# Some Novel Cosine Similarity Measures Based on Complex Hesitant Fuzzy Sets and Their Applications

Authors: Ronnason Chinram; Tahir Mahmood; Ubaid Ur Rehman; Zeeshan Ali; Aiyared Iampan
Journal: Journal of Mathematics
 (2021)

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## Abstract

The theory of complex hesitant fuzzy set (CHFS) is a modification technique of the complex fuzzy set (CFS) to cope with awkward and unreliable information’s in daily life issues. CHFS contains the grade of truth in the form of complex number, whose real and imaginary parts are in the form of the finite subset of the unit interval. CHFS is the mixture of hesitant fuzzy set (HFS) and CFS, which handles the complex and uncertain information in real-world issues which is compared with fuzzy sets and complex fuzzy sets. The positive membership in CHFS is in the form a polar coordinate belonging to unit disc in the complex plane. The aims of this manuscript are to explore some similarity measures (SMs), weighted SMs (WSMs) such as cosine SMs, weighted cosine SMs, SMs based on cosine function, WSMs based on cosine function, SMs based on tangent function, and WSMs based on tangent function of CHFS. Some special cases of the presented measures are discussed in detail. Moreover, we use our described SMs and weighted SMs of CHFS in the environment of medical diagnosis and pattern recognition to assess the practicality and competence of the described SMs. Finally, to find the validity and proficiency of the investigated measures based on CHFSs, the comparison between explored measures with some already defined measures and their graphical representations are also discussed in detail.


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## Body

## 1. Introduction

The fuzzy set (FS) is the modification of crisp set which was given by Zadeh [1] to manage the vagueness and uncertainty in the information in real-life decisions. In the theory of FS, the positive grade belongs to closed interval [0, 1], where greatest value designated greatest positive grade. FS has numerous applications in various fields [2–4]. Bustince et al. [5] operated on FSs and their models, extensions portrayal, and aggregation. SMs between FSs play an essential role in the theory of FS, which attracted a lot of attention from the authors. SMs have a lot of applications in real-world problems and are extremely useful in numerous fields [6, 7]. Chen [8] interpreted the similarity function to find the similarity degree among FSs. Pedrycz [9] presented fuzzy control and fuzzy systems. FSs in pattern recognition, methodology, and methods are also presented by Pedrycz [10]. Rangel-Valdez et al. [11] described parallel designs for metaheuristics that solve portfolio selection problems using fuzzy outranking relations. Mahmood [12] described a novel approach towards bipolar soft set and their applications.Numerous authors mentioned the issue, what will be the impact when we alter the range of FS into a unit circle of a complex plane. To deal with such sorts of circumstance, Ramot et al. [13] described the notion of CFS as a modification of FS to handle the complex and tricky data in real-world. The idea of CFS is represented by complex-valued positive grade, which carries two-dimensional data in a particular set. Moreover, Tamir et al. [14] presented the Cartesian form of CFS and the Cartesian complex fuzzy positive grade where both real and imaginary parts carry the fuzzy data. In polar portrayal, the fuzzy data carries the phase value and absolute value of complex positive grade. The complex fuzzy number is not the same as the CFS. The δ-equalities and operation properties of CFS were introduced by Zhang et al. [15].HFS are the significant expansions of the theory of FS. Torra [16, 17] described the notion of HFS. An HFS is represented by positive grade which is in the shape of a finite subset of closed interval 0,1. Torra and Narukawa [17] characterized some fundamental operations on HFS. Rodriguez et al. [18] built up the idea of hesitant fuzzy linguistic term sets. Farhadinia [19] interpreted the idea of similarity and distance measures for higher order HFS. The notion of hesitant fuzzy data aggregation in decision-making (DM) was described by Xia and Xu [20]. Wei et al. [21] interpreted the idea of hesitant fuzzy Choquet aggregation operators and their applications to multiple attribute DM (MADM). Zhang [22] characterized hesitant fuzzy aggregation operators and their application to MADM. Xu and Xia [23] explored the idea of separation and correlation measures of HFS. Zhu et al. [24] gave the idea of hesitant fuzzy geometric Bonferroni means. Herrera et al. [25] described HFSs, an emerging tool in decision-making. A review of HFSs, quantitative and qualitative extensions, was explored by Rodriguez et al. [26]. Li et al. [27] described the consistency of hesitant fuzzy linguistic preference relations. Muhiuddin et al. [28] interpreted the generalized hesitant fuzzy ideals in semigroups.The idea of similarity is a fundamental idea in human cognizance. Similarity has a key role in recognition, taxonomy, and several different fields. There are numerous aspects of the notion of the similarity that have escaped formalization. As per (HFS) detailing of a substantial, broadly useful deﬁnition of similarity is a difficult issue. There does not exist a legitimate, universally useful deﬁnition of similarity. There exist numerous specific deﬁnitions that have been utilized with accomplishment in diagnostics, classification, cluster analysis, and recognition. There are a few comparability measures that are interpreted and utilized for different purposes [29]. The SMs are categorized into 3 classifications: (1) measures based on implicators. (2) Measure based on metric. (3) Measure based on set-theoretic. While managing SMs based on distance, examples have been developed for perceptual similarity where each distance adage is obviously damaged by dissimilarity measures, especially the triangle inequality [17], and thusly the relating SM ignores transitivity. This model hypothesizes that the perceptual distance fulfils the metric adages, the observational legitimacy of which has been tentatively tested by a few authors, especially the triangle inequality (for subtleties see [16] and [17, 30, 31]). Thus, in the event of set-theoretic SMs, it is seen that crisp transitivity is a lot more grounded condition to be put upon SM. Set-theoretic SMs are additionally partitioned in three gatherings: (i) measures dependent on crisp logic; (ii) measures dependent on fuzzy logic; (iii) measures dependent on HFSs.In this paper, we present complex HFSs. The inspiration is that when characterizing the positive grade of the element, the struggle of establishing the positive grade is not as we have margin of error (as in complex intuitionistic FS [32]), or some chance circulations (as in type 2 CFSs) on the probable values. In the existing theories, numerous scholars have faced several troubles. When a decision-maker provides such types of information for the grade of truth in the form of 0.22e(0.3) and 0.5e(0.31), this circumstance can emerge in a multicriteria DM. Basically, the theory of complex hesitant fuzzy set contains the grade of truth in the form of complex number, whose real and imaginary parts are in the form of the finite subset of the unit interval. In this unique situation, rather than considering only an aggregation operator [12], it is helpful to manage all the possible values. This circumstance, as we will talk about later, can be demonstrated utilizing multisets. Therefore, the existing theories are not able to cope with such types of troubles. The investigated ideas are more able to cope with it effectivelyg.Due to this and preserving the advantages of the SMs, in this manuscript, the notion of CHFS is explored, which is the fusion of HFS and CFS to manage the uncertainty and complicated data in real world. The positive membership in CHFS is in the form of a finite subset of unit disc in the complex plane. Moreover, in this manuscript, we interpreted some similarity measures (SMs) and weighted SMs (WSMs). Additionally, we use our explored SMs and weighted SMs of CHFS in the environment of medical diagnosis and pattern recognition to assess the practicality and competence of the described SMs. The comparison between explored measures with some already defined measures and their graphical representations are also discussed in detail.The structure of this manuscript is given as follows: in Section2 of this manuscript, we present preliminaries. In Section 3, the notion of the CHFS and its fundamental properties are explored. In Section 4 of this manuscript, we explore some similarity measures (SMs) and weighted SMs (WSMs) of CHFS. In Section 5, we use proposed SMs and weighted SMs in the environment of medical diagnosis and pattern recognition. The comparison between explored measure with some already defined measures and their graphical representations are also discussed in detail in Section 6. In Section 7, we discuss the conclusion of the article.


## 2. Preliminaries

In this section, we revise fundamental definitions such as FS, CFS, and HFS. Throughout this paper,x denotes the fix set.Definition 1.
(see [1]). An FS E is of the shape,(1)E=x,μΕx|x∈χ,with a condition 0≤μΕx≤1, where μEx stands for the grade of membership. Throughout this paper, the family of all FSs on X are designated by FSX. The pair E=x,μEx is said to be fuzzy number (FN).Definition 2.
(see [13]). A CFS E is of the shape,(2)E=x,μEx|x∈χ,where μEx=γEx.ei2πωγEx stands for the complex-valued membership grade in the shape of polar coordinate, where γEx,ωγEx∈0,1. Moreover, the pair E=x,γEx.ei2πωγEx is said to be complex fuzzy number (CFN).Definition 3.
(see [16, 17]). An HFS E is of the shape,(3)E=x,μΕx|x∈χ,where μEx is a finite subset of 0,1 standing for the grade of membership for every element x∈χ. Moreover, the pair E=x,μEx is said to be hesitant fuzzy number (HFN).Definition 4.
(see [29]). For any two HFSs E and F, the SM DE,F fulfils the following axioms:(1)
0≤SE,F≤1;(2)
SE,F=1⟺E=F;(3)
SE,F=SF,E.Definition 5.
(see [29]). For any two HFSs E and F, the distance measure dE,F fulfils the following properties:(1)
0≤DE,F≤1;(2)
DE,F=1⟺E=F;(3)
DE,F=DF,E.
From the discussion we did above, we get that theSE,F=1=DE,F.


