# A Novel Approach of Complex Dual Hesitant Fuzzy Sets and Their Applications in Pattern Recognition and Medical Diagnosis

Authors: Ubaid Ur Rehman; Tahir Mahmood; Zeeshan Ali; Thammarat Panityakul
Journal: Journal of Mathematics
 (2021)

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## Abstract

Complex dual hesitant fuzzy set (CDHFS) is an assortment of complex fuzzy set (CFS) and dual hesitant fuzzy set (DHFS). In this manuscript, the notion of the CDHFS is explored and its operational laws are discussed. The new methodology of the complex interval-valued dual hesitant fuzzy set (CIvDHFS) and its necessary laws are introduced and are also defensible with the help of examples. Further, the antilogarithmic and with-out exponential-based similarity measures, generalized similarity measures, and their important characteristics are also developed. These similarity measures are applied in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established measures. We also solved some numerical examples using the established measures to examine the reliability and validity of the proposed measures by comparing these with existing measures. To strengthen the proposed study, the comparative analysis is made and it is conferred that the proposed study is much more superior to the existing studies.


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## Body

## 1. Introduction

Zadeh [1] presented the theory of fuzzy sets (FSs), which contain the grade of truth belonging to unit interval. But, in some cases, the theory of FSs has been failed. For instance, when a decision-maker faced information in the form of truth and falsity grades, then the FSs are not able to cope with it. In reality, these sets have given different approaches to allot the participation degree or the nonmembership level of a component to a given set portrayed by various properties. IFSs [2], otherwise called IVFSs from a numerical perspective, can be demonstrated with two capacities that characterize a stretch to mirror some vulnerability on the enrollment work of the components. IVFSs are the speculation of FSs and can demonstrate vulnerability for the need of data, in which a closed subinterval of [0, 1] is relegated to the participation degree. Atanassov and Gargov [3] demonstrated that IFSs and IVFSs are equipollent speculations of FSs and proposed the thought of IVIFS, which has been examined and utilized broadly [4–8].T2FSs, depicted by enrollment work that is portrayed by more boundaries, grant the fuzzy enrollment as a fuzzy set improving the demonstrating ability compared to the original one. Scientifically, IFSs can be viewed as a specific instance of T2FSs, where the participant’s work restores a lot of fresh stretches. In spite of the wide uses of T2FSs [9–12], they experience issues in building up the optional enrollment capacities and troubles in control [13–15]. FMSs are also the extension of the FSs, which contains the grade of truth in the form of the finite subset of the unit interval. Note that in spite of the fact that the highlights of FMSs permit the application to data recovery on the internet, where a web index recovers different events of the same subjects with conceivable various degrees of significance [16], they have issues with the fundamental tasks, for example, the definitions for association and crossing point, which do not sum up the ones for FSs. Rickard [17] gave an elective definition that underlines the value of a commutative property between a set activity and an α-cut, settling this issue. HFSs were initially presented by Torra [4]. The theory of hesitant fuzzy information is also reliable and more efficient to cope with complicated and vague information in realistic issues. Torra [4] inspected IFSs and FMSs and drew examinations and made characteristic associations among them. The theories of hesitant fuzzy information and fuzzy multisets are both the same concepts because the resultant values of both sets are in the form of the finite subset of the unit interval. HFSs were esteemed IFSs when the HFS is a nonempty shut span. In light of the connections between IFSs and HFSs, Torra [4] gave a definition comparing the envelope of HFS. Xu and Xia [18] explored the conglomeration administrators, separation, what’s more, comparability measures for HFSs, and applied them to decision-making problems [19–21].Recently, Zhu et al. [22] explored the DHFS which encompass fuzzy sets, IFSs, HFSs, and fuzzy multisets as special cases. DHFSs have described in terms of the following two functions: the membership hesitancy function and the nonmembership hesitancy function. Wang et al. [23] defined distance and similarity measures of DHFSs with their applications to multiple attribute decision making. Taking into account such functions provides us with more exemplary and flexible access to assign values for each element in the domain. Apparently, DHFSs can reflect the human’s hesitance more objectively than the other existing extensions of the fuzzy set (AIFSs, IVAIFSs, HFSs, etc.). However, in the spirit of what has been done for IVFSs, Farhadinia [24] introduced a dual interval‐valued hesitant fuzzy set (DIVHFS) where its fundamental characteristic is that the values of the membership function and nonmembership function are set of intervals rather than a set of exact numbers. Certain scholars have asked a question, when we changed the range of the fuzzy set into unit disc in complex plane, what will be the result. Ramot et al. [25] introduced the idea of complex FS (CFS), which contains the truth grade in the form of a complex number by a member of a unit disc in the complex plane. CFS deals with two dimensions in a single set. CFS is a powerful procedure to illustrate the belief of a human being in the formation of grades. The complex fuzzy set considers only the membership degree, but does not weight on the nonmembership portion of the data entities, which likewise assume an equal part in assessing the object in the decision-making process. However, in the real world, it is regularly hard to express the estimation of the membership degree by an exact value in a fuzzy set. In such cases, it might be easier to depict vagueness and uncertainty in the real world using a 2-dimensional information instead of a single one. Consequently, an extension of the existing theories might be extremely valuable to depict the uncertainties because of his/her reluctant judgment in complex decision-making problems. Many researchers have utilized it in the environment of different fields.In real‐life issues, we run over numerous circumstances where we have to measure the vulnerability existing in the information to settle on ideal choices. Exponential-based similitude measures and with-out exponential-based comparability measures are significant apparatuses for taking care of dubious data present in our day-to-day life issues. Various measures, for example, similitude, exponential, separation, entropy, and incorporation, process the questionable data and empower us to arrive at some resolution. As of late, these measures have increased a lot of considerations from numerous creators because of their wide applications in different fields, for example, design acknowledgment, clinical determination, grouping examination, and picture portion. All the current methodologies of chiefs, in light of exponential-based closeness measures and with-out exponential-based likeness measures, in FS, CFS, and HFS speculations, manage participation capacities having a place with a unit span as a subset in the idea of HFS. In the CDHFS hypothesis, enrollment and falsity degrees are perplexing esteemed and are spoken to in polar directions. When a decision-maker provides such sorts of information, whose membership and non-membership grades in the form of a finite subset of complex numbers belonging to complex plane with a rule that is the sum of the maximum of the real parts (also for imaginary parts) of the membership and non-membership grades is restricted to the unit interval. For instance,0.9ei2π0.3,0.7ei2π0.6,0.3ei2π0.2,0.1ei2π0.2 and 0.09ei2π0.03,0.07ei2π0.06,0.03ei2π0.02,0.01ei2π0.02, then all the existing notions are failed. For coping with such kinds of problems, the CDHFS is a proficient technique to resolve realistic decision problems in the environment of the FS theory. CDHFS is more powerful and more general than existing notions such as HFS, CFS, and FS to cope with awkward and complicated information in real-life decisions. Because all these notions are the special cases of the explored CDHFS, the advantages of the presented CDHFS are discussed below:(1)
When we choose the imaginary parts of the CDHFS as zero, then the CDHFS is reduced into DHFS which is in the form of0.9,0.7,0.3,0.1 and 0.09,0.07,0.03,0.01.(2)
When we choose the CDHFS as a singleton set, then the CDHFS is reduced into CIFS which is in the form of0.9ei2π0.3 and 0.09ei2π0.03.(3)
When we choose the CDHFS as a singleton set and the imaginary parts as zero, then the CDHFS is reduced into IFS which is in the form of0.9 and 0.09.In real‐life problems, we come across many situations where we need to quantify the uncertainty existing in the data to make optimal decisions. Information measures are important tools for handling uncertain information present in our day-to-day life problems. Different measures of information, such as similarity, distance, entropy, and inclusion, process the uncertain information and enable us to attain some conclusions. Recently, these measures have gained much attention from many authors due to their wide applications in various fields, such as pattern recognition, medical diagnosis, clustering analysis, and image segment. All the existing approaches of decision making, based on information measures, in FS and DHFS theories, deal with membership and nonmembership functions, which are real valued. In CDHFS theory, membership and nonmembership degrees are complex-valued and are represented in polar coordinates. The amplitude term corresponding to membership (nonmembership) degree gives the extent of belongingness (not belongingness) of an object in a CDHFS, and the phase term associated with membership (nonmembership) degree gives the additional information, generally related with periodicity. The phase terms are novel parameters of the membership and nonmembership degrees and these are the parameters which distinguish the CDHFS and traditional DHFS theories. DHFS theory deals with only one dimension at a time, which results in information loss in some instances. However, in real life, we come across complex natural phenomena where it becomes essential to add the second dimension to the expression of membership and nonmembership grades. By introducing this second dimension, the complete information can be projected in one set, and hence, loss of information can be avoided. To illustrate the significance of phase term, we give an example. Suppose XYZ company decides to set up biometric‐based attendance devices (BBADs) in all of its offices spread all over the country. For this, the company consults an expert who gives the information regarding (i) models of BBADs and (ii) production dates of BBADs. The company wants to select the most optimal model of BBADs with its production date simultaneously. Here, the problem is two-dimensional, namely, the model of BBADs and production date of BBADs. This type of problem cannot be modeled accurately using traditional DHFS theory as DHFS theory cannot tackle with both the dimensions simultaneously. The best way to represent all of the information provided by the expert is by using CDHFS theory. The amplitude terms in CDHFS may be used to give company’s decision regarding the model of BBADs and the phase terms may be used to represent company’s judgment in respect of production date of BBADs. Motivated by the abovementioned challenges and keeping the advantages of the CDHFS, in this manuscript, some key contributions are made:(1)
Complex dual hesitant fuzzy set (CDHFS) is a combination of two modifications, called complex fuzzy set (CFS) and dual hesitant fuzzy set (DHFS). CDHFS composes two degrees, called truth valued and falsity valued in the form of a finite subset of a unit disc in a complex plane and is a proficient technique to cope with uncertain and unpredictable information in real-life decisions. The aim of this manuscript is to explore the notion of a CDHFS and its operational laws.(2)
The novel approach of CIvDHFS and its fundamental laws are explored and also justified with the help of examples.(3)
Further, the antilogarithmic and with-out exponential-based similarity measures and generalized similarity measures and their important characteristics are also explored.(4)
These similarity measures are applied in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established measures. We also solved some numerical examples using the established measures.(5)
Examining the reliability and validity of the proposed measures by comparing it with existing measures.(6)
The advantages, comparative analysis, and graphical representation of the explored measures and existing measures are also discussed in detail.The remainder of this manuscript is as follows. In Section2, we review some fundamental definitions such as FS, CFS, HFS, DHFS, and IvDHFS. In Section 3, we explore the notion of a CDHFS and its operational laws. The novel approach of CIvDHFS and its fundamental laws are also explored and also justified with the help of examples. In Section 4, the antilogarithmic and with-out exponential-based similarity measures and generalized similarity measures and their important characteristics are also explored. In Section 5, these similarity measures are applied in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established measures. We also solved some numerical examples using the established measures to examine the reliability and validity of the proposed measures by comparing it with existing measures given in Section 6. The advantages, comparative analysis, and graphical representation of the explored measures and existing measures are also discussed in detail. The conclusion of this manuscript is discussed in Section 7.


## 2. Preliminaries

In this section, we review fundamental definitions such as FS, CFS, HFS, DHFS, and IvDHFS. Through this article,X speaks to a fix set.Definition 1 (see [1]).
A FSA is of the following form:(1)A=x,μAx|x∈X,with a condition 0≤μAx≤1, where μAx represents the grade of truth. Through this article, the collection of all FSs on X are represented by FSX. The pair A=x,μAx is known as fuzzy number (FN).Definition 2 (see [25]).
A CFSA is of the following form:(2)A=x,μAx|x∈X,where μAx=γAx.ei2πωγAx represents the complex-valued truth grade in the form of polar coordinate, where γAx,ωγAx∈0,1. Additionally, the pair A=x,γAx⋅ei2πωγAx is known as complex fuzzy number (CFN).Definition 3 (see [4]).
A HFSA is of the following form:(3)A=x,μAx|x∈X,where μAxis the set of different finite values in 0,1, representing the grade of truth for each element x∈X. Further, the pair A=x,μAx is known as hesitant fuzzy number (HFN).Definition 4 (see [23]).
A DHFSA is of the following form:(4)A=x,μAx,νAx|x∈X,where μAx and νAx are the two finite subsets in 0,1, representing the membership grade and nonmembership grade of the component x∈X, respectively, with the conditions 0≤γAx,δAx≤1 and 0≤γA+x+δA+x≤1, where γAx∈μAx, δAx∈νAx, γA+x∈μA+x=∪γAx∈μAxmaxγAx, and δA+x∈νA+x=∪δAx∈νAxmaxδAx.Definition 5 (see [24]).
For any two DHFSsA and ℬ, the similarity measure sA,ℬ satisfies the following axioms:(1)
0≤SA,ℬ≤1(2)
SA,ℬ=1⟺A=ℬ(3)
SA,ℬ=Sℬ,ADefinition 6 (see [24]).
For any two DHFSsA and ℬ, the distance measure dA,ℬ satisfies the following axioms:(1)
0≤dA,ℬ≤1(2)
dA,ℬ=1⟺A=ℬ(3)
dA,ℬ=dℬ,A
From the abovementioned analysis, we get that thesA,ℬ=1−dA,ℬ.Definition 7 (see [26]).
A IvDHFSA is of the following form:(5)A=x,μAx,νAx|x∈X,where μAx=∪γALxγAUx∈μAxmaxγALx,γAUx and νAx=∪δALx,δAUx∈νAxmaxδALx,δAUx are two finite subsets of some interval-values in 0,1, representing the membership grade and nonmembership grade of the component x∈X, respectively, with the conditions γALxγAUx,δALx,δAUx⊆0,1, and 0≤γAUx++δAUx+≤1, where γALxγAUx∈μAx, δALx,δAUx∈νAx, γAUx+∈μA+x=∪γALxγAUx∈μAxmaxγAUx, and δAUx+∈νA+x=∪δALx,δAUx∈νAxmaxδAx, for allx∈X.


