# Correlation Coefficient and Entropy Measures Based on Complex Dual Type-2 Hesitant Fuzzy Sets and Their Applications

Authors: Tahir Mahmood; Zeeshan Ali; Harish Garg; Lemnaouar Zedam; Ronnason Chinram
Journal: Journal of Mathematics
 (2021)

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## Abstract

The theory of complex dual type-2 hesitant fuzzy sets (CDT-2HFSs) is a blend of two different modifications of fuzzy sets (FSs), called complex fuzzy sets (CFSs) and dual type-2 hesitant fuzzy sets (DT-2HFSs). CDT-2HFS is a proficient technique to cope with unpredictable and awkward information in realistic decision problems. CDT-2HFS is composed of the grade of truth and the grade of falsity, and the grade of truth (also for grade of falsity) contains the grade of primary and secondary parts in the form of polar coordinates with the condition that the sum of the maximum of the real part (also for the imaginary part) of the primary grade (also for the secondary grade) cannot exceed the unit interval [0, 1]. The aims of this manuscript are to discover the novel approach of CDT-2HFS and its operational laws. These operational laws are also justified with the help of an example. Additionally, based on a novel CDT-2HFS, we explored the correlation coefficient (CC) and entropy measures (EMs), and their special cases are also discussed. TOPSIS method based on CDT-2HFS is also explored. Then, we applied our explored measures based on CDT-2HFSs in the environment of the TOPSIS method, medical diagnosis, pattern recognition, and clustering algorithm to cope with the awkward and complicated information in realistic decision issues. Finally, some numerical examples are given to examine the proficiency and validity of the explored measures. Comparative analysis, advantages, and graphical interpretation of the explored measures with some other existing measures are also discussed.


