Zadeh [1] proposed fuzzy sets for the first time to deal with fuzzy information in uncertain problems. Atanassow [2] further extended the fuzzy set and proposed the intuitionistic fuzzy set, which is described by a membership degree and a nonmembership degree. Smarandache [3] proposed the concept of a neutrosophic set (NS) considering the truth, falsity, and determinacy membership degrees. NS shows its main merit in dealing with indeterminate and inconsistent information. Hence, NSs have been widely used in image segmentation [4,5], decision making [6], clustering analysis [7], and so on. Wang et al. [8] introduced the concept of a single-valued neutrosophic set (SVNS) within the real interval [0,1] to more effectively solve practical problems. For example, some similarity measures of SVNSs were applied in multiple attribute decision-making (MADM) [9,10] and clustering analysis [11]. Then, the cross entropy of SVNSs was utilized for MADM [12] and object tracking [13]. Some aggregation operators of SVNSs and correlation coefficients of SVNSs were used for MADM [14,15]. As the subclass of NS, some aggregation operators of simplified NSs (containing SVNSs and interval NSs) [16–18] and similarity measures of simplified NSs [19] were presented and used for MADM.

When considering neutrosophic multi-valued/hesitant information, some similarity measures of single-valued neutrosophic multisets (SVNMs) were proposed and used for medical diagnosis [20] and MADM [21]. Multi-valued/hesitant NSs were introduced and applied in decision-making [22–29]. However, MVNS loses some identical neutrosophic values due to hesitant characteristics, thus the information aggregation of MVNSs may produce the union of multiple aggregated values in MADM problems, which may lead to computational complexity [24]. Furthermore, converting single-valued neutrosophic sequences in SVNM into SVNS [21] only contains the average value in the transformation process, which may lead to the loss of useful information (e.g., standard deviation). To solve these issues, Ye et al. [30] proposed a neutrosophic multi-valued set (NMVS), which contains identical and/or different true, false and indeterminate values, and defined a new method that converts NMVS into consistent single-valued neutrosophic sets (CSVNSs) in view of the mean and consistent degree of the true, false and indeterminate sequences. They also introduced the correlation coefficients of CSVNSs and applied them to MADM.

However, a similarity measure for CSVNSs was studied in the literature [30] as it is a key mathematical tool for MADM problems in the setting of NMVSs. Therefore, we should propose new similarity measures of CSVNSs to perform MADM in the case of NMVSs. In this paper, two cotangent similarity measures of NMVSs are proposed and applied to the purchase decision issue of potential cars. The rest of the paper consists of the following sections. Section 2 introduces some concepts of NMVSs and CSVNSs. The cotangent similarity measures of the CSVNSs are established from the cotangent function, and the properties of the cotangent similarity measures are demonstrated in Section 3. Section 4 introduces the MADM algorithm with respect to the established cotangent similarity measures of CSVNSs. Section 5 presents an example of purchase decision issues of potential cars and the comparative results of the related approach to prove the effectiveness and rationality of the new approach. Finally, conclusions and further works are put forward in Section 6.

This section introduces some concepts of NMVSs and CSVNSs presented by Ye et al. [30].

Definition 2.1 [30]. Let D = {d₁, d₂, …, ds} be a finite set. A NMVS Y defined on D is given by

(1)Y={<dj,NTY(dj),NIY(dj),NFY(dj)>|djϵD}where NF_Y (d_j), NI_Y (d_j) and NT_Y (d_j) are the falsity-membership function, the indeterminacy membership function, and the truth membership function, respectively, which are described by three multi-valued sequences NTY(dj)=(φY1(dj),φY2(dj),…,φYqj(dj)), NIY(dj)=(λY1(dj),λY2(dj),…,λYqj(dj)) and NFY(dj)=(βY1(dj),βY2(dj),…,βYqj(dj)) with different and/or identical fuzzy values, such that φYk(dj),λYk(dj),βYk(dj)∈[0,1] (k = 1, 2, …, q_j) and 0 ≤ sup NT_Y(d_j)+ sup NI_Y(d_j) + sup NF_Y(d_j) ≤ 3 for d_j ∈ D (j = 1, 2, …, s).

The basic element <d_j, NT_Y(d_j), NI_Y(d_j), NF_Y(d_j)> (j = 1, 2, …, s) in Y is simply denoted as yj=⟨NTj,NIj,NFj⟩=⟨(φj1,φj2,…,φjqj),(λj1,λj2,…,λjqj),(βj1,βj2,…,βjqj)⟩, which is named a neutrosophic multi-valued element (NMVE).

