Arccosine and arctangent similarity measures of refined simplified neutrosophic indeterminate sets and their multicriteria decision-making method

Abstract

In indeterminate and inconsistent setting, existing simplified neutrosophic indeterminate set (SNIS) can be depicted by the neutrosophic number (NN) functions of the truth, falsity and indeterminacy. Then, the three NN functions in SNIS lack their refined expressions and then the simplified neutrosophic indeterminate decision making (DM) method cannot carry out the multicriteria DM problems with both criteria and sub-criteria in the setting of SNISs. To overcome the flaws, this study first proposes a new notion of a refined simplified neutrosophic indeterminate set (RSNIS), which is described by the refined truth, falsity and indeterminate NN information regarding both elements and sub-elements in a universe set, as the extension of SNIS. Next, we propose the arccosine and arctangent similarity measures of RSNISs and their multicriteria DM method with various indeterminate risk ranges so as to carry out multicriteria DM problems with weight values of both criteria and sub-criteria in RSNIS setting. Lastly, the proposed DM method is applied to a multicriteria DM example of slope design schemes for an open pit mine to illustrate its application in the indeterminate DM problem with RSNISs. The decision results and comparative analysis indicate the rationality and efficiency of the proposed DM method with different indeterminate risk ranges.

Keywords

Refined simplified neutrosophic indeterminate set arccosine similarity measure arctangent similarity measure multicriteria decision making Slope design scheme

1 Introduction

Due to the indeterminacy and inconsistent of human cognition in the real world issues, a neutrosophic set presented by Smarandache [1] can be depicted by the truth, falsity, indeterminacy membership degrees in the real standard interval [0, 1] or nonstandard interval]⁻0, 1⁺ [. Regarding the real standard interval [0, 1], Ye [2] proposed a simplified neutrosophic set (SNS), implying the interval-valued neutrosophic set (IVNS) [3] and single-valued neutrosophic set (SVNS) [4], and two weighted aggregation operators of SNSs for easy decision making (DM) applications. Afterword, various measures and algorithms of SNSs have been used for many DM problems [5 –18]. As the generalization of SNS, Smarandache [19] further put forward an n-valued refined neutrosophic set (RNS) by the refined/splitted truth, indeterminacy, and falsity membership functions in SNS. Then, some researchers proposed the similarity measures of RSNSs (refined-SVNSs and/or refined-IVNSs) for DM [20, 21] and the correlation measures of RSNSs for medical diagnosis [22]. However, existing refined neutrosophic DM methods cannot carry out the multicriteria DM problems with both criteria and sub-criteria in the setting of RSNSs. Thus, Ye and Smarandache [23] defined another form of refined single-valued neutrosophic sets (RSVNSs) and their similarity measure to handle the multicriteria DM problems with both criteria and sub-criteria in the setting of RSVNSs. Further, the cosine measures [24], vector similarity measures [25] and correlation coefficients [26] of RSVNSs and/or RIVNSs were presented and used for solving the multicriteria DM problems with weight values of both criteria and sub-criteria in the environment of RSNSs.

Indeterminacy and risks widely exist in the mine design, development and operation since the indeterminate information results from the incompleteness and vagueness of human observation and estimations in the real world. Then, the concept of a neutrosophic number (NN) N = c + uI for c, u ∈ $R$ and I ∈ [inf I, sup I] [10 , 28] is comprised of both its certain part c and its indeterminate part uI with the indeterminate coefficient u and indeterminacy I ∈ [inf I, sup I]. Since NN indicates a changeable interval number depending on different indeterminate ranges of I ∈ [inf I, sup I], it very suits the indeterminate information expression and flexible DM problems in the indeterminate DM risk ranges [29]. However, there exist some indeterminate factors in the assessment process of slope design schemes (SDSs) for an open pit mine (OPM) due to some insufficiencies and vagueness of human cognition in slope design problems. Thus, we need to propose a reasonable and effective DM method of SDSs to obtain the best SDS, which makes the risks and benefits obtain optimal design requirements in the mining process [30]. Hence, Du et al. [31] first introduced NNs into the truth, indeterminacy and falsity membership functions in SNS and proposed a simplified neutrosophic indeterminate set (SNIS), including an interval-valued neutrosophic indeterminate set (IVNIS) and a single-valued neutrosophic indeterminate set (SVNIS), based on the truth, indeterminacy, and falsity NN functions, and then developed some simplified neutrosophic indeterminate weighted aggregation operators and their multicriteria DM method with decision makers’ indeterminate ranges in SNIS setting. However, the three NN functions in SNIS lack their refined expressions and then the simplified neutrosophic indeterminate DM method [31] cannot also handle the multicriteria DM problems with the weight values of both criteria and sub-criteria in the setting of SNISs. To overcome the flaws, this study first proposes a refined simplified neutrosophic indeterminate set (RSNIS), including a refined interval-valued neutrosophic indeterminate set (RIVNIS) and a refined single-valued neutrosophic indeterminate set (RSVNIS), as the extension of SNIS. RSNIS is described by the refined truth, indeterminacy, and falsity NN functions corresponding to each element and sub-elements in a universe set, which can indicate their changeable interval numbers/single values within [0, 1] depending on the indeterminate ranges/values of I ∈ [inf I, sup I]. Then, we propose arccosine and arctangent similarity measures between RSNISs and their multicriteria DM method with indeterminate risk ranges/values to solve indeterminate multicriteria DM problems with the weight values of both criteria and sub-criteria in the environment of RSNISs.

In this original study, its main contributions and motivation are summarized below:

The proposed RSNIS concept can express the indeterminate membership information of the refined truth, falsity and indeterminacy NN functions regarding both elements and sub-elements in a universe set and overcome the difficult expression problem of SNIS in the setting of RSNISs.

The presented arccosine and arctangent similarity measures of RSNISs provide effective mathematical tools for indeterminate multicriteria DM problems with the weight values of both criteria and sub-criteria in the environment of RSNISs.

The developed DM method with various indeterminate risk ranges can solve the decision problem of SDSs for OPM with the weight values of both criteria and sub-criteria and make the decision results more flexible and more effective than existing DM methods in indeterminate DM problems.

This article consists of the following sections. Section 2 simply introduces preliminaries of RSNSs and SNISs. Section 3 presents RSNISs and their basic relations. In Section 4, arccosine and arctangent similarity measures of RSNISs are proposed based on the Hamming distance of RSNISs. Section 5 develops a multicriteria DM method using the arccosine and arctangent similarity measures of RSNISs with various indeterminate risk ranges/values of I ∈ [inf I, sup I] in RSNIS setting. In Section 6, the developed DM method is utilized in a multicriteria DM example of SDS for OPM with decision makers’ various indeterminate risk ranges to illustrate the rationality and flexibility of the developed DM method in RSNIS setting. Conclusions and further research are summarized in Section 7.

2 Preliminaries of RSNSs and SNISs

As the extention of the SNS notion, Chen et al. [25] presented RSNSs, including RSVNSs and RIVNSs. Then, a RSNS M in a universe set Z = {z₁, z₂, ... , z_n} and the sets of sub-elements z_j = {z_j₁, z_j₂, ... , z_jq_(j)} (j = 1, 2, ... , n) is expressed as $M = {〈 z_{j}, (T_{M} (z_{j 1}), T_{M} (z_{j 2}), . . ., T_{M} (z_{jq (j)})), (I_{M} (z_{j 1}), I_{M} (z_{j 2}), . . ., I_{M} (z_{jq (j)})), (F_{M} (z_{j 1}), F_{M} (z_{j 2}), . . ., F_{M} (z_{jq (j)})) 〉 | z_{j} \in Z, z_{jk} \in z_{j}},$ which is described by its refined truth, indeterminacy, and falsity membership functions T_M(z_jk), I_M(z_jk), F_M(z_jk) ⊆ [0, 1] (k = 1, 2, ... , q(j)) for RIVNS or T_M(z_jk), I_M(z_jk), F_M(z_jk) ∈ [0, 1] (k = 1, 2, ... , q(j)) for RSVNS.

For expressional convenience, a refined simplified neutrosophic element (RSNE)< (T_M(z_j1), T_M(z_j2), ... , T_M(z_jq(j))), (I_M(z_j1), I_M(z_j2), ... , I_M(z_jq(j))), (F_M(z_j1), F_M(z_j2), ... , F_M(z_jq(j)))>in M is simply expressed as m_j = < (T_j1, T_j2, ... , T_jq(j)), (I_j1, I_j2, ... , I_jq(j)), (F_j1, F_j2, ... , F_jq(j))> (j = 1, 2, ... , n).

