#### 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.

#### 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, and , 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 of and .(2)When we choose the CDHFS as a singleton set, then the CDHFS is reduced into CIFS which is in the form of and .(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 of and .

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 Section 2, 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, speaks to a fix set.

*Definition 1 (see [1]). *A FS is of the following form:with a condition , where represents the grade of truth. Through this article, the collection of all FSs on are represented by . The pair is known as fuzzy number (FN).

*Definition 2 (see [25]). *A CFS is of the following form:where represents the complex-valued truth grade in the form of polar coordinate, where . Additionally, the pair is known as complex fuzzy number (CFN).

*Definition 3 (see [4]). *A HFS is of the following form:where is the set of different finite values in , representing the grade of truth for each element . Further, the pair is known as hesitant fuzzy number (HFN).

*Definition 4 (see [23]). *A DHFS is of the following form:where and are the two finite subsets in , representing the membership grade and nonmembership grade of the component , respectively, with the conditions and , where , , , and .

*Definition 5 (see [24]). *For any two DHFSs and , the similarity measure satisfies the following axioms:(1)(2)(3)

*Definition 6 (see [24]). *For any two DHFSs and , the distance measure satisfies the following axioms:(1)(2)(3)From the abovementioned analysis, we get that the .

*Definition 7 (see [26]). *A IvDHFS is of the following form:where and are two finite subsets of some interval-values in , representing the membership grade and nonmembership grade of the component , respectively, with the conditions , and , where , , , and , for all.

#### 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 CDHFS is of the following form:whererepresented the complex-valued membership grade and nonmembership grade, which are subsets of a unit disc in complex plane with a condition , , and , where , , and , , for and . Further, is called complex dual hesitant fuzzy number (CDHFN).

*Definition 9. *Let and be two CDHFNs. Then, their complement, union, and intersection are defined as follows:(1)(2)(3)

*Example 1. *Let be two CDHFSs. Then,(1)(2)(3)

##### 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 CIvDHFS is of the following form:whererepresented 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 , , and , where , , , and , for and . Further, is called complex interval-valued dual hesitant fuzzy number (CIvDHFN).

*Definition 11. *Let be two CIvDHFNs. Then, their complement, union, and intersection are defined as follows:(1)(2)(3)

*Example 2. *Letbe two CIvDHFNs. Then,(1)(2)(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. *Let and be two CDHFSs on set . Then, similarity measure (SM) between and is indicated by , which holds the following axioms:(1)(2)if and only if (3)

*Definition 13. *Let and be two CDHFSs on set . Then, distance measure between and is indicated by , which holds the following axioms:(1)(2)if and only if (3)By investigating Definitions 12 and 13, we can observe that the higher the similarity, the littler the distance between the two CDHFSs will be. It is noted that . 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. *Let and be two CDHFS on set . Then, the antilogarithmic-based generalized SM between and is given aswhere and represents the maximum operation.

Theorem 1. *Let and be two CDHES on set . Then, the SM hold the following axioms:*(1)*(2)**if and only if *(3)

*Proof. *(1)Since , and , then This implies that, for , we obtain For , By continuing this process, we get(2)By Definition 14,