Refined Simplified Neutrosophic Similarity Measures Based on Trigonometric Function and Their Application in Construction Project Decision-Making

Journal of Soft Computing in Civil Engineering 2-3 (2018) 01-12
How to cite this article: Solang U, Ye J. Refined simplified neutrosophic similarity measures based on trigonometric function and
their application in construction project decision-making. J Soft Comput Civ Eng 2018;2(3):01–12.
https://doi.org/10.22115/scce.2018.126129.1056.
2588-2872/ © 2018 The Authors. Published by Pouyan Press.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Refined Simplified Neutrosophic Similarity Measures Based on
Trigonometric Function and Their Application in Construction
Project Decision-Making
U. Solang 1
, J. Ye1,2*
1. Department of Civil Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang Province
312000, P.R. China
2. Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road,
Shaoxing, Zhejiang Province 312000, P.R. China
Corresponding author: yehjun@aliyun.com, yejun@usx.edu.cn
https://doi.org/10.22115/SCCE.2018.126129.1056
ARTICLE INFO ABSTRACT
Article history:
Received: 03 April 2018
Revised: 07 May 2018
Accepted: 07 May 2018
Refined simplified neutrosophic sets (RSNSs) are
appropriately used in decision-making problems with sub-
attributes considering their truth components, indeterminacy
components, and falsity components independently. This
paper presents the similarity measures of RSNSs based on
tangent and cotangent functions. When the weights of each
element/attribute and each sub-element/sub-attribute in
RSNSs are considered according to their importance, we
propose the weighted similarity measures of RSNSs and their
multiple attribute decision-making (MADM) method with
RSNS information. In the MADM process, the developed
method gives the ranking order and the best selection of
alternatives by getting the weighted similarity measure
values between alternatives and the ideal solution according
to the given attribute weights and sub-attribute weights.
Then, an illustrative MADM example in a construction
project with RSNS information is presented to show the
effectiveness and feasibility of the proposed MADM method
under RSNS environments. This study extends existing
methods and provides a new way for the refined simplified
neutrosophic MADM problems containing both the attribute
weight and the sub-attribute weights.
Keywords:
Refined simplified neutrosophic
set;
Tangent function;
Cotangent function;
Similarity measure;
Construction project;
Multiple attribute decision-
making.

2 U. Solang, J. Ye/ Journal of Soft Computing in Civil Engineering 2-3 (2018) 01-12
1. Introduction
The fuzzy set [1] is represented by the membership function for a fuzzy problem. However, the
fuzzy set cannot be described by the non-membership function for a fuzzy problem. As the
generalization of fuzzy sets, Atanassov [2] presented an intuitionistic fuzzy set (IFS), which is
characterized by the membership and non-membership functions. However, IFS can only handle
incomplete and uncertain information but not inconsistent and indeterminate information. Thus, a
neutrosophic set (NS) was introduced by Smarandache [3], where the indeterminacy is quantified
explicitly. In NS, the components of the truth, indeterminacy, and falsity are denoted as T, I, F,
and then they are expressed independently by the truth, falsity, indeterminacy membership
functions defined in the real standard interval [0, 1] or non-standard interval ]−
0,1+
[. After that,
Simplified NSs introduced by Ye [4] are the subclasses of NSs and contain two concepts of
single-valued neutrosophic sets (SVNSs) and interval neutrosophic sets (INSs), then they were
applied in decision making [5–17]. Even though they have been applied in real MADM
problems, however, their decision-making methods cannot handle the problems of both the
attributes and the sub-attributes. Smarandache [18] first defined n-value/refined neutrosophic set,
which is composed of its n-subcomponents represented by p truth sub-membership degrees, r
indeterminacy sub-membership degrees, and s falsity sub-membership degrees satisfying p + r +
s = n. Next, n-value/refined neutrosophic sets/multisets were applied to medical diagnoses and
MADM [19–22]. Further, Ye and Smarandache [23] particularized the n-value/refined
neutrosophic set to a refined single-valued neutrosophic set (RSVNS), where its components T, I,
F and the sub-components T1, T2, ..., Tq and I1, I2, ..., Iq, and F1, F2, ..., Fq are constructed as an
RSVNS, and then they introduced the similarity measure using the union and the intersection of
RSVNSs to deal with MADM problems with both attributes and sub-attributes. Then, the Dice,
Jaccard and cosine similarity measures of refined simplified NSs (RSNS) have been proposed
[24], along with their applications in MADM problems. Thereafter, cosine measures of refined
interval NSs (RINS) were introduced by Fan and Ye [25] as an extension of RSVNS and used for
MADM problems.
It is well known that a similarity measure in decision-making theory is an important
mathematical tool. So, Ye [15] put forward similarity measures of SVNS corresponding to
cotangent function, then Mondal and Pramanik [22] presented the tangent function-based
similarity measure of refined NSs (i.e., neutrosophic multi-sets) for the MADM problem without
sub-attributes. However, their MADM methods [15,19–22] cannot handle the MADM problems
with both attributes and sub-attributes. In fact, there are no tangent and cotangent similarity
measures for RSNSs in existing literature. Therefore, we introduce new similarity measures of
RSNSs corresponding to tangent function and cotangent function to extend the existing decision-
making methods of multiple attributes to MADM problems with attributes and sub-attributes,
and then the developed method is applied in an MADM example on a construction project with
both attributes and sub-attributes in RSNS (RSVNS and RINS) setting.
The rest of this article is constructed as the following. In the second section, we present the
extended tangent function similarity measure and cotangent function similarity measure for
SNSs in existing literature. The third section, the similarity measures of RSNSs were introduced