## 3. Complex Dual Hesitant Fuzzy Sets and Complex Interval-Valued Dual Hesitant Fuzzy Sets

The aim of this section is to propose the novel of CDHFS and its operational laws. We also proposed the novel of CIvDHFS and its operational laws. We verified these operation laws with the help of numerical examples.


### 3.1. Complex Dual Hesitant Fuzzy Sets

We defined the notion of CDHFS which is the combination of CFS and DHFS and discussed its operational laws. Additionally, we verified its operational laws with the help of example.Definition 8.
A CDHFSA is of the following form:(6)A=x,μAx,νAx|x∈X,where(7)μAx=x,γAJx.ei2πωγAJx,J=1,2,3,…,g,νAx=x,δAJx.ei2πϖδAJx,J=1,2,3,…,h,represented the complex-valued membership grade and nonmembership grade, which are subsets of a unit disc in complex plane with a condition γAJx,ωγAJx,δAJx,ϖδAJx∈0,1, 0≤γA+x+δAJ+x≤1, and 0≤ωγA+x+ϖδA+x≤1, where γA+x=∪γAJx∈μAxmaxγAJx, ωγA+x=∪ωγAJx∈μAxmaxωγAJx, and δA+x=∪δAJx∈νAxmaxδAJx, ϖνA+x=∪ϖνAJx∈νAxmaxϖνAJx, for J=1,2,…,g and J=1,2,…,h. Further, A=x,γAJx⋅ei2πωγAJx,δAJx⋅ei2πϖδAJx is called complex dual hesitant fuzzy number (CDHFN).Definition 9.
LetA=x,γAJx⋅ei2πωγAJx,δAJx⋅ei2πϖδAJxand ℬ=x,γℬJx⋅ei2πωγℬJx,δℬJx⋅ei2πϖδℬJx be two CDHFNs. Then, their complement, union, and intersection are defined as follows:(1)
Ac=x,νAx,μAx(2)
A∪ℬ=x,maxγAjx,γℬjx⋅ei2πmaxωγAjx,ωγℬjx,minδAjx,δℬjx⋅ei2πminϖδAjx,ϖδℬjx(3)
A∩ℬ=x,minγAJx,γℬJx.ei2πminωγAJx,ωγℬJx,maxδAJx,δℬJx.ei2πmaxϖδAJx,ϖδℬJxExample 1.
Let(8)A=x1,0.8ei2π0.6,0.5ei2π0.4,0.1ei2π0.1,x2,0.6ei2π0.7,0.3ei2π0.2,x3,0.5ei2π0.6,0.4ei2π0.5,0.7ei2π0.4,0.1ei2π0.3,0.2ei2π0.1,x4,0.6ei2π0.50.3ei2π0.4,0.3ei2π0.2,0.2ei2π0.4,ℬ=x1,0.6ei2π0.8,0.5ei2π0.55,0.8ei2π0.45,0.15ei2π0.1,0.1ei2π0.15,0.2ei2π0.1,x2,0.2ei2π0.30.7ei2π0.65,0.3ei2π0.5,x3,0.65ei2π0.35,0.5ei2π0.65,0.25ei2π0.15,0.3ei2π0.3,x4,0.55ei2π0.45,0.45ei2π0.25,be two CDHFSs. Then,(1)
Ac=x1,0.1ei2π0.1,0.8ei2π0.6,0.5ei2π0.4,x2,0.3ei2π0.2,0.6ei2π0.7,x3,0.1ei2π0.3,0.2ei2π0.1,0.5ei2π0.6,0.4ei2π0.5,0.7ei2π0.4,x4,0.3ei2π0.2,0.2ei2π0.4,0.6ei2π0.5,0.3ei2π0.4(2)
A∪ℬ=x1,0.8ei2π0.8,0.5ei2π0.55,0.8ei2π0.45,0.1ei2π0.1,x2,0.6ei2π0.70.3ei2π0.2,x3,0.65ei2π0.6,0.5ei2π0.650.7ei2π0.4,0.1ei2π0.15,0.2ei2π0.1,x4,0.6ei2π0.650.3ei2π0.4,0.3ei2π0.2(3)
A∩ℬ=x1,0.6ei2π0.60.5ei2π0.4,0.15ei2π0.3,0.2ei2π0.150.2ei2π0.1,x2,0.2ei2π0.30.7ei2π0.65,0.3ei2π0.5,x3,0.5ei2π0.35,0.4ei2π0.5,0.25ei2π0.3,0.3ei2π0.3,x4,0.55ei2π0.45,0.45ei2π0.250.2ei2π0.4


### 3.2. Complex Interval-Valued Dual Hesitant Fuzzy Sets

In this subsection, we explored the novel of CIvDHFS and discussed its operational laws. Additionally, We verified its operational laws with the help of example.Definition 10.
A CIvDHFSA is of the following form:(9)A=x,μAx,νAx|x∈X,where(10)μAx=μALx,μAUx=x,γAJLx,γAJUx.ei2πωγAJLx,ωγAJUx,J=1,2,3,…,g,νAx=νALx,νAUx=x,δAJLx,δAJUx.ei2πϖδAJLx,ϖδAJUx,J=1,2,3,…,h,represented the complex-valued membership grade and nonmembership grade, which are finite subsets of different interval-values of a unit disc in complex plane with a condition γAJLx,γAJUx,ωγAJLx,ωγAJUx,δAJLx,δAJUx,ϖδAJLx,ϖδAJUx∈0,1, 0≤γAUx++δAUx+≤1, and 0≤ωγAUx++ϖδAUx+≤1, where γAUx+=∪γAJLx,γAJUx∈μAxmaxγAJUx, ωγAUx+=∪ωγAJLx,ωγAJUx∈μAxmaxωγAJUx, δAUxU=∪δAJLx,δAJUx∈νAxmaxδAJUx, and ϖνAUx+=∪ϖδAJLx,ϖδAJUx∈νAxmaxϖνAJUx, for J=1,2,…,g and J=1,2,…,h. Further, A=x,γAJLx,γAJUx⋅ei2πωγAJLx,ωγAJUx,δAJLx,δAJUx⋅ei2πϖδAJLx,ϖδAJUx is called complex interval-valued dual hesitant fuzzy number (CIvDHFN).Definition 11.
Let(11)A=x,γAJLx,γAJUx⋅ei2πωγAJLx,ωγAJUx,δAJLx,δAJUx⋅ei2πϖδAJLx,ϖδAJUx,ℬ=x,γℬJLx,γℬJUx⋅ei2πωγℬJLx,ωγℬJUx,δℬJLx,δℬJUx⋅ei2πϖδℬJLx,ϖδℬJUx,be two CIvDHFNs. Then, their complement, union, and intersection are defined as follows:(1)
Ac=x,νAx,μAx(2)
A∪ℬ=x,maxγAJLx,γℬJLx,maxγAJUx,γℬJUx⋅ei2πmaxωγAJLx,ωγℬJLx,maxωγAJUx,ωγℬJUx,minδAJLx,δℬJLx,minδAJUx,δℬJUx⋅ei2πminϖδAJLx,ϖδℬJLx,minϖδAJUx,ϖδℬJUx(3)
A∩ℬ=x,minγAJLx,γℬJLx,minγAJUx,γℬJUx⋅ei2πminωγAJLx,ωγℬJLx,minωγAJUx,ωγℬJUx,maxδAJLx,δℬJLx,maxδAJUx,δℬJUx⋅ei2πmaxϖδAJLx,ϖδℬJLx,maxϖδAJUx,ϖδℬJUxExample 2.
Let(12)A=x1,0.4,0.8ei2π0.2,0.6,0.2,0.5ei2π0.1,0.4,0.05,0.1ei2π0.1,0.2,x2,0.6,0.7ei2π0.4,0.70.15,0.2ei2π0.1,0.25,x3,0.3,0.5ei2π0.5,0.6,0.4,0.6ei2π0.5,0.55,0.2,0.7ei2π0.1,0.4,0.1,0.2ei2π0.15,0.30.2,0.25ei2π0.1,0.2,x4,0.4,0.6ei2π0.4,0.50.3,0.5ei2π0.2,0.40.1,0.3ei2π0.2,0.30.2,0.25ei2π0.1,0.4,ℬ=x1,0.2,0.6ei2π0.3,0.60.5,0.7ei2π0.3,0.550.3,0.8ei2π0.45,0.6,0.05,0.1ei2π0.1,0.2,0.1,0.15ei2π0.15,0.30.15,0.2ei2π0.2,0.3,x2,0.1,0.2ei2π0.2,0.30.5,0.7ei2π0.3,0.60.3,0.6ei2π0.2,0.5,x3,0.4,0.65ei2π0.35,0.5,0.5,0.6ei2π0.5,0.65,0.2,0.25ei2π0.1,0.15,0.15,0.3ei2π0.2,0.3,x4,0.3,0.55ei2π0.4,0.45,0.2,0.45ei2π0.1,0.25,be two CIvDHFNs. Then,(1)
Ac=x10.05,0.1ei2π0.1,0.2,0.4,0.8ei2π0.2,0.6,0.2,0.5ei2π0.1,0.4,x2,0.15,0.2ei2π0.1,0.25,0.6,0.7ei2π0.4,0.7,x3,0.1,0.2ei2π0.15,0.30.2,0.25ei2π0.1,0.2,0.3,0.5ei2π0.5,0.6,0.4,0.6ei2π0.5,0.55,0.2,0.7ei2π0.1,0.4,x4,0.1,0.3ei2π0.2,0.3,0.2,0.25ei2π0.1,0.4,0.4,0.6ei2π0.4,0.50.3,0.5ei2π0.2,0.4(2)
A∪ℬ=x1,0.4,0.8ei2π0.3,0.6,0.5,0.7ei2π0.3,0.55,0.3,0.8ei2π0.45,0.6,0.05,0.1ei2π0.1,0.2,x2,0.6,0.7ei2π0.4,0.70.15,0.2ei2π0.1,0.25,x30.4,0.65ei2π0.5,0.6,0.5,0.6ei2π0.5,0.650.2,0.7ei2π0.1,0.4,0.1,0.2ei2π0.1,0.15,0.15,0.25ei2π0.1,0.2,x4,0.4,0.6ei2π0.4,0.50.3,0.5ei2π0.2,0.4,0.1,0.3ei2π0.1,0.25(3)
A∩ℬ=x10.2,0.6ei2π0.2,0.60.2,0.5ei2π0.1,0.4,0.05,0.1ei2π0.1,0.2,0.1,0.15ei2π0.15,0.30.15,0.2ei2π0.2,0.3,x2,0.1,0.2ei2π0.2,0.30.5,0.7ei2π0.3,0.60.3,0.6ei2π0.2,0.5,x30.3,0.5ei2π0.35,0.5,0.4,0.6ei2π0.5,0.55,0.2,0.25ei2π0.15,0.13,0.2,0.3ei2π0.2,0.3,x4,0.3,0.55ei2π0.4,0.45,0.2,0.45ei2π0.2,0.3