Then, Chen et al. [25] proposed basic relations between two RSNEs m₁ = < (T₁₁, T₁₂, ... , T_1q), (I₁₁, I₁₂, ... , I_1q), (F₁₁, F₁₂, ... , F_1q)>and m₂=< (T₂₁, T₂₂, ... , T_2q), (I₂₁, I₂₂, ... , I_2q), (F₂₁, F₂₂, ... , F_2q)>below:

m₁ ⊆ m₂, if and only if inf T_1k≤inf T_2k, sup T_1k≤sup T_2k, inf I_1k≥inf I_2k, sup I_1k≥sup I_2k, inf F_1k≥inf F_2k, and sup F_1k≥sup F_2k for k = 1, 2, ... , q;

m₁ = m₂, if and only if m₁ ⊆ m₂ and m₂ ⊆ m₁, i.e., T_1k = T_2k, I_1k = I_2k, and F_1k = F_2k for k = 1, 2, ... , q;

m₁∪ m₂ = 〈 (T₁₁ ∨ T₂₁, T₁₂ ∨ T₂₂, . . . , T_1q ∨ T_2q) , (I₁₁ ∧ I₂₁, I₁₂ ∧ I₂₂, . . . , I_1q ∧ I_2q) , (F₁₁ ∧ F₂₁, F₁₂ ∧ F₂₂, . . . , F_1q ∧ F_2q) 〉;

m₁∩ m₂ = 〈 (T₁₁ ∧ T₂₁, T₁₂ ∧ T₂₂, . . . , T_1q ∧ T_2q) , (I₁₁ ∨ I₂₁, I₁₂ ∨ I₂₂, . . . , I_1q ∨ I_2q) , (F₁₁ ∨ F₂₁, F₁₂ ∨ F₂₂, . . . , F_1q ∨ F_2q) 〉.

Regarding two RSNSs M₁ = {m₁₁, m₁₂, ... , m_1n} and M₂ = {m₂₁, m₂₂, ... , m_2n}, where m₁ _j = < (T_1j1, T_1j2, ... , T_1jq (j)), (I_1j1, I_1j2, ... , I_1jq (j)), (F_1j1, F_1j2, ... , F_1jq (j))>and m₂ _j = < (T_2j1, T_2j2, ... , T_2jq (j)), (I_2j1, I_2j2, ... , I_2jq (j)), (F_2j1, F_2j2, ... , F_2jq (j))> (j = 1, 2, ... , n) are two group of RSNEs. If the importance of elements m₁ _j and m₂ _j in M₁ and M₂ and sub-elements in m₁ _j and m₂ _j is specified by the weight vectors V = (v₁, v₂, ... , v_n) and v_j = (v_j₁, v_j₂, ... , v_jq_(j)) with v_j, v_jk ∈ [0, 1], $\sum_{j = 1}^{n} v_{j} = 1$ , and $\sum_{k = 1}^{q (j)} v_{jk} = 1$ (j = 1, 2, ... , n; k = 1, 2, ... , q(j)), the weighted cosine similarity measure between M₁ and M₂ is introduced as the following formula [24]: $C_{w} (M_{1}, M_{2}) = \sum_{j = 1}^{n} v_{j} {\sum_{k = 1}^{q_{j}} v_{jk} cos [\frac{π}{12} (\begin{matrix} | inf T_{1 jk} - inf T_{2 jk} | + | sup T_{1 jk} - sup T_{2 jk} | + | inf I_{1 jk} - inf I_{2 jk} | \\ + | sup I_{1 jk} - sup I_{2 jk} | + | inf F_{1 jk} - inf F_{2 jk} | + | sup F_{1 jk} - sup F_{2 jk} | \end{matrix})]} .$ (1)

It is clear that Equation (1) is reduced to the weighted cosine similarity measures of RSVNSs when there exist inf T_1jk = sup T_1jk, inf I_1jk = sup I_1jk, inf F_1jk = sup F_1jk, inf T_2jk = sup T_2jk, inf I_2jk = sup I_2jk, and inf F_2jk = sup F_2jk.

Based on the SNIS notion [31], a SNIS S ={ 〈 z_j, TN_S (z_j, I) , IN_S (z_j, I) , FN_S (z_j, I) 〉 |z_j ∈ Z } in a universe set Z = {z₁, z₂, ... , z_n} is described by its truth, indeterminacy, and falsity NN functions TN_S(z_j, I) = T_cj + T_ujI ⊆ [0, 1], IN_S(z_j, I) = I_cj + I_ujI ⊆ [0, 1], and FN_S(z_j, I) = F_cj + F_ujI ⊆ [0, 1] (j = 1, 2, ... , n) for z_j ∈ Z, T_cj, T_uj, I_cj, I_uj, F_cj, F_uj∈ $R$ , and I ∈ [inf I, sup I], which contains IVNIS and SVNIS depending on the indeterminate range and value of I.

For expressional convenience, a simplified neutrosophic indeterminate element (SNIE) 〈z_j, TN_S (z_j, I) , IN_S (z_j, I) , FN_S (z_j, I)〉 in S is simply expressed as s_j(I) = < (TN_j(I), IN_j(I), FN_j(I)>, where TN_j(I) = T_cj + T_ujI ⊆ [0, 1], IN_j(I) = I_cj + I_ujI ⊆ [0, 1], and FN_j(I) = F_cj + F_ujI ⊆ [0, 1] (j = 1, 2, ... , n) for z_j ∈ Z, T_cj, T_uj, I_cj, I_uj, F_cj, F_uj ∈ $R$ , and I ∈ [inf I, sup I] are the truth, indeterminacy and falsity NN functions, respectively.

Then, Du et al. [31] proposed the basic relations between two SNIEs s₁(I) = < (TN₁(I), IN₁(I), FN₁(I)>and s₂(I) = < TN₂(I), IN₂(I), FN₂(I)>for TN₁(I), IN₁(I), FN₁(I), TN₂(I), IN₂(I), FN₁(I) ⊆ [0, 1] and I ∈ [inf I, sup I] below:

s₁(I) ⊆ s₂(I), if and only if inf TN₁(I) ≤inf TN₂(I), sup TN₁(I) ≤sup TN₂(I), inf IN₁(I)≥inf IN₂(I), sup IN₁(I)≥sup IN₂(I), inf FN₁(I)≥inf FN₂(I), and sup FN₁(I)≥sup FN₂(I);

s₁(I) = s₂(I), if and only if s₁(I) ⊆ s₂(I) and s₂(I) ⊆ s₁(I), i.e., TN₁(I) = TN₂(I), IN₁(I) = IN₂(I), FN₁(I) = FN₂(I);

s₁ (I)∪ s₂ (I) = 〈 TN₁ (I) ∨ TN₂ (I) , IN₁ (I) ∧ IN₂ (I) , FN₁ (I) ∧ FN₂ (I) 〉;

s₁ (I)∩ s₂ (I) = 〈 TN₁ (I) ∧ TN₂ (I) , IN₁ (I) ∨ IN₂ (I) , FN₁ (I) ∨ FN₂ (I) 〉.

3 Refined simplified neutrosophic indeterminate sets

Regarding a SNIS S ={ 〈 z_j, TN_S (z_j, I) , IN_S (z_j, I) , FN_S (z_j, I) 〉 |z_j ∈ Z } in a universe set Z = {z₁, z₂, ... , z_n}, if the components TN_S(z_j, I), IN_S(z_j, I), FN_S(z_j, I) in the SNIS S can be refined (splitted) into (TN_S(z_j₁, I), TN_S(z_j₂, I),..., TN_S(z_jq_(j), I)), (IN_S(z_j₁, I), IN_S(z_j₂, I),..., IN_S(z_jq_(j), I)), and (FN_S(z_j₁, I), FN_S(z_j₂, I),..., FN_S(z_jq_(j), I)), respectively, for z_j ∈ Z, z_j = {z_j₁, z_j₂,..., z_jq_(j)} (j = 1, 2,..., n), and a positive integer q(j), which can be defined as RSNIS.