Definition 2.2 [30]. Set Y = {y₁, y₂, …, y_s} as NMVS, where yj=⟨NTj,NIj,NFj⟩ = ⟨(φj1,φj2,…,φjqj),(λj1,λj2,…,λjqj),(βj1,βj2,…,βjqj)⟩ (j = 1, 2, …, s) is the jth NMVE. By means of the average values of NT_j, NI_j, and NF_j and the consistent degrees of NT_j, NI_j, and NF_j (j = 1, 2, …, s), the CSVNS G = {g₁, g₂, …, g_s} is transformed from NMVS. The consistent single-valued neutrosophic element (CSVNE) of CSVNS is represented by g_j = < (n_NTj, b_NTj), (n_NIj, b_NIj), (n_NFj, b_NFj) > (j = 1, 2, …, s), where n_NTj, n_NIj, n_NFj ∈[0,1] are the average values of NT_j, NI_j, and NF_j and b_NTj, b_NIj, b_NFj ∈[0,1] are the consistent degrees of NT_j, NI_j, and NF_j. Then, the average values of NT_j, NI_j, and NF_j and the consistent degrees of NT_j, NI_j, and NF_j are obtained by the following formulae.

(2)nNTj=1qj∑k=1qjφjk

(3)nNIj=1qj∑k=1qjλjk

(4)nNFj=1qj∑k=1qjβjk

(5)bNTj=1−δNTj=1−1qj−1∑k=1qj(φjk−nNTj)2

(6)bNIj=1−δNIj=1−1qj−1∑k=1qj(λjk−nNIj)2

(7)bNFj=1−δNFj=1−1qj−1∑k=1qj(βjk−nNFj)2where δ_NTj, δ_NIj, δ_NFj ∈[0,1] are the standard deviations corresponding to NT_j, NI_j, and NF_j (j = 1, 2, …, s), respectively, and q_j is the number of fuzzy values in NT_j, NI_j, and NF_j.

For two CSVNEs g_j = <(n_NTj, b_NTj), (n_NIj, b_NIj), (n_NFj, b_NFj)> (j = 1, 2), both contain the following relationships:

(1) If g₁ ⊆ g₂, there are n_NT1 ≤ n_NT2, n_NI1 ≥ n_NI2, n_NF1 ≥ n_NF2, b_NT1 ≤ b_NT2, b_NI1 ≥ b_NI2, and b_NF1 ≥ b_NF2;

(2) If g₁ ⊆ g₂ and g₂ ⊆ g₁, there are n_NT1 = n_NT2, n_NI1 = n_NI2, n_NF1 = n_NF2, b_NT1 = b_NT2, b_NI1 = b_NI2, and b_NF1 = b_NF2.

By means of the average values of NT_j, NI_j and NF_j and the consistent degrees of NT_j, NI_j, and NF_j, weighted correlation coefficients between CSVNSs are introduced below [30].

Definition 2.3 [30]. Set Y₁ = {y₁₁, y₁₂, …, y_1s} and Y₂ = {y₂₁, y₂₂, …, y_2s} as two NMVSs, where yij=<NTij,NIij,NFij>=<(φij1,φij2,…,φijqij),(λij1,λij2,…,λijqij),(βij1,βij2,…,βijqij)> (i = 1, 2; j = 1, 2, …, s) are NMVEs. Suppose that the weight of y_ij (i = 1, 2; j = 1, 2, …, s) is w_j with w_j ∈[0,1] and ∑j=1swj=1. Based on the consistent degrees and the average values of NT_j, NI_j, and NF_j, the weighted correlation coefficients between CSVNSs are presented by the following formulae:

(8)WS1(Y1,Y2)=WS1(G1,G2)=∑j=1swj[nNT1jnNT2j+nNI1jnNI2j+nNF1jnNF2j]+∑j=1swj[bNT1jbNT2j+bNI1jbNI2j+bNF1jbNF2j](∑j=1swj[(nNT1j)2+(nNI1j)2+(nNF1j)2+(bNT1j)2+(bNI1j)2+(bNF1j)2]×∑j=1swj[(nNT2j)2+(nNI2j)2+(nNF2j)2+(bNT2j)2+(bNI2j)2+(bNF2j)2])