## 3. Complex Hesitant Fuzzy Sets

In this section, we explored the notion of complex hesitant fuzzy sets (CHFSs) and some of its properties.Definition 6.
A CHFSE is of the shape,(4)E=x,μEx|x∈X,where(5)μΕx=γEjx.ei2πωγEjx,j=1,2,3,…,n=γE1x.ei2πωγE1x,γE2x.ei2πωγE2x,……,γEnx.ei2πωγEnx,expressed the complex-valued grade of membership which is the subset of unit disc in complex plane with acondition γEjx,ωγEjx∈0,1. Further, E=x,γEjx.ei2πωγEjx is known as the complex hesitant fuzzy number (CHFN).Definition 7.
LetE=x,γEjx.ei2πωγEjx and F=x,γEjx.ei2πωγEjx be two CHFNs. Then,(1)
cγEx=x,1−γEjx.ei2π1−ωγEjx;(2)
E∪F=x,maxγEjx,γFjx.ei2πmaxωγEjx,ωγEj;(3)
E∩F=x,minγEjx,γFjx.ei2πminωγEjx,ωγEj.
The theory of CHFS is a powerful tool to deal with unsure and complicated data in real-world issues. The CHFS holds the grade of membership in the shape of a finite subset of the unit disc in the complex plane, whose entities are in the shape of polar coordinates. Essentially, the CHFS holds two-dimensional data in a particular set. The explored CHFS is more general than the existing notions such as FS, CFS, and HFS.Example 1.
LetE=x1,0.8ei2π0.9,0.6ei2π0.5,x2,0.5ei2π0.7,0.2ei2π1,1ei2π0.5,x3,0.1ei2π0.2x4,0.4ei2π0.5,0.5ei2π0.6,0.3ei2π0.4,x5,0.1ei2π0.3,0.3ei2π0.5 and F=x1,0.5ei2π0.8,0.7ei2π0.4,1ei2π0.5,x2,0.4ei2π0.6,x3,0.8ei2π0.6,0.5ei2π0.8x4,0.9ei2π0.6,0.4ei2π0.3,x5,0.5ei2π0.7,0.3ei2π0.6,0.2ei2π0.5, be two CHFSs. Then,(1)
Ec=0.2ei2π0.1,0.4ei2π0.5,0.5ei2π0.3,0.8ei2π0.0,0.0ei2π0.5,0.9ei2π0.80.6ei2π0.5,0.5ei2π0.4,0.7ei2π0.6,0.9ei2π0.7,0.7ei2π0.5;(2)
E∪F=0.8ei2π0.9,0.7ei2π0.5,0.5ei2π0.7,0.2ei2π1,1ei2π0.5,0.8ei2π0.6,0.5ei2π0.80.9ei2π0.6,0.5ei2π0.6,0.3ei2π0.4,0.5ei2π0.7,0.3ei2π0.6,0.2ei2π0.5;(3)
E∩F=0.5ei2π0.8,0.6ei2π0.4,0.4ei2π0.6,0.1ei2π0.20.4ei2π0.5,0.4ei2π0.3,0.1ei2π0.3,0.3ei2π0.5;


## 4. Similarity Measures Based on the Cosine Function for CHFSs

In this section, we interpreted some SMs such as cosine SMs for CHFSs, SMs of CHFSs based on cosine function, and SMs of CHFSs based on cotangent function.Definition 8.
LetE and F be two CHFSs on set X. Then, SM between E and F is represented by ScE,F, which fulfils the following postulate:(1)
0≤ScE,F≤1;(2)
ScE,F=1 if and only if E=F;(3)
ScE,F=ScF,E.


### 4.1. Cosine Similarity Measures for CHFS

LetE be a CHFS on a set X. Then, the elements contained in CHFS can be presented as the function of membership degree μEx, which is a subset of a unit disc in a complex plane. Consequently, a cosine SM and weighted cosine SM with CHF data are expressed similarly to the cosine SM based on Bhattacharya’s distance [33].Definition 9.
LetE and F be two CHFSs on a set X. Then, the cosine SM between E and F can be presented as(6)Sc1E,F=1n∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑jLωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ.
In Definition9, if we assume the imaginary parts zero, then the interpreted SM transforms for HFS. Likewise, if we assume the CHFS as a singleton set, then the interpreted SM transforms for CFS. Moreover, if we assume the CHFS as a singleton set and the imaginary part zero, then the interpreted SM transforms for FS. Its structure makes it important and expert to deal with unknown and undependable data in real decision theory.Theorem 1.
The SMSc1E,F fulfils the following postulates:(1)
0≤Sc1E,F≤1;(2)
Sc1E,F=1 if E=F;(3)
Sc1E,F=Sc1F,E.Proof.
(1)
Since1/L∑j=1LγEjxκ⋅γFjxκ∈0,1, 1/L∑jLωγEjxκ⋅ωγFjxκ∈0,1, 1/L∑j=1LγEj2xκ∈0,1, 1/L∑j=1LωγEj2xκ∈0,1, and 1/L∑j=1LγEj2xκ∈0,1, 1/L∑j=1LωγEj2xκ∈0,1 and denominator will always remain greater than the nominator. So, for κ=1, we have(7)1/L∑j=1LγEjx1⋅γFjx1+1/L∑j=1LωγEjx1⋅ωγFjx11/L∑j=1LγEj2x1+1/L∑j=1LωγEj2x11/L∑j=1LγEj2x1+1/L∑j=1LωγEj2x1∈0,1.Forκ=2, we have(8)1/L∑j=1LγEjx2⋅γFjx2+1/L∑j=1LωγEjx2⋅ωγFjx21/L∑j=1LγEj2x2+1/L∑j=1LωγEj2x21/L∑j=1LγEj2x2+1/L∑j=1LωγEj2x2∈0,1.By continuing this procedure, we obtain(9)∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ∈n0,1.This implies that(10)0≤∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ≤n,0≤12∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ≤1,which implies that(11)0≤Sc1xκ≤1.(2)
We have(12)Sc1E,F=1n∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ.Now, asE=F⟹μExκ=μFxκ, for κ=1,2,…,n⟹γEjxκei2πωEjxκ=γFjxκei2πωFjxκ for κ=1,2,…,n⟹γEjxκ=γFjxκ and ei2πωEjxκ=ei2πωFjxκ for κ=1,2,…,n. Then,(13)Sc1E,F=1n∑κ=1n1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ2.Sc1E,F=1n∑κ=1n1/LγE12xκ+γE22xκ+⋯+γEL2xκ+1/LωγE12xκ+ωγE22xκ+⋯+ωγEL2xκ1/LγE12xκ+γE22xκ+⋯+γEL2xκ+1/LωγE12xκ+ωγE22xκ+⋯+ωγEL2xκ.(14)Sc1E,F=1n1/LγE12x1+γE22x1+⋯+γEL2x1+1/LωγE12x1+ωγE22x1+⋯+ωγEL2x11/LγE12x1+γE22x1+⋯+γEL2x1+1/LωγE12x1+ωγE22x1+⋯+ωγEL2x1+1/LγE12x2+γE22x2+⋯+γEL2x2+1/LωγE12x2+ωγE22x2+⋯+ωγEL2x21/LγE12x2+γE22x2+⋯+γEL2x2+1/LωγE12x2+ωγE22x2+⋯+ωγEL2x2+⋯+1/LγE12xn+γE22xn+⋯+γEL2xn+1/LωγE12xn+ωγE22xn+⋯+ωγEL2xn1/LγE12xn+γE22xn+⋯+γEL2xn+1/LωγE12xn+ωγE22xn+⋯+ωγEL2xn,Sc1E,F=1.(3)
We have(15)Sc1E,F=1n∑K=1n1/L∑j=1LγEjxK.γFjxK+1/L∑jLωγEjxK.ωγFjxK1/L∑j=1LγEj2xK+1/L∑j=1LωγEj2xK1/L∑j=1LγFj2xK+1/L∑j=1LωγFj2xK=1n∑K=1n1/L∑j=1LγFjxK.γEjxK+1/L∑jLωγFjxK.ωγEjxK1/L∑j=1LγFj2xK+1/L∑j=1LωγFj2xK1/L∑j=1LγEj2xK+1/L∑j=1LωγEj2xK,=Sc1F,E.
We defined distance measure of the angle asdE,F=arccosSc1E,F. It holds the following axioms:(1)
dE,F≥0 if 0≤Sc1F,E≤1;(2)
dE,F=arccos1=0 if Sc1E,F=1;(3)
dE,F=dF,E if ScE,F=ScF,E.Definition 10.
LetE and F be two CHFSs on a set X. Then, the weighted cosine SM between E and F can be presented as(16)Scw1E,F=∑K=1nwK1/L∑j=1LγEjxK.γFjxK+1/L∑jLωγEjxK.ωγFjxK1/L∑j=1LγEj2xK+1/L∑j=1LωγEj2xK1/L∑j=1LγFj2xK+1/L∑j=1LωγFj2xK,where w=w1,w2,…,wnT represents the weight vector of every element xKK=1,2,…,.n included in CHFS and the weight vector satisfies wK∈0,1 for every K=1,2,…,n, ∑K=1nwK=1. When we suppose the weight vector to be w=1/n,1/n,…,1/nT, the weighted cosine SM will transform into cosine SM. Otherwise speaking, when wK=1/n,K=1,2,3,…,n, the Scw1E,F=Sc1E,F.