U. Solang, J. Ye/ Journal of Soft Computing in Civil Engineering 2-3 (2018) 01-12 3
corresponding to tangent and cotangent functions. In the fourth section, we present the MADM
method using the tangent and cotangent similarity measures of RSNSs. In the fifth section, an
illustrative example on the decision-making of a construction project with attributes and sub-
attributes is given in RSNS (RSVNS and RINS) setting. Lastly, this article is concluded in the
sixth section.
2. Trigonometric function-based similarity measures
NS [3] is described by the three components T, I, F, which are defined independently as the
membership degrees of the truth, indeterminacy and falsity within a real standard interval [0,1]
or a nonstandard interval ]-
0,1+
[. For its application in real science and engineering, this can be
constrained in the real standard interval [0,1]. Thereby, as a simplified form or a subclass of an
NS Ye [4] presented the concept of simplified NS. The simplified NS contains SVNS and INS. A
simplified NS P in a universe of discourse X with the element x is denoted as P = {<x, TP(x),
IP(x), FP(x)>|xX} where each membership function is considered as a singleton or a sub-
interval in the real standard [0,1], such that TP(x), IP(x), FP(x) ∈ [0,1] for SVNS or TP(x), IP(x),
FP(x)⊆[0,1] for INS. An element <x, TP(x), IP(x), FP(x)> in the simplified NS P is called a
simplified neutrosophic number (SNN), simply denoted as p = 〈tp, ip, fp〉, which contains single-
valued and neutrosophic interval numbers.
Similarity measures mainly describe the similarity degree between different objects. Assume two
simplified NSs in the universe of discourse X are P = {p1, p2, …, pn} for pj ∈ P {j = 1, 2, …, n}
and Q = {q1, q2,…, qn} for qj ∈ Q (j = 1, 2, …, n) such that pj = <tpj, ipj, fpj> and qj = <tqj, iqj, fqj>.
Then, the tangent function and cotangent function-based similarity measures between two
SVNSs P and Q are expressed below [15,22]:
S1(P,Q) =
1
𝑛
∑ 1 − tan[(|𝑡𝑝𝑗 − 𝑡𝑞𝑗| + |𝑖𝑝𝑗 − 𝑖𝑞𝑗| + |𝑓𝑝𝑗 − 𝑓𝑞𝑗|)
𝑛
𝑗=1
𝜋
3×4
], (1)
S2(P,Q)=
1
𝑛
∑ cot[
𝜋
4
+ (|𝑡𝑝𝑗 − 𝑡𝑞𝑗| + |𝑖𝑝𝑗 − 𝑖𝑞𝑗| + |𝑓𝑝𝑗 − 𝑓𝑞𝑗|)
𝑛
𝑗=1
𝜋
3×4
]. (2)
For INSs, the tangent function and cotangent function-based similarity measures between two
INSs P and Q are presented by
𝑆3(𝑃, 𝑄) =
1
𝑛
∑ 1 − tan[(|inf 𝑡𝑝𝑗 − inf 𝑡𝑞𝑗| + |sup 𝑡𝑝𝑗 − sup 𝑡𝑞𝑗| + |inf 𝑖𝑝𝑗 − inf 𝑖𝑞𝑗| +
𝑛
𝑗=1
|sup 𝑖𝑝𝑗 − sup 𝑖𝑞𝑗| + |inf 𝑓𝑝𝑗 − inf 𝑓𝑞𝑗| + |sup 𝑓𝑝𝑗 − sup 𝑓𝑞𝑗|)
𝜋
6×4
], (3)
𝑆4(𝑃, 𝑄) =
1
𝑛
∑ cot[
𝜋
4
+ (|inf 𝑡𝑝𝑗 − inf 𝑡𝑞𝑗| + |sup 𝑡𝑝𝑗 − sup 𝑡𝑞𝑗| + |inf 𝑖𝑝𝑗 − inf 𝑖𝑞𝑗| +
𝑛
𝑗=1
|sup 𝑖𝑝𝑗 − sup 𝑖𝑞𝑗| + |inf 𝑓𝑝𝑗 − inf 𝑓𝑞𝑗| + |sup 𝑓𝑝𝑗 − sup 𝑓𝑞𝑗|)
𝜋
6×4
]. (4)