(9)WS2(Y1,Y2)=WS2(G1,G2)=∑j=1swj[nNT1jnNT2j+nNI1jnNI2j+nNF1jnNF2j]+∑j=1swj[bMT1jbMT2j+bMI1jbMI2j+bMF1jbMF2j]max(∑j=1swj[(nNT1j)2+(nNI1j)2+(nNF1j)2+(bNT1j)2+(bNI1j)2+(bNF1j)2],∑j=1swj[(nNT2j)2+(nNI2j)2+(nNF2j)2+(bNT2j)2+(bNI2j)2+(bNF2j)2])where b_NTij, b_NIij, b_NFij, n_NTij, n_NIij, and n_NFij are the consistent degrees and average values of NT_ij, NI_ij, and NF_ij (i = 1, 2; j = 1, 2, …, s), which are produced by Eqs. (2)–(7).

This section introduces cotangent similarity measures and weighted cotangent similarity measures between CSVNSs and their properties.

Definition 3.1. Let Y₁ = {y₁₁, y₁₂, …, y_1s} and Y₂ = {y₂₁, y₂₂, …, y_2s} be two NMVSs, where yij=<NTij,NIij,NFij>=<(φij1,φij2,…,φijqij),(λij1,λij2,…,λijqij),(δij1,δij2,…,δijqij)> (i = 1, 2; j = 1, 2, …, s) are NMVEs. Then, cotangent similarity measures between CSVNSs are proposed as follows:

(10)Cot1(Y1,Y2)=Cot1(G1,G2)=1s∑j=1scot⁡[π4+π4max(|nNT1j−nNT2j|,|nNI1j−nNI2j|,|nNF1j−nNF2j|,|bNT1j−bNT2j|,|bNI1j−bNI2j|,|bNF1j−bNF2j|)]

(11)Cot2(Y1,Y2)=Cot2(G1,G2)=1s∑j=1scot⁡[π4+π24(|nNT1j−nNT2j|+|nNI1j−nNI2j|+|nNF1j−nNF2j|+|bNT1j−bNT2j|+|bNI1j−bNI2j|+|bNF1j−bNF2j|)]where b_NTj, b_NIj, b_NFj, n_NTj, n_NIj, and n_NFj are the consistent degrees and average values of NT_ij, NI_ij, and NF_ij, which are produced by Eqs. (2)–(7).

Proposition 3.1. The cotangent similarity measures Cot₁(G₁, G₂) and Cot₂(G₁, G₂) in CSVNSs have the following properties:

(Z1) Cot₁(G₁, G₂) = Cot₁(G₂, G₁) and Cot₂(G₁, G₂) = Cot₂(G₂, G₁);

(Z2) 0 ≤ Cot₁(G₁, G₂), Cot₂(G₁, G₂) ≤ 1;

(Z3) Cot₁(G₁, G₂) = 1 if only if G₁ = G₂;

(Z4) For any CSVNS G₃ and G₁ ⊆ G₂ ⊆ G₃, Cot₁(G₁, G₂) ≥ Cot₁(G₁, G₃) and Cot₂(G₂, G₃) ≥ Cot₂(G₁, G₃).

Proof: (Z1) It is obvious that the proof of the property (Z1) is straightforward.

(Z2) Since the values of |n_NT1j − n_NT2j|, |n_NI1j − n_NI2j|, |n_NF1j − n_NF2j|, |b_NT1j − b_NT2j|, |b_NI1j − b_NI2j| and |b_NF1j − b_NF2j| for j = 1, 2, …, s are between 0 and 1 and the value of cotangent function falls in the interval [π/4, π/2], the cotangent values in Eqs. (10) and (11) are also located between 0 and 1. Hence, there is 0 ≤ Cot_i(G₁, G₂) ≤ 1 for i = 1, 2.

(Z3) For the two CSVNSs G₁ and G₂, if G₁ = G₂, this implies n_NT1j = n_NT2j, n_NI1j = n_NI2j, n_NF1j = n_NF2j, b_NT1j = b_NT2j, b_NI1j = b_NI2j and b_NF1j = b_NF2j for j = 1, 2, …, s. Hence |n_NT1j − n_NT2j| = 0, |n_NI1j − n_NI2j| = 0, |n_NF1j − n_NF2j| = 0, |b_NT1j − b_NT2j| = 0, |b_NI1j − b_NI2j| = 0, and |b_NF1j − b_NF2j| = 0. Thus Cot_i(G₁, G₂) = 1 for i = 1, 2.