### 4.2. Similarity Measures of CHFSs Based on Cosine Function

In this part of the paper, we interpreted SMs of CHFSs based on cosine function and studied their properties.Definition 11.
LetE and F be two CHFSs on a set X. Then, the SMs based on the cosine function between E and F can be presented as(17)Sc2E,F=1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where Sc2E,F means the SM based on the cosine function between E and F, which considers the maximum distance based on the amplitude and phase terms.(18)Sc3E,F=1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,where Sc3E,F means the SM based on the cosine function between E and F, which considers the sum of the distance based on the amplitude and phase terms.Theorem 2.
The SMSc2E,F fulfils the following postulates:(1)
0≤Sc2E,F≤1;(2)
Sc2E,F=1 if E=F;(3)
Sc2E,F=Sc2F,E.Proof .
1.
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑jLωγEjxK−ωγFjxK∈0,1, this implies that max1/L∑j=1LγEjxK−γFjxK,1/L∑j=1LωγEjxK−ωγFjxK∈0,1. So, for K=1, we have(19)Cosπ2max1L∑j=1LγEjx1−γFjx1,1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(20)Cosπ2max1L∑j=1LγEjx2−γFjx2,1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(21)∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(22)0≤∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(23)0≤Sc2xK≤1.(2)
We have(24)Sc2E,F=1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK, for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(25)Sc2E,F=1.(3)
We have(26)Sc2E,F=1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,=1n∑K=1nCosπ2max1L∑j=1LγFjxK−γEjxK,1L∑j=1LωγFjxK−ωγExK,=Sc2F,E.Theorem 3.
The SMSc3E,F fulfils the following postulates:(4)
0≤Sc3E,F≤1;(5)
Sc3E,F=1 if E=F;(6)
Sc3E,F=Sc3F,E.Proof .
(1)
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑j=1LωEjxK−ωFjxK∈0,1, this implies that, 1/2max1/L∑j=1LγEjxK−γFjxK∈0,1,1/L∑j=1LωEjxK−ωFjxK∈0,1. So, for K=1, we have(27)Cosπ41L∑j=1LγEjx1−γFjx1+1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(28)Cosπ41L∑j=1LγEjx2−γFjx2+1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(29)∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(30)0≤∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(31)0≤Sc3xK≤1.(2)
We have(32)Sc3E,F=1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK. for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(33)Sc3E,F=1.(3)
We have(34)Sc3E,F=1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,=1n∑K=1nCosπ41LγFjxK−γEjxK,1L∑j=1LωγFjxK−ωγExK,=Sc3F,E.Definition 12.
LetE and F be two CHFSs on a set X. Then, the weighted SMs based on the cosine function between E and F can be presented as(35)Scw2E,F=∑K=1nwKCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,Scw3E,F=∑K=1nwKCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,where w=w1,w2,…,wnT represents the weight vector of every element xKK=1,2,,,.n carried in CHFS and the weight vector satisfies wK∈0,1 for every K=1,2,3,,,,.n, ∑K=1nwK=1. When we assume the weight vector to be w=1/n,1/n,…,1/nT, the weighted SMs based on the cosine function will transform into SMs based on the cosine function. Otherwise speaking, when wK=1/n,K=1,2,3,…,n, the ScwmE,F=ScmE,Fm=2,3


### 4.3. Similarity Measures of CHFSs Based on Cotangent Function

In this section, according to the cotangent function, we interpreted some cotangent SMs between CHFSs and studied their properties.Definition 13.
LetE and F be two CHFSs on a set X. Then, the cotangent SMs between E and F can be presented as(36)Sc4E,F=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where Sc4E,F means the cotangent SM between E and F, which considers the maximum distance based on the amplitude and phase terms.(37)Sc5E,F=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where Sc5E,F means the cotangent SM between E and F, which considers the sum of distance based on the amplitude and phase terms.Theorem 4.
The SMSc4E,F fulfils the following postulates:(7)
0≤Sc4E,F≤1;(8)
Sc4E,F=1 if E=F;(9)
Sc4E,F=Sc2F,E.Proof .
1.
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑j=1LωEjxK−ωFjxK∈0,1, this implies that max1/L∑j=1LγEjxK−γFjxK,1/L∑j=1LωEjxK−ωFjxK∈0,1. So, for K=1, we have(38)Cotπ4+π4max1L∑j=1LγEjx1−γFjx1,1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(39)Cotπ4+π4max1L∑j=1LγEjx2−γFjx2,1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(40)∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(41)0≤∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(42)0≤Sc4xK≤1.(2)
We have(43)Sc4E,F=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK, for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(44)Sc4E,F=1.(3)
We have(45)Sc4E,F=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK=Sc4F,E.Theorem 5.
The SMSc5E,F fulfils the following postulates:(10)
0≤Sc5E,F≤1;(11)
Sc5E,F=1 if E=F;(12)
Sc5E,F=Sc5F,E.Proof .
(1)
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑j=1LωEjxK−ωFjxK∈0,1, this implies that, 1/2max1/L∑j=1LγEjxK−γFjxK,∑j=1LωEjxK−ωFjxK∈0,1. So, for K=1, we have(46)Cotπ4+π81L∑j=1LγEjx1−γFjx1+1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(47)Cotπ4+π81L∑j=1LγEjx2−γFjx2+1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(48)∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(49)0≤∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(50)0≤Sc5xK≤1.(2)
We have(51)Sc5E,F=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK, for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(52)Sc5E,F=1.(3)
We have(53)Sc5E,F=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,=Sc5F,E.Definition 14.
LetE and F be two CHFSs on a set X. Then, the weighted cotangent SMs between E and F can be presented as(54)Scw4E,F=∑K=1nwKCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,Scw5E,F=∑K=1nwKCotπ4+π81L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where w=w1,w2,…,wnT represents the weight vector of every element xKK=1,2,,,.n carried in CHFS and the weight vector satisfies wK∈0,1 for every K=1,2,3,,,,.n, ∑K=1nwK=1. When we assume the weight vector to be w=1/n,1/n,…,1/nT, the weighted cotangent SMs will transform into cotangent SMs. Otherwise speaking, when wK=1/n,K=1,2,…,n, the ScwmE,F=ScmE,Fm=4,5.