According to the tangent and cotangent similarity measure properties in [15,22], the cotangent
and tangent similarity measure Sk(P, Q) (k = 1, 2, 3, 4) between two simplified NSs P and Q also
have the following properties:
(R1) 0 ≤ Sk(P, Q) ≤ 1;
(R2) Sk(P, Q) = Sk(Q, P);
(R3) Sk(P, Q) = 1 if and only if P = Q;
(R4) Suppose M is also a simplified NS in the universe X, If P⊆ Q⊆M, then Sk(P, M) ≤ Sk(P, Q)
and Sk(P, M) ≤ Sk(Q, M).
3. Tangent and cotangent similarity measures of RSNSs
This section presents the tangent and cotangent similarity measures between RSNSs and the
weighted tangent and cotangent similarity measures of RSNSs containing both the weights of
their elements and the weights of their sub-elements, which are more suitable for solving
MADM problems with sub-attributes.
If a simplified NS (SVNS or INS) P = {p1, p2, p3, …, pn}for pj∈P (j = 1, 2, …, n) is refined, pj
=<tj, ij, fj> consists of the sub-components such as pji =<tj1, tj2, …,ij1, ij2,…, fj1, fj2,…>. Then
RSNS contains RSVNS with the components tj1, tj2, …∈ [0,1], ij1, ij2,…∈ [0,1], fj1, fj2,…∈ [0,1]
and 0 ≤ tji + iji + fji ≤ 3 and RINS with tj1, tj2, … ⊆ [0,1], ij1, ij2,…⊆ [0,1], fj1, fj2,…⊆ [0,1] and
0 ≤ suptji + supiji + supfji≤3.
Then, we introduce the tangent and cotangent functions to similarity measures of RSNSs.
Assume we consider two RSNS P = {p1, p2,…, pn} and Q = {q1, q2,…, qn} for pj ∈ P and qj∈Q (j
= 1, 2, …, n), where pji = <(tpj1, tpj2, …,tpjr(j)),(ipj1, ipj2,…,ipjr(j)),(fpj1, fpj2,…, fpjr(j))> and qji = <(tqj1,
tqj2, …,tqjr(j)),(iqj1, iqj2,…,iqjr(j)),(fqj1, fqj2,…, fqjr(j))> for pji∈pj and qji∈qj (i = 1, 2,…, r(j); j = 1,
2, …, n). Thus, the similarity measures between two RSVNSs and between two RINSs based on
the trigonometric functions of tangent and cotangent are given as follows:
(a) Similarity measures of RSVNSs
𝑇1(𝑃, 𝑄) =
1
𝑛
∑
1
𝑟(𝑗)
∑ {1 − tan[(|𝑡𝑝𝑗𝑟(𝑗) − 𝑡𝑞𝑗𝑟(𝑗)| + |𝑖𝑝𝑗𝑟(𝑗) − 𝑖𝑞𝑗𝑟(𝑗)| + |𝑓𝑝𝑗𝑟(𝑗) −
𝑟(𝑗)
𝑖=1
𝑛
𝑗=1
𝑓𝑞𝑗𝑟(𝑗)|)
𝜋
3×4
]}, (5)
𝑇2(𝑃, 𝑄) =
1
𝑛
∑
1
𝑟(𝑗)
∑ cot[
𝜋
4
+ (|𝑡𝑝𝑗𝑟(𝑗) − 𝑡𝑞𝑗𝑟(𝑗)| + |𝑖𝑝𝑗𝑟(𝑗) − 𝑖𝑞𝑗𝑟(𝑗)| + |𝑓𝑝𝑗𝑟(𝑗) −
𝑟(𝑗)
𝑖=1
𝑛
𝑗=1
𝜋
3×4
] . (6)
(b) Similarity measures of RINSs