If Cot_i(G₁, G₂) = 1 for i = 1, 2, this implies cot(π/4) = 1 and |n_NT1j - n_NT2j| = 0, |n_NI1j - n_NI2j| = 0, |n_NF1j − n_NF2j| = 0, |b_NT1j − b_NT2j| = 0, |b_NI1j − b_NI2j| = 0, and |b_NF1j − b_NF2j| = 0 for j = 1, 2, …, s. Then, there are n_NT1j = n_NT2j, n_NI1j = n_NI2j, n_NF1j = n_NF2j, b_NT1j = b_NT2j, b_NI1j = b_NI2j and b_NF1j = b_NF2j for j = 1, 2, …, s. Therefore, there is G₁ = G₂.

(Z4) Since there exists G₁ ⊆ G₂ ⊆ G₃, there are |n_NT1j − n_NT3j| ≥ |n_NT1j − n_NT2j|, |n_NT1j − n_NT3j| ≥ |n_NT2j − n_NT3j|, |n_NI1j − n_NI3j| ≥ |n_NI1j − n_NI2j|, |n_NI1j − n_NI3j| ≥ |n_NI2j − n_NI3j|, |n_NF1j − n_NF3j| ≥ |n_NF1j − n_NF2j|, |n_NF1j − n_NF3j| ≥ |n_NF2j − n_NF3j|, |b_NT1j − b_NT3j| ≥ |b_NT1j − b_NT2j|, |b_NT1j − b_NT3j| ≥ |b_NT2j − b_NT3j|, |b_NI1j − b_NI3j| ≥ |b_NI1j −b_NI2j|, |b_NI1j − b_NI3j| ≥ |b_NI2j − b_NI3j|, |b_NF1j − b_NF3j| ≥ |b_NF1j − b_NF2j|, and |b_NF1j − b_NF3j| ≥ |b_NF2j − b_NF3j|. Since the cotangent function is a decreasing function within the interval [π/4, π/2], there are Cot₁(G₁, G₂) ≥ Cot₁(G₁, G₃) and Cot₂(G₂, G₃) ≥ Cot₂(G₁, G₃).

When the importance of each y_ij is different, the weight of y_ij (i = 1, 2; j = 1, 2, …, s) is specified as w_j with w_j ∈ [0,1] and ∑j=1swj=1. Thus, we present the weighted cotangent similarity measures of CSVNSs.

(12)WCot1(Y1,Y2)=WCot1(G1,G2)=∑j=1swjcot⁡[π4+π4max(|nNT1j−nNT2j|,|nNI1j−nNI2j|,|nNF1j−nNF2j|,|bNT1j−bNT2j|,|bNI1j−bNI2j|,|bNF1j−bNF2j|)]

(13)WCot2(Y1,Y2)=WCot2(G1,G2)=∑j=1swjcot⁡[π4+π24(|nNT1j−nNT2j|+|nNI1j−nNI2j|+|nNF1j−nNF2j|+|bNT1j−bNT2j|+|bNI1j−bNI2j|+|bNF1j−bNF2j|)]

When w_j =1/s (j = 1, 2, …, s), Eqs. (12) and (13) are reduced to Eqs. (10) and (11), which are special cases of Eqs. (12) and (13).

This part introduces a MADM approach corresponding to the proposed cotangent similarity measures in the setting of NMVSs. When dealing with a MADM problem, there is often a set of multiple alternatives T = {T₁, T₂, …, T_q}, evaluated by a group of multiple attributes E = {e₁, e₂, …, e_s}. The weight vector of E is specified as w = (w₁, w₂, …, w_s). Each alternative T_i (i = 1, 2, …, q) is evaluated over the attributes e_j by a NMVE yij=<NTij,NIij,NFij>=<(φij1,φij2,…,φijqij),(λij1,λij2,…,λijqij),(δij1,δij2,…,δijqij)> (i = 1, 2, …, q; j = 1, 2, …, s) for φijk,λijk,βijk∈[0,1] and 0≤φijk+λijk+βijk≤3 (k = 1, 2, …, q_ij). Hence, the NMVS Y_i = {y_i1, y_i2, …, y_is} and the decision matrix of NMVSs Y = (y_ij)_{q × s} are established by the obtained NMVEs. Therefore, we can perform the MADM problem through the following steps:

Step 1: Using Eqs. (2)–(7), the NMVS Y_i = {y_i1, y_i2, …, y_is} and the decision matrix Y = (y_ij)_q.s are transformed into the CSVNS G_i = {g_i1, g_i2, …, g_is} and the decision matrix of CSVNSs G = (g_ij)_q.s, respectively.