## 4.1. Cosine Similarity Measures for CHFS

LetE be a CHFS on a set X. Then, the elements contained in CHFS can be presented as the function of membership degree μEx, which is a subset of a unit disc in a complex plane. Consequently, a cosine SM and weighted cosine SM with CHF data are expressed similarly to the cosine SM based on Bhattacharya’s distance [33].Definition 9.
LetE and F be two CHFSs on a set X. Then, the cosine SM between E and F can be presented as(6)Sc1E,F=1n∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑jLωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ.
In Definition9, if we assume the imaginary parts zero, then the interpreted SM transforms for HFS. Likewise, if we assume the CHFS as a singleton set, then the interpreted SM transforms for CFS. Moreover, if we assume the CHFS as a singleton set and the imaginary part zero, then the interpreted SM transforms for FS. Its structure makes it important and expert to deal with unknown and undependable data in real decision theory.Theorem 1.
The SMSc1E,F fulfils the following postulates:(1)
0≤Sc1E,F≤1;(2)
Sc1E,F=1 if E=F;(3)
Sc1E,F=Sc1F,E.Proof.
(1)
Since1/L∑j=1LγEjxκ⋅γFjxκ∈0,1, 1/L∑jLωγEjxκ⋅ωγFjxκ∈0,1, 1/L∑j=1LγEj2xκ∈0,1, 1/L∑j=1LωγEj2xκ∈0,1, and 1/L∑j=1LγEj2xκ∈0,1, 1/L∑j=1LωγEj2xκ∈0,1 and denominator will always remain greater than the nominator. So, for κ=1, we have(7)1/L∑j=1LγEjx1⋅γFjx1+1/L∑j=1LωγEjx1⋅ωγFjx11/L∑j=1LγEj2x1+1/L∑j=1LωγEj2x11/L∑j=1LγEj2x1+1/L∑j=1LωγEj2x1∈0,1.Forκ=2, we have(8)1/L∑j=1LγEjx2⋅γFjx2+1/L∑j=1LωγEjx2⋅ωγFjx21/L∑j=1LγEj2x2+1/L∑j=1LωγEj2x21/L∑j=1LγEj2x2+1/L∑j=1LωγEj2x2∈0,1.By continuing this procedure, we obtain(9)∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ∈n0,1.This implies that(10)0≤∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ≤n,0≤12∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ≤1,which implies that(11)0≤Sc1xκ≤1.(2)
We have(12)Sc1E,F=1n∑κ=1n1/L∑j=1LγEjxκ⋅γFjxκ+1/L∑j=1LωγEjxκ⋅ωγFjxκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ.Now, asE=F⟹μExκ=μFxκ, for κ=1,2,…,n⟹γEjxκei2πωEjxκ=γFjxκei2πωFjxκ for κ=1,2,…,n⟹γEjxκ=γFjxκ and ei2πωEjxκ=ei2πωFjxκ for κ=1,2,…,n. Then,(13)Sc1E,F=1n∑κ=1n1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ1/L∑j=1LγEj2xκ+1/L∑j=1LωγEj2xκ2.Sc1E,F=1n∑κ=1n1/LγE12xκ+γE22xκ+⋯+γEL2xκ+1/LωγE12xκ+ωγE22xκ+⋯+ωγEL2xκ1/LγE12xκ+γE22xκ+⋯+γEL2xκ+1/LωγE12xκ+ωγE22xκ+⋯+ωγEL2xκ.(14)Sc1E,F=1n1/LγE12x1+γE22x1+⋯+γEL2x1+1/LωγE12x1+ωγE22x1+⋯+ωγEL2x11/LγE12x1+γE22x1+⋯+γEL2x1+1/LωγE12x1+ωγE22x1+⋯+ωγEL2x1+1/LγE12x2+γE22x2+⋯+γEL2x2+1/LωγE12x2+ωγE22x2+⋯+ωγEL2x21/LγE12x2+γE22x2+⋯+γEL2x2+1/LωγE12x2+ωγE22x2+⋯+ωγEL2x2+⋯+1/LγE12xn+γE22xn+⋯+γEL2xn+1/LωγE12xn+ωγE22xn+⋯+ωγEL2xn1/LγE12xn+γE22xn+⋯+γEL2xn+1/LωγE12xn+ωγE22xn+⋯+ωγEL2xn,Sc1E,F=1.(3)
We have(15)Sc1E,F=1n∑K=1n1/L∑j=1LγEjxK.γFjxK+1/L∑jLωγEjxK.ωγFjxK1/L∑j=1LγEj2xK+1/L∑j=1LωγEj2xK1/L∑j=1LγFj2xK+1/L∑j=1LωγFj2xK=1n∑K=1n1/L∑j=1LγFjxK.γEjxK+1/L∑jLωγFjxK.ωγEjxK1/L∑j=1LγFj2xK+1/L∑j=1LωγFj2xK1/L∑j=1LγEj2xK+1/L∑j=1LωγEj2xK,=Sc1F,E.
We defined distance measure of the angle asdE,F=arccosSc1E,F. It holds the following axioms:(1)
dE,F≥0 if 0≤Sc1F,E≤1;(2)
dE,F=arccos1=0 if Sc1E,F=1;(3)
dE,F=dF,E if ScE,F=ScF,E.Definition 10.
LetE and F be two CHFSs on a set X. Then, the weighted cosine SM between E and F can be presented as(16)Scw1E,F=∑K=1nwK1/L∑j=1LγEjxK.γFjxK+1/L∑jLωγEjxK.ωγFjxK1/L∑j=1LγEj2xK+1/L∑j=1LωγEj2xK1/L∑j=1LγFj2xK+1/L∑j=1LωγFj2xK,where w=w1,w2,…,wnT represents the weight vector of every element xKK=1,2,…,.n included in CHFS and the weight vector satisfies wK∈0,1 for every K=1,2,…,n, ∑K=1nwK=1. When we suppose the weight vector to be w=1/n,1/n,…,1/nT, the weighted cosine SM will transform into cosine SM. Otherwise speaking, when wK=1/n,K=1,2,3,…,n, the Scw1E,F=Sc1E,F.


## 4.2. Similarity Measures of CHFSs Based on Cosine Function

In this part of the paper, we interpreted SMs of CHFSs based on cosine function and studied their properties.Definition 11.
LetE and F be two CHFSs on a set X. Then, the SMs based on the cosine function between E and F can be presented as(17)Sc2E,F=1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where Sc2E,F means the SM based on the cosine function between E and F, which considers the maximum distance based on the amplitude and phase terms.(18)Sc3E,F=1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,where Sc3E,F means the SM based on the cosine function between E and F, which considers the sum of the distance based on the amplitude and phase terms.Theorem 2.
The SMSc2E,F fulfils the following postulates:(1)
0≤Sc2E,F≤1;(2)
Sc2E,F=1 if E=F;(3)
Sc2E,F=Sc2F,E.Proof .
1.
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑jLωγEjxK−ωγFjxK∈0,1, this implies that max1/L∑j=1LγEjxK−γFjxK,1/L∑j=1LωγEjxK−ωγFjxK∈0,1. So, for K=1, we have(19)Cosπ2max1L∑j=1LγEjx1−γFjx1,1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(20)Cosπ2max1L∑j=1LγEjx2−γFjx2,1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(21)∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(22)0≤∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(23)0≤Sc2xK≤1.(2)
We have(24)Sc2E,F=1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK, for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(25)Sc2E,F=1.(3)
We have(26)Sc2E,F=1n∑K=1nCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,=1n∑K=1nCosπ2max1L∑j=1LγFjxK−γEjxK,1L∑j=1LωγFjxK−ωγExK,=Sc2F,E.Theorem 3.
The SMSc3E,F fulfils the following postulates:(4)
0≤Sc3E,F≤1;(5)
Sc3E,F=1 if E=F;(6)
Sc3E,F=Sc3F,E.Proof .
(1)
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑j=1LωEjxK−ωFjxK∈0,1, this implies that, 1/2max1/L∑j=1LγEjxK−γFjxK∈0,1,1/L∑j=1LωEjxK−ωFjxK∈0,1. So, for K=1, we have(27)Cosπ41L∑j=1LγEjx1−γFjx1+1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(28)Cosπ41L∑j=1LγEjx2−γFjx2+1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(29)∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(30)0≤∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(31)0≤Sc3xK≤1.(2)
We have(32)Sc3E,F=1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK. for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(33)Sc3E,F=1.(3)
We have(34)Sc3E,F=1n∑K=1nCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,=1n∑K=1nCosπ41LγFjxK−γEjxK,1L∑j=1LωγFjxK−ωγExK,=Sc3F,E.Definition 12.
LetE and F be two CHFSs on a set X. Then, the weighted SMs based on the cosine function between E and F can be presented as(35)Scw2E,F=∑K=1nwKCosπ2max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,Scw3E,F=∑K=1nwKCosπ41L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,where w=w1,w2,…,wnT represents the weight vector of every element xKK=1,2,,,.n carried in CHFS and the weight vector satisfies wK∈0,1 for every K=1,2,3,,,,.n, ∑K=1nwK=1. When we assume the weight vector to be w=1/n,1/n,…,1/nT, the weighted SMs based on the cosine function will transform into SMs based on the cosine function. Otherwise speaking, when wK=1/n,K=1,2,3,…,n, the ScwmE,F=ScmE,Fm=2,3