## 3.1. Complex Dual Hesitant Fuzzy Sets

We defined the notion of CDHFS which is the combination of CFS and DHFS and discussed its operational laws. Additionally, we verified its operational laws with the help of example.Definition 8.
A CDHFSA is of the following form:(6)A=x,μAx,νAx|x∈X,where(7)μAx=x,γAJx.ei2πωγAJx,J=1,2,3,…,g,νAx=x,δAJx.ei2πϖδAJx,J=1,2,3,…,h,represented the complex-valued membership grade and nonmembership grade, which are subsets of a unit disc in complex plane with a condition γAJx,ωγAJx,δAJx,ϖδAJx∈0,1, 0≤γA+x+δAJ+x≤1, and 0≤ωγA+x+ϖδA+x≤1, where γA+x=∪γAJx∈μAxmaxγAJx, ωγA+x=∪ωγAJx∈μAxmaxωγAJx, and δA+x=∪δAJx∈νAxmaxδAJx, ϖνA+x=∪ϖνAJx∈νAxmaxϖνAJx, for J=1,2,…,g and J=1,2,…,h. Further, A=x,γAJx⋅ei2πωγAJx,δAJx⋅ei2πϖδAJx is called complex dual hesitant fuzzy number (CDHFN).Definition 9.
LetA=x,γAJx⋅ei2πωγAJx,δAJx⋅ei2πϖδAJxand ℬ=x,γℬJx⋅ei2πωγℬJx,δℬJx⋅ei2πϖδℬJx be two CDHFNs. Then, their complement, union, and intersection are defined as follows:(1)
Ac=x,νAx,μAx(2)
A∪ℬ=x,maxγAjx,γℬjx⋅ei2πmaxωγAjx,ωγℬjx,minδAjx,δℬjx⋅ei2πminϖδAjx,ϖδℬjx(3)
A∩ℬ=x,minγAJx,γℬJx.ei2πminωγAJx,ωγℬJx,maxδAJx,δℬJx.ei2πmaxϖδAJx,ϖδℬJxExample 1.
Let(8)A=x1,0.8ei2π0.6,0.5ei2π0.4,0.1ei2π0.1,x2,0.6ei2π0.7,0.3ei2π0.2,x3,0.5ei2π0.6,0.4ei2π0.5,0.7ei2π0.4,0.1ei2π0.3,0.2ei2π0.1,x4,0.6ei2π0.50.3ei2π0.4,0.3ei2π0.2,0.2ei2π0.4,ℬ=x1,0.6ei2π0.8,0.5ei2π0.55,0.8ei2π0.45,0.15ei2π0.1,0.1ei2π0.15,0.2ei2π0.1,x2,0.2ei2π0.30.7ei2π0.65,0.3ei2π0.5,x3,0.65ei2π0.35,0.5ei2π0.65,0.25ei2π0.15,0.3ei2π0.3,x4,0.55ei2π0.45,0.45ei2π0.25,be two CDHFSs. Then,(1)
Ac=x1,0.1ei2π0.1,0.8ei2π0.6,0.5ei2π0.4,x2,0.3ei2π0.2,0.6ei2π0.7,x3,0.1ei2π0.3,0.2ei2π0.1,0.5ei2π0.6,0.4ei2π0.5,0.7ei2π0.4,x4,0.3ei2π0.2,0.2ei2π0.4,0.6ei2π0.5,0.3ei2π0.4(2)
A∪ℬ=x1,0.8ei2π0.8,0.5ei2π0.55,0.8ei2π0.45,0.1ei2π0.1,x2,0.6ei2π0.70.3ei2π0.2,x3,0.65ei2π0.6,0.5ei2π0.650.7ei2π0.4,0.1ei2π0.15,0.2ei2π0.1,x4,0.6ei2π0.650.3ei2π0.4,0.3ei2π0.2(3)
A∩ℬ=x1,0.6ei2π0.60.5ei2π0.4,0.15ei2π0.3,0.2ei2π0.150.2ei2π0.1,x2,0.2ei2π0.30.7ei2π0.65,0.3ei2π0.5,x3,0.5ei2π0.35,0.4ei2π0.5,0.25ei2π0.3,0.3ei2π0.3,x4,0.55ei2π0.45,0.45ei2π0.250.2ei2π0.4


## 3.2. Complex Interval-Valued Dual Hesitant Fuzzy Sets

In this subsection, we explored the novel of CIvDHFS and discussed its operational laws. Additionally, We verified its operational laws with the help of example.Definition 10.
A CIvDHFSA is of the following form:(9)A=x,μAx,νAx|x∈X,where(10)μAx=μALx,μAUx=x,γAJLx,γAJUx.ei2πωγAJLx,ωγAJUx,J=1,2,3,…,g,νAx=νALx,νAUx=x,δAJLx,δAJUx.ei2πϖδAJLx,ϖδAJUx,J=1,2,3,…,h,represented the complex-valued membership grade and nonmembership grade, which are finite subsets of different interval-values of a unit disc in complex plane with a condition γAJLx,γAJUx,ωγAJLx,ωγAJUx,δAJLx,δAJUx,ϖδAJLx,ϖδAJUx∈0,1, 0≤γAUx++δAUx+≤1, and 0≤ωγAUx++ϖδAUx+≤1, where γAUx+=∪γAJLx,γAJUx∈μAxmaxγAJUx, ωγAUx+=∪ωγAJLx,ωγAJUx∈μAxmaxωγAJUx, δAUxU=∪δAJLx,δAJUx∈νAxmaxδAJUx, and ϖνAUx+=∪ϖδAJLx,ϖδAJUx∈νAxmaxϖνAJUx, for J=1,2,…,g and J=1,2,…,h. Further, A=x,γAJLx,γAJUx⋅ei2πωγAJLx,ωγAJUx,δAJLx,δAJUx⋅ei2πϖδAJLx,ϖδAJUx is called complex interval-valued dual hesitant fuzzy number (CIvDHFN).Definition 11.
Let(11)A=x,γAJLx,γAJUx⋅ei2πωγAJLx,ωγAJUx,δAJLx,δAJUx⋅ei2πϖδAJLx,ϖδAJUx,ℬ=x,γℬJLx,γℬJUx⋅ei2πωγℬJLx,ωγℬJUx,δℬJLx,δℬJUx⋅ei2πϖδℬJLx,ϖδℬJUx,be two CIvDHFNs. Then, their complement, union, and intersection are defined as follows:(1)
Ac=x,νAx,μAx(2)
A∪ℬ=x,maxγAJLx,γℬJLx,maxγAJUx,γℬJUx⋅ei2πmaxωγAJLx,ωγℬJLx,maxωγAJUx,ωγℬJUx,minδAJLx,δℬJLx,minδAJUx,δℬJUx⋅ei2πminϖδAJLx,ϖδℬJLx,minϖδAJUx,ϖδℬJUx(3)
A∩ℬ=x,minγAJLx,γℬJLx,minγAJUx,γℬJUx⋅ei2πminωγAJLx,ωγℬJLx,minωγAJUx,ωγℬJUx,maxδAJLx,δℬJLx,maxδAJUx,δℬJUx⋅ei2πmaxϖδAJLx,ϖδℬJLx,maxϖδAJUx,ϖδℬJUxExample 2.
Let(12)A=x1,0.4,0.8ei2π0.2,0.6,0.2,0.5ei2π0.1,0.4,0.05,0.1ei2π0.1,0.2,x2,0.6,0.7ei2π0.4,0.70.15,0.2ei2π0.1,0.25,x3,0.3,0.5ei2π0.5,0.6,0.4,0.6ei2π0.5,0.55,0.2,0.7ei2π0.1,0.4,0.1,0.2ei2π0.15,0.30.2,0.25ei2π0.1,0.2,x4,0.4,0.6ei2π0.4,0.50.3,0.5ei2π0.2,0.40.1,0.3ei2π0.2,0.30.2,0.25ei2π0.1,0.4,ℬ=x1,0.2,0.6ei2π0.3,0.60.5,0.7ei2π0.3,0.550.3,0.8ei2π0.45,0.6,0.05,0.1ei2π0.1,0.2,0.1,0.15ei2π0.15,0.30.15,0.2ei2π0.2,0.3,x2,0.1,0.2ei2π0.2,0.30.5,0.7ei2π0.3,0.60.3,0.6ei2π0.2,0.5,x3,0.4,0.65ei2π0.35,0.5,0.5,0.6ei2π0.5,0.65,0.2,0.25ei2π0.1,0.15,0.15,0.3ei2π0.2,0.3,x4,0.3,0.55ei2π0.4,0.45,0.2,0.45ei2π0.1,0.25,be two CIvDHFNs. Then,(1)
Ac=x10.05,0.1ei2π0.1,0.2,0.4,0.8ei2π0.2,0.6,0.2,0.5ei2π0.1,0.4,x2,0.15,0.2ei2π0.1,0.25,0.6,0.7ei2π0.4,0.7,x3,0.1,0.2ei2π0.15,0.30.2,0.25ei2π0.1,0.2,0.3,0.5ei2π0.5,0.6,0.4,0.6ei2π0.5,0.55,0.2,0.7ei2π0.1,0.4,x4,0.1,0.3ei2π0.2,0.3,0.2,0.25ei2π0.1,0.4,0.4,0.6ei2π0.4,0.50.3,0.5ei2π0.2,0.4(2)
A∪ℬ=x1,0.4,0.8ei2π0.3,0.6,0.5,0.7ei2π0.3,0.55,0.3,0.8ei2π0.45,0.6,0.05,0.1ei2π0.1,0.2,x2,0.6,0.7ei2π0.4,0.70.15,0.2ei2π0.1,0.25,x30.4,0.65ei2π0.5,0.6,0.5,0.6ei2π0.5,0.650.2,0.7ei2π0.1,0.4,0.1,0.2ei2π0.1,0.15,0.15,0.25ei2π0.1,0.2,x4,0.4,0.6ei2π0.4,0.50.3,0.5ei2π0.2,0.4,0.1,0.3ei2π0.1,0.25(3)
A∩ℬ=x10.2,0.6ei2π0.2,0.60.2,0.5ei2π0.1,0.4,0.05,0.1ei2π0.1,0.2,0.1,0.15ei2π0.15,0.30.15,0.2ei2π0.2,0.3,x2,0.1,0.2ei2π0.2,0.30.5,0.7ei2π0.3,0.60.3,0.6ei2π0.2,0.5,x30.3,0.5ei2π0.35,0.5,0.4,0.6ei2π0.5,0.55,0.2,0.25ei2π0.15,0.13,0.2,0.3ei2π0.2,0.3,x4,0.3,0.55ei2π0.4,0.45,0.2,0.45ei2π0.2,0.3


## 4. The Antilogarithmic and Nonexponential-Based Generalized Distance and Similarity Measures for CDHFS

In this section, we have two subsections in which we interpreted some antilogarithmic and nonexponential-based generalized distance and SMs for CDHFSs.Definition 12.
LetA and ℬ be two CDHFSs on set X. Then, similarity measure (SM) between A and ℬis indicated by SCDA,ℬ, which holds the following axioms:(1)
0≤SCDA,ℬ≤1(2)
SCDA,ℬ=1if and only if A=ℬ(3)
SCDA,ℬ=SCDℬ,ADefinition 13.
LetA and ℬ be two CDHFSs on set X. Then, distance measure between A and ℬis indicated by dCDA,ℬ, which holds the following axioms:(1)
0≤dCDA,ℬ≤1(2)
dCDA,ℬ=0if and only if A=ℬ(3)
dCDA,ℬ=dCDℬ,A
By investigating Definitions12 and 13, we can observe that the higher the similarity, the littler the distance between the two CDHFSs will be. It is noted that dCDA,ℬ=1−SCDA,ℬ. In like manner, we predominantly discuss the SMs for CDHFSs in this paper, and the relating distance measures can be acquired without any problem.


