{matrix} (\begin{matrix} [{T_{1 S_{m}}^{L}}^{\max}, {T_{1 S_{m}}^{U}}^{\max}], \\ [{T_{2 S_{m}}^{L}}^{\max}, {T_{2 S_{m}}^{U}}^{\max}], \\ \dots, \\ [{T_{k_{i} S_{m}}^{L}}^{\max}, {T_{k_{i} S_{m}}^{U}}^{\max}], \end{matrix}), \\ (\begin{matrix} [{I_{1 S_{m}}^{L}}^{\min}, {I_{1 S_{m}}^{U}}^{\min}], \\ [{I_{2 S_{m}}^{L}}^{\min}, {I_{2 S_{m}}^{U}}^{\min}], \\ \dots, \\ [{I_{k_{i} S_{m}}^{L}}^{\min}, {I_{k_{i} S_{m}}^{U}}^{\min}], \end{matrix}), \\ (\begin{matrix} [{F_{1 S_{m}}^{L}}^{\min}, {F_{1 S_{m}}^{U}}^{\min}], \\ [{F_{2 S_{m}}^{L}}^{\min}, {F_{2 S_{m}}^{U}}^{\min}], \\ \dots, \\ [{F_{k_{i} S_{m}}^{L}}^{\min}, {F_{k_{i} S_{m}}^{U}}^{\min}] \end{matrix}) \end{matrix} 〉, for i = 1, 2, \dots, n .$ (7)

So we can get the ideal alternative $S^{*} = {S_{1}^{*}, S_{2}^{*}, \dots, S_{n}^{*}}$ .

Step 2: When the weights of attributes are given by w = (w₁, w₂, …, w_n) with w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . The cosine measure between each alternative S_r (r = 1, 2, …, m) and the ideal alternative S^* can be calculated according to the formula (5). Then we can obtain the values of W (S_r, S^*) for r = 1, 2, …, m.

Step 3: According the values of W (S_r, S^*) for r = 1, 2, …, m, we can rank the alternatives in a descending order, then the alternative of biggest W (S_r, S^*) value is the best choice.

Step 4: End.

5 Illustrative example

Now, we discuss the decision-making problem adapted from the literature [4]. A construction company wants to determine which construction projects could be selected. Now four construction projects were provided by decision makers, so we can get a set of four alternatives S = {S₁, S₂, S₃, S₄}. Then, the selection of these construction projects is dependent on three main attributes, and these attributes can be divided into seven sub-attributes: (1) financial state (G₁): budget control (G₁1) and risk/return ratio (G₁2); (2) environmental protection (G₂): public relation (G₂1), geographical location (G₂2), and health and safety (G₂3); (3) technology (G₃): technical knowhow (G₃1) and technological capability (G₃2).

In the following, we use the proposed two decision-making methods to get the best chose of this example under R-SVNS and R-INS environments.

5.1 Decision-making problem with R-SVNSs

For this example, decision makers ask some experts to evaluate the value of these four possible alternatives under the above attributes. When these experts give some single values, we can construct the R-SVNS decision matrix D with these values, which is shown in Table 3.

Table 3
The R-SVNS decision matrix D [4] for four alternatives

G₁ (G₁₁, G₁₂) G₂ (G₂₁, G₂₂, G₂₃) G₃ (G₃₁, G₃₂)

S 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)〉

S 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)〉

S ₂ 〈 (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)〉

S 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)〉

	G₁ (G₁₁, G₁₂)	G₂ (G₂₁, G₂₂, G₂₃)	G₃ (G₃₁, G₃₂)
S 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)〉
S 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)〉
S ₂	〈 (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)〉
S 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)〉

From the R-SVNS decision matrix D and the formula (6), we can obtain the ideal solution (ideal alternative): $\begin{matrix} S^{*} & = & {< (0.8, 0.8), (0.1, 0.1), (0.2, 0.2) >, \\ < (0.9, 0.8, 0.8), (0.1, 0.1, 0.1), \\ (0.1, 0.1, 0.1) >, < (0.8, 0.8), \\ (0.1, 0.2), (0.1, 0.1) >} . \end{matrix}$

With the weight vector of the three attributes by w = (0.4, 0.3, 0.3) on the opinion of the experts and formula (3), we can obtain the weighted cosine measure values between each alternative S_r (r = 1, 2, 3, 4) and the ideal alternation S^*, the measure values are listed as follows: $\begin{matrix} W (S_{1}, S^{*}) & = & 0.9848, W (S_{2}, S^{*}) = 0.9938, \\ W (S_{3}, S^{*}) & = & 0.9858, W (S_{4}, S^{*}) = 0.9879 . \end{matrix}$