## 4.3. Similarity Measures of CHFSs Based on Cotangent Function

In this section, according to the cotangent function, we interpreted some cotangent SMs between CHFSs and studied their properties.Definition 13.
LetE and F be two CHFSs on a set X. Then, the cotangent SMs between E and F can be presented as(36)Sc4E,F=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where Sc4E,F means the cotangent SM between E and F, which considers the maximum distance based on the amplitude and phase terms.(37)Sc5E,F=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where Sc5E,F means the cotangent SM between E and F, which considers the sum of distance based on the amplitude and phase terms.Theorem 4.
The SMSc4E,F fulfils the following postulates:(7)
0≤Sc4E,F≤1;(8)
Sc4E,F=1 if E=F;(9)
Sc4E,F=Sc2F,E.Proof .
1.
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑j=1LωEjxK−ωFjxK∈0,1, this implies that max1/L∑j=1LγEjxK−γFjxK,1/L∑j=1LωEjxK−ωFjxK∈0,1. So, for K=1, we have(38)Cotπ4+π4max1L∑j=1LγEjx1−γFjx1,1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(39)Cotπ4+π4max1L∑j=1LγEjx2−γFjx2,1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(40)∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(41)0≤∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(42)0≤Sc4xK≤1.(2)
We have(43)Sc4E,F=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK, for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(44)Sc4E,F=1.(3)
We have(45)Sc4E,F=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK=1n∑K=1nCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK=Sc4F,E.Theorem 5.
The SMSc5E,F fulfils the following postulates:(10)
0≤Sc5E,F≤1;(11)
Sc5E,F=1 if E=F;(12)
Sc5E,F=Sc5F,E.Proof .
(1)
Since1/L∑j=1LγEjxK−γFjxK∈0,1, 1/L∑j=1LωEjxK−ωFjxK∈0,1, this implies that, 1/2max1/L∑j=1LγEjxK−γFjxK,∑j=1LωEjxK−ωFjxK∈0,1. So, for K=1, we have(46)Cotπ4+π81L∑j=1LγEjx1−γFjx1+1L∑j=1LωγEjx1−ωγFjx1∈0,1.ForK=2, we have(47)Cotπ4+π81L∑j=1LγEjx2−γFjx2+1L∑j=1LωγEjx2−ωγFjx2∈0,1.By continuing this procedure, we obtain(48)∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK∈n0,1.This implies that(49)0≤∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤n,0≤1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK≤1,which implies that(50)0≤Sc5xK≤1.(2)
We have(51)Sc5E,F=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK.Now, asE=F⟹μExK=μFxK, for K=1,2,…,n⟹γEjxKei2πωEjxK=γFjxKei2πωFjxKforK=1,2,…,n⟹γEjxK=γFjxK and ei2πωEjxK=ei2πωFjxKforK=1,2,…,n. Then, γEjxK−γFjxK=0 and ωγEjxK−ωγFjxK=0 for K=1,2,…,n. This implies that(52)Sc5E,F=1.(3)
We have(53)Sc5E,F=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,=1n∑K=1nCotπ4+π81L∑j=1LγEjxK−γFjxK+1L∑j=1LωγEjxK−ωγFjxK,=Sc5F,E.Definition 14.
LetE and F be two CHFSs on a set X. Then, the weighted cotangent SMs between E and F can be presented as(54)Scw4E,F=∑K=1nwKCotπ4+π4max1L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,Scw5E,F=∑K=1nwKCotπ4+π81L∑j=1LγEjxK−γFjxK,1L∑j=1LωγEjxK−ωγFjxK,where w=w1,w2,…,wnT represents the weight vector of every element xKK=1,2,,,.n carried in CHFS and the weight vector satisfies wK∈0,1 for every K=1,2,3,,,,.n, ∑K=1nwK=1. When we assume the weight vector to be w=1/n,1/n,…,1/nT, the weighted cotangent SMs will transform into cotangent SMs. Otherwise speaking, when wK=1/n,K=1,2,…,n, the ScwmE,F=ScmE,Fm=4,5.


## 5. Applications

In this section, we gave two applications about cosine SM, SMs based on cosine function, and cotangent SM under CHF environment. The interpreted SMs are applied to pattern recognition and medical diagnosis to express the usefulness of these SMs.


### 5.1. Pattern Recognition

Example 2.
Without any hesitancy, the quantity of construction usually relies on the standard of building materials. Accordingly, building material scrutiny is the assumption of good engineering standards. The selection of material must be strictly controlled. Scrutiny authorizes the builders to correctly recognize qualified materials and upgrade the standard of the project. To resolve the abovementioned issues, we choose the building meterialsEj(j = 1, 2, 3, 4, 5), which are discussed as follows:(55)E1=x1,0.6ei2π1,0.5ei2π0.5,x2,0.7ei2π0.4,x3,0.6ei2π0.8,0.4ei2π0.7x4,0.8ei2π0.9,0.2ei2π0.7,x5,0.2ei2π0.3,0.6ei2π0.5,0.4ei2π0.6,E2=x1,0.1ei2π0.4,x2,0.5ei2π0.1,0.1ei2π0.6,x3,0.2ei2π0.6,0.7ei2π0.4x4,0.1ei2π0.4,0.3ei2π0.1,x5,0.5ei2π0.6,E3=x1,1ei2π0.8,0.6ei2π0.8,0.5ei2π0.9,x2,0.5ei2π0.7,x3,0.8ei2π1,0.7ei2π0.9x4,0.9ei2π0.8,0.7ei2π0.6,,x5,0.7ei2π0.5,0.2ei2π0.4,0.3ei2π0.7,E4=x1,0.3ei2π0.9,1ei2π1,,x2,0.4.ei2π0.2,0.2ei2π0.5,x3,0.2ei2π1x4,0.8ei2π0.6,x5,0.5ei2π0.1,0.6ei2π0.3,0.8ei2π0.5,E5=x1,0.4ei2π0.2,0.2ei2π0.5,x2,0.4.ei2π0.2,0.4ei2π0.1,x3,0.2ei2π0.4,0.3ei2π0.1x4,0.6ei2π0.7,0.3ei2π0.50.6ei2π0.1,x5,0.1ei2π0.3.
To resolve the abovementioned issue, we choose the complex hesitant fuzzy set in the form of unknown materials.(56)E=x1,0.9ei2π0.8,0.7ei2π0.4,1ei2π0.8,x2,0.4ei2π0.6,x3,0.8ei2π0.6,0.5ei2π0.8x4,0.9ei2π0.6,0.4ei2π0.3x5,0.5ei2π0.7,0.3ei2π0.9,0.2ei2π0.5,.
The aim of this issue is to categorize the unspecified building materialE in one of the categories Ejj=1,2,3,4,5. For it, the cosine SM, SMs based on cosine function, and cotangent SMs which are explored in this paper have been used to determine the similarity from E to Ejj=1,…,5 and calculations are introduced in Tables 1 and 2.
As stated by the above-computed calculations given in Table1, we simply note that the degree of similarity between E and E3 is the greatest one as an extract by all five SMs. This specifies that all five SMs assign the unspecified building material E to the specified building material E3 based on the principle of the maximum degree of similarity. Ranking of the explored cosine and cotangent SMs between E and Ejj=1,…,5 is also introduced in Table 1. The graphical representation of the interpreted SMs between E and Ejj=1,…,5 is indicated in Figure 1.
The weight of elements has great significance to suppose in real decision-making problems. If we suppose the weight of elementsxKK=1,2,3,4,5 to be wK=0.15,0.1,0.25,0.2,0.3, respectively, then the interpreted WSMs (weighted cosine SMs and weighted cotangent SMs) have been used to determine the similarity from E to Ejj=1,…,5 and calculations are introduced in Table 2.
As stated by the above-computed calculations given in Table2, we simply note that the degree of similarity between E and E3 is the greatest one as an extract by all five WSMs. This specifies that all five WSMs assign the unspecified building material E to the specified building material E3 based on the principle of the maximum degree of similarity. Ranking of the explored weighted cosine SMs, weighted SMs based on cosine function, and weighted cotangent SMs between E and Ejj=1,…,5 is also introduced in Table 2. The graphical representation of the interpreted WSMs between E and Ejj=1,…,5 is indicated in Figure 2.Table 1
The explored SMs betweenE and Ejj=1,2,3,4,5.
SMsE,E1E,E2E,E3E,E4E,E5RankingSc1E,Ej0.47330.28290.5170.34290.2844E3≥E1≥E4≥E5≥E2Sc2E,Ej0.86740.63720.93210.60470.7278E3≥E1≥E5≥E2≥E4Sc3E,Ej0.90220.7960.9590.74050.8054E3≥E1≥E5≥E2≥E4Sc4E,Ej0.60560.37770.69880.37620.4605E3≥E1≥E5≥E2≥E4Sc5E,Ej0.65280.52640.75630.47460.7479E3≥E5≥E1≥E2≥E4Table 2
The explored WSMs betweenE and Ejj=1,2,3,4,5.
SMsE,E1E,E2E,E3E,E4E,E5RankingScw1E,Ej0.42470.29330.45570.31280.2826E3≥E1≥E4≥E2≥E5Scw2E,Ej0.88290.67050.92190.6720.7072E3≥E1≥E5≥E4≥E2Scw3E,Ej0.9110.8160.95510.76030.7964E3≥E1≥E2≥E5≥E4Scw4E,Ej0.6280.40110.67370.42070.4359E3≥E1≥E5≥E4≥E2Scw5E,Ej0.66640.54340.74310.49330.7386E3≥E5≥E1≥E2≥E4Figure 1
The graphical representation of interpreted SMs.Figure 2
The graphical representation of interpreted WSMs.