### 4.1. The Antilogarithmic-Based Generalized SMs for CDHFS

In this subsection, we defined some antilogarithmic-based generalized SMs for CDHFS.Definition 14.
LetA and ℬ be two CDHFS on set X. Then, the antilogarithmic-based generalized SM between A and ℬ is given as(13)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ,where λ>0 and ∨ represents the maximum operation.Theorem 1.
LetA and B be two CDHES on set X. Then, the SM SCD1A,ℬ hold the following axioms:(1)
0≤SCD1A,ℬ≤1(2)
SCD1A,ℬ=1if and only if A=ℬ(3)
SCD1A,ℬ=SCD1ℬ,AProof.
(1)
Since1/g∑J=1gγAJxk−γℬJxkλ∈0,1,1/g∑J=1gωγAJxk−ωγℬJxkλ∈0,1,1/h∑J=1hδAJxk−δℬJxkλ∈0,1, and 1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1, then 1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1 This implies that, for k=1, we obtain(14)21−1/g∑J=1gγAJx1−γℬJx1λ∨1/g∑J=1gωγAJx1−ωγℬJx1λ∨1/h∑J=1hδAJx1−δℬJx1λ∨1/h∑J=1hϖδAJx1−ϖδℬJx1λ−1∈0,1.Fork=2,(15)21−1/g∑J=1gγAJx2−γℬJx2λ∨1/g∑J=1gωγAJx2−ωγℬJx2λ∨1/h∑J=1hδAJx2−δℬJx2λ∨1/h∑J=1hϖδAJx2−ϖδℬJx2λ−1∈0,1.By continuing this process, we get(16)∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1∈n0,1⟹0≤∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤n⟹0≤1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤1⟹0≤1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ≤1⟹0≤SCD1A,ℬ≤1.(2)
By Definition14,(17)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ⟺SCD2A,ℬ1n∑k=1n21−1/41/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ+1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ+1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxkλ+1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ−11/λ⟺SCD1A,ℬ1n21−1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λ+1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ+1/hδA1x1−δℬ1λ+δA2x1−δℬ2λ+⋯+δAhx1−δℬhx1λ+1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ−1+211/41/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ+1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ+1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ+1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ−1+⋯+211/41/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ+1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ+1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ+1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ−11/λ.Now, asA=ℬ⟺μAxk=μℬxk and νAxk=νℬxk for k=1,2,…,n⟺γAJxkei2πωγAJxk=γℬJxkei2πωγℬJxkandδAJxkei2πϖδAJxk=δℬJxkei2πϖδℬJxkfork=1,2,…,n⟺γAJxk=γℬJxk, δAJxk=δℬJxk, ei2πωγAJxk=ei2πωγℬJxkandei2πϖδAJxk=ei2πϖδℬJxkfork=1,2,…,n, then(18)⟺SCD1A,ℬ=1n21−0−1+21−0−1+…+21−0−11/λ⟺SCD1A,ℬ=1.(3)
We have(19)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ=1n∑k=1n21−1/g∑J=1gγℬJxk−γAJxkλ∨1/g∑J=1gωγℬJxk−ωγAJxkλ∨1/h∑J=1hδℬJxk−δAJxkλ∨1/h∑J=1hϖδℬJxk−ϖδAJxkλ−11/λ=SCD1ℬ,A.
In the following definition, we defined another type of antilogarithmic-based generalized SM.Definition 15.
LetA and ℬ be two CDHFS on set X. Then, the antilogarithmic-based generalized SM between A and ℬ is given as(20)SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ.Theorem 2.
LetA and B be two CDHES on set X. Then, the SM SCD2A,ℬ hold the following axioms:(1)
0≤SCD2A,ℬ≤1(2)
SCD2A,ℬ=1if and only if A=ℬ(3)
SCD2A,ℬ=SCD2ℬ,AProof.
(1)
Since1/g∑J=1gγAJxk−γℬJxkλ,1/g∑J=1gωγAJxk−ωγℬJxkλ,1/h∑J=1hδAJxk−δℬJxkλ∈0,1, and 1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1, then 1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1; this implies that for k=1, we obtain(21)21−1/41/g∑J=1gγAJx1−γℬJx1λ+1/g∑J=1gωγAJx1−ωγℬJx1λ+1/h∑J=1hδAJx1−δℬJx1λ+1/h∑J=1hϖδAJx1−ϖδℬJx1λ−1∈0,1.Fork=2,(22)21−1/41/g∑J=1gγAJx2−γℬJx2λ+1/g∑J=1gωγAJx2−ωγℬJx2λ+1/h∑J=1hδAJx2−δℬJx2λ+1/h∑J=1hϖδAJx2−ϖδℬJx2λ−1∈0,1.By continuing this process, we get(23)∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1∈n0,10≤∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤n0≤1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤10≤1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ≤10≤SCD2A,ℬ≤1.(2)
By Definition15,(24)SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ⟺SCD2A,ℬ1n∑k=1n21−1/41/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ+1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ+1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxhλ+1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ−11/λ⟺SCD2A,ℬ1n21−1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λ+1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ+1/hδA1x1−δℬ1λ+δA2x1−δℬ2λ+⋯+δAhx1−δℬhx1λ+1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ−1+211/41/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ+1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ+1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ+1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ−1+…+211/41/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ+1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ+1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ+1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ−11/λ.Now, asA=ℬ⟺μAxk=μℬxk and νAxk=νℬxk for k=1,2,…,n⟺γAJxkei2πωγAJxk=γℬJxkei2πωγℬJxkandδAJxkei2πϖδAJxk=δℬJxkei2πϖδℬJxkfork=1,2,…,n⟺γAJxk=γℬJxk, δAJxk=δℬJxk, ei2πωγAJxk=ei2πωγℬJxkandei2πϖδAJxk=ei2πϖδℬJxkfork=1,2,…,n, then(25)⟺SCD2A,ℬ=1n21−0−1+21−0−1+…+21−0−11/λ⟺SCD2A,ℬ=1.(3)
We have(26)SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ=1n∑k=1n21−1/41/g∑J=1gγℬJxk−γAJxkλ+1/g∑J=1gωγℬJxk−ωγAJxkλ+1/h∑J=1hδℬJxk−δAJxkλ+1/h∑J=1hϖδℬJxk−ϖδAJxkλ−11/λ=SCD2ℬ,A.Remark 1.
Ifλ=1, then the antilogarithmic-based generalized SMs between A and ℬ become(27)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1,SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1.Definition 16.
By using the CDHFs and their weight vector, the anti-logarithmic-based weighted generalized SMs between two CDHFSsA and ℬ are given as(28)SCDw1A,ℬ=∑k=1nwk21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ,SCDw2A,ℬ=∑k=1nwk21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ,where w=w1,w2,…,wnT represents the weight vector of each element xkk=1,2,3,…,n contained in CDHFS and the weight vector satisfies wk∈0,1 for each k=1,2,3,…,n, ∑k=1nwk=1. When we assume the weight vector to be w=1/n,1/n,…,1/nT, then the antilogarithmic-based weighted generalized SMs will transform into antilogarithmic-based generalized SMs. Otherwise speaking, when wk=1/n,k=1,2,3,…,n, the SCDwqA,ℬ=SCDqA,ℬq=1,2.Remark 2.
Ifλ=1, then the antilogarithmic-based weighted generalized SM between A and ℬ becomes(29)SCDw1A,ℬ=∑k=1nwk21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1,SCDw2A,ℬ=∑k=1nwk21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1.


### 4.2. The Nonexponential-Based Generalized SMs for CDHFS

In this subsection, we defined some nonexponential-based generalized SMs for CDHFS.Definition 17.
LetA and ℬ be two CDHFS on set X. Then, we defined following three nonexponential-based generalized SMs between A and ℬ as(30)SCD3A,ℬ=1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ,SCD4A,ℬ=1n∑k=1n1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/λ,SCD5A,ℬ=1n∑k=1n12g∑J=1gηccγAJxk∧γℬJxkλγAJxk∨γℬJxkλ+ωγAJxk∧ωγℬJxkλ1+ωγAJxk∨ωγℬJxkλ+12h∑J=1hϕccδAJxk∧δℬJxkλδAJxk∨δℬJxkλ+ϖδAJxk∧ϖδℬJxkλ1+ϖδAJxk∨ϖδℬJxkλ1/λ.Where λ>0andηcc,ϕcc∈0,1 such that ηcc+ϕcc=1.Theorem 3.
LetA and B be two CDHES on set X. Then, the SM SCD3A,ℬ satisfies the following properties:(1)
0≤SCD3A,ℬ≤1(2)
SCD3A,ℬ=1if and only if A=ℬ(3)
SCD3A,ℬ=SCD3ℬ,AProof.
(1)
Since1/g∑J=1gγAJxk−γℬJxkλ∈0,1,1/g∑J=1gωγAJxk−ωγℬJxkλ∈0,1,1/h∑J=1hδAJxk−δℬJxkλ∈0,1, and 1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1, then 1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1 and the denominator will always remain greater than the numerator, then for k=1, we obtain(31)1−1/g∑J=1gγAJx1−γℬJx1λ∨1/g∑J=1gωγAJx1−ωγℬJx1λ∨1/h∑J=1hδAJx1−δℬJx1λ∨1/h∑J=1hϖδAJx1−δδℬJx1λ1+1/g∑J=1gγAJx1−γℬJx1λ∨1/g∑J=1gωγAJx1−ϖγℬJx1λ∨1/h∑J=1hδAJx1−δℬJx1λ∨1/h∑J=1hϖδAJx1−ϖδℬJx1λ∈0,1.Fork  = 2,(32)1−1/g∑J=1gγAJx2−γℬJx2λ∨1/g∑J=1gωγAJx2−ωγℬJx2λ∨1/h∑J=1hδAJx2−δℬJx2λ∨1/h∑J=1hϖδAJx2−δδℬJx2λ1+1/g∑J=1gγAJx2−γℬJx2λ∨1/g∑J=1gωγAJx2−ϖγℬJx2λ∨1/h∑J=1hδAJx2−δℬJx2λ∨1/h∑J=1hϖδAJx2−ϖδℬJx2λ∈0,1.By continuing this process, we get(33)∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ∈n0,1⟹0≤∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ≤n⟹0≤1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ≤1⟹0≤1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1/λ≤1⇒SCD3A,ℬ.(2)
By Definition17, we have(34)SCD3A,ℬ1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ⟺SCD3A,ℬ1n∑k=1n1−1/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ∨1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ∨1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxkλV1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ1+1/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ∨1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ∨1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxkλV1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ1/λ⟺SCD3A,ℬ=1n1−1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λV1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ∨1/hδA1x1−δℬ1x1λ+δA2x1−δℬ2x1λ+⋯+δAhx1−δℬhx1λV1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ1+1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λV1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ∨1/hδA1x1−δℬ1x1λ+δA2x1−δℬ2x1λ+⋯+δAhx1−δℬhx1λV1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ+1−1/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ∨1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ∨1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ∨1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ1+1/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ∨1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ∨1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ∨1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ+1−1/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ∨1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ∨1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ∨1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ1+1/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ∨1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ∨1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ∨1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ1/λ.Now, asA=ℬ⟺μAxk=μℬxk and νAxk=νℬxk for k=1,2,…,n⟺γAJxkei2πωγAJxk=γℬJxkei2πωγℬJxkandδAJxkei2πϖδAJxk=δℬJxkei2πϖδℬJxkfork=1,2,…,n⟺γAJxk=γℬJxk, δAJxk=δℬJxk, ei2πωγAJxk=ei2πωγℬJxkandei2πϖδAJxk=ei2πϖδℬJxkfork=1,2,…,n, then(35)⟺SCD3A,ℬ=1n1+1+⋯+11/λ⟺SCD3A,ℬ=1.(3)
We have(36)SCD3A,ℬ=1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ=1n∑k=1n1−1/g∑J=1gγℬJxk−γAJxkλ∨1/g∑J=1gωγℬJxk−ωγAJxkλ∨1/h∑J=1hδℬJxk−δAJxkλ∨1/h∑J=1hϖδℬJxk−δδAJxkλ1+1/g∑J=1gγℬJxk−γAJxkλ∨1/g∑J=1gωγℬJxk−ωγAJxkλ∨1/h∑J=1hδℬJxk−δAJxkλ∨1/h∑J=1hϖδℬJxk−δδAJxkλ1/λ=SCD3ℬ,A.Theorem 4.
LetA and B be two CDHES on set X. Then, the SM SCD4A,ℬ hold the following axioms:(1)
0≤SCD4A,ℬ≤1(2)
SCD4A,ℬ=1if and only if A=ℬ(3)
SCD4A,ℬ=SCD4ℬ,AProof.
Straight forward (similar to the proof of Theorem3.),Theorem 5.
LetA and B be two CDHES on set X. Then, the SM SCD5A,ℬ hold the following axioms:(1)
0≤SCD5A,ℬ≤1(2)
SCD5A,ℬ=1if and only if A=ℬ(3)
SCD5A,ℬ=SCD5ℬ,AProof.
Straight forward (similar to the proof of Theorem3.),Remark 3.
Ifλ=1, then nonexponential-based generalized SMs between A and ℬ becomes(37)SCD3A,ℬ=1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ,SCD4A,ℬ=1n∑k=1n1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∨γℬJxkλ+1/g∑J=1gωγAJxk∨ωγℬJxkλ∨1/h∑J=1hδAJxk∨δℬJxkλ+1/h∑J=1hϖδAJxk∨ϖδℬJxkλ1/λ,SCD5A,ℬ=1n∑k=1n12gηccγAJxk∧γℬJxkγAJxk∨γℬJxk+ωγAJxk∧ωγℬJxk1+ωγAJxk∨ωγℬJxk+12hϕccδAJxk∧δℬJxkδAJxk∨δℬJxk+ϖγAJxk∧ϖγℬJxk1+ϖγAJxk∨ϖγℬJxk1/λ,where ηcc,ϕcc∈0,1 such that ηcc+ϕcc=1.Definition 18.
Let the weighting vector of the elements in CDHFS. Then, the nonexponential-based weighted generalized SMs between two CDHFSs A and B are given as(38)SCDw3A,ℬ=∑k=1nwk1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ,SCDw4A,ℬ=∑k=1nwk1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∨γℬJxkλ+1/g∑J=1gωγAJxk∨ωγℬJxkλ∨1/h∑J=1hδAJxk∨δℬJxkλ+1/h∑J=1hϖδAJxk∨ϖδℬJxkλ1/λ,SCDw5A,ℬ=∑k=1n12gηccγAJxk∧γℬJxkγAJxk∨γℬJxk+ωγAJxk∧ωγℬJxk1+ωγAJxk∨ωγℬJxk+12hϕccδAJxk∧δℬJxkδAJxk∨δℬJxk+ϖγAJxk∧ϖγℬJxk1+ϖγAJxk∨ϖγℬJxk1/λ,where w=w1,w2,…,wnT represents the weight vector of each element xkk=1,2,3,…,n contained in CDHFS and the weight vector satisfies wk∈0,1 for each k=1,2,3,…,n, ∑k=1nwk=1. If we choose the weight vector in the form of w=1/n,1/n,…,1/nT, then the nonexponential-based weighted generalized SMs will transform into nonexponential-based generalized SMs. Otherwise speaking, when wk=1/n,k=1,2,3,…,n, the SCDwqA,ℬ=SCDqA,ℬq=3,4,5.Remark 4.
Ifλ=1, then nonexponential-based weighted generalized SMs between A and ℬ becomes(39)SCDw3A,ℬ=∑k=1nwk1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ,SCDw4A,ℬ=∑k=1nwk1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∨γℬJxkλ+1/g∑J=1gωγAJxk∨ωγℬJxkλ∨1/h∑J=1hδAJxk∨δℬJxkλ+1/h∑J=1hϖδAJxk∨ϖδℬJxkλ,SCDw5A,ℬ=∑k=1nwk12g∑J=1gηccγAJxk∧γℬJxkγAJxk∨γℬJxk+ωγAJxk∧ωγℬJxk1+ωγAJxk∨ωγℬJxk+12h∑J=1hϕccδAJxk∧δℬJxkδAJxk∨δℬJxk+ϖδAJxk∧ϖδℬJxk1+ϖδAJxk∨ϖδℬJxk.