𝑇3(𝑃, 𝑄) =
1
𝑛
∑
1
𝑟(𝑗)
∑ {1 − tan[(|𝑖𝑛𝑓 𝑡𝑝𝑗𝑟(𝑗) − 𝑖𝑛𝑓 𝑡𝑞𝑗𝑟(𝑗)| + |𝑠𝑢𝑝 𝑡𝑝𝑗𝑟(𝑗) − 𝑠𝑢𝑝 𝑡𝑞𝑗𝑟(𝑗)|
𝑟(𝑗)
𝑖=1
+
𝑛
𝑗=1
|𝑖𝑛𝑓 𝑖𝑝𝑗𝑟(𝑗) − 𝑖𝑛𝑓 𝑖𝑞𝑗𝑟(𝑗)| + |𝑠𝑢𝑝 𝑖𝑝𝑗𝑟(𝑗) − 𝑠𝑢𝑝 𝑖𝑞𝑗𝑟(𝑗)| + |𝑖𝑛𝑓 𝑓𝑝𝑗𝑟(𝑗) − 𝑖𝑛𝑓 𝑓𝑞𝑗𝑟(𝑗)| +
|𝑠𝑢𝑝 𝑓𝑝𝑗𝑟(𝑗) − 𝑠𝑢𝑝 𝑓𝑞𝑗𝑟(𝑗)|)
𝜋
6×4
]}, (7)
𝑇4(𝑃, 𝑄) =
1
𝑛
∑
1
𝑟(𝑗)
∑ cot[
𝜋
4
+ (|𝑖𝑛𝑓 𝑡𝑝𝑗𝑟(𝑗) − 𝑖𝑛𝑓 𝑡𝑞𝑗𝑟(𝑗)| + |𝑠𝑢𝑝 𝑡𝑝𝑗𝑟(𝑗) − 𝑠𝑢𝑝 𝑡𝑞𝑗𝑟(𝑗)|
𝑟(𝑗)
𝑖=1
+
𝑛
𝑗=1
𝜋
6×4
]. (8)
Similar to the properties (R1)-(R4) of the similarity measures discussed above, the simplified
neutrosophic similarity measures Tk(P, Q) (k = 1, 2, 3, 4) based on cotangent and tangent
functions also contain the following properties:
(R1) 0 ≤ Tk(P, Q) ≤ 1;
(R2) Tk(P, Q) = Tk(Q, P);
(R3) Tk(P, Q) = 1 if and only if P = Q;
(R4) Suppose M is also an RSNS in the universe X, If P⊆ Q⊆M, then Tk(P, M) ≤ Tk(P, Q) and
Tk(P, M) ≤ Tk(Q, M).
To apply them efficiently in decision-making, we need to consider the weights of elements in
RSNS as w = (w1, w2, …,wn) and the weights of sub-elements in RSNS as wj = (w11, w22, …,
wnr(j)) (i = 1, 2, …, r(j); j =1, 2, …, n). Thus, the weighted similarity measures are presented as
follows:
(a) The weighted similarity measures between RSVNSs
𝑊1(𝑃, 𝑄) = ∑ 𝑤𝑗 ∑ 𝑤𝑗𝑖{1 − tan[(|𝑡𝑝𝑗𝑟(𝑗) − 𝑡𝑞𝑗𝑟(𝑗)| + |𝑖𝑝𝑗𝑟(𝑗) − 𝑖𝑞𝑗𝑟(𝑗)| + |𝑓𝑝𝑗𝑟(𝑗) −
𝑟(𝑗)
𝑖=1
𝑛
𝑗=1
𝜋
3×4
]}, (9)
𝑊2(𝑃, 𝑄) = ∑ 𝑤𝑗 ∑ 𝑤𝑗𝑖cot[
𝜋
4
+ (|𝑡𝑝𝑗𝑟(𝑗) − 𝑡𝑞𝑗𝑟(𝑗)| + |𝑖𝑝𝑗𝑟(𝑗) − 𝑖𝑞𝑗𝑟(𝑗)| + |𝑓𝑝𝑗𝑟(𝑗) −
𝑟(𝑗)
𝑖=1
𝑛
𝑗=1
𝜋
3×4
] ; (10)
(b) The weighted similarity measures between RINSs
𝑊3(𝑃, 𝑄) = ∑ 𝑤𝑗 ∑ 𝑤𝑗𝑖{1 − tan[(|𝑖𝑛𝑓 𝑡𝑝𝑗𝑟(𝑗) − 𝑖𝑛𝑓 𝑡𝑞𝑗𝑟(𝑗)| + |𝑠𝑢𝑝 𝑡𝑝𝑗𝑟(𝑗) −
𝑟(𝑗)
𝑖=1
𝑛
𝑗=1
𝑠𝑢𝑝 𝑡𝑞𝑗𝑟(𝑗)| + |𝑖𝑛𝑓 𝑖𝑝𝑗𝑟(𝑗) − 𝑖𝑛𝑓 𝑖𝑞𝑗𝑟(𝑗)| + |𝑠𝑢𝑝 𝑖𝑝𝑗𝑟(𝑗) − 𝑠𝑢𝑝 𝑖𝑞𝑗𝑟(𝑗)| + |𝑖𝑛𝑓 𝑓𝑝𝑗𝑟(𝑗) −
𝑖𝑛𝑓 𝑓𝑞𝑗𝑟(𝑗)| + |𝑠𝑢𝑝 𝑓𝑝𝑗𝑟(𝑗) − 𝑠𝑢𝑝 𝑓𝑞𝑗𝑟(𝑗)|)
𝜋
6×4
]}, (11)