### 5.2. Medical Diagnosis

Symptoms of every diseases are almost different. To examine that the victim is suffering from what type of diseases, the medical diagnosis relies on the victim’s symptoms. The victim’s symptoms are a set of symptoms and unspecified diseases will be a set of diagnostic diseases. The interpreted SMs are illustrated by a following numerical example of medical diagnosis.Example 3.
Let a set of diagnosisD=D1Typhoid,D2Flu,D3Heart problem,D4Pneumonia,D5Coronavirus and set of symptoms X=x1fever,x2cough,x3heart pain,x4loss of ppetite,x5short of breath. The victim’s symptoms are represented in the form of CHFSs as follows:(57)P=x1,0.8ei2π0.9,0.6ei2π0.5,x2,0.5ei2π0.70.9ei2π1,1ei2π0.5,x3,0.1ei2π0.2x4,0.4ei2π0.5,0.5ei2π0.6,0.7ei2π0.4,x5,0.1ei2π0.3,0.3ei2π0.5.
The indications of each diseaseDjj=1,2,3,4,5 are represented in the form CHFSs as follows:(58)D1=x1,0.7ei2π1,0.9ei2π0.8,x2,1ei2π0.8,0.5ei2π0.6,0.6ei2π0.9,x3,0.4ei2π0.6,x4,0.9ei2π0.8,0.7ei2π0.6,0.5ei2π0.7,x5,0.2ei2π0.4,0.3ei2π0.4,D2=x1,0.5ei2π0.6,0.9ei2π0.8,x2,0.8ei2π1,0.7ei2π0.8,x3,0.1ei2π0.05x4,0.2ei2π0.1,0.5ei2π0.2,x5,0.5ei2π0.3,D3=x1,0.6ei2π0.5,x2,0.2ei2π0.2,0.4ei2π0.1,x3,0.8ei2π1,1ei2π1,0.7ei2π0.9x4,0.5ei2π0.7,0.3ei2π0.4,,x5,0.7ei2π0.6,0.2ei2π0.7,D4=x1,0.6ei2π0.9,0.7ei2π0.8,0.4ei2π0.7,x2,0.5ei2π0.7,0.7ei2π0.3,0.1ei2π0.6,x3,0.1ei2π0.4,x4,0.6ei2π0.8,x5,0.4ei2π0.1,0.2ei2π0.4,D5=x1,0.8ei2π0.4,0.5ei2π0.7,x2,0.6ei2π0.7,0.7ei2π0.9,x3,0.1ei2π0.4,0.3ei2π0.2x4,0.3ei2π0.4,x5,0.8ei2π0.7,0.9ei2π1,1ei2π0.7.
The aim of this issue is to find the disease of the victimP in one of the diseases Djj=1,2,3,4,5. For it, the cosine SM, SMs based on cosine function, and cotangent SMs which are explored in this paper have been utilized to determine the similarity from P to Djj=1,…,5 and calculations are introduced in Tables 3 and 4.
As stated by the above-computed calculations described in Table3, we simply note that the degree of similarity between P and D1 is the greatest one as an extract by five SMs. This specifies that all five SMs express that the victim P has typhoid based on the principle of the maximum similarity degree. Ranking of the explored cosine and cotangent SMs between P and Djj=1,…,5 is also introduced in Table 3. Next, the graphical representation of the interpreted SMs between P and Djj=1,…,5 is indicated in Figure 3.
The weight of elements has great significance to suppose in real decision-making problems. If we suppose the weight of elementsxKK=1,2,3,4,5 to be wK=0.15,0.1,0.25,0.2,0.3, respectively, then the interpreted WSMs (weighted cosine SMs and weighted cotangent SMs) have been used to determine the similarity from P to Djj=1,…,5 and calculations are introduced in Table 4.
As stated by the above-computed calculations described in Table3, we simply note that the degree of similarity between P and D1 is the greatest one as an extract by WSMs, except Scw5. This specifies that WSMs Scw1, Scw2, Scw3, and Scw4 show that the victim P has typhoid based on the principle of the maximum similarity degree. We also note that degree of similarity between P and D5 is the highest one as an extract by WSM Scw5. This specifies that the WSM Scw5 shows that the victim P has coronavirus. Ranking of the explored weighted cosine SMs, weighted SMs based on cosine function, and weighted cotangent SMs between P and Djj=1,…,5 is also introduced in Table 4. Next, we have the graphical representation of the interpreted WSMs between P and Djj=1,…,5 in Figure 4.Table 3
The explored SMs betweenP and Djj=1,2,3,4,5.
Similarity measuresP,D1P,D2P,D3P,D4P,D5RankingSC1P,Dj0.51320.42790.30190.42160.3039D1≥D2≥D4≥D5≥D3SC2P,Dj0.88330.84650.63630.87010.7434D1≥D4≥D2≥D5≥D3SC3P,Dj0.91190.88430.67360.89850.8249D1≥D4≥D2≥D5≥D3SC4P,Dj0.63670.57770.39530.60440.4858D1≥D4≥D2≥D5≥D3SC5P,Dj0.72880.67970.6361043140.6621D1≥D2≥D5≥D3≥D4Table 4
The explored WSMs betweenP and Djj=1,2,3,4,5
Similarity measuresP,D1P,D2P,D3P,D4P,D5RankingScw1P,Dj0.55750.3930.30890.45980.2974D1≥D4≥D2≥D3≥D5Scw2P,Dj0.89720.86520.63690.88560.7065D1≥D4≥D2≥D5≥D3Scw3P,Dj0.92240.89920.67850.91110.7898D1≥D4≥D2≥D5≥D3Scw4P,Dj0.66350.6010.40260.63020.465D1≥D4≥D2≥D5≥D3Scw5P,Dj0.70580.66180.44330.68930.7526D5≥D1≥D4≥D2≥D3Figure 3
The graphical representation of interpreted SMs.Figure 4
The graphical representation of interpreted WSMs.


## 5.1. Pattern Recognition

Example 2.
Without any hesitancy, the quantity of construction usually relies on the standard of building materials. Accordingly, building material scrutiny is the assumption of good engineering standards. The selection of material must be strictly controlled. Scrutiny authorizes the builders to correctly recognize qualified materials and upgrade the standard of the project. To resolve the abovementioned issues, we choose the building meterialsEj(j = 1, 2, 3, 4, 5), which are discussed as follows:(55)E1=x1,0.6ei2π1,0.5ei2π0.5,x2,0.7ei2π0.4,x3,0.6ei2π0.8,0.4ei2π0.7x4,0.8ei2π0.9,0.2ei2π0.7,x5,0.2ei2π0.3,0.6ei2π0.5,0.4ei2π0.6,E2=x1,0.1ei2π0.4,x2,0.5ei2π0.1,0.1ei2π0.6,x3,0.2ei2π0.6,0.7ei2π0.4x4,0.1ei2π0.4,0.3ei2π0.1,x5,0.5ei2π0.6,E3=x1,1ei2π0.8,0.6ei2π0.8,0.5ei2π0.9,x2,0.5ei2π0.7,x3,0.8ei2π1,0.7ei2π0.9x4,0.9ei2π0.8,0.7ei2π0.6,,x5,0.7ei2π0.5,0.2ei2π0.4,0.3ei2π0.7,E4=x1,0.3ei2π0.9,1ei2π1,,x2,0.4.ei2π0.2,0.2ei2π0.5,x3,0.2ei2π1x4,0.8ei2π0.6,x5,0.5ei2π0.1,0.6ei2π0.3,0.8ei2π0.5,E5=x1,0.4ei2π0.2,0.2ei2π0.5,x2,0.4.ei2π0.2,0.4ei2π0.1,x3,0.2ei2π0.4,0.3ei2π0.1x4,0.6ei2π0.7,0.3ei2π0.50.6ei2π0.1,x5,0.1ei2π0.3.
To resolve the abovementioned issue, we choose the complex hesitant fuzzy set in the form of unknown materials.(56)E=x1,0.9ei2π0.8,0.7ei2π0.4,1ei2π0.8,x2,0.4ei2π0.6,x3,0.8ei2π0.6,0.5ei2π0.8x4,0.9ei2π0.6,0.4ei2π0.3x5,0.5ei2π0.7,0.3ei2π0.9,0.2ei2π0.5,.
The aim of this issue is to categorize the unspecified building materialE in one of the categories Ejj=1,2,3,4,5. For it, the cosine SM, SMs based on cosine function, and cotangent SMs which are explored in this paper have been used to determine the similarity from E to Ejj=1,…,5 and calculations are introduced in Tables 1 and 2.
As stated by the above-computed calculations given in Table1, we simply note that the degree of similarity between E and E3 is the greatest one as an extract by all five SMs. This specifies that all five SMs assign the unspecified building material E to the specified building material E3 based on the principle of the maximum degree of similarity. Ranking of the explored cosine and cotangent SMs between E and Ejj=1,…,5 is also introduced in Table 1. The graphical representation of the interpreted SMs between E and Ejj=1,…,5 is indicated in Figure 1.
The weight of elements has great significance to suppose in real decision-making problems. If we suppose the weight of elementsxKK=1,2,3,4,5 to be wK=0.15,0.1,0.25,0.2,0.3, respectively, then the interpreted WSMs (weighted cosine SMs and weighted cotangent SMs) have been used to determine the similarity from E to Ejj=1,…,5 and calculations are introduced in Table 2.
As stated by the above-computed calculations given in Table2, we simply note that the degree of similarity between E and E3 is the greatest one as an extract by all five WSMs. This specifies that all five WSMs assign the unspecified building material E to the specified building material E3 based on the principle of the maximum degree of similarity. Ranking of the explored weighted cosine SMs, weighted SMs based on cosine function, and weighted cotangent SMs between E and Ejj=1,…,5 is also introduced in Table 2. The graphical representation of the interpreted WSMs between E and Ejj=1,…,5 is indicated in Figure 2.Table 1
The explored SMs betweenE and Ejj=1,2,3,4,5.
SMsE,E1E,E2E,E3E,E4E,E5RankingSc1E,Ej0.47330.28290.5170.34290.2844E3≥E1≥E4≥E5≥E2Sc2E,Ej0.86740.63720.93210.60470.7278E3≥E1≥E5≥E2≥E4Sc3E,Ej0.90220.7960.9590.74050.8054E3≥E1≥E5≥E2≥E4Sc4E,Ej0.60560.37770.69880.37620.4605E3≥E1≥E5≥E2≥E4Sc5E,Ej0.65280.52640.75630.47460.7479E3≥E5≥E1≥E2≥E4Table 2
The explored WSMs betweenE and Ejj=1,2,3,4,5.
SMsE,E1E,E2E,E3E,E4E,E5RankingScw1E,Ej0.42470.29330.45570.31280.2826E3≥E1≥E4≥E2≥E5Scw2E,Ej0.88290.67050.92190.6720.7072E3≥E1≥E5≥E4≥E2Scw3E,Ej0.9110.8160.95510.76030.7964E3≥E1≥E2≥E5≥E4Scw4E,Ej0.6280.40110.67370.42070.4359E3≥E1≥E5≥E4≥E2Scw5E,Ej0.66640.54340.74310.49330.7386E3≥E5≥E1≥E2≥E4Figure 1
The graphical representation of interpreted SMs.Figure 2
The graphical representation of interpreted WSMs.