## 4.1. The Antilogarithmic-Based Generalized SMs for CDHFS

In this subsection, we defined some antilogarithmic-based generalized SMs for CDHFS.Definition 14.
LetA and ℬ be two CDHFS on set X. Then, the antilogarithmic-based generalized SM between A and ℬ is given as(13)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ,where λ>0 and ∨ represents the maximum operation.Theorem 1.
LetA and B be two CDHES on set X. Then, the SM SCD1A,ℬ hold the following axioms:(1)
0≤SCD1A,ℬ≤1(2)
SCD1A,ℬ=1if and only if A=ℬ(3)
SCD1A,ℬ=SCD1ℬ,AProof.
(1)
Since1/g∑J=1gγAJxk−γℬJxkλ∈0,1,1/g∑J=1gωγAJxk−ωγℬJxkλ∈0,1,1/h∑J=1hδAJxk−δℬJxkλ∈0,1, and 1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1, then 1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1 This implies that, for k=1, we obtain(14)21−1/g∑J=1gγAJx1−γℬJx1λ∨1/g∑J=1gωγAJx1−ωγℬJx1λ∨1/h∑J=1hδAJx1−δℬJx1λ∨1/h∑J=1hϖδAJx1−ϖδℬJx1λ−1∈0,1.Fork=2,(15)21−1/g∑J=1gγAJx2−γℬJx2λ∨1/g∑J=1gωγAJx2−ωγℬJx2λ∨1/h∑J=1hδAJx2−δℬJx2λ∨1/h∑J=1hϖδAJx2−ϖδℬJx2λ−1∈0,1.By continuing this process, we get(16)∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1∈n0,1⟹0≤∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤n⟹0≤1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤1⟹0≤1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ≤1⟹0≤SCD1A,ℬ≤1.(2)
By Definition14,(17)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ⟺SCD2A,ℬ1n∑k=1n21−1/41/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ+1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ+1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxkλ+1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ−11/λ⟺SCD1A,ℬ1n21−1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λ+1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ+1/hδA1x1−δℬ1λ+δA2x1−δℬ2λ+⋯+δAhx1−δℬhx1λ+1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ−1+211/41/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ+1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ+1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ+1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ−1+⋯+211/41/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ+1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ+1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ+1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ−11/λ.Now, asA=ℬ⟺μAxk=μℬxk and νAxk=νℬxk for k=1,2,…,n⟺γAJxkei2πωγAJxk=γℬJxkei2πωγℬJxkandδAJxkei2πϖδAJxk=δℬJxkei2πϖδℬJxkfork=1,2,…,n⟺γAJxk=γℬJxk, δAJxk=δℬJxk, ei2πωγAJxk=ei2πωγℬJxkandei2πϖδAJxk=ei2πϖδℬJxkfork=1,2,…,n, then(18)⟺SCD1A,ℬ=1n21−0−1+21−0−1+…+21−0−11/λ⟺SCD1A,ℬ=1.(3)
We have(19)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ=1n∑k=1n21−1/g∑J=1gγℬJxk−γAJxkλ∨1/g∑J=1gωγℬJxk−ωγAJxkλ∨1/h∑J=1hδℬJxk−δAJxkλ∨1/h∑J=1hϖδℬJxk−ϖδAJxkλ−11/λ=SCD1ℬ,A.
In the following definition, we defined another type of antilogarithmic-based generalized SM.Definition 15.
LetA and ℬ be two CDHFS on set X. Then, the antilogarithmic-based generalized SM between A and ℬ is given as(20)SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ.Theorem 2.
LetA and B be two CDHES on set X. Then, the SM SCD2A,ℬ hold the following axioms:(1)
0≤SCD2A,ℬ≤1(2)
SCD2A,ℬ=1if and only if A=ℬ(3)
SCD2A,ℬ=SCD2ℬ,AProof.
(1)
Since1/g∑J=1gγAJxk−γℬJxkλ,1/g∑J=1gωγAJxk−ωγℬJxkλ,1/h∑J=1hδAJxk−δℬJxkλ∈0,1, and 1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1, then 1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1; this implies that for k=1, we obtain(21)21−1/41/g∑J=1gγAJx1−γℬJx1λ+1/g∑J=1gωγAJx1−ωγℬJx1λ+1/h∑J=1hδAJx1−δℬJx1λ+1/h∑J=1hϖδAJx1−ϖδℬJx1λ−1∈0,1.Fork=2,(22)21−1/41/g∑J=1gγAJx2−γℬJx2λ+1/g∑J=1gωγAJx2−ωγℬJx2λ+1/h∑J=1hδAJx2−δℬJx2λ+1/h∑J=1hϖδAJx2−ϖδℬJx2λ−1∈0,1.By continuing this process, we get(23)∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1∈n0,10≤∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤n0≤1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1≤10≤1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ≤10≤SCD2A,ℬ≤1.(2)
By Definition15,(24)SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ⟺SCD2A,ℬ1n∑k=1n21−1/41/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ+1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ+1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxhλ+1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ−11/λ⟺SCD2A,ℬ1n21−1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λ+1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ+1/hδA1x1−δℬ1λ+δA2x1−δℬ2λ+⋯+δAhx1−δℬhx1λ+1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ−1+211/41/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ+1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ+1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ+1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ−1+…+211/41/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ+1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ+1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ+1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ−11/λ.Now, asA=ℬ⟺μAxk=μℬxk and νAxk=νℬxk for k=1,2,…,n⟺γAJxkei2πωγAJxk=γℬJxkei2πωγℬJxkandδAJxkei2πϖδAJxk=δℬJxkei2πϖδℬJxkfork=1,2,…,n⟺γAJxk=γℬJxk, δAJxk=δℬJxk, ei2πωγAJxk=ei2πωγℬJxkandei2πϖδAJxk=ei2πϖδℬJxkfork=1,2,…,n, then(25)⟺SCD2A,ℬ=1n21−0−1+21−0−1+…+21−0−11/λ⟺SCD2A,ℬ=1.(3)
We have(26)SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ=1n∑k=1n21−1/41/g∑J=1gγℬJxk−γAJxkλ+1/g∑J=1gωγℬJxk−ωγAJxkλ+1/h∑J=1hδℬJxk−δAJxkλ+1/h∑J=1hϖδℬJxk−ϖδAJxkλ−11/λ=SCD2ℬ,A.Remark 1.
Ifλ=1, then the antilogarithmic-based generalized SMs between A and ℬ become(27)SCD1A,ℬ=1n∑k=1n21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1,SCD2A,ℬ=1n∑k=1n21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1.Definition 16.
By using the CDHFs and their weight vector, the anti-logarithmic-based weighted generalized SMs between two CDHFSsA and ℬ are given as(28)SCDw1A,ℬ=∑k=1nwk21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ,SCDw2A,ℬ=∑k=1nwk21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−11/λ,where w=w1,w2,…,wnT represents the weight vector of each element xkk=1,2,3,…,n contained in CDHFS and the weight vector satisfies wk∈0,1 for each k=1,2,3,…,n, ∑k=1nwk=1. When we assume the weight vector to be w=1/n,1/n,…,1/nT, then the antilogarithmic-based weighted generalized SMs will transform into antilogarithmic-based generalized SMs. Otherwise speaking, when wk=1/n,k=1,2,3,…,n, the SCDwqA,ℬ=SCDqA,ℬq=1,2.Remark 2.
Ifλ=1, then the antilogarithmic-based weighted generalized SM between A and ℬ becomes(29)SCDw1A,ℬ=∑k=1nwk21−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1,SCDw2A,ℬ=∑k=1nwk21−1/41/g∑J=1gγAJxk−γℬJxkλ+1/g∑J=1gωγAJxk−ωγℬJxkλ+1/h∑J=1hδAJxk−δℬJxkλ+1/h∑J=1hϖδAJxk−ϖδℬJxkλ−1.