𝑊4(𝑃, 𝑄) = ∑ 𝑤𝑗 ∑ 𝑤𝑗𝑖cot[
𝜋
4
+ (|𝑖𝑛𝑓 𝑡𝑝𝑗𝑟(𝑗) − 𝑖𝑛𝑓 𝑡𝑞𝑗𝑟(𝑗)| + |𝑠𝑢𝑝 𝑡𝑝𝑗𝑟(𝑗) −
𝑟(𝑗)
𝑖=1
𝑛
𝑗=1 𝑠𝑢𝑝 𝑡𝑞𝑗𝑟(𝑗)| +
𝜋
6×4
]. (12)
The above-weighted similarity measures Wk(P, Q) (k = 1, 2, 3, 4) based on tangent and cotangent
also contain the following properties:
(R1) 0 ≤ Wk(P, Q) ≤ 1;
(R2) Wk(P, Q) = Wk(Q, P);
(R3) Wk(P, Q) = 1 if and only if P = Q;
(R4) Suppose M is also a simplified NS in the universe X, If P⊆ Q⊆M, then Wk(P, M) ≤ Wk(P,
Q) and Wk(P, M) ≤ Wk(Q, M).
4. MADM method based on the proposed similarity measures of RSNSs
In an MADM problem that has multiple attributes with their sub-attributes, this section proposes
MADM method using the proposed similarity measures of RSNSs.
Let’s consider a set of m alternatives P = {P1, P2, …, Pm} to be judged under attributes Z = {z1,
z2, …, zn} with their sub-attributes zj = {zj1, zj2, …, zjr(j)} for j = 1, 2, …, n. Then, we give the
evaluation of the alternatives over the attributes and sub-attributes by RSNSs (RSVNSs and
RINSs). Table 1 shows the relative evaluation values between alternatives and the attributes and
sub-attributes, known as the RSNS decision matrix D=(psji)mn, where psji (i = 1, 2, …, r(j); j = 1,
2, …, n; s = 1, 2, …, m) represents the evaluation value of Pj regarding each sub-attribute zjr(j).
Table 1
The RSNS decision matrix D=(psji)mn.
z1 z2 … zn
z11, z12, …, z1r(1) z21, z22, …, z2r(2) … zn1, zn2, …, znr(n)
P1 p11r(1) p12r(2) … p1nr(n)
P2 p21r(1) p22r(2) … p2nr(n)
P3 p31r(1) p32r(2) … p3nr(n)
… … … … …
Pm pm1r(1) pm2r(2) … pmnr(n)
In Table 1, each alternative Ps in the set P = {P1, P2, …, Pm} is evaluated under all attributes Z =
{z1, z2, …, zn} and sub-attributes zj = {zj1, zj2,…, zjr(j)} by RSNN psji = <(tsj1, tsj2, …,tsjr(j)),(isj1,
isj2,…,isjr(j)),(fsj1, fsj2,…, fsjr(j))> (i = 1, 2, …, r(j); j = 1, 2, …, n; s = 1, 2, …, m). The importance
of attributes and sub-attributes is presented as the weight vectors w = (w1,w2,…,wn) for the set Z
= {z1, z2, …, zn} and wj = (wj1, wj2,…, wjr(j)) for the set of sub-attributes such that ∑ 𝑤𝑗
𝑛
𝑗=1 =1 and
∑ 𝑤𝑗𝑖
𝑟(𝑗)
𝑖=1 =1 with wj,wji∈ [0,1].