Definition 1. Set Z = {z₁, z₂, ... , z_n} and z_j = {z_j₁, z_j₂,..., z_jq_(j)} (j = 1, 2,..., n) as a universe set and a sub-universe set (a set of a sub-elements for z_j). Then a RSNIS R in Z can be defined as $R = {〈 \begin{matrix} z_{j}, ({TN}_{R} (z_{j 1}, I), {TN}_{R} (z_{j 2}, I), . . ., {TN}_{R} (z_{jq (j)}, I)), \\ ({IN}_{R} (z_{j 1}, I), {IN}_{R} (z_{j 2}, I), . . ., {IN}_{R} (z_{jq (j)}, I)), \\ ({FN}_{R} (z_{j 1}, I), {FN}_{R} (z_{j 2}, I), . . ., {FN}_{R} (z_{jq (j)}, I)) \end{matrix} 〉 | z_{j} \in Z, z_{jk} \in z_{j}, I \in [inf I, sup I]},$ where TN_R(z_j₁, I), TN_R(z_j₂, I),..., TN_R(z_jq_(j), I), IN_R(z_j₁, I), IN_R(z_j₂, I),..., IN_R(z_jq_(j), I), and FN_R(z_j₁, I), FN_R(z_j₂, I),..., FN_R(z_jq_(j), I) for z_j ∈ Z, z_jk ∈ z_j (j = 1, 2, ... , n; k = 1, 2, ... , q(j)), and I ∈ [inf I, sup I] are indeterminate subintervals/subsets in the real standard interval [0, 1], such that TN_R(z_jk, I) = TC_jk + TU_jkI: Z ⟶ [0, 1], IN_R(z_jk, I) = IC_jk + IU_jkI: Z ⟶ [0, 1], and FN_R(z_jk, I) = FC_jk + FU_jkI: Z ⟶ [0, 1].

Then, the RSNIS R implies the following two notions:

(1) If TN_R(z_jk, I) = TC_jk + TU_jkI ∈ [0, 1], IN_R(z_jk, I) = IC_jk + IU_jkI ∈ [0, 1], and FN_R(z_jk, I) = FC_jk + FU_jkI ∈ [0, 1] (j = 1, 2, ... , n; k = 1, 2, ... , q(j)) in R for z_j ∈ Z, z_jk ∈ z_j, and I = inf I = sup I are single values, R is reduced to RSVNIS, along with the condition 0≤TN_R(z_jk, I) + IN_R(z_jk, I) + FN_R(z_jk, I) ≤3 (j = 1, 2, ... , n; k = 1, 2, ... , q(j));

(2) If TN_R(z_jk, I) = TC_jk + TU_jkI ⊆ [0, 1], IN_R(z_jk, I) = IC_jk + IU_jkI ⊆ [0, 1], and FN_R(z_jk, I) = FC_jk + FU_jkI ⊆ [0, 1] (j = 1, 2, ... , n; k = 1, 2, ... , q(j)) in R for z_j ∈ Z, z_jk ∈ z_j, and I ∈ [inf I, sup I] are interval values, R is reduced to RIVNIS, along with the condition 0≤sup TN_R(z_jk, I) + sup IN_R(z_jk, I) + sup FN_R(z_jk, I) ≤3 (j = 1, 2, ... , n; k = 1, 2, ... , q(j)).

It is clearly that RSVNIS and RIVNIS are subsets of RSNIS. Then, RSNIS can imply a group of RIVNSs or RSVNSs depending upon a group of indeterminate ranges or values of I ∈ [inf I, sup I].

For expressional convenience, a refined simplified neutrosophic indeterminate element (RSNIE)< (z_j, (TN_R(z_j1, I), TN_R(z_j2, I),..., TN_R(z_jq (j), I)), (IN_R(z_j1, I), IN_R(z_j2, I),..., IN_R(z_jq (j), I)), (FN_R(z_j1, I), FN_R(z_j2, I),..., FN_R(z_jq (j), I))> in R is simply represented as r_j(I) = < (TN_j1(I), TN_j2(I), ... , TN_jq (j)(I)), (IN_j1, IN_j2, ... , IN_jq (j)(I)), (FN_j1, FN_j2, ... , FN_jq (j)(I))> .

Set two RSNIEs as r₁(I) = < (TN₁₁(I), TN₁₂(I), ... , TN_1q(I)), (IN₁₁(I), IN₁₂(I), ... , IN_1q(I)), (FN₁₁(I), FN₁₂(I), ... , FN_1q(I))> and r₂(I) = < (TN₂₁(I), TN₂₂(I), ... , TN_2q(I)), (IN₂₁(I), IN₂₂(I), ... , IN_2q(I)), (FN₂₁(I), FN₂₂(I), ... , FN_2q(I))> for TN_1k(I), TN_2k(I), IN_1k(I), IN_2k(I), FN_1k(I), FN_2k(I) ⊆ [0, 1] (k = 1, 2, ... , q). Then, there are the following relations between r₁(I) and r₂(I):

r₁(I) ⊆ r₂(I), if and only if inf TN_1k(I) ≤inf TN_2k(I), sup TN_1k(I) ≤sup TN_2k(I), inf IN_1k(I)≥inf IN_2k(I), sup IN_1k(I)≥sup IN_2k(I), inf FN_1k(I)≥inf FN_2k(I), and sup FN_1k(I)≥sup FN_2k(I) (k = 1, 2, ... , q);

r₁(I) = r₂(I), if and only if r₁(I) ⊆ r₂(I) and r₂(I) ⊆ r₁(I), i.e., TN_1k(I) = TN_2k(I), IN_1k(I) = IN_2k(I), FN_1k(I) = FN_2k(I) (k = 1, 2, ... , q);

$r_{1} (I) \cup r_{2} (I) = 〈 \begin{matrix} ({TN}_{11} (I) \lor {TN}_{21} (I), {TN}_{12} (I) \lor {TN}_{22} (I), . . ., {TN}_{1 q} (I) \lor {TN}_{2 q} (I)), \\ ({IN}_{11} (I) \land {IN}_{21} (I), {IN}_{12} (I) \land {IN}_{22} (I), . . ., {IN}_{1 q} (I) \land {IN}_{2 q} (I)), \\ ({FN}_{11} (I) \land {FN}_{21} (I), {FN}_{12} (I) \land {FN}_{22} (I), . . ., {FN}_{1 q} (I) \land {FN}_{2 q} (I)) \end{matrix} 〉$ ;

$r_{1} (I) \cap r_{2} (I) = 〈 \begin{matrix} ({TN}_{11} (I) \land {TN}_{21} (I), {TN}_{12} (I) \land {TN}_{22} (I), . . ., {TN}_{1 q} (I) \land {TN}_{2 q} (I)), \\ ({IN}_{11} (I) \lor {IN}_{21} (I), {IN}_{12} (I) \lor {IN}_{22} (I), . . ., {IN}_{1 q} (I) \lor {IN}_{2 q} (I)), \\ ({FN}_{11} (I) \lor {FN}_{21}, (I) {FN}_{12} (I) \lor {FN}_{22} (I), . . ., {FN}_{1 q} (I) \lor {FN}_{2 q} (I)) \end{matrix} 〉$ .

4 Arccosine and arctangent similarity measures of RSNISs

This section proposes arccosine and arctangent similarity measures based on the Hamming distance between RSNISs.

Definition 2. Set any two RSNISs as R₁ = {r₁₁(I), r₁₂(I), ... , r_1n(I)} and R₂ = {r₂₁(I), r₂₂(I), ... , r_2n(I)}, where r₁ _j(I) = < (TN_1j1(I), TN_1j2(I), ... , TN_1jq (j)(I)), (IN_1j1(I), IN_1j2(I), ... , IN_1jq (j)(I)), (FN_1j1(I), FN_1j2(I), ... , FN_1jq (j)(I))>and r₂ _j(I) = < (TN_2j1(I), TN_2j2(I), ... , TN_2jq (j)(I)), (IN_2j1(I), IN_2j2(I), ... , IN_2jq (j)(I)), (FN_2j1(I), FN_2j2(I), ... , FN_2jq (j)(I))> are RSNIEs for TN_1jk(I) = TC_1jk + TU_1jkI ⊆ [0, 1], TN_2jk(I) = TC_2jk + TU_2kI ⊆ [0, 1], IN_1jk(I) = IC_1jk + IU_1jkI ⊆ [0, 1], IN_2jk (I) = IC_2jk + IU_2jkI ⊆ [0, 1], FN_1jk(I) = FC_1jk + FU_1jkI ⊆ [0, 1], FN_2jk(I) = FC_2jk + FU_2jkI ⊆ [0, 1] (j = 1, 2, ... , n; k = 1, 2, ... , q(j)), and I ∈ [inf I, sup I]. Then, arccosine and arctangent similarity measures based on the Hamming distance between R₁ and R₂ are defined as the following formulae: $\begin{matrix} \begin{matrix} AC (R_{1}, R_{2}) = \frac{1}{n} \sum_{j = 1}^{n} {\frac{2}{π q (j)} \sum_{k = 1}^{q (j)} arc cos [\frac{1}{6} (\begin{matrix} | inf {TN}_{1 jk} (I) - inf {TN}_{2 jk} (I) | + | sup {TN}_{1 jk} (I) - sup {TN}_{2 jk} (I) | \\ + | inf {IN}_{1 jk} (I) - inf {IN}_{2 jk} (I) | + | sup {IN}_{1 jk} (I) - sup {IN}_{2 jk} (I) | \\ + | inf {FN}_{1 jk} (I) - inf {FN}_{2 jk} (I) | + | sup {FN}_{1 jk} (I) - sup {FN}_{2 jk} (I) | \end{matrix})]} \\ = \frac{1}{n} \sum_{j = 1}^{n} {\frac{2}{π q (j)} \sum_{k = 1}^{q (j)} arc cos [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2 jk} - {IU}_{2 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | \end{matrix})]} \end{matrix} \end{matrix},$ (2) $\begin{matrix} \begin{matrix} AT (R_{1}, R_{2}) = 1 - \frac{1}{n} \sum_{j = 1}^{n} {\frac{4}{π q (j)} \sum_{k = 1}^{q (j)} arc tan [\frac{1}{6} (\begin{matrix} | inf {TN}_{1 jk} (I) - inf {TN}_{2 jk} (I) | + | sup {TN}_{1 jk} (I) - sup {TN}_{2 jk} (I) | \\ + | inf {IN}_{1 jk} (I) - inf {IN}_{2 jk} (I) | + | sup {IN}_{1 jk} (I) - sup {IN}_{2 jk} (I) | \\ + | inf {FN}_{1 jk} (I) - inf {FN}_{2 jk} (I) | + | sup {FN}_{1 jk} (I) - sup {FN}_{2 jk} (I) | \end{matrix})]} \\ = 1 - \frac{1}{n} \sum_{j = 1}^{n} {\frac{4}{π q (j)} \sum_{k = 1}^{q (j)} arc tan [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2, jk} - {IU}_{2 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | \end{matrix})]} \end{matrix} \end{matrix},$ (3)