## 4.2. The Nonexponential-Based Generalized SMs for CDHFS

In this subsection, we defined some nonexponential-based generalized SMs for CDHFS.Definition 17.
LetA and ℬ be two CDHFS on set X. Then, we defined following three nonexponential-based generalized SMs between A and ℬ as(30)SCD3A,ℬ=1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ,SCD4A,ℬ=1n∑k=1n1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/λ,SCD5A,ℬ=1n∑k=1n12g∑J=1gηccγAJxk∧γℬJxkλγAJxk∨γℬJxkλ+ωγAJxk∧ωγℬJxkλ1+ωγAJxk∨ωγℬJxkλ+12h∑J=1hϕccδAJxk∧δℬJxkλδAJxk∨δℬJxkλ+ϖδAJxk∧ϖδℬJxkλ1+ϖδAJxk∨ϖδℬJxkλ1/λ.Where λ>0andηcc,ϕcc∈0,1 such that ηcc+ϕcc=1.Theorem 3.
LetA and B be two CDHES on set X. Then, the SM SCD3A,ℬ satisfies the following properties:(1)
0≤SCD3A,ℬ≤1(2)
SCD3A,ℬ=1if and only if A=ℬ(3)
SCD3A,ℬ=SCD3ℬ,AProof.
(1)
Since1/g∑J=1gγAJxk−γℬJxkλ∈0,1,1/g∑J=1gωγAJxk−ωγℬJxkλ∈0,1,1/h∑J=1hδAJxk−δℬJxkλ∈0,1, and 1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1, then 1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ∈0,1 and the denominator will always remain greater than the numerator, then for k=1, we obtain(31)1−1/g∑J=1gγAJx1−γℬJx1λ∨1/g∑J=1gωγAJx1−ωγℬJx1λ∨1/h∑J=1hδAJx1−δℬJx1λ∨1/h∑J=1hϖδAJx1−δδℬJx1λ1+1/g∑J=1gγAJx1−γℬJx1λ∨1/g∑J=1gωγAJx1−ϖγℬJx1λ∨1/h∑J=1hδAJx1−δℬJx1λ∨1/h∑J=1hϖδAJx1−ϖδℬJx1λ∈0,1.Fork  = 2,(32)1−1/g∑J=1gγAJx2−γℬJx2λ∨1/g∑J=1gωγAJx2−ωγℬJx2λ∨1/h∑J=1hδAJx2−δℬJx2λ∨1/h∑J=1hϖδAJx2−δδℬJx2λ1+1/g∑J=1gγAJx2−γℬJx2λ∨1/g∑J=1gωγAJx2−ϖγℬJx2λ∨1/h∑J=1hδAJx2−δℬJx2λ∨1/h∑J=1hϖδAJx2−ϖδℬJx2λ∈0,1.By continuing this process, we get(33)∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ∈n0,1⟹0≤∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ≤n⟹0≤1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ≤1⟹0≤1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ϖγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−ϖδℬJxkλ1/λ≤1⇒SCD3A,ℬ.(2)
By Definition17, we have(34)SCD3A,ℬ1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ⟺SCD3A,ℬ1n∑k=1n1−1/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ∨1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ∨1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxkλV1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ1+1/gγA1xk−γℬ1xkλ+γA2xk−γℬ2xkλ+⋯+γAgxk−γℬgxkλ∨1/gωγA1xk−ωγℬ1xkλ+ωγA2xk−ωγℬ2xkλ+⋯+ωγAgxk−ωγℬgxkλ∨1/hδA1xk−δℬ1xkλ+δA2xk−δℬ2xkλ+⋯+δAhxk−δℬhxkλV1/hϖδA1xk−ϖδℬ1xkλ+ϖδA2xk−ϖδℬ2xkλ+⋯+ϖδAhxk−ϖδℬhxkλ1/λ⟺SCD3A,ℬ=1n1−1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λV1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ∨1/hδA1x1−δℬ1x1λ+δA2x1−δℬ2x1λ+⋯+δAhx1−δℬhx1λV1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ1+1/gγA1x1−γℬ1x1λ+γA2x1−γℬ2x1λ+⋯+γAgx1−γℬgx1λV1/gωγA1x1−ωγℬ1x1λ+ωγA2x1−ωγℬ2x1λ+⋯+ωγAgx1−ωγℬgx1λ∨1/hδA1x1−δℬ1x1λ+δA2x1−δℬ2x1λ+⋯+δAhx1−δℬhx1λV1/hϖδA1x1−ϖδℬ1x1λ+ϖδA2x1−ϖδℬ2x1λ+⋯+ϖδAhx1−ϖδℬhx1λ+1−1/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ∨1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ∨1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ∨1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ1+1/gγA1x2−γℬ1x2λ+γA2x2−γℬ2x2λ+⋯+γAgx2−γℬgx2λ∨1/gωγA1x2−ωγℬ1x2λ+ωγA2x2−ωγℬ2x2λ+⋯+ωγAgx2−ωγℬgx2λ∨1/hδA1x2−δℬ1x2λ+δA2x2−δℬ2x2λ+⋯+δAhx2−δℬhx2λ∨1/hϖδA1x2−ϖδℬ1x2λ+ϖδA2x2−ϖδℬ2x2λ+⋯+ϖδAhx2−ϖδℬhx2λ+1−1/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ∨1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ∨1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ∨1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ1+1/gγA1xn−γℬ1xnλ+γA2xn−γℬ2xnλ+⋯+γAgxn−γℬgxnλ∨1/gωγA1xn−ωγℬ1xnλ+ωγA2xn−ωγℬ2xnλ+⋯+ωγAgxn−ωγℬgxnλ∨1/hδA1xn−δℬ1xnλ+δA2xn−δℬ2xnλ+⋯+δAhxn−δℬhxnλ∨1/hϖδA1xn−ϖδℬ1xnλ+ϖδA2xn−ϖδℬ2xnλ+⋯+ϖδAhxn−ϖδℬhxnλ1/λ.Now, asA=ℬ⟺μAxk=μℬxk and νAxk=νℬxk for k=1,2,…,n⟺γAJxkei2πωγAJxk=γℬJxkei2πωγℬJxkandδAJxkei2πϖδAJxk=δℬJxkei2πϖδℬJxkfork=1,2,…,n⟺γAJxk=γℬJxk, δAJxk=δℬJxk, ei2πωγAJxk=ei2πωγℬJxkandei2πϖδAJxk=ei2πϖδℬJxkfork=1,2,…,n, then(35)⟺SCD3A,ℬ=1n1+1+⋯+11/λ⟺SCD3A,ℬ=1.(3)
We have(36)SCD3A,ℬ=1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ=1n∑k=1n1−1/g∑J=1gγℬJxk−γAJxkλ∨1/g∑J=1gωγℬJxk−ωγAJxkλ∨1/h∑J=1hδℬJxk−δAJxkλ∨1/h∑J=1hϖδℬJxk−δδAJxkλ1+1/g∑J=1gγℬJxk−γAJxkλ∨1/g∑J=1gωγℬJxk−ωγAJxkλ∨1/h∑J=1hδℬJxk−δAJxkλ∨1/h∑J=1hϖδℬJxk−δδAJxkλ1/λ=SCD3ℬ,A.Theorem 4.
LetA and B be two CDHES on set X. Then, the SM SCD4A,ℬ hold the following axioms:(1)
0≤SCD4A,ℬ≤1(2)
SCD4A,ℬ=1if and only if A=ℬ(3)
SCD4A,ℬ=SCD4ℬ,AProof.
Straight forward (similar to the proof of Theorem3.),Theorem 5.
LetA and B be two CDHES on set X. Then, the SM SCD5A,ℬ hold the following axioms:(1)
0≤SCD5A,ℬ≤1(2)
SCD5A,ℬ=1if and only if A=ℬ(3)
SCD5A,ℬ=SCD5ℬ,AProof.
Straight forward (similar to the proof of Theorem3.),Remark 3.
Ifλ=1, then nonexponential-based generalized SMs between A and ℬ becomes(37)SCD3A,ℬ=1n∑k=1n1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ,SCD4A,ℬ=1n∑k=1n1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∨γℬJxkλ+1/g∑J=1gωγAJxk∨ωγℬJxkλ∨1/h∑J=1hδAJxk∨δℬJxkλ+1/h∑J=1hϖδAJxk∨ϖδℬJxkλ1/λ,SCD5A,ℬ=1n∑k=1n12gηccγAJxk∧γℬJxkγAJxk∨γℬJxk+ωγAJxk∧ωγℬJxk1+ωγAJxk∨ωγℬJxk+12hϕccδAJxk∧δℬJxkδAJxk∨δℬJxk+ϖγAJxk∧ϖγℬJxk1+ϖγAJxk∨ϖγℬJxk1/λ,where ηcc,ϕcc∈0,1 such that ηcc+ϕcc=1.Definition 18.
Let the weighting vector of the elements in CDHFS. Then, the nonexponential-based weighted generalized SMs between two CDHFSs A and B are given as(38)SCDw3A,ℬ=∑k=1nwk1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1/λ,SCDw4A,ℬ=∑k=1nwk1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∨γℬJxkλ+1/g∑J=1gωγAJxk∨ωγℬJxkλ∨1/h∑J=1hδAJxk∨δℬJxkλ+1/h∑J=1hϖδAJxk∨ϖδℬJxkλ1/λ,SCDw5A,ℬ=∑k=1n12gηccγAJxk∧γℬJxkγAJxk∨γℬJxk+ωγAJxk∧ωγℬJxk1+ωγAJxk∨ωγℬJxk+12hϕccδAJxk∧δℬJxkδAJxk∨δℬJxk+ϖγAJxk∧ϖγℬJxk1+ϖγAJxk∨ϖγℬJxk1/λ,where w=w1,w2,…,wnT represents the weight vector of each element xkk=1,2,3,…,n contained in CDHFS and the weight vector satisfies wk∈0,1 for each k=1,2,3,…,n, ∑k=1nwk=1. If we choose the weight vector in the form of w=1/n,1/n,…,1/nT, then the nonexponential-based weighted generalized SMs will transform into nonexponential-based generalized SMs. Otherwise speaking, when wk=1/n,k=1,2,3,…,n, the SCDwqA,ℬ=SCDqA,ℬq=3,4,5.Remark 4.
Ifλ=1, then nonexponential-based weighted generalized SMs between A and ℬ becomes(39)SCDw3A,ℬ=∑k=1nwk1−1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ1+1/g∑J=1gγAJxk−γℬJxkλ∨1/g∑J=1gωγAJxk−ωγℬJxkλ∨1/h∑J=1hδAJxk−δℬJxkλ∨1/h∑J=1hϖδAJxk−δδℬJxkλ,SCDw4A,ℬ=∑k=1nwk1/g∑J=1gγAJxk∧γℬJxkλ+1/g∑J=1gωγAJxk∧ωγℬJxkλ∨1/h∑J=1hδAJxk∧δℬJxkλ+1/h∑J=1hϖδAJxk∧ϖδℬJxkλ1/g∑J=1gγAJxk∨γℬJxkλ+1/g∑J=1gωγAJxk∨ωγℬJxkλ∨1/h∑J=1hδAJxk∨δℬJxkλ+1/h∑J=1hϖδAJxk∨ϖδℬJxkλ,SCDw5A,ℬ=∑k=1nwk12g∑J=1gηccγAJxk∧γℬJxkγAJxk∨γℬJxk+ωγAJxk∧ωγℬJxk1+ωγAJxk∨ωγℬJxk+12h∑J=1hϕccδAJxk∧δℬJxkδAJxk∨δℬJxk+ϖδAJxk∧ϖδℬJxk1+ϖδAJxk∨ϖδℬJxk.


## 5. Applications

In this part of the article, we describe two applications, i.e., pattern recognition and medical diagnosis, to show the effectiveness and usefulness of the interpreted SMS. We applied the interpreted SMs to the environment of CDHFSs in pattern recognition and medical diagnosis.


### 5.1. Pattern Recognition

The instruments of similarity measures have applications in pattern classiﬁcation. By using the known and unknown alternatives, we examine the similarity measures based on the investigated measures to examine the reliability and validity of the explored approaches. During this section, the similarity measures grew thus far in Section4 are applied to a pattern recognition (building pattern recognition) downside wherever the category of an unknown building substance has been assessed. The outcomes acquired utilizing the SMs of CDHFSs are then investigated for the depiction of the upsides of proposed SMs and the confinements of existing work. To clarify the marvel, the following example is discussed.Example 3.
Its clear that quantity of constructions by a company is directly proportional to the standard of building substances they use. Proper inspection of the building substance before construction is proof of good engineering standards. The construction substances to be used should be strictly checked before applying. The proper check and balance system of inspection authorizes the builders to use the right substances for constructions to improve the standard of their project. Let four known building substancesAJJ=1,2,3,4, which are given in the shape of CDHFSs, be as follows:(40)A1=x1,0.1ei2π0.2,0.2ei2π0.3,0.4ei2π0.5,0.4ei2π0.3,0.3ei2π0.2,x2,0.4ei2π0.3,0.5ei2π0.6,0.2ei2π0.3,x3,0.1ei2π0.6,0.4ei2π0.2,x4,0.2ei2π0.05,0.65ei2π0.75,0.15ei2π0.05,A2=x1,0.15ei2π0.25,0.75ei2π0.6,0.4ei2π0.3,x2,0.4ei2π0.35,0.55ei2π0.45,0.3ei2π0.6,0.35ei2π0.25,0.2ei2π0.3,x3,0.4ei2π0.5,0.4ei2π0.35,x4,0.33ei2π0.6,0.45ei2π0.5,0.5ei2π0.2,0.3ei2π0.3,A3=x1,0.7ei2π0.5,0.4ei2π0.8,0.2ei2π0.1,x2,0.5ei2π0.6,0.4ei2π0.3,x3,0.3ei2π0.4,0.5ei2π0.6,0.4ei2π0.2,0.4ei2π0.3,0.3ei2π0.2,x4,0.5ei2π0.3,0.7ei2π0.6,0.1ei2π0.1,0.2ei2π0.3,A4=x1,0.2ei2π0.3,0.45ei2π0.2,0.3ei2π0.4,0.1ei2π0.5,x2,0.7ei2π0.45,0.6ei2π0.6,0.2ei2π0.4,x3,0.4ei2π0.2,0.3ei2π0.7,x4,0.5ei2π0.7,0.3ei2π0.25,0.2ei2π0.15.
Now, let an unknown building substance, which needs to be identified, be(41)A=x1,0.8ei2π0.6,0.5ei2π0.4,0.1ei2π0.1,x2,0.6ei2π0.7,0.3ei2π0.2,x3,0.5ei2π0.6,0.4ei2π0.5,0.7ei2π0.4,0.1ei2π0.3,0.2ei2π0.1,x4,0.6ei2π0.5,0.3ei2π0.4,0.3ei2π0.2,0.2ei2π0.4.
The intention of this matter is to sort the unknown building substanceA in one of the classifications AJJ=1,2,3,4. For this, the interpreted antilogarithmic and nonexponential SMs are utilized to ascertain the similarity from A to AJJ=1,2,3,4 and estimations are provided in Tables 1 and 2.
According to the above-computed estimations supplied in Table1, we essentially mention that the similarity level between A and A3 is the best one as an infusion by all SMs. This defines that all interpreted SMs dole out the unknown building substance A to the known building substance A3, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between A and AJJ=1,2,3,4 is likewise provided in Table 1. The graphical portrayal of the interpreted SMs between A and AJJ=1,2,3,4 is demonstrated in Figure 1.
The heaviness of components has extraordinary noteworthiness to consider genuine dynamic issues. By using the four attributes whose expressions are in the form ofxkk=1,2,3,4 be wk0.3,0.2,0.1,0.4, respectively. At this point, the deciphered WSMs are utilized to decide the similarity from A and AJJ=1,2,3,4 and estimations are provided in Table 2.
According to the above-computed estimations supplied in Table2, we essentially mention that the similarity level between A and A3 is the best one as an infusion by all WSMs. This defines that all interpreted WSMs dole out the unknown building substance A to the known building substance A3, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between A and AJJ=1,2,3,4 is likewise provided in Table 2. The graphical portrayal of the interpreted SMs between A and AJJ=1,2,3,4 is demonstrated in Figure 2.Table 1
Calculation of explored SMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresA1A2A3A4RankingSCD1A,AJ0.47320.39680.74270.5277A3≥A4≥A1≥A2SCD2A,AJ0.52280.6490.82510.6735A3≥A4≥A2≥A1SCD3A,AJ0.38860.42870.67120.4415A3≥A4≥A2≥A1SCD4A,AJ0.31050.37150.69830.4156A3≥A4≥A2≥A1SCD5A,AJ0.12880.12410.3060.185A3≥A4≥A2≥A1Table 2
Calculation of explored WSMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresA1A2A3A4RankingSCDw1A,AJ0.4850.49070.6380.5016A3≥A4≥A2≥A1SCDw2A,AJ0.51750.69790.73610.6346A3≥A2≥A4≥A1SCDw3A,AJ0.39950.42130.56850.4238A3≥A4≥A2≥A1SCDw4A,AJ0.30930.390.61330.4174A3≥A4≥A2≥A1SCDw5A,AJ0.12570.13290.24990.1755A3≥A4≥A2≥A1Figure 1
The graphical portrayal of the interpreted SMs betweenA and AJJ=1,2,3,4.Figure 2
The graphical portrayal of the interpreted WSMs betweenA and AJJ=1,2,3,4.