Step 2: The ideal solution G∗=(g1∗,g2∗,…,gs∗) is determined by the ideal CSVNE:

(14)gj∗=⟨(nTNj∗,bTNj∗),(nINj∗,bINj∗),(nFNj∗,bFNj∗)⟩=⟨(maxi(nTNij),maxi(bTNij),mini(nINij),mini(bINij)),(mini(nFNij),mini(bFNij))⟩

Step 3: The weighted cotangent similarity measures of the CSVNSs G_i and G ^* for T_i are obtained by the following equations:

(15)WCot1(Gi,G∗)=∑j=1swjcot⁡[π4+π4max(|nNTij−nNTj∗|,|nNIij−nNIj∗|,|nNFij−nNFj∗|,|bNTij−bNTj∗|,|bNIij−bNIj∗|,|bNFij−bNFj∗|)]or

(16)WCot2(Gi,G∗)=∑j=1swjcot⁡[π4+π24(|nNTij−nNTj∗|+|nNIij−nNIj∗|+|nNFij−nNFj∗|+|bNTij−bNTj∗|+|bNIij−bNIj∗|+|bNFij−bNFj∗|)]

Step 4: According to the values of the weighted cotangent similarity measure, the alternatives are arranged in descending order, and the best one is selected.

Step 5: End.

To easily compare the proposed MADM approach with existing relevant MADM methods, this section presents an illustrative example on the selection purchase of potential cars in [21] to illustrate the effectiveness and rationality of the proposed MADM approach.

Customers want to choose a suitable car according to their own living needs and driving habits. There are four types of potential cars, represented by the set of alternatives T = {T₁, T₂, T₃, T₄}. They must satisfy the requirements of four indexes/attributes: (1) e₁ is the fuel economy; (2) e₂ is the price; (3) e₃ is the amenity; (4) e₄ is the safety. The weight vector of E = {e₁, e₂, …, e_s} is expressed as w = (0.5, 0.25, 0.125, 0.125). The evaluation values of the four attributes for each alternative are expressed by the NMVE yij=<NTij,NIij,NFij>=<(φij1,φij2,⋯,φijqij),(λij1,λij2,⋯,λijqij),(δij1,δij2,⋯,δijqij)> (i, j = 1, 2, 3, 4) for φijk,λijk,δijk∈[0,1]. Table 1 shows the decision matrix of NMVEs.

In this section, the proposed MADM approach based on the cotangent similarity measures is compared with the MADM method in [21,30] to prove the feasibility and effectiveness of the proposed MADM method.

To enhance the comparability, we use the same weight vector w = (0.5, 0.25, 0.125, 0.125) for various MADM methods and present their decision results in Table 3. In Table 3, the ranking order of the proposed MADM approach using Eq. (15) is mostly consistent with the MADM method using Eq. (9) [30]. Two Eqs. (15) and (9) [30] using the parameter ‘max’, which has the same result. It shows that the results are reasonable. Then, the ranking order of the MADM method proposed by Eq. (16) is calculated with the average, which is the same as the MADM method using the cosine measure [21]. The comparison results show the efficiency of the proposed MADM approach. However, the MADM method using the cosine measure [21] does not consider the consistent degree in the MADM process, which may lead to the loss of some useful information and the difference in sorting results. Furthermore, the consistent degree of multi-valued sequences can affect the sorting results of alternatives, revealing the importance of the consistent information in MADM problems, and making the decision results more credible and reasonable.

The advantages of the cotangent similarity measure in this study can make up for some deficiencies of the cosine similarity measure in the vector space to a certain extent. Using two different methods of cotangent similarity measures, different ranking results can be obtained. It shows that the results have excellent plasticity and rationality. At the same time, the method also provides a new way of thinking about the multi-valued decision-making problem.

Based on the concepts of NMVS and CSVNSs, the cotangent similarity measures of CSVNSs are proposed by the cotangent function. The proposed MADM method using the cotangent similarity measures of CSVNSs is applied to perform the MADM problem with NMVSs. Then, the proposed MADM method is applied to the selection problem of potential cars to verify the effectiveness of the proposed MADM method. Compared with other MADM methods, the proposed MADM method shows its high efficiency and rationality. In future research, we will further develop other new similarity measures of CSVNSs and apply them in the fields of medical diagnosis/ assessment and image processing in the setting of NMVSs.