The ideal RSNN or the ideal solution is given from the RSNS decision matrix D=(psji)mn as
follows:
ppji = <(tpj1, tpj2, …,tpjr(j)), (ipj1, ipj2,…,ipjr(j)), (fpj1, fpj2,…, fpjr(j))> = <(maxs(tsj1), maxs(tsj2), …,maxs(t
sjr(j))), (mins(isj1), mins(isj2), …,mins(isjr(j))), (mins(fsj1), mins(fsj2),…, mins(fsjr(j)))> for RSVNNs
(13)
or ppji = <(tpj1, tpj2, …,tpjr(j)),(ipj1, ipj2,…,ipjr(j)),(fpj1, fpj2,…, fpjr(j))> = <([maxs(inf tsj1), maxs(sup tsj
1)], [maxs(inf tsj2), maxs(sup tsj2)], …, [maxs(inf tsjr(j))), maxs(sup tsjr(j))]), ([mins(inf isj1), mins(sup
isj1)], [mins(inf isj2), mins(sup isj2)], …, [mins(inf isjr(j)), mins(sup isjr(j))]), ([mins(inf fsj1), mins(sup f
sj1)], [mins(inf fsj2), mins(sup fsj2)], …, [mins(inf fsjr(j)), mins(sup fsjr(j))])> for RINNs. (14)
Then, the ideal solution/alternative is presented as P*
= {𝑝1
∗
, 𝑝2
∗
, …, 𝑝𝑛
∗
}, where 𝑝𝑗
∗
= (ppj1,
ppj1, …, ppjr(j)) for j = 1, 2, …, n.
Thus, we use the equations (9) and (11) or (10) and (12) to get the values of Wk(Ps, P*
) (k = 1, 3
or 2, 4; s = 1, 2, …, m). By the similarity measure values between the ideal solution P*
and each
alternative set Ps, all the alternatives are ranked, and the best one is determined based on the one
with biggest weighted similarity measure value given by Wk(Ps, P*
) among the alternatives.
5. Illustrative example
A successful project can be achieved by many interacted factors as presented in previous
literatures [7,8,17], which mainly depends on the decision-making method. Hence, the manager
has to effectively make an accurate and reliable decision according to the presented requirements
or objective attributes with their highly subjective judgmental factors to select the best
alternative for some project.
In a construction project, the manager has to select the best alternative in the decision set of the
alternatives P = {P1, P2, P3, P4} suggested by different personalities or departments like
administration department, technical department, finance department, etc. to meet the
requirements and the objectives of the project from the contractor company, as well as the
contracting company. The following two cases composed of the suggested alternatives with their
attributes set Z = {z1, z2, z3} and sub-attributes set zj = {zj1, zj2, zjr(j)} (j = 1, 2, 3) in a construction
project are presented to describe the applicability of the proposed method. Here, the attributes
and sub-attributes of alternatives are shown in Table 2.
Table 2
The attributes and sub-attributes.
z1: Budget z2: Quality z3: Delivery
z11: Human resource cost
z21: Experience or
performance
z31: Schedule
z12: Materials and
equipment cost
z22: Technology z32: Communication
z13: Facilities z33: Risk and uncertainties