Table 4

The R-INS decision matrix D for four alternatives

		G₁ (G₁₁, G₁₂)	…
S 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])〉	…
S 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])〉	…
S ₂		〈 ([0.6, 0.7] , [0.8, 0.9]) , ([0.1, 0.2] , [0.3, 0.4]) , ([0.3, 0.4] , [0.4, 0.5])〉	…
S 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])〉	…
	…	G₂ (G₂₁, G₂₂, G₂₃)	…
S ₁	…	〈 ([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])〉	…
S ₂	…	〈 ([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])〉	…
S ₂	…	〈 ([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])〉	…
S 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])〉	…
	…	G₃ (G₃₁, G₃₂)
S ₁	…	〈 ([0.6, 0.7] , [0.8, 0.9]) , ([0.3, 0.4] , [0.2, 0.3]) , ([0.3, 0.4] , [0.4, 0.5])〉
S 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])〉
S ₂	…	〈 ([0.8, 0.9] , [0.7, 0.8]) , ([0.4, 0.5] , [0.3, 0.4]) , ([0.2, 0.3] , [0.1, 0.2])〉
S ₄	…	〈 ([0.7, 0.8] , [0.7, 0.8]) , ([0.2, 0.3] , [0.3, 0.4]) , ([0.2, 0.3] , [0.3, 0.4])〉

For the measure values are W (S₂, S^*) > W (S₄, S^*) > W (S₃, S^*) > W (S₁, S^*), the ranking order of four alternatives is S₂ ≻ S₄ ≻ S₃ ≻ S₁. Therefore, the alternative S₂ is the best choice among all alternatives.

Comparing with the method of literature [4], we can see that the proposed cosine measure between two R-SVNSs is relatively simpler and easier, and then the proposed decision-making method can obtains the same choice as in [4] through the weighted cosine measure values between each alternative S_r (r = 1, 2, 3, 4) and the ideal alternation S^*.

5.2 Decision-making problem with R-INSs

For the same example, decision makers ask some experts to evaluate the value of these four possible alternatives under the above attributes. When these experts give interval values, we can construct the R-INS decision matrix D with these values, which is shown in Table 4.

From the R-INS decision matrix D and the formula (7), we can obtain the ideal solution (ideal alternative): $\begin{matrix} S^{*} & = & {< ([0.8, 0.9], [0.8, 0.9]), ([0.1, 0.2], \\ [0.1, 0.2]), ([0.2, 0.3], [0.2, 0.3]) >, \\ < ([0.8, 0.9], [0.8, 0.9], [0.8, 0.9]), \\ ([0.1, 0.2], [0.1, 0.2], [0.1, 0.2]), \\ ([0.1, 0.2], [0.1, 0.2], [0.1, 0.2]) >, \\ < ([0.8, 0.9], [0.8, 0.9]), ([0.1, 0.2], \\ [0.2, 0.3]), ([0.1, 0.2], [0.1, 0.2]) >} . \end{matrix}$

Then, we use formula (5), we can obtain the weighted cosine measure values between each alternative S_r (r = 1, 2, 3, 4) and the ideal alternation S^*, the measure values are listed as follows: $\begin{matrix} W (S_{1}, S^{*}) & = & 0.9848, W (S_{2}, S^{*}) = 0.9932, \\ W (S_{3}, S^{*}) & = & 0.9868, W (S_{4}, S^{*}) = 0.9886 . \end{matrix}$

From the result of above, we can see that the proposed cosine measure between two R-INSs can obtains the best choice through the weighted cosine measure values between each alternative S_r (r = 1, 2, 3, 4) and the ideal alternation S^*.

6 Conclusions

In this paper, we introduced R-INS as an extension of R-SVNS and used the cosine measures of R-SVNSs and R-INSs in the decision-making problems with refined neutrosophic information. Then, we proposed the cosine measure-based multiple attribute decision-making methods under R-SVNS and R-INS environments. Through the cosine measure between each alternative and the ideal alternative, we can determine the ranking order of all alternatives and the best alternative can be selected. Finally, we presented an actual example adapted from the literature [4] under R-SVNS and R-INS environments, and the ranking orders in the example with cosine measures are agreement with the ranking results of reference [4], then the methods proposed in this paper are suitable for actual applications in decision-making problems with refined-single valued neutrosophic information and refined-interval neutrosophic information. In the future, we shall continue studying in the application of the cosine measures of R-SVNSs and R-INSs and extending the proposed decision making methods to other domains.

References

Smarandache

, Neutrosophy: Neutrosophic probability, set, and logic, American Research Press, Rehoboth, USA, 1998.

Wang

, Smarandache

, Zhang

Y.Q.

and Sunderraman

, Single valued neutrosophic, sets, Multispace and Multi Structure4 (2010), 410–413.

Wang

, Smarandache

, Zhang

Y.Q.

and Sunderraman

, Interval neutrosophic sets and logic: Theory and applications in computing, Hexis, Phoenix, AZ, 2005.

and Smarandache

, Similarity measure of refined single-valued neutrosophic sets and its multicriteria decision making method, Neutrosophic Sets and Systems12 (2016), 41–44.

, Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses, Artificial Intelligence in Medicine63(3) (2015), 171–179.

Mondal

and Pramanik

, Neutrosophic refined similarity measure based on cotangent function and its application to multi-attribute decision making, Global Journal of Advanced Research2(2) (2015), 486–496.

Broumi

and Smarandache

, Neutrosophic refined similarity measure based on cosine function, Neutrosophic Sets and Systems6 (2014), 42–48.

Broumi

and Deli

, Correlation measure for neutrosophic refined sets and its application in medical diagnosis, Palestine Journal of Mathematics3(1) (2014), 11–19.

and Ye

, Dice similarity measure between single valued neutrosophic multisets and its application in medical diagnosis, Neutrosophic sets and System6 (2014), 49–54.