## 5.2. Medical Diagnosis

Symptoms of every diseases are almost different. To examine that the victim is suffering from what type of diseases, the medical diagnosis relies on the victim’s symptoms. The victim’s symptoms are a set of symptoms and unspecified diseases will be a set of diagnostic diseases. The interpreted SMs are illustrated by a following numerical example of medical diagnosis.Example 3.
Let a set of diagnosisD=D1Typhoid,D2Flu,D3Heart problem,D4Pneumonia,D5Coronavirus and set of symptoms X=x1fever,x2cough,x3heart pain,x4loss of ppetite,x5short of breath. The victim’s symptoms are represented in the form of CHFSs as follows:(57)P=x1,0.8ei2π0.9,0.6ei2π0.5,x2,0.5ei2π0.70.9ei2π1,1ei2π0.5,x3,0.1ei2π0.2x4,0.4ei2π0.5,0.5ei2π0.6,0.7ei2π0.4,x5,0.1ei2π0.3,0.3ei2π0.5.
The indications of each diseaseDjj=1,2,3,4,5 are represented in the form CHFSs as follows:(58)D1=x1,0.7ei2π1,0.9ei2π0.8,x2,1ei2π0.8,0.5ei2π0.6,0.6ei2π0.9,x3,0.4ei2π0.6,x4,0.9ei2π0.8,0.7ei2π0.6,0.5ei2π0.7,x5,0.2ei2π0.4,0.3ei2π0.4,D2=x1,0.5ei2π0.6,0.9ei2π0.8,x2,0.8ei2π1,0.7ei2π0.8,x3,0.1ei2π0.05x4,0.2ei2π0.1,0.5ei2π0.2,x5,0.5ei2π0.3,D3=x1,0.6ei2π0.5,x2,0.2ei2π0.2,0.4ei2π0.1,x3,0.8ei2π1,1ei2π1,0.7ei2π0.9x4,0.5ei2π0.7,0.3ei2π0.4,,x5,0.7ei2π0.6,0.2ei2π0.7,D4=x1,0.6ei2π0.9,0.7ei2π0.8,0.4ei2π0.7,x2,0.5ei2π0.7,0.7ei2π0.3,0.1ei2π0.6,x3,0.1ei2π0.4,x4,0.6ei2π0.8,x5,0.4ei2π0.1,0.2ei2π0.4,D5=x1,0.8ei2π0.4,0.5ei2π0.7,x2,0.6ei2π0.7,0.7ei2π0.9,x3,0.1ei2π0.4,0.3ei2π0.2x4,0.3ei2π0.4,x5,0.8ei2π0.7,0.9ei2π1,1ei2π0.7.
The aim of this issue is to find the disease of the victimP in one of the diseases Djj=1,2,3,4,5. For it, the cosine SM, SMs based on cosine function, and cotangent SMs which are explored in this paper have been utilized to determine the similarity from P to Djj=1,…,5 and calculations are introduced in Tables 3 and 4.
As stated by the above-computed calculations described in Table3, we simply note that the degree of similarity between P and D1 is the greatest one as an extract by five SMs. This specifies that all five SMs express that the victim P has typhoid based on the principle of the maximum similarity degree. Ranking of the explored cosine and cotangent SMs between P and Djj=1,…,5 is also introduced in Table 3. Next, the graphical representation of the interpreted SMs between P and Djj=1,…,5 is indicated in Figure 3.
The weight of elements has great significance to suppose in real decision-making problems. If we suppose the weight of elementsxKK=1,2,3,4,5 to be wK=0.15,0.1,0.25,0.2,0.3, respectively, then the interpreted WSMs (weighted cosine SMs and weighted cotangent SMs) have been used to determine the similarity from P to Djj=1,…,5 and calculations are introduced in Table 4.
As stated by the above-computed calculations described in Table3, we simply note that the degree of similarity between P and D1 is the greatest one as an extract by WSMs, except Scw5. This specifies that WSMs Scw1, Scw2, Scw3, and Scw4 show that the victim P has typhoid based on the principle of the maximum similarity degree. We also note that degree of similarity between P and D5 is the highest one as an extract by WSM Scw5. This specifies that the WSM Scw5 shows that the victim P has coronavirus. Ranking of the explored weighted cosine SMs, weighted SMs based on cosine function, and weighted cotangent SMs between P and Djj=1,…,5 is also introduced in Table 4. Next, we have the graphical representation of the interpreted WSMs between P and Djj=1,…,5 in Figure 4.Table 3
The explored SMs betweenP and Djj=1,2,3,4,5.
Similarity measuresP,D1P,D2P,D3P,D4P,D5RankingSC1P,Dj0.51320.42790.30190.42160.3039D1≥D2≥D4≥D5≥D3SC2P,Dj0.88330.84650.63630.87010.7434D1≥D4≥D2≥D5≥D3SC3P,Dj0.91190.88430.67360.89850.8249D1≥D4≥D2≥D5≥D3SC4P,Dj0.63670.57770.39530.60440.4858D1≥D4≥D2≥D5≥D3SC5P,Dj0.72880.67970.6361043140.6621D1≥D2≥D5≥D3≥D4Table 4
The explored WSMs betweenP and Djj=1,2,3,4,5
Similarity measuresP,D1P,D2P,D3P,D4P,D5RankingScw1P,Dj0.55750.3930.30890.45980.2974D1≥D4≥D2≥D3≥D5Scw2P,Dj0.89720.86520.63690.88560.7065D1≥D4≥D2≥D5≥D3Scw3P,Dj0.92240.89920.67850.91110.7898D1≥D4≥D2≥D5≥D3Scw4P,Dj0.66350.6010.40260.63020.465D1≥D4≥D2≥D5≥D3Scw5P,Dj0.70580.66180.44330.68930.7526D5≥D1≥D4≥D2≥D3Figure 3
The graphical representation of interpreted SMs.Figure 4
The graphical representation of interpreted WSMs.