Case1. Under RSVNS environment, the evaluation values of the decision set of the alternatives
P = {P1, P2, P3, P4} over the attributes and sub-attributes must belong to the interval [0,1]. The
weight vector of the given attribute set Z = {z1, z2, z3} is w = (0.3, 0.4, 0.3) and the weight
vectors of the sub-attribute sets {z11, z12, z13}, {z21, z22} and {z31, z32, z33} are given respectively
as w1 = (0.5, 0.3, 0.2), w2 = (0.6, 0.4) and w3 = (0.4, 0.2, 0.4). Thus, the RSVNS decision matrix
D=(psji)43 corresponding to the alternatives with respect to the three attributes with their given
sub-attributes is given in Table 3.
Table 3
The RSVNS decision matrix D=(psji)43.
z1 z2 z3
z11,z12,z13 z21,z22 z31,z32, z33
P1 〈(0.5,0.5,0.6), (0.3,0.4,0.2), (0.2,0.1,0.2)〉 〈(0.7,0.8),(0.1,0.2),(0.1,0.2)〉 〈(0.9,0.8,0.5), (0.1,0.1,0.3), (0,0.1,0.2)〉
P2 〈(0.7,0.6,0.5), (0.2,0.2,0.3), (0.1,0.2,0.2)〉 〈(0.9,0.5),(0.1,0.3),(0.2,0.2)〉 〈(0.7,0.6,0.8), (0.1,0.3,0.1), (0.2,0.1,0.1)〉
P3 〈(0.8,0.6,0.8),(0,0.3,0.1), (0.2,0.1,0.1)〉 〈(0.7,0.6),(0.2,0.1),(0.3.0.1)〉 〈(0.5,0.6,0.6), (0.2,0.2,0.3), (0.3,0.2,0.1)〉
P4 〈(0.6,0.7,0.7), (0.2,0.2,0.1), (0.2,0.1,0.2)〉 〈(0.5,0.8), (0.3,0.1), (0.1,0)〉 〈(0.8,0.8,0.6), (0.1,0.2,0.2), (0.1,0,0.2)〉
To get the value of RSVNS for P*
we apply the formula (13) to obtain the following ideal
solution:
𝑃∗
= {〈(0.8,0.7,0.8), (0,0.2,0.1), (0.1,0.1,0.1)〉, 〈(0.9,0.8), (0.1,0.1). (0.1,0)〉,
〈(0.9,0.8,0.8), (0.1,0.1,0.1), (0,0,0.1)〉}.
For the RSVNS we use the equations (9)-(10) to get the following results of similarity measures
between the alternatives Ps (s = 1, 2, 3, 4) and the ideal solution P*
in Table 4.
Table 4
The similarity measure values between Ps and P*
.
Measure Similarity measure value Ranking order The best choice
W1(Ps, P*
)
W1(P1, P*
) = 0.9106
W1(P2, P*
) = 0.9176
W1(P3, P*
) = 0.9012
W1(P4, P*
) = 0.9186
P4>P2>P1>P3 P4
W2(Ps, P*
)
W2(P1, P*
) = 0.8414
W2(P2, P*
) = 0.8536
W2(P3, P*
) = 0.8254
W2(P4, P*
) = 0.8557
P4>P2>P1>P3 P4
In Table 4, the attribute P4 is considered as the best choice, which is the best alternative under
RSVNS environment.

Case2. Under RINS environment, the evaluation values of the set of the alternatives P = {P1, P2,
P3, P4} over the attributes and sub-attributes are the sub-interval of the interval [0,1]. The weight
vector of the given attribute set Z = {z1, z2, z3} is w = (0.3, 0.4, 0.3) and the weight vectors of the
sub-attribute sets {z11, z12, z13}, {z21, z22} and {z31, z32, z33} are given as w1 = (0.5, 0.3, 0.2), w2 =
(0.6, 0.4), and w3 = (0.4, 0.2, 0.4), respectively. Then, the refined interval neutrosophic decision
matrix D=(psji)43 corresponding to the alternatives over the three attributes with three groups of
the sub-attributes is given in Table 5.
Table 5
The RINS decision matrix D=(psji)43.
z1 z2 z3
z11,z12,z13 z21,z22 z31,z32, z33
P1
     
 
     
 
     
 
0.5,0.6 , 0.5,0.6 , 0.6,0.7 ,
0.3,0.4 , 0.4,0.5 , 0.2,0.3 ,
0.2,0.3 , 0.1,0.2 , 0.2,0.3
   
 
   
 
   
 
0.7,0.8 , 0.8,0.9 ,
0.1,0.2 , 0.2,0.3 ,
0.1,0.2 , 0.2,0.3
     
 
     
 
     
 
0.8,0.9 , 0.8,0.9 , 0.5,0.6 ,
0.1,0.2 , 0.1,0.2 , 0.3,0.4 ,
0,0.1 , 0.1,0.2 , 0.2,0.3
P2
     
 
     
 
     
 
0.7,0.8 , 0.6,0.7 , 0.5,0.6 ,
0.2,0.3 , 0.2,0.3 , 0.3,0.4 ,
0.1,0.2 , 0.2,0.3 , 0.2,0.3
   
 
   
 
   
 
0.8,0.9 , 0.5,0.6 ,
0.1,0.2 , 0.3,0.4 ,
0.2,0.3 , 0.2,0.3
     
 
     