### 5.2. Medical Diagnosis

Distinct diseases have various signs. The medical diagnosis trust in the casualty’s signs to inspect what sort of disease the casualty has. The casualty’s signs are a set of signs and unknown diseases will be a set of diagnostic diseases. The deciphered SMs are outlined by an after numerical example of medical diagnosis.Example 4.
Let a set of diagnoses beD=D1HepatitisA,D2Heart problem,D3Malaria,D4Flu and a set of signs be X=x1Abdominal pian,x2Heart pain,x3Fever,x4Cough. The casualty’s signs can be given in the shape of CDHFSs as follows:(42)PCausalty=x1,0.6ei2π0.8,0.5ei2π0.55,0.8ei2π0.45,0.15ei2π0.1,0.1ei2π0.15,0.2ei2π0.1,x2,0.2ei2π0.30.7ei2π0.65,0.3ei2π0.5,x3,0.65ei2π0.35,0.5ei2π0.65,0.25ei2π0.15,0.3ei2π0.3,x4,0.55ei2π0.45,0.45ei2π0.25.
The signs of each diseaseDJJ=1,2,3,4 can be represented in CDHFSs as follows:(43)D1HepatitisA=x1,0.55ei2π0.7,0.6ei2π0.65,0.5ei2π0.7,0.1ei2π0.25,0.15ei2π0.2,0.35ei2π0.3,x2,0.4ei2π0.30.5ei2π0.45,0.3ei2π0.6,x3,0.45ei2π0.75,0.6ei2π0.55,0.25ei2π0.2,0.3ei2π0.15,x4,0.4ei2π0.5,0.4ei2π0.3,D2Heart problem=x1,0.1ei2π0.5,0.3ei2π0.5,0.2ei2π0.1,x2,0.7ei2π0.65,0.6ei2π0.6,0.45ei2π0.55,0.2ei2π0.15,0.25ei2π0.1,0.25ei2π0.3x3,0.15ei2π0.05,0.3ei2π0.15,x4,0.05ei2π0.35,0.25ei2π0.25,D3Malaria=x1,0.3ei2π0.45,0.5ei2π0.35,x2,0.15ei2π0.05,0.2ei2π0.55,x3,0.5ei2π0.6,0.6ei2π0.8,0.45ei2π0.7,0.35ei2π0.1,0.2ei2π0.15,x4,0.5ei2π0.4,0.6ei2π0.7,0.1ei2π0.2,0.3ei2π0.25,D4Flu=x1,0.3ei2π0.40.35ei2π0.2,x2,0.4ei2π0.20.5ei2π0.45,0.45ei2π0.6,x3,0.7ei2π0.5,0.6ei2π0.4,0.2ei2π0.4,0.1ei2π0.3,x4,0.4ei2π0.7,0.6ei2π0.65,0.75ei2π0.55,0.2ei2π0.25,0.15ei2π0.1,0.25ei2π0.2.
The intention of this matter is to sort the disease of the casualtyP in one of the diseases, DJJ=1,2,3,4. For this, the interpreted antilogarithmic and nonexponential SMs are utilized to ascertain the similarity from P to DJJ=1,2,3,4, and estimations are provided in Tables 3 and 4.
According to the above-computed estimations supplied in Table2, we essentially mention that the similarity level between P and D1 is the best one as an infusion by all SMs. This defines that all interpreted SMs dole out that the casualty P has hepatitis A, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between P and DJJ=1,2,3,4 is likewise provided in Table 3. The graphical portrayal of the interpreted SMs between P and DJJ=1,2,3,4 is demonstrated in Figure 3.
The heaviness of components has extraordinary noteworthiness to consider genuine dynamic issues. By using the four attributes whose expressions are in the form ofxkk=1,2,3,4 be wk0.3,0.2,0.1,0.4, respectively. At this point, the deciphered WSMs are utilized to decide the similarity from P and DJJ=1,2,3,4 and estimations are provided in Table 4.
According to the above-computed estimations supplied in Table4, we essentially mention that the similarity level between P and D3 is the best one as an infusion by all WSMs. This defines that all interpreted WSMs dole out that the casualty P has hepatitis A, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between P and DJJ=1,2,3,4 is likewise provided in Table 4. The graphical portrayal of the interpreted SMs between P and DJJ=1,2,3,4 is demonstrated in Figure 4.Table 3
Calculation of explored SMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresD1D2D3D4RankingSCD1P,DJ0.7570.2870.49770.5695D1≥D4≥D3≥D2SCD2P,DJ0.85280.60240.64820.7046D1≥D4≥D3≥D2SCD3P,DJ0.68620.30880.41270.4927D1≥D4≥D3≥D2SCD4P,DJ0.75930.27820.3430.3488D1≥D4≥D3≥D2SCD5P,DJ0.3330.1180.10680.1681D1≥D4≥D2≥D3Table 4
Calculation of explored WSMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresD1D2D3D4RankingSCDw1P,DJ0.77810.34270.43810.4283D1≥D3≥D4≥D2SCDw2P,DJ0.86150.57890.56440.581D1≥D4≥D2≥D3SCDw3P,DJ0.71070.27350.36240.3579D1≥D3≥D4≥D2SCDw4P,DJ0.77320.31090.27780.2981D1≥D2≥D4≥D3SCDw5P,DJ0.36020.14050.08550.0963D1≥D2≥D4≥D3Figure 3
The graphical portrayal of the interpreted SMs betweenP and DJJ=1,2,3,4.Figure 4
The graphical portrayal of the interpreted WSMs betweenP and DJJ=1,2,3,4.


## 5.1. Pattern Recognition

The instruments of similarity measures have applications in pattern classiﬁcation. By using the known and unknown alternatives, we examine the similarity measures based on the investigated measures to examine the reliability and validity of the explored approaches. During this section, the similarity measures grew thus far in Section4 are applied to a pattern recognition (building pattern recognition) downside wherever the category of an unknown building substance has been assessed. The outcomes acquired utilizing the SMs of CDHFSs are then investigated for the depiction of the upsides of proposed SMs and the confinements of existing work. To clarify the marvel, the following example is discussed.Example 3.
Its clear that quantity of constructions by a company is directly proportional to the standard of building substances they use. Proper inspection of the building substance before construction is proof of good engineering standards. The construction substances to be used should be strictly checked before applying. The proper check and balance system of inspection authorizes the builders to use the right substances for constructions to improve the standard of their project. Let four known building substancesAJJ=1,2,3,4, which are given in the shape of CDHFSs, be as follows:(40)A1=x1,0.1ei2π0.2,0.2ei2π0.3,0.4ei2π0.5,0.4ei2π0.3,0.3ei2π0.2,x2,0.4ei2π0.3,0.5ei2π0.6,0.2ei2π0.3,x3,0.1ei2π0.6,0.4ei2π0.2,x4,0.2ei2π0.05,0.65ei2π0.75,0.15ei2π0.05,A2=x1,0.15ei2π0.25,0.75ei2π0.6,0.4ei2π0.3,x2,0.4ei2π0.35,0.55ei2π0.45,0.3ei2π0.6,0.35ei2π0.25,0.2ei2π0.3,x3,0.4ei2π0.5,0.4ei2π0.35,x4,0.33ei2π0.6,0.45ei2π0.5,0.5ei2π0.2,0.3ei2π0.3,A3=x1,0.7ei2π0.5,0.4ei2π0.8,0.2ei2π0.1,x2,0.5ei2π0.6,0.4ei2π0.3,x3,0.3ei2π0.4,0.5ei2π0.6,0.4ei2π0.2,0.4ei2π0.3,0.3ei2π0.2,x4,0.5ei2π0.3,0.7ei2π0.6,0.1ei2π0.1,0.2ei2π0.3,A4=x1,0.2ei2π0.3,0.45ei2π0.2,0.3ei2π0.4,0.1ei2π0.5,x2,0.7ei2π0.45,0.6ei2π0.6,0.2ei2π0.4,x3,0.4ei2π0.2,0.3ei2π0.7,x4,0.5ei2π0.7,0.3ei2π0.25,0.2ei2π0.15.
Now, let an unknown building substance, which needs to be identified, be(41)A=x1,0.8ei2π0.6,0.5ei2π0.4,0.1ei2π0.1,x2,0.6ei2π0.7,0.3ei2π0.2,x3,0.5ei2π0.6,0.4ei2π0.5,0.7ei2π0.4,0.1ei2π0.3,0.2ei2π0.1,x4,0.6ei2π0.5,0.3ei2π0.4,0.3ei2π0.2,0.2ei2π0.4.
The intention of this matter is to sort the unknown building substanceA in one of the classifications AJJ=1,2,3,4. For this, the interpreted antilogarithmic and nonexponential SMs are utilized to ascertain the similarity from A to AJJ=1,2,3,4 and estimations are provided in Tables 1 and 2.
According to the above-computed estimations supplied in Table1, we essentially mention that the similarity level between A and A3 is the best one as an infusion by all SMs. This defines that all interpreted SMs dole out the unknown building substance A to the known building substance A3, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between A and AJJ=1,2,3,4 is likewise provided in Table 1. The graphical portrayal of the interpreted SMs between A and AJJ=1,2,3,4 is demonstrated in Figure 1.
The heaviness of components has extraordinary noteworthiness to consider genuine dynamic issues. By using the four attributes whose expressions are in the form ofxkk=1,2,3,4 be wk0.3,0.2,0.1,0.4, respectively. At this point, the deciphered WSMs are utilized to decide the similarity from A and AJJ=1,2,3,4 and estimations are provided in Table 2.
According to the above-computed estimations supplied in Table2, we essentially mention that the similarity level between A and A3 is the best one as an infusion by all WSMs. This defines that all interpreted WSMs dole out the unknown building substance A to the known building substance A3, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between A and AJJ=1,2,3,4 is likewise provided in Table 2. The graphical portrayal of the interpreted SMs between A and AJJ=1,2,3,4 is demonstrated in Figure 2.Table 1
Calculation of explored SMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresA1A2A3A4RankingSCD1A,AJ0.47320.39680.74270.5277A3≥A4≥A1≥A2SCD2A,AJ0.52280.6490.82510.6735A3≥A4≥A2≥A1SCD3A,AJ0.38860.42870.67120.4415A3≥A4≥A2≥A1SCD4A,AJ0.31050.37150.69830.4156A3≥A4≥A2≥A1SCD5A,AJ0.12880.12410.3060.185A3≥A4≥A2≥A1Table 2
Calculation of explored WSMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresA1A2A3A4RankingSCDw1A,AJ0.4850.49070.6380.5016A3≥A4≥A2≥A1SCDw2A,AJ0.51750.69790.73610.6346A3≥A2≥A4≥A1SCDw3A,AJ0.39950.42130.56850.4238A3≥A4≥A2≥A1SCDw4A,AJ0.30930.390.61330.4174A3≥A4≥A2≥A1SCDw5A,AJ0.12570.13290.24990.1755A3≥A4≥A2≥A1Figure 1
The graphical portrayal of the interpreted SMs betweenA and AJJ=1,2,3,4.Figure 2
The graphical portrayal of the interpreted WSMs betweenA and AJJ=1,2,3,4.