## 6. Comparison

In this section of the paper, we expressed the effectiveness and advantages of the interpreted SMs by comparing with some already defined SMs.Example 4.
Without any hesitancy, the quantity of construction usually relies on the standard of building materials. Accordingly, building material scrutiny is the assumption of good engineering standards. The selection of material must be strictly controlled. Scrutiny authorizes the builders to correctly recognize qualified materials and upgrade the standard of the project. Suppose pattern recognition problem about the categorization of building materials, Let five specified building materialsEjj=1,2,3,4,5 which are represented in the form of HFSs as follows:(59)E1=x1,0.6,0.5,x2,0.7,x3,0.6,0.4x4,0.8,0.2,x5,0.2,0.6,0.4,E2=x1,0.1,x2,0.5,0.1,x3,0.2,0.7x4,0.1,0.3,x5,0.5,E3=x1,1,0.6,0.5,x2,0.5,x3,0.8,0.7x4,0.9,0.7,,x5,0.7,0.2,0.3,E4=x1,0.3,1,x2,0.4,0.2,x3,0.2x4,0.8,x5,0.5,0.6,0.8,E5=x1,0.4,0.2,x2,0.4,0.4,x3,0.2,0.3x4,0.6,0.3,0.6,x5,0.1.
Next, let an unspecified building materialE in the form of CHFS which needs to be recognized be(60)E=x1,0.9,0.7,1,x2,0.4,x3,0.8,0.5x4,0.9,0.4x5,0.5,0.3,0.2,.
We convert the HFSs in the CHFSs by taking1=e0 as follows:(61)E1=x1,0.6ei2π0.0,0.5ei2π0.0,x2,0.7ei2π0.0,x3,0.6ei2π0.0,0.4ei2π0.0x4,0.8ei2π0.0,0.2ei2π0.0,x5,0.2ei2π0.0,0.6ei2π0.0,0.4ei2π0.0,E2=x1,0.1ei2π0.0,x2,0.5ei2π0.0,0.1ei2π0.0,x3,0.2ei2π0.0,0.7ei2π0.0x4,0.1ei2π0.0,0.2ei2π0.0,x5,0.5ei2π0.0,E3=x1,1ei2π0.0,0.6ei2π0.0,0.5ei2π0.0,x2,0.5ei2π0.0,x3,0.8ei2π0.0,0.7ei2π0.0x4,0.9ei2π0.0,0.7ei2π0.0x5,0.7ei2π0.0,0.2ei2π0.0,0.3ei2π0.0,E4=x1,0.3ei2π0.0,1ei2π0.0,x2,0.4ei2π0.0,0.2ei2π0.0,x3,0.2eiπ0.0x4,0.8ei2π0.0x5,0.5eiπ0.0,0.6ei2π0.0,0.8ei2π0.0,and(62)E=x1,0.9ei2π0.0,0.7ei2π0.0,1ei2π0.0,x2,0.4ei2π0.0,x3,0.8ei2π0.0,0.5ei2π0.0x4,0.9ei2π0.0,0.4ei2π0.0x5,0.5ei2π0.0,0.3ei2π0.0,0.2ei2π0.0.
For Example4, we need to find that the unknown building material E belongs to which of the specified building material Ejj=1,2,3,4,5. In Example 4, the data are in the shape of HFSs. We found similarity between E and Ejj=1,…,5 through some already defined SMs for HFSs, as shown in Table 5. As 1=e0, then the data given in Example 4 are transformed into CHFSs. Then, through interpreted SMs, we found the similarity between E and Ejj=1,2,3,4,5 which is also given in Table 5. Our interpreted SMs showed that unspecified building material E belongs to the specified building material E3 because the similarity between E and E3 is the greatest one. Ranking of the interpreted and already defined SMs is also described in Table 5. Next, we have the graphical representation of the comparison of the proposed and already defined SMs which is represented in Figure 5.
Now, we discuss the comparison between interpreted and already defined SMs for Example2. In Example 2, the data are in the shape of CHFSs. We know that no SM exists in the literature to solve this kind of data. The existing SMs are ineffective to find the similarity between E and Ejj=1,…,5 as demonstrated in Table 6. From Table 6, we observe that the data given in Example 2 are solvable by the interpreted SMs. The interpreted SMs get the similarity between E and Ejj=1,…,5, as demonstrated in Table 6. Our interpreted SMs showed that unspecified building material E belongs to the specified building material E3 because the similarity between E and E3 is the greatest one. Ranking of the explored SMs is also introduced in Table 6. Next, we have the graphical representation of the comparison of proposed and already defined SMs which is represented in Figure 6.
From the above discussion, our explored SMs can represent extra fuzzy information and put it broadly in circumstances in real-life problems. Based on CHFS, we explored the SMs; our SMs are more satisfactory for real-life problems, and the existing SMs and our SMs are more general than the existing SMs.Table 5
The comparison between interpreted and some already defined SMs of Example4.
MethodScore valueRankingXu and Xia [29]SE,E1=0.7667, SE,E2=0.5967SE,E3=0.7733, SE,E4=0.3467SE,E5=0.7067E3≥E1≥E5≥E2≥E4Zeng et al. [30]ShE,E1=0.8367, ShE,E2=0.5283, ShE,E3=0.845, ShE,E4=0.415, ShE,E4=0.6872E3≥E1≥E5≥E2≥E4Jun [31]CosHFSE,E1=0.963, CosHFSE,E2=0.841, CosHFSE,E3=0.911, CosHFSE,E4=0.883, CosHFSE,E5=0.925E1≥E5≥E3≥E4≥E2Proposed SMSc1E,E1=0.5006, Sc1E,E2=0.3334, Sc1E,E3=0.5234Sc1E,E4=0.453, Sc1E,E5=0.3147E3≥E1≥E4≥E2≥E5Proposed SMSc2E,E1=0.8914, Sc2E,E2=0.7565, Sc2E,E3=0.9719, Sc2E,E4=0.7922, Sc2E,E5=0.8036E3≥E1≥E5≥E4≥E2Proposed SMSc3E,E1=0.9721, Sc3E,E2=0.9342, Sc3E,E3=0.9929, Sc3E,E4=0.9456, Sc3E,E5=0.9486E3≥E1≥E5≥E4≥E2Proposed SMSc4E,E1=0.6513, Sc4E,E2=0.5454, Sc4E,E3=0.7994, Sc4E,E4=0.5352, Sc4E,E5=0.5395E3≥E1≥E2≥E5≥E4Proposed SMSc5E,E1=0.8105, Sc5E,E2=0.7457, Sc5E,E3=0.8953, Sc5E,E4=0.7418, Sc5E,E5=0.8671E3≥E5≥E1≥E2≥E4Figure 5
The graphical representation of interpreted SMs with some existing SMs for Example4.Table 6
The comparison between interpreted and some already defined SMs of Example2.
MethodScore valueRankingXu and Xia [29]UnsuccessfulUnsuccessfulZeng et al. [30]UnsuccessfulUnsuccessfulJun [31]UnsuccessfulUnsuccessfulProposed SMSc1E,E1=0.4733, Sc1E,E2=0.2829Sc1E,E3=0.517Sc1E,E4=0.3429, Sc1E,E5=0.2844E3≥E1≥E4≥E5≥E2Proposed SMSc2E,E1=0.8674, Sc2E,E2=0.6372, Sc2E,E3=0.9321, Sc2E,E4=0.6047, Sc2E,E5=0.7278E3≥E1≥E5≥E2≥E4Proposed SMSc3E,E1=0.9022, Sc3E,E2=0.796, Sc3E,E3=0.959, Sc3E,E4=0.7405, Sc3E,E5=0.8054E3≥E1≥E5≥E2≥E4Proposed SMSc4E,E1=0.6056, Sc4E,E2=0.3777, Sc4E,E3=0.6988, Sc4E,E4=0.3762, Sc4E,E5=0.4605E3≥E1≥E5≥E2≥E4Proposed SMSc5E,E1=0.6528, Sc5E,E2=0.5264, Sc5E,E3=0.7563, Sc5E,E4=0.4746, Sc5E,E5=0.7479E3≥E5≥E1≥E2≥E4Figure 6
The graphical representation of interpreted SMs with some existing SMs for Example2.


## 7. Conclusion

The CHFS is one of the enlargements of the CFS in which the possibility of the enrollment work is stretched out from the subset of the genuine number to the unit disc which is interpreted. In this article, we explored another type of similarity measure (SM) which relies on the cosine and cotangent functions. At that stage, we use our introduced SMs and weighted SMs (based on the cosine and cotangent functions) between CHFSs to manage pattern recognition and medical diagnosis problems including design acknowledgment and plan choice. Finally, two numerical models are given to represent the logic and effectiveness of the likeness measures for design acknowledgment and conspire choice. The comparison between explored measure with some existing measures and their graphical representations are also discussed in detail.Consequently, the measures defined in this manuscript can be utilized in a larger range of applications. In future research, we will extend this work to suppose the two facts: (1) similarity measures and aggregation operators [34–42]; (2) methods [43].



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