 
     
 
0.7,0.8 , 0.6,0.7 , 0.8,0.9 ,
0.1,0.2 , 0.3,0.4 , 0.1,0.2 ,
0.2,0.3 , 0.1,0.2 , 0.1,0.2
P3
     
 
     
 
     
 
0.8,0.9 , 0.6,0.7 , 0.8,0.9 ,
0,0.1 , 0.3,0.4 , 0.1,0.2 ,
0.2,0.3 , 0.1,0.2 , 0.1,0.2
   
 
   
 
   
 
0.7,0.8 , 0.6,0.7 ,
0.2,0.3 , 0.1,0.2 ,
0.3,0.4 , 0.1,0.2
     
 
     
 
     
 
0.5,0.6 , 0.6,0.7 , 0.6,0.7 ,
0.2,0.3 , 0.2,0.3 , 0.3,0.4 ,
0.3,0.4 , 0.2,0.3 . 0.1,0.2
P4
     
 
     
 
     
 
0.6,0.7 , 0.7,0.8 , 0.7,0.8 ,
0.2,0.3 , 0.2,0.3 , 0.1,0.2 ,
0.2,0.3 , 0.1,0.2 , 0.2,0.3
   
 
   
 
   
 
0.5,0.6 , 0.8,0.9 ,
0.3,0.4 , 0.1,0.2 ,
0.1,0.2 , 0,0.1
     
 
     
 
     
 
0.8,0.9 , 0.8,0.9 , 0.6,0.7 ,
0.1,0.2 , 0.2,0.3 , 0.2,0.3 ,
0.1,0.2 , 0,0.1 , 0.2,0.3
From Table 5 we get the RINS P*
for the ideal solution by the formula (14) as follow:
     
     
     
   
   
   
     
     
   
( 0.8,0.9 , 0.7,0.8 , 0.8,0.9 ), ( 0.8,0.9 , 0.8,0.9 ), ( 0.8,0.9 , 0.8,0.9 , 0.8,0.9 ),
* ( 0,0.1 , 0.2,0.3 , 0.1,0.2 ), ( 0.1,0.2 , 0.1,0.2 ), ( 0.1,0.2 , 0.1,0.2 , 0.1,0.2 ),
( 0.1,0.2 , 0.1,0.2 , 0.1,0.2 ) ( 0.1,0.2 , 0,0.1 ) ( 0,0.1 , 0,0.1 , 0.1,0.
P 
 
2 )
 
 
 
 
 
 
 
Thus, we use the equations (11)-(12) to get the values of similarity measures between the ideal
solution P*
and the alternatives Ps (s = 1, 2, 3, 4) and decision results, which are shown in Table
6.

Table 6
The similarity measure values between Ps and P*
and decision results.
Measure Similarity measure value Ranking order The best choice
W3(Ps, P*
)
W3(P1, P*
) = 0.9169
W3(P2, P*
) = 0.9208
W3(P3, P*
) = 0.9109
W3(P4, P*
) = 0.9282
P4>P2>P1>P3 P4
W4(Ps, P*
)
W4(P1, P*
) = 0.8531
W4(P2, P*
) = 0.8589
W4(P3, P*
) = 0.8328
W4(P4, P*
) = 0.8713
P4>P2>P1>P3 P4
In Table 6, the alternative P4 is considered as the best choice, which is the best one.
However, the same ranking orders are shown in the two cases under RSNS environments.
However, existing literature [15,19–22] cannot deal with such two cases with both attributes and
sub-attributes in RSNS setting.
6. Conclusion
This study presented the tangent and cotangent functions-based similarity measures of RSNSs,
and then proposed their decision-making method, which is more suitable for the problems that
have multiple attributes with sub-attributes, along with both the attribute weights and the sub-
attribute weights.
By the similarity measure values between alternatives and the ideal solution, we can rank
alternatives and choose the best one. Then, an illustrative example of the decision-making
problem of a construction project was provided to indicate the feasibility and effectiveness of the
proposed method in RSNS (RSVNS and RINS) setting. Obviously, this study extends existing
methods and provides a new way for the refined simplified neutrosophic MADM problems
containing both the attribute weight and the sub-attribute weights. For the future study, the
presented method will be extended to the similarity measures based on logarithm function for
group decision-making.
Acknowledgments
This paper was supported by the National Natural Science Foundation of China (Nos. 71471172,
61703280).
Conflict of interest
The authors declare no conflict of interest.

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