According to the similarity measure properties [24], the arccosine and arctangent similarity measures of RSNISs imply the proposition below.

Proposition 1. The arccosine and arctangent similarity measures AC(R₁, R₂) and AT(R₁, R₂) include the following properties:

AC(R₁, R₂) = AT(R₁, R₂) = 1 iff R₁ = R₂;

AC(R₁, R₂) = AC(R₂, R₁) and AT(R₁, R₂) = AT(R₂, R₁);

AC(R₁, R₂), AT(R₁, R₂) ∈ [0, 1];

If R₁ ⊆ R₂ ⊆ R₃ for any RSNIS R₃, then AC(R₁, R₂)≥AT(R₁, R₃) and AC(R₂, R₃)≥AT(R₁, R₃).

Proof. The properties (S1) and (S2) are straightforward. Hence, we only verify the properties (S3) and (S4).

(S3) Since there exist TN_1jk(I) = TC_1jk + TU_1jkI ⊆ [0, 1], IN_1jk(I) = IC_1jk + IU_1jkI ⊆ [0, 1], FN_1jk(I) = FC_1jk + FU_1jkI ⊆ [0, 1], TN_2jk(I) = TC_2jk + TU_2jkI ⊆ [0, 1], IN_2jk(I) = IC_2jk + IU_2jkI ⊆ [0, 1], and FN_2jk(I) = FC_2jk + FU_2jkI ⊆ [0, 1] (j = 1, 2, ... , n; k = 1, 2, ... , q(j)) for I ∈ [inf I, sup I], there is the following inequality: $0 ⩽ (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2 jk} - {IU}_{2 jk} inf I | + | {IC}_{1, jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | \end{matrix}) ⩽ 6 .$

Thus, there exists AC(R₁, R₂), AT(R₁, R₂) ∈ [0, 1] based on Equations (2) and (3) since π/2≥arccos (x)≥0 and 0≤arctan (x) ≤π/4 for 0≤x≤1 can hold.

(S4) Since R₁ ⊆ R₂ ⊆ R₃, there are TN_1jk(I) ⊆ TN_2jk(I) ⊆ TN_3jk(I), IN_1jk(I) ⊇ IN_2jk(I) ⊇ IN_3jk(I), and FN_1jk(I) ⊇ FN_2jk(I) ⊇ FN_3jk(I) (j = 1, 2, ... , n; k = 1, 2, ... , q(j)), i.e., TN_3jk(I) = TC_3jk + TU_3jkinf I≥TN_2jk(I) = TC_2jk + TU_2jkinf I≥TN_1jk(I) = TC_1jk + TU_1jkinf I, TN_3jk(I) = TC_3jk + TU_3jksup I≥TN_2jk(I) = TC_2jk + TU_2jksup I≥TN_1jk(I) = TC_1jk + TU_1jksup I, IN_1jk(I) = IC_1jk + IU_1jkinf I≥IN_2jk(I) = IC_2jk + IU_2jkinf I≥IN_3jk(I) = IC_3jk + IU_3jkinf I, IN_1jk(I) = IC_1jk + IU_1jksup I≥IN_2jk(I) = IC_2jk + IU_2jksup I≥IN_3jk(I) = IC_3jk + IU_3jksup I, FN_1jk(I) = FC_1jk + FU_1jkinf I≥FN_2jk(I) = FC_2jk + FU_2jkinf I≥FN_3jk(I) = FC_3jk + FU_3jkinf I, and FN_1jk(I) = FC_1jk + FU_1jksup I≥FN_2jk(I) = FC_2jk + FU_2jksup I≥FN_3jk(I) = FC_3jk + FU_3jksup I for I ∈ [inf I, sup I]. Thus, there exist the following inequalities: $| {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | ⩽ | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I |,$ $| {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | ⩽ | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I |$ $| {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2 jk} - {IU}_{2 jk} inf I | ⩽ | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I |$ $| {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | ⩽ | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I |,$ $| {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | ⩽ | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I |,$ $| {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | ⩽ | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I |,$ $| {TC}_{2 jk} + {TU}_{2 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I | ⩽ | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I |,$ $| {TC}_{2 jk} + {TU}_{2 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I | ⩽ | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I |,$ $| {IC}_{2 jk} + {IU}_{2 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I | ⩽ | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I |,$ $| {IC}_{2 jk} + {IU}_{2 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I | ⩽ | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I |,$ $| {FC}_{2 jk} + {FU}_{2 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I | ⩽ | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I |,$ $| {FC}_{2 jk} + {FU}_{2 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I | ⩽ | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I | .$

Since π/2≥arccos (x)≥0 and 0≤arctan (x)≤π/4 for 0≤x≤1 are a decreasing function and an increasing function respectively, corresponding to the above inequalities, we can obtain the following inequalities: $\begin{matrix} \begin{matrix} \frac{2}{π q (j)} \sum_{k = 1}^{q (j)} arc cos [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2 jk} - {IU}_{2 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | \end{matrix})] \\ ⩾ \frac{2}{π q (j)} \sum_{k = 1}^{q (j)} arc cos [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I | \end{matrix})] \end{matrix} \end{matrix},$ $\begin{matrix} \frac{2}{π q (j)} \sum_{k = 1}^{q (j)} arc cos [\frac{1}{6} (\begin{matrix} | {TC}_{2 jk} + {TU}_{2 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I | + | {TC}_{2 jk} + {TU}_{2 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I | \\ + | {IC}_{2 jk} + {IU}_{2 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I | + | {IC}_{2 jk} + {IU}_{2 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I | \\ + | {FC}_{2 jk} + {FU}_{2 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I | + | {FC}_{2 jk} + {FU}_{2 jk} sup I - {FC}_{3, jk} - {FU}_{3 jk} sup I | \end{matrix})] \\ ⩾ \frac{2}{π q (j)} \sum_{k = 1}^{q (j)} arc cos [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I | \end{matrix})], \end{matrix}$ $\begin{matrix} \frac{4}{π q (j)} \sum_{k = 1}^{q (j)} arc tan [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2 jk} - {IU}_{2 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | \end{matrix})] \\ ⩽ \frac{4}{π q (j)} \sum_{k = 1}^{q (j)} arc tan [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1, jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I | \end{matrix})], \end{matrix}$ $\begin{matrix} \begin{matrix} \frac{4}{π q (j)} \sum_{k = 1}^{q (j)} arc tan [\frac{1}{6} (\begin{matrix} | {TC}_{2 jk} + {TU}_{2 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I | + | {TC}_{2 jk} + {TU}_{2 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I | \\ + | {IC}_{2 jk} + {IU}_{2 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I | + | {IC}_{2 jk} + {IU}_{2 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I | \\ + | {FC}_{2 jk} + {FU}_{2 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I | + | {FC}_{2 jk} + {FU}_{2 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I | \end{matrix})] \\ ⩽ \frac{4}{π q (j)} \sum_{k = 1}^{q (j)} arc tan [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{3 jk} - {TU}_{3 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{3 jk} - {TU}_{3 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{3 jk} - {IU}_{3 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{3 jk} - {IU}_{3 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{3 jk} - {FU}_{3 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{3 jk} - {FU}_{3 jk} sup I | \end{matrix})] \end{matrix} \end{matrix} .$

Based on the above results and Equations (2) and (3), it is obvious that there exist AC(R₁, R₂)≥AT(R₁, R₃) and AC(R₂, R₃)≥AT(R₁, R₃).