## 5.2. Medical Diagnosis

Distinct diseases have various signs. The medical diagnosis trust in the casualty’s signs to inspect what sort of disease the casualty has. The casualty’s signs are a set of signs and unknown diseases will be a set of diagnostic diseases. The deciphered SMs are outlined by an after numerical example of medical diagnosis.Example 4.
Let a set of diagnoses beD=D1HepatitisA,D2Heart problem,D3Malaria,D4Flu and a set of signs be X=x1Abdominal pian,x2Heart pain,x3Fever,x4Cough. The casualty’s signs can be given in the shape of CDHFSs as follows:(42)PCausalty=x1,0.6ei2π0.8,0.5ei2π0.55,0.8ei2π0.45,0.15ei2π0.1,0.1ei2π0.15,0.2ei2π0.1,x2,0.2ei2π0.30.7ei2π0.65,0.3ei2π0.5,x3,0.65ei2π0.35,0.5ei2π0.65,0.25ei2π0.15,0.3ei2π0.3,x4,0.55ei2π0.45,0.45ei2π0.25.
The signs of each diseaseDJJ=1,2,3,4 can be represented in CDHFSs as follows:(43)D1HepatitisA=x1,0.55ei2π0.7,0.6ei2π0.65,0.5ei2π0.7,0.1ei2π0.25,0.15ei2π0.2,0.35ei2π0.3,x2,0.4ei2π0.30.5ei2π0.45,0.3ei2π0.6,x3,0.45ei2π0.75,0.6ei2π0.55,0.25ei2π0.2,0.3ei2π0.15,x4,0.4ei2π0.5,0.4ei2π0.3,D2Heart problem=x1,0.1ei2π0.5,0.3ei2π0.5,0.2ei2π0.1,x2,0.7ei2π0.65,0.6ei2π0.6,0.45ei2π0.55,0.2ei2π0.15,0.25ei2π0.1,0.25ei2π0.3x3,0.15ei2π0.05,0.3ei2π0.15,x4,0.05ei2π0.35,0.25ei2π0.25,D3Malaria=x1,0.3ei2π0.45,0.5ei2π0.35,x2,0.15ei2π0.05,0.2ei2π0.55,x3,0.5ei2π0.6,0.6ei2π0.8,0.45ei2π0.7,0.35ei2π0.1,0.2ei2π0.15,x4,0.5ei2π0.4,0.6ei2π0.7,0.1ei2π0.2,0.3ei2π0.25,D4Flu=x1,0.3ei2π0.40.35ei2π0.2,x2,0.4ei2π0.20.5ei2π0.45,0.45ei2π0.6,x3,0.7ei2π0.5,0.6ei2π0.4,0.2ei2π0.4,0.1ei2π0.3,x4,0.4ei2π0.7,0.6ei2π0.65,0.75ei2π0.55,0.2ei2π0.25,0.15ei2π0.1,0.25ei2π0.2.
The intention of this matter is to sort the disease of the casualtyP in one of the diseases, DJJ=1,2,3,4. For this, the interpreted antilogarithmic and nonexponential SMs are utilized to ascertain the similarity from P to DJJ=1,2,3,4, and estimations are provided in Tables 3 and 4.
According to the above-computed estimations supplied in Table2, we essentially mention that the similarity level between P and D1 is the best one as an infusion by all SMs. This defines that all interpreted SMs dole out that the casualty P has hepatitis A, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between P and DJJ=1,2,3,4 is likewise provided in Table 3. The graphical portrayal of the interpreted SMs between P and DJJ=1,2,3,4 is demonstrated in Figure 3.
The heaviness of components has extraordinary noteworthiness to consider genuine dynamic issues. By using the four attributes whose expressions are in the form ofxkk=1,2,3,4 be wk0.3,0.2,0.1,0.4, respectively. At this point, the deciphered WSMs are utilized to decide the similarity from P and DJJ=1,2,3,4 and estimations are provided in Table 4.
According to the above-computed estimations supplied in Table4, we essentially mention that the similarity level between P and D3 is the best one as an infusion by all WSMs. This defines that all interpreted WSMs dole out that the casualty P has hepatitis A, dependent on the principle of the greatest level of similarity. The positioning of the investigated antilogarithmic and nonexponential SMs between P and DJJ=1,2,3,4 is likewise provided in Table 4. The graphical portrayal of the interpreted SMs between P and DJJ=1,2,3,4 is demonstrated in Figure 4.Table 3
Calculation of explored SMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresD1D2D3D4RankingSCD1P,DJ0.7570.2870.49770.5695D1≥D4≥D3≥D2SCD2P,DJ0.85280.60240.64820.7046D1≥D4≥D3≥D2SCD3P,DJ0.68620.30880.41270.4927D1≥D4≥D3≥D2SCD4P,DJ0.75930.27820.3430.3488D1≥D4≥D3≥D2SCD5P,DJ0.3330.1180.10680.1681D1≥D4≥D2≥D3Table 4
Calculation of explored WSMs based onλ=1 and ηcc,ϕcc=0.5,0.5.
Similarity measuresD1D2D3D4RankingSCDw1P,DJ0.77810.34270.43810.4283D1≥D3≥D4≥D2SCDw2P,DJ0.86150.57890.56440.581D1≥D4≥D2≥D3SCDw3P,DJ0.71070.27350.36240.3579D1≥D3≥D4≥D2SCDw4P,DJ0.77320.31090.27780.2981D1≥D2≥D4≥D3SCDw5P,DJ0.36020.14050.08550.0963D1≥D2≥D4≥D3Figure 3
The graphical portrayal of the interpreted SMs betweenP and DJJ=1,2,3,4.Figure 4
The graphical portrayal of the interpreted WSMs betweenP and DJJ=1,2,3,4.


## 6. Comparison

In this section, we demonstrated the viability and points of interest of the deciphered SMs by contrasting some current SMs.Example 5.
Its clear that quantity of constructions by a company is directly proportional to the standard of building substances they use. Proper inspection of the building substance before construction is proof of good engineering standards. The construction substances to be used should be strictly checked before applying. The proper check and balance system of inspection authorizes the builders to use the right substances for constructions to improve the standard of their project. Let four known building substancesAJJ=1,2,3,4, which are given in the shape of DHFSs, be as follows:(44)A1=x1,0.1,0.2,0.4,0.4,0.3,x2,0.4,0.5,0.2,x3,0.1,0.4,x4,0.2,0.65,0.15,A2=x1,0.15,0.75,0.4,x2,0.4,0.55,0.3,0.35,0.2,x3,0.4,0.4,x4,0.33,0.45,0.5,0.3,A3=x1,0.7,0.4,0.2,x2,0.5,0.4,x3,0.3,0.5,0.4,0.4,0.3,x4,0.5,0.7,0.1,0.2,A4=x1,0.2,0.45,0.3,0.1,x2,0.7,0.6,0.2,x3,0.4,0.3,x4,0.5,0.3,0.2.
Now, let an unknown building substance, which needs to be identified, be(45)A=x1,0.8,0.5,0.1,x2,0.6,0.3,x3,0.5,0.4,0.7,0.1,0.2,x4,0.6,0.3,0.3,0.2.
We transform the DHFSs in the shape of CDHFSs by taking1=e0 as follows:(46)A1=x1,0.1ei2π0.0,0.2ei2π0.0,0.4ei2π0.0,0.4ei2π0.0,0.3ei2π0.0,x2,0.4ei2π0.0,0.5ei2π0.0,0.2ei2π0.0,x3,0.1ei2π0.0,0.4ei2π0.0,x4,0.2ei2π0.0,0.65ei2π0.0,0.15ei2π0.0,A2=x1,0.15,0.75ei2π0.0,0.4ei2π0.0,x2,0.4ei2π0.0,0.55ei2π0.00.3ei2π0.0,0.35ei2π0.0,0.2ei2π0.0,x3,0.4ei2π0.0,0.4ei2π0.0,x4,0.33ei2π0.0,0.45ei2π0.0,0.5ei2π0.0,0.3ei2π0.0,A3=x1,0.7ei2π0.0,0.4ei2π0.0,0.2ei2π0.0,x2,0.5ei2π0.0,0.4ei2π0.0,x3,0.3ei2π0.0,0.5ei2π0.0,0.4ei2π0.0,0.4ei2π0.0,0.3ei2π0.0,x4,0.5ei2π0.0,0.7ei2π0.0,0.1,0.2,A4=x1,0.2ei2π0.0,0.45ei2π0.0,0.3ei2π0.0,0.1ei2π0.0,x2,0.7ei2π0.0,0.6ei2π0.0,0.2ei2π0.0,x3,0.4ei2π0.0,0.3ei2π0.0,x4,0.5ei2π0.0,0.3ei2π0.0,0.2ei2π0.0,A=x1,0.8ei2π0.0,0.5ei2π0.0,0.1ei2π0.0,x2,0.6ei2π0.0,0.3ei2π0.0,x3,0.5ei2π0.0,0.4ei2π0.0,0.7ei2π0.0,0.1ei2π0.0,0.2ei2π0.0,x4,0.6ei2π0.0,0.3ei2π0.0,0.3ei2π0.0,0.2ei2π0.0.
By using the information of Example 5, we examine the similarity measures between the family of unknown with the family of known to find the supremacy of the investigated approaches. The information in Example5 presented in the shape of DHFSs. By some existing SMs for DHFSs, we detected the similarity between A and AJJ=1,2,3,4 as appeared in Table 5. As we have 1=e0, then the information given in Example 5 changed into CDHFSs. At this point through deciphered SMs, we detected the similarity between A and AJJ=1,2,3,4 likewise appeared in Table 5. Three of our deciphered SMs (SCD1,SCD4,and SCD5) demonstrated that unknown building substance A has a place with predefined building substance A3 on the grounds that the similarity among A and A3 is most prominent one and two of our deciphered SMs (SCD2 and SCD3) demonstrated that unknown building substance A has a place with predefined building substance A4 on the grounds that the similarity among A and A3 is most prominent one. Ranking of the deciphered and existing SMs is likewise introduced in Table 5. The graphical portrayal of the comparison of proposed and existing SMs is spoken in Figure 5.
Presently, we examine the comparison among deciphered and existing SMs for Example3. For Example 3, the information is given in the structure of CDHFSs. We realize that no SM exists in the writing to fathom this sort of information. The existing SMs are inadequate to discover the similarity among A and AJJ=1,2,3,4 as appeared in Table 6. From Table 6, we see that the information presented in Example 3 is only feasible by the explored SMs. The deciphered SMs discover the similarity among A and AJJ=1,2,3,4 as appeared in Table 6. Our deciphered SMs demonstrated that vague building material A has a place with the predetermined building material A3 on the grounds that the similarity among A and A3 is most noteworthy one. The positioning of the interpreted SMs is additionally introduced in Table 6. The geometrical expressions of the investigated measures are discussed in the form of Figure 6.
In this comparison, we contrasted our investigated SMs with some existing SMs characterized by Wang et al. [27]. From the above conversation, unmistakably, our proposed SMs can speak extra fuzzy information and put it comprehensively in circumstances in real-life issues. In the view of CDHFS, we explored the SMs; our SMs are progressively agreeable for genuine issues than the existing SMs and our SMs are broader than the existing SMs.Table 5
Comparison between explored and existing SMs for Example5 when λ=1 and ηcc,ϕcc=0.5,0.5.
MethodScore valueRankingWang et al. [27]s1A,A1=0.7265, s1A,A2=0.7506s1A,A3=0.8537, s1A,A4=0.874A4≥A3≥A2≥A1Wang et al. [27]s2A,A1=0.525, s2A,A2=0.62s2A,A3=0.775, s2A,A4=0.7A3≥A4≥A2≥A1Wang et al. [27]s3A,A1=0.7261, s3A,A2=0.6486s3A,A3=0.526, s3A,A4=0.7031A3≥A4≥A2≥A1Explored SMSCD1A,A1=0.5362, SCD1A,A2=0.3968SCD1A,A3=0.7887, SCD1A,A4=0.6056A3≥A4≥A2≥A1Explored SMSCD2A,A1=0.7979, SCD2A,A2=0.7972SCD2A,A3=0.8566, SCD2A,A4=0.903A4≥A3≥A1≥A2Explored SMSCD3A,A1=0.4545, SCD3A,A2=0.4582SCD3A,A3=0.5215, SCD3A,A4=0.7257A4≥A3≥A2≥A1Explored SMSCD4A,A1=0.3035, SCD4A,A2=0.3266SCD4A,A3=0.6849, SCD4A,A4=0.479A3≥A4≥A2≥A1Explored SMSCD5A,A1=0.0916, SCD5A,A2=0.0778SCD5A,A3=0.2215, SCD5A,A4=0.1396A3≥A4≥A1≥A2Figure 5
The graphical portrayal of the comparison of proposed and existing SMs for Example5.Table 6
Comparison between explored and existing SMs for Example3 when λ=1 and ηcc,ϕcc=0.5,0.5.
MethodScore valueRankingWang et al. [27]FailedFailedWang et al. [27]FailedFailedWang et al. [27]FailedFailedExplored SMSCD1A,A1=0.4732, SCD1A,A2=0.3968SCD1A,A3=0.5277, SCD1A,A4=0.5277A3≥A4≥A1≥A2Explored SMSCD2A,A1=0.6228, SCD2A,A2=0.649SCD2A,A3=0.8251, SCD2A,A4=0.6735A3≥A4≥A2≥A1Explored SMSCD3A,A1=0.3885., SCD3A,A2=0.429SCD3A,A3=0.6712, SCD3A,A4=0.4415A3≥A4≥A2≥A1Explored SMSCD4A,A1=0.3105, SCD4A,A2=0.3715SCD4A,A3=0.6983, SCD4A,A4=0.4156A3≥A4≥A2≥A1Explored SMSCD5A,A1=0.1288., SCD5A,A2=0.124SCD5A,A3=0.306, SCD5A,A4=0.185A3≥A4≥A2≥A1Figure 6
The graphical portrayal of the comparison of proposed and existing SMs for Example3.


## 7. Conclusion

The aims of this article are to explore the novel approaches of complex dual hesitant fuzzy sets and their fundamental laws and also verify with the help of some numerical examples. The main purpose of this manuscript is discussed as follows:(1)
The idea of complex dual hesitant fuzzy sets is found, which is the mixture of complex fuzzy sets and dual hesitant fuzzy sets to cope with complicated information in realistic issues.(2)
The novel approach of complex interval-valued dual hesitant fuzzy set and its fundamental laws are also explored and also justified with the help of examples.(3)
Some antilogarithmic and with-out exponential-based similarity measures and generalized similarity measures and their important characteristics are also explored.(4)
These similarity measures are applied in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established measures.(5)
To illustrate some numerical examples using the established measures to examine the reliability and validity of the proposed measures by comparing it with existing measures.(6)
The advantages, comparative analysis, and graphical representation of the explored measures and existing measures are also discussed in detail.In the future, there is a scope of extending the proposed approach to the different environments, such as complex Pythagorean fuzzy sets, linguistic features, decision making [28–34], its application to the various fields of the pattern recognition, and decision theory.



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*Source: 6611782-2021-04-28.xml*