Therefore, the proof is completed. □

Especially when I ∈ [inf I, sup I] is specified as a single value (I = inf I = sup I) or an interval range, Equations (2) and (3) are reduced to the arccosine and arctangent similarity measures of RSVNSs or RIVNSs. However, the arccosine similarity measure of RSNISs or the arctangent similarity measure of RSNISs implies a group of arccosine similarity measures of RSNSs or a group of arctangent similarity measures of RSNSs corresponding to a group of indeterminate ranges or values of I, which shows the flexibility of the two similarity measures.

When the importance of elements and sub-elements in R₁ and R₂ is specified by the weight vectors V = (v₁, v₂, ... , v_n) and v_j = (v_j₁, v_j₂, ... , v_jq_(j)) with v_j, v_jk ∈ [0, 1], $\sum_{j = 1}^{n} v_{j} = 1$ , and $\sum_{k = 1}^{q (j)} v_{jk} = 1$ (j = 1, 2, ... , n; k = 1, 2, ... , q(j)), the weighted arccosine and arctangent similarity measures between R₁ and R₂ are defined as the following formulae: $\begin{matrix} \begin{matrix} {AC}_{w} (R_{1}, R_{2}) = \sum_{j = 1}^{n} v_{j} {\frac{2}{π} \sum_{k = 1}^{q (j)} v_{jk} arc cos [\frac{1}{6} (\begin{matrix} | inf {TN}_{1 jk} (I) - inf {TN}_{2 jk} (I) | + | sup {TN}_{1 jk} (I) - sup {TN}_{2 jk} (I) | \\ + | inf {IN}_{1 jk} (I) - inf {IN}_{2 jk} (I) | + | sup {IN}_{1 jk} (I) - sup {IN}_{2 jk} (I) | \\ + | inf {FN}_{1 jk} (I) - inf {FN}_{2 jk} (I) | + | sup {FN}_{1 jk} (I) - sup {FN}_{2 jk} (I) | \end{matrix})]} \\ = \sum_{j = 1}^{n} v_{j} {\frac{2}{π} \sum_{k = 1}^{q (j)} v_{jk} arc cos [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2 jk} - {IU}_{2 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | \end{matrix})]} \end{matrix} \end{matrix},$ (4) $\begin{matrix} \begin{matrix} {AT}_{w} (R_{1}, R_{2}) = 1 - \sum_{j = 1}^{n} v_{j} {\frac{4}{π} \sum_{k = 1}^{q (j)} v_{jk} arc tan [\frac{1}{6} (\begin{matrix} | inf {TN}_{1 jk} (I) - inf {TN}_{2 jk} (I) | + | sup {TN}_{1 jk} (I) - sup {TN}_{2 jk} (I) | \\ + | inf {IN}_{1 jk} (I) - inf {IN}_{2 jk} (I) | + | sup {IN}_{1 jk} (I) - sup {IN}_{2 jk} (I) | \\ + | inf {FN}_{1 jk} (I) - inf {FN}_{2 jk} (I) | + | sup {FN}_{1 jk} (I) - sup {FN}_{2 jk} (I) | \end{matrix})]} \\ = 1 - \sum_{j = 1}^{n} v_{j} {\frac{4}{π} \sum_{k = 1}^{q (j)} v_{jk} arc tan [\frac{1}{6} (\begin{matrix} | {TC}_{1 jk} + {TU}_{1 jk} inf I - {TC}_{2 jk} - {TU}_{2 jk} inf I | + | {TC}_{1 jk} + {TU}_{1 jk} sup I - {TC}_{2 jk} - {TU}_{2 jk} sup I | \\ + | {IC}_{1 jk} + {IU}_{1 jk} inf I - {IC}_{2 jk} - {IU}_{2 jk} inf I | + | {IC}_{1 jk} + {IU}_{1 jk} sup I - {IC}_{2 jk} - {IU}_{2 jk} sup I | \\ + | {FC}_{1 jk} + {FU}_{1 jk} inf I - {FC}_{2 jk} - {FU}_{2 jk} inf I | + | {FC}_{1 jk} + {FU}_{1 jk} sup I - {FC}_{2 jk} - {FU}_{2 jk} sup I | \end{matrix})]} \end{matrix} \end{matrix} .$ (5) Similarly, the weighted arccosine and arctangent similarity measures of RSNISs also contain the proposition below.

Proposition 2. The weighted arccosine and arctangent similarity measures AC_w(R₁, R₂) and AT_w(R₁, R₂) imply the following properties:

AC_w(R₁, R₂) = AT_w(R₁, R₂) = 1 iff R₁ = R₂;

AC_w(R₁, R₂) = AC_w(R₂, R₁) and AT_w(R₁, R₂) = AT_w(R₂, R₁);

AC_w(R₁, R₂), AT_w(R₁, R₂) ∈ [0, 1];

If R₁ ⊆ R₂ ⊆ R₃ for any RSNIS R₃, then AC_w(R₁, R₂)≥AT_w(R₁, R₃) and AC_w(R₂, R₃)≥AT_w(R₁, R₃).

Based on the similar proof way of Proposition 1, one can easily verify the properties (S1)-(S4) of the weighted arccosine and arctangent similarity measures, which are not repeated here.

5 Multicriteria DM method using the arccosine and arctangent similarity measures of RSNISs with indeterminate risk ranges

In this part, a multicriteria DM method based on the arccosine and arctangent similarity measures of RSNISs with indeterminate risk ranges is proposed regarding indeterminate multicriteria DM problems with weight values of both criteria and sub-criteria in the setting of RSNISs.

Suppose there is a set of s alternatives R = {R₁, R₂, ... , R_s} to be assessed under a set of n criteria C = {c₁, c₂, ... , c_n} containing a set of sub-criteria c_j = {c_j₁, c_j₂, ... , c_jq_(j)} (j = 1, 2, ... , n) in a multicriteria DM problem. Then, decision makers are invited to assess the alternatives over both criteria and sub-criteria by RSNIEs r_ij(I) = < (TN_ij1(I), TN_ij2(I), ... , TN_ijq (j)(I)), (IN_ij1(I), IN_ij2(I), ... , IN_ijq (j)(I)), (FN_1j1(I), FN_1j2(I), ... , FN_ijq (j)(I))> for the specifical indeterminacy I ∈ [inf I, sup I], TN_ijk(I) = TC _ijk + TU _ijk I ⊆ [0, 1], IN_ijk(I) = IC _ijk + IU _ijk I ⊆ [0, 1], and FN_ijk(I) = FC _ijk + FU _ijk I ⊆ [0, 1] (i = 1, 2, ... , s; j = 1, 2, ... , n; k = 1, 2, ... , q(j)). Thus, all the given RSNIEs can be constructed as the decision matrix of RSNISs R = (r_ij(I)) _s _×n, which is shown in Table 1.

Table 1
The decision matrix of RSNISs R = (r_ij(I)) _s _×n for I ∈ [inf I, sup I]

C = {c₁, c₂, ... , c_n}

c₁ = {c₁₁, c₁₂, ... , c_1q (1)} c₂ = {c₂₁, c₂₂, ... , c_2q (2)} ... c_n = {c_n₁, c_n₂, ... , c_nq_(n)}

R ₁ r₁₁(I) r₁₂(I) ... r_1n(I)

R ₂ r₂₁(I) r₂₂(I) ... r_2n(I)

... ... ... ... ...

R_s r_s₁(I) r_s₂(I) .... r_sn(I)

	C = {c₁, c₂, ... , c_n}
R ₁	r₁₁(I)	r₁₂(I)	...	r_1n(I)
R ₂	r₂₁(I)	r₂₂(I)	...	r_2n(I)
...	...	...	...	...
R_s	r_s₁(I)	r_s₂(I)	....	r_sn(I)

Whereas the importance of the criteria and sub-criteria is specified by the weight vectors V = (v₁, v₂, ... , v_n) and v_j = (v_j₁, v_j₂, ... , v_jq_(j)) with v_j, v_jk ∈ [0, 1], $\sum_{j = 1}^{n} v_{j} = 1$ , and $\sum_{k = 1}^{q (j)} v_{jk} = 1$ (j = 1, 2, ... , n; k = 1, 2, ... , q(j)) in the multicriteria DM problem with RSNIS information, the multicriteria DM method with indeterminate risk ranges is proposed and depicted by the following decision algorithm:

Step 1: From the decision matrix of RSNISs R = (r_ij(I)) _s _×n for I ∈ [inf I, sup I], we give a ideal alternative R^* = {r₁^*, r₂^*, ... , r_n^*} by the following formula:

r_j^*(I) = < ([max_i(inf TN_ij1(I)), max_i(sup TN_ij1(I))], [max_i(inf TN_ij2(I), max_i(sup TN_ij2(I))], ... , [max_i(inf TN_ijq (j)(I)), max_i(sup TN_ijq (j)(I))]), ([min_i(inf IN_ij1(I)), min_i(sup IN_ij1(I))], [min_i(inf IN_ij2(I)), min_i(sup IN_ij2(I))], ... , [min_i(inf IN_ijq (j)(I)), min_i(sup IN_ijq (j)(I))]), ([min_i(inf FN_ij1(I)), min_i(sup FN_ij1(I))], [min_i(inf FN_ij2(I)), min_i(sup FN_ij2(I))], ... , [min_i(inf FN_ijq (j)(I)), min_i(sup FN_ijq (j)(I))]>for i = 1, 2, ... , s and j = 1, 2, ... , n. (6)

Step 2: By using Equation (4) or (5), we get the values of AC_w(R_i, R^*) or AT_w(R_i, R^*) with indeterminate risk ranges of I ∈ [inf I, sup I].

Step 3: Alternatives are ranked based on the values of AC_w(R_i, R^*) or AT_w(R_i, R^*) (i = 1, 2, ... , s) and then the best one can be chosen within the set of alternatives.

Step 4: End.

6 Multicriteria DM example of SDSs for OPM and Comparison with existing related DM methods

6.1 Multicriteria DM example of SDSs for OPM

Since a reasonable SDS for OPM is a critical problem in the mine design, development and operation, mining developers/investors firstly need to ensure mining workers against accidents caused by slope failure in mining process [30]. Hence, mining developers/investors require designers to present a group of preliminary/potential SDSs for OPM, and then request that experts/decision makers make an optimal decision of potential SDSs over satisfying specifical indices/criteria and sub-indices/sub-criteria to help mining developers/investors avoid risks and improve benefits.

Suppose designers provide four preliminary/potential SDSs denoted by a set of four alternatives R = {R₁, R₂, R₃, R₄} for some OPM, which must satisfy the requirements of three specifical indices/criteria and seven sub-indices/sub-criteria:

Economy (c₁) is refined into two sub-criteria: cost (c₁₁) and construction period (c₁₂);

Technology (c₂) is refined into three sub-criteria: technological capability (c₂₁), reliability (c₂₂), safety (c₂₃);

Environment (c₃) is refined into two sub-criteria: geographical location (c₃₁) and climate condition (c₃₂).

Then, the weight vector of the three criteria is specified as v = (0.3, 0.4, 0.3) and the weight vectors of the three group of sub-criteria sets {c₁₁, c₁₂}, {c₂₁, c₂₂, c₂₃}, and {c₃₁, c₃₂} are specified, respectively, by v₁ = (0.6, 0.4), v₂ = (0.3, 0.4, 0.3), and v₃ = (0.55, 0.45).

Since there are the inconsistent and indeterminacy of human cognition in the complicated DM problem, decision-makers/experts are invited to assess the four potential alternatives over the three criteria and seven sub-criteria by suitability judgments under the indeterminate environment of RSNIEs, then their evaluation values are represented by RSNIEs r_ij(I) = < (TN_ij1(I), TN_ij2(I), ... , TN_ijq (j)(I)), (IN_ij1(I), IN_ij2(I), ... , IN_ijq (j)(I)), (FN_1,j1(I), FN_1,j2(I), ... , FN_ijq (j)(I))>for TN_ijk(I) = TC _ijk + TU _ijk I ⊆ [0, 1], IN_ijk(I) = IC _ijk + IU _ijk I ⊆ [0, 1], and FN_ijk(I) = FC _ijk + FU _ijk I ⊆ [0, 1] (i = 1, 2, 3, 4; j = 1, 2, 3; k = 1, 2, ... , q(j); q(1) = 2, q(2) = 3, q(3) = 2) for the specifical indeterminacy I ∈ [0, 1]. Thus, the decision matrix of RSNISs can be established as R = (r_ij(I))_4 ×3 and shown in Table 2.

Table 2
The decision matrix of RSNISs R = (r_ij(I))_4 ×3 for I ∈ [0, 1]

C = {c₁, c₂, c₃

c₁ = {c₁₁, c₁₂} c₂ = {c₂₁, c₂₂, c₂₃} c₃ = {c₃₁, c₃₂}

R ₁ < (0.6 + 0.2I, 0.7 + 0.1I), (0.2 + 0.1I, 0.3 + 0.1I) < (0.7 + 0.2I, 0.7 + 0.1I, 0.7 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I, 0.2 + 0.1I) < (0.6 + 0.3I, 0.8 + 0.1I), (0.3 + 0.2I, 0.2 + 0.1I)

(0.1 + 0.1I, 0.2 + 0.1I)> (0.2 + 0.1I, 0.2 + 0.1I, 0.1 + 0.2I)> (0.3 + 0.1I, 0.4 + 0.1I)>

R ₂ < (0.8 + 0.15I, 0.7 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I) < (0.8 + 0.1I, 0.7 + 0.1I, 0.8 + 0.1I), (0.2 + 0.2I, 0.1 + 0.1I, 0.3 + 0.1I) < (0.7 + 0.2I, 0.7 + 0.1I), (0.2 + 0.2I, 0.1 + 0.1I)

(0.2 + 0.1I, 0.1 + 0.1I)> (0.2 + 0.2I, 0.3 + 0.1I, 0.3 + 0.2I)> (0.1 + 0.1I, 0.1 + 0.1I)>

R ₃ < (0.7 + 0.1I, 0.6 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I) < (0.7 + 0.15I, 0.7 + 0.1I, 0.8 + 0.1I), (0.1 + 0.2I, 0.1 + 0.1I, 0.1 + 0.1I) < (0.7 + 0.2I, 0.7 + 0.1I), (0.2 + 0.2I, 0.3 + 0.1I)

(0.1 + 0.1I, 0.1 + 0.2I)> (0.1 + 0.2I, 0.1 + 0.2I, 0.1 + 0.2I)> (0.2 + 0.1I, 0.2 + 0.1I)>

R ₄ < (0.8 + 0.1I, 0.8 + 0.1I), (0.1 + 0.1I, 0.1 + 0.2I) < (0.7 + 0.1I, 0.6 + 0.2I, 0.7 + 0.1I), (0.1 + 0.2I, 0.1 + 0.2I, 0.1 + 0.1I) < (0.8 + 0.15I, 0.8 + 0.1I), (0.2 + 0.2I, 0.2 + 0.1I)

(0.1 + 0.1I, 0.1 + 0.2I)> (0.2 + 0.2I, 0.1 + 0.2I, 0.2 + 0.2I)> (0.2 + 0.2I, 0.2 + 0.2I)>

	C = {c₁, c₂, c₃
R ₁	< (0.6 + 0.2I, 0.7 + 0.1I), (0.2 + 0.1I, 0.3 + 0.1I)	< (0.7 + 0.2I, 0.7 + 0.1I, 0.7 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I, 0.2 + 0.1I)	< (0.6 + 0.3I, 0.8 + 0.1I), (0.3 + 0.2I, 0.2 + 0.1I)
	(0.1 + 0.1I, 0.2 + 0.1I)>	(0.2 + 0.1I, 0.2 + 0.1I, 0.1 + 0.2I)>	(0.3 + 0.1I, 0.4 + 0.1I)>
R ₂	< (0.8 + 0.15I, 0.7 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I)	< (0.8 + 0.1I, 0.7 + 0.1I, 0.8 + 0.1I), (0.2 + 0.2I, 0.1 + 0.1I, 0.3 + 0.1I)	< (0.7 + 0.2I, 0.7 + 0.1I), (0.2 + 0.2I, 0.1 + 0.1I)
	(0.2 + 0.1I, 0.1 + 0.1I)>	(0.2 + 0.2I, 0.3 + 0.1I, 0.3 + 0.2I)>	(0.1 + 0.1I, 0.1 + 0.1I)>
R ₃	< (0.7 + 0.1I, 0.6 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I)	< (0.7 + 0.15I, 0.7 + 0.1I, 0.8 + 0.1I), (0.1 + 0.2I, 0.1 + 0.1I, 0.1 + 0.1I)	< (0.7 + 0.2I, 0.7 + 0.1I), (0.2 + 0.2I, 0.3 + 0.1I)
	(0.1 + 0.1I, 0.1 + 0.2I)>	(0.1 + 0.2I, 0.1 + 0.2I, 0.1 + 0.2I)>	(0.2 + 0.1I, 0.2 + 0.1I)>
R ₄	< (0.8 + 0.1I, 0.8 + 0.1I), (0.1 + 0.1I, 0.1 + 0.2I)	< (0.7 + 0.1I, 0.6 + 0.2I, 0.7 + 0.1I), (0.1 + 0.2I, 0.1 + 0.2I, 0.1 + 0.1I)	< (0.8 + 0.15I, 0.8 + 0.1I), (0.2 + 0.2I, 0.2 + 0.1I)
	(0.1 + 0.1I, 0.1 + 0.2I)>	(0.2 + 0.2I, 0.1 + 0.2I, 0.2 + 0.2I)>	(0.2 + 0.2I, 0.2 + 0.2I)>

In the setting of RSNISs, the proposed DM method with various indeterminate risk ranges of I ∈ [0, 1] is utilized for the indeterminate multicriteria DM problem of SDSs. The decision procedures are indicated corresponding to the specifical indeterminate risk ranges I = [0, 0] (no indeterminacy/single value), [0, 0.3] (the smaller risk range), [0, 0.5] (the moderate risk range), [0, 1] (the bigger risk range) as follows:

By Equation (6), we first give the ideal alternative from the decision matrix of Table 2 as follows:

R^* = <r₁^*, r₂^*, r₃^*> = < ([0.8, 0.95], [0.8, 0.9]), ([0.1, 0.2], [0.1, 0.3]), ([0.1, 0.2], [0.1, 0.2])>,< ([0.8, 0.9], [0.7, 0.8], [0.8, 0.9]), ([0.1, 0.3], [0.1, 0.2], [0.1, 0.2]), ([0.1, 0.3], [0.1, 0.3], [0.1, 0.3])>,< ([0.8, 0.95], [0.8, 0.9]), ([0.2, 0.4], [0.1, 0.2]), ([0.1, 0.2], [0.1, 0.2])>.

By using Equation (4) or (5), we obtain the weighted similarity measure values of AC_w(R_i, R^*) or AT_w(R_i, R^*) (i = 1, 2, 3, 4) regarding the specifical indeterminate risk ranges I = [0, 0], [0, 0.3], [0, 0.5], [0, 1]. Then, all the decision results are shown in Table 3.

Table 3

Decision results based on the arccosine or arctangent similarity measures between R_i and R^*

I = [inf I, sup I]	AC_w(R_i, R^*)	AT_w(R_i, R^*)	Ranking	The best one
I = [0, 0]	0.9122, 0.9226, 0.9196, 0.9286	0.8556, 0.8759, 0.8701, 0.8878	R₄ > R₂ > R₃ > R₁	R ₄
I = [0, 0.3]	0.9201, 0.9310, 0.9306, 0.9394	0.8711, 0.8925, 0.8918, 0.9091	R₄ > R₂ > R₃ > R₁	R ₄
I = [0, 0.5]	0.9254, 0.9366, 0.9379, 0.9466	0.8814, 0.9036, 0.9063, 0.9233	R₄ > R₃ > R₂ > R₁	R ₄
I = [0, 1]	0.9325, 0.9417, 0.9465, 0.9585	0.8955, 0.9137, 0.9232, 0.9471	R₄ > R₃ > R₂ > R₁	R ₄

Corresponding to the decision results in Table 3, the ranking orders based on the arccosine and arctangent similarity measures are identical in the DM example regarding the specifical indeterminate risk ranges I = [0, 0], [0, 0.3], [0, 0.5], [0, 1]. Then, the best SDS for OPM is R₄. However, the different indeterminate risk ranges can affect the ranking orders of the four alternatives, then the final decision result depends on the indeterminate risk range selected by decision makers. Therefore, the proposed DM method with the different indeterminate risk ranges makes the decision results more flexible and more objective by specifying various ranges/values of the indeterminacy I in the setting of RSNISs, which demonstrate its highlighting advantage.

6.2 Comparison with existing related DM methods

To compare the proposed DM method with the DM methods presented in [24, 31], we only consider the multicriteria DM problem with RIVNSs as a special case of the above DM example. In this case, if we only specify I = [0, 1], the decision matrix of RSNISs in Table 2 is reduced to the decision matrix of RIVNSs, which is show in Table 4.

Table 4
The decision matrix of RIVNSs

C = {c₁, c₂, c₃}

c₁ = {c₁₁, c₁₂} c₂ = {c₂₁, c₂₂, c₂₃} c₃ = {c₃₁, c₃₂}

R ₁ < ([0.6, 0.8], [0.7, 0.8]), ([0.2, 0.3], [0.3, 0.4]) < ([0.7, 0.9], [0.7, 0.8], [0.7, 0.9]), ([0.1, 0.3], [0.2, 0.3], [0.2, 0.3]) < ([0.6, 0.9], [0.8, 0.9]), ([0.3, 0.5], [0.2, 0.3])

([0.1, 0.2], [0.2, 0.3])> ([0.2, 0.3], [0.2, 0.3], [0.1, 0.3])> ([0.3, 0.4], [0.4, 0.5])>

R ₂ < ([0.8, 0.95], [0.7, 0.9]), ([0.1, 0.3], [0.2, 0.3) < (0.8, 0.9], [0.7, 0.8], [0.8, 0.9]), ([0.2, 0.4], [0.1, 0.2], [0.3, 0.4]) < ([0.7, 0.9], [0.7, 0.8]), ([0.2, 0.4], [0.1, 0.2])

([0.2, 0.3], [0.1, 0.2])> ([0.2, 0.4], [0.3, 0.4], [0.3, 0.5])> ([0.1, 0.2], [0.1, 0.2])>

R ₃ < ([0.7, 0.8], [0.6, 0.8]), ([0.1, 0.3], [0.2, 0.3]) < ([0.7, 0.85], [0.7, 0.8], [0.8, 0.9]), ([0.1, 0.3], [0.1, 0.2], [0.1, 0.2]) < ([0.7, 0.9], [0.7, 0.8]), ([0.2, 0.4], [0.3, 0.4])

([0.1, 0.2], [0.1, 0.3])> ([0.1, 0.3], [0.1, 0.3], [0.1, 0.3])> ([0.2, 0.3], [0.2, 0.3])>

R ₄ < ([0.8,0.9], [0.8, 0.9]), ([0.1, 0.2], [0.1, 0.3]) < ([0.7, 0.8], [0.6, 0.8], [0.7, 0.8]), ([0.1, 0.3], [0.1, 0.3], [0.1, 0.2]) < ([0.8, 0.95], [0.8, 0.9]), ([0.2, 0.4], [0.2, 0.3])

([0.1,0.2], [0.1, 0.3])> ([0.2, 0.4], [0.1, 0.3], [0.2, 0.4])> ([0.2, 0.4], [0.2, 0.4])>

	C = {c₁, c₂, c₃}
R ₁	< ([0.6, 0.8], [0.7, 0.8]), ([0.2, 0.3], [0.3, 0.4])	< ([0.7, 0.9], [0.7, 0.8], [0.7, 0.9]), ([0.1, 0.3], [0.2, 0.3], [0.2, 0.3])	< ([0.6, 0.9], [0.8, 0.9]), ([0.3, 0.5], [0.2, 0.3])
	([0.1, 0.2], [0.2, 0.3])>	([0.2, 0.3], [0.2, 0.3], [0.1, 0.3])>	([0.3, 0.4], [0.4, 0.5])>
R ₂	< ([0.8, 0.95], [0.7, 0.9]), ([0.1, 0.3], [0.2, 0.3)	< (0.8, 0.9], [0.7, 0.8], [0.8, 0.9]), ([0.2, 0.4], [0.1, 0.2], [0.3, 0.4])	< ([0.7, 0.9], [0.7, 0.8]), ([0.2, 0.4], [0.1, 0.2])
	([0.2, 0.3], [0.1, 0.2])>	([0.2, 0.4], [0.3, 0.4], [0.3, 0.5])>	([0.1, 0.2], [0.1, 0.2])>
R ₃	< ([0.7, 0.8], [0.6, 0.8]), ([0.1, 0.3], [0.2, 0.3])	< ([0.7, 0.85], [0.7, 0.8], [0.8, 0.9]), ([0.1, 0.3], [0.1, 0.2], [0.1, 0.2])	< ([0.7, 0.9], [0.7, 0.8]), ([0.2, 0.4], [0.3, 0.4])
	([0.1, 0.2], [0.1, 0.3])>	([0.1, 0.3], [0.1, 0.3], [0.1, 0.3])>	([0.2, 0.3], [0.2, 0.3])>
R ₄	< ([0.8,0.9], [0.8, 0.9]), ([0.1, 0.2], [0.1, 0.3])	< ([0.7, 0.8], [0.6, 0.8], [0.7, 0.8]), ([0.1, 0.3], [0.1, 0.3], [0.1, 0.2])	< ([0.8, 0.95], [0.8, 0.9]), ([0.2, 0.4], [0.2, 0.3])
	([0.1,0.2], [0.1, 0.3])>	([0.2, 0.4], [0.1, 0.3], [0.2, 0.4])>	([0.2, 0.4], [0.2, 0.4])>

Then by applying Equation (1) [24], we can obtain the weighted cosine similarity measure values of C_w(R_i, R^*): $\begin{matrix} C_{w} (R_{1}, R^{*}) = 0.8503, C_{w} (R_{2}, R^{*}) = 0.9567, \\ C_{w} (R_{3}, R^{*}) = 0.9620, and C_{w} (R_{4}, R^{*}) \\ = 0.9724 . \end{matrix}$

Thus, the ranking order of the four alternatives is R₄ > R₃ > R₂ > R₁ and the best one is R₄.

Obviously, this ranking order and the best one are the same as the results of the proposed DM method with the indeterminate risk ranges of I = [0, 0.5], [0, 1], which show the effectiveness of the proposed DM method. Then, the ranking orders of the proposed DM method regarding the indeterminate risk ranges of I = [0, 0], [0, 0.3] and the DM method presented in [24] indicate their difference. The reason is that existing DM method [24] only indicates the unique decision result without considering various indeterminate risk ranges, which lacks the indeterminacy and flexibility of the decision process due to the vagueness and indeterminacy of human cognition in complicated DM problems; while the proposed DM method can overcome the insufficiencies and show that the different indeterminate risk ranges can affect the ranking orders of alternatives in the indeterminate decision process. Then, the final decision result of the proposed DM method depends on the indeterminate risk range selected by decision makers, which the existing DM method [24] cannot do. It is obvious that the proposed DM method demonstrates its highlighting advantages in the refined indeterminate information expression and decision flexibility by comparison with existing DM method based on the cosine similarity measure of RSNSs [24]. Therefore, the proposed DM method with different indeterminate risk ranges makes the decision results more reasonable and more effective than the existing DM method [24] in indeterminate DM problems.

Compared with existing DM method with different indeterminate risk ranges [31], the proposed DM method with the weight values of both criteria and sub-criteria not only can express the RSNIE assessment information regarding both criteria and sub-criteria in indeterminate DM problems but also can deal with such a DM problem with the weight values of both criteria and sub-criteria in the setting of RSNISs, while existing DM method with weight values of unique criteria [31] cannot do them in the setting of RSNISs because existing DM method [31] can only express the SNIS assessment information and deal with such a multicriteria DM problem with the weight values of unique criteria in the setting of SNISs. Obviously, the proposed DM method with the weight values of both criteria and sub-criteria is more objective and more effective than the existing DM method [31] in indeterminate DM problems with various indeterminate risk ranges.

Furthermore, existing various neutrosophic DM methods [2 , 5–18] cannot express the information of RSNISs and also cannot carry out the indeterminate DM problems with the weight values of both criteria and sub-criteria, which imply their insufficiencies. Then, the proposed DM method can overcome these insufficiencies and reveal the flexible information expression and decision advantage regarding various indeterminate risk ranges under indeterminate DM environment.

7 Conclusion

To refine the truth, falsity and indeterminacy NN functions in SNIS under indeterminate and inconsistent environment, this paper presented a RSNIS concept as the extension of SNIS, which flexibly expresses the refined simplified neutrosophic indeterminate information. Then, we proposed the arccosine and arctangent similarity measures of RSNISs to utilize them for indeterminate DM problems along with various indeterminate risk ranges of decision makers in RSNIS setting. Further, the developed DM method was applied in a DM example of SDSs for OPM with weight values of both criteria and sub-criteria in the environment of RSNISs, then decision results and comparative analysis demonstrated the rationality and effectiveness of the developed DM method and the influence of different indeterminate risk ranges on the ranking of alternatives. Generally, the highlights of the original study are summarized as follows:

(i) The presented RSNIS solved the indeterminate information expression problems of the refined truth, falsity and indeterminacy NN functions.

(ii) The proposed arccosine and arctangent similarity measures of RSNISs provided the effective mathematical tools for the flexible multicriteria DM method regarding indeterminate DM problems with weight values of both criteria and sub-criteria.

(iii) The developed DM method with various indeterminate risk ranges carried out the indeterminate DM problem of SDSs with more flexible decision results and indicated its practicality and suitability under the environment of RSNISs.

In the future research, this original study will be extended to propose some aggregation operations, new similarity measures, and correlation coefficients for RSNISs so as to solve indeterminate problems like medical diagnosis, image processing, data analysis, group DM, slope stability evaluation and so on under the environment of RSNISs.

Conflicts of interest

The authors declare no conflict of interest.

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	C = {c₁, c₂, ... , c_n}
	c₁ = {c₁₁, c₁₂, ... , c_1q (1)}	c₂ = {c₂₁, c₂₂, ... , c_2q (2)}	...	c_n = {c_n₁, c_n₂, ... , c_nq_(n)}
R ₁	r₁₁(I)	r₁₂(I)	...	r_1n(I)
R ₂	r₂₁(I)	r₂₂(I)	...	r_2n(I)
...	...	...	...	...
R_s	r_s₁(I)	r_s₂(I)	....	r_sn(I)

	C = {c₁, c₂, c₃
	c₁ = {c₁₁, c₁₂}	c₂ = {c₂₁, c₂₂, c₂₃}	c₃ = {c₃₁, c₃₂}
R ₁	< (0.6 + 0.2I, 0.7 + 0.1I), (0.2 + 0.1I, 0.3 + 0.1I)	< (0.7 + 0.2I, 0.7 + 0.1I, 0.7 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I, 0.2 + 0.1I)	< (0.6 + 0.3I, 0.8 + 0.1I), (0.3 + 0.2I, 0.2 + 0.1I)
	(0.1 + 0.1I, 0.2 + 0.1I)>	(0.2 + 0.1I, 0.2 + 0.1I, 0.1 + 0.2I)>	(0.3 + 0.1I, 0.4 + 0.1I)>
R ₂	< (0.8 + 0.15I, 0.7 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I)	< (0.8 + 0.1I, 0.7 + 0.1I, 0.8 + 0.1I), (0.2 + 0.2I, 0.1 + 0.1I, 0.3 + 0.1I)	< (0.7 + 0.2I, 0.7 + 0.1I), (0.2 + 0.2I, 0.1 + 0.1I)
	(0.2 + 0.1I, 0.1 + 0.1I)>	(0.2 + 0.2I, 0.3 + 0.1I, 0.3 + 0.2I)>	(0.1 + 0.1I, 0.1 + 0.1I)>
R ₃	< (0.7 + 0.1I, 0.6 + 0.2I), (0.1 + 0.2I, 0.2 + 0.1I)	< (0.7 + 0.15I, 0.7 + 0.1I, 0.8 + 0.1I), (0.1 + 0.2I, 0.1 + 0.1I, 0.1 + 0.1I)	< (0.7 + 0.2I, 0.7 + 0.1I), (0.2 + 0.2I, 0.3 + 0.1I)
	(0.1 + 0.1I, 0.1 + 0.2I)>	(0.1 + 0.2I, 0.1 + 0.2I, 0.1 + 0.2I)>	(0.2 + 0.1I, 0.2 + 0.1I)>
R ₄	< (0.8 + 0.1I, 0.8 + 0.1I), (0.1 + 0.1I, 0.1 + 0.2I)	< (0.7 + 0.1I, 0.6 + 0.2I, 0.7 + 0.1I), (0.1 + 0.2I, 0.1 + 0.2I, 0.1 + 0.1I)	< (0.8 + 0.15I, 0.8 + 0.1I), (0.2 + 0.2I, 0.2 + 0.1I)
	(0.1 + 0.1I, 0.1 + 0.2I)>	(0.2 + 0.2I, 0.1 + 0.2I, 0.2 + 0.2I)>	(0.2 + 0.2I, 0.2 + 0.2I)>