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## Body

## 1. Introduction

The present decision-making is one of the genuinely basic movements in individuals’ everyday life, the reason for existing of which is to rank the limited arrangement of options regarding that they are so solid to the choice maker(s). Multiattribute decision-making (MADM) is a part of decision-making and is viewed as an intellectual-based human movement. People unavoidably are confronted with different decision-making issues, which include numerous fields [1–3]. The idea of the fuzzy set (FS) proposed by Zadeh [4] modified the method of measuring the vulnerability/fuzziness. Before the development of the FS hypothesis by Zadeh [4], the likelihood hypothesis was the customary instrument to quantify the vulnerability. Be that as it may, to gauge the vulnerability utilizing likelihood, it ought to have been communicated as exact numbers which are its primary constraints. The obscure terms, for instance, “without doubt” and “marginally,” could not be measured utilizing the likelihood hypothesis. To gauge the vulnerability/fuzziness related to such unclear terms, the FS hypothesis has ended up being a successful apparatus. In the FS hypothesis, every component relating to a specific universe of talk has been appointed an enrollment degree lying somewhere in the range of 0 and 1, which indicates its level of belongingness to the set being referred to called FS. By the goodness of its reasonableness in genuine issues, FSs increased a lot of prevalence with analysts around the world. Endeavors were made to additionally sum up the idea by numerous creators to make it progressively versatile for viable issues.Notwithstanding, in certain issues including etymological factors, for example, exceptionally low, low, medium, high, and extremely high, the assurance of the participation capacity may not be simple; that is, in an issue, dubious participation capacity might be experienced. To survive such circumstances, the idea of type‐2 FSs (T-2FSs) was presented by Zadeh [5], as a distinction from common FSs. Many researchers have utilized T-2FSs in different areas [6–8]. The tale structures, which are speculations and expansions of the FSs, have been proposed by numerous analysts since Zadeh presented the FSs. The fundamental motivation behind these structures is to take out vulnerabilities and to guarantee that specialists settle on choices in a way that is without blunder or with not many mistakes. One of these structures is the idea of hesitant FS (HFS) characterized by Torra [9]. Feng et al. [10] presented the type-2 hesitant fuzzy set (T-2HFS). The idea of dual HFS (DHFS) was first characterized as a speculation of the HFSs characterized by Zhu et al. [11]. A DHFS is distinguished as two distinct capacities called enrollment and nonmembership capacity. This structure permits the leader to make more adaptable, precise, and reasonable remarks about the components under the reluctant zone. In this manner, it limits the blunder edge by giving more solid outcomes than the current structures, as HFSs and interval‐valued HFSs. Alcantud et al. [12] characterized the idea of the double broadened HFSs and applied it to a decision‐making issue under dual extended hesitant fuzzy data.As for the above existing examinations, it has been dissected that they have researched the decision-making issues under the FS, IFS, or its speculations, which are just ready to manage the vulnerability and dubiousness existing in the information. These models cannot speak to the fractional obliviousness of the information and its changes at a given period of time. Be that as it may, in complex informational collections, vulnerability and ambiguity in the information happen simultaneously with changes to the stage (periodicity) of the information. Instances of complex informational indexes incorporate a lot of information that is created from clinical research, just as government databases for biometric and facial acknowledgments, sound, and pictures, all of which may contain a lot of deficient, dubious, and ambiguous data. To deal with these kinds of issues, the theory of complex FS (CFS) was discovered by Ramot et al. [13]. CFS contains the grade of membership in the form of a complex number belonging to a unit disc in a complex plane. Various scholars utilized CFS in different fields [14–16].Correlation examination shows a direct connection between two sets and it has a very significant spot for dynamics. In this way, numerous researchers in various fields have considered the relationship coefficients. Additionally, the FS and its speculations have a significant job in dynamics, so CCs have drawn in the consideration of scientists examining the FS and its speculations. For instance, Chiang and Lin [17] and Chaudhuri and Bhattacharya [18] examined the correlation between two FSs. Gerstenkorn and Mańko [19] worked the relationship and CC of the intuitionistic FSs (IFSs). The entropy of FSs is a proportion of fuzziness between FSs. De Luca and Termini [20] first presented the aphorism development for the entropy of FSs concerning Shannon’s likelihood entropy. Yager [21] characterized fuzziness proportions of FSs as far as a need of differentiation between the FS and its nullification based on Lp standard. Kosko [22] gave a proportion of fuzziness between FSs utilizing a proportion of separation between the FS and its closest set to the separation between the FS and its farthest set. Xuecheng [23] gave some aphorism definitions of entropy and furthermore characterized a σ-entropy. Pal and Pal [24] proposed exponential entropy. Meanwhile Fan and Ma [25] gave some new fuzzy entropy equations. The technique for establishing order preference by similarity to the ideal solution (TOPSIS) technique as a strategy for building up request inclination by likeness to the perfect arrangement, started by Hwang and Yoon [26], is one of the best and beneficial methods for decision‐making. The basic idea of TOPSIS strategy is to pick the elective that has the briefest good way from the positive perfect arrangement (PIS) and the greatest good way from the negative perfect arrangement (NIS). There exists a tremendous writing including study and utilization of TOPSIS hypothesis in a wide scope of MCDM just as multicriteria group decision‐making (MCGDM) issues [27–29].Dual type-2 hesitant fuzzy set contains the grade of truth and the grade of falsity in the form of the subset of the unit interval with the condition that the sum of the maximum of the truth grade and the maximum of the falsity grade cannot exceed the unit interval. The complex dual type-2 hesitant fuzzy set is a generalization of the dual type-2 hesitant fuzzy set, in which the amplitude term provides the extent of belonging of an object, while the phase term describes the periodicity. These phase terms distinguish the complex dual type-2 hesitant fuzzy set from the traditional dual type-2 hesitant fuzzy set theories. In dual type-2 hesitant fuzzy set theory, the data are managed with the compensation of only the degree of belonging, while the part of periodicity is completely ignored. Hence, this may result in the loss of information during the decision-making processes in some certain cases. To further illustrate the concept of phase terms, we take an example. Suppose that a person wants to purchase a car under crucial factors such as its model and its production date. Since the model of each car moves with the evolution of the production dates, to make a selection or decision regarding choosing the optimal car is a decision-making process taking these two factors into account simultaneously. Moreover, it is quite obvious that such types of problems cannot be modeled accurately with traditional theories. However, complex dual type-2 hesitant fuzzy set theory is well suited for such classes of problems, where the amplitude terms may be used to provide a decision about the model of a car, while the phase term concerns its production dates. Henceforth, a complex dual type-2 hesitant fuzzy set is a more generalized continuation of the existing theories, such as type-2 hesitant fuzzy sets and dual type-2 hesitant fuzzy sets.When a decision-maker gives0.4ei2π0.3,0.3ei2π0.2 and 0.41ei2π0.31,0.31ei2π0.21 for the grade of complex-valued supporting and the grade of complex-valued supporting against in the form of primary and secondary information with the condition that the sum of the maximum of the real part (also for the imaginary part) of the complex-valued supporting (also for supporting against) grade for primary (also for secondary) information cannot exceed the unit interval, There exist notions like FSs, T-2FSs, HFSs, DHFSs, CFSs, and DT-2HFSs. Handling such kind of issues is very difficult, but when a decision-maker provides such kind of information in the form of the finite subset of unit interval, then it is very complicated for a decision-maker to handle it. For coping with such kind of issues, in this manuscript, the novel approach of CDT-2HFS, which is a mixture of two different modifications of FS, that is, CFS and DT-2HFS, is explored. CDT-2HFS is a proficient technique to cope with unpredictable and awkward information in realistic decision problems. CDT-2HFS composes the grade of truth and the grade of falsity, and the grade truth (also for falsity grade) contains the grade of primary and secondary parts in the form of polar coordinates with the condition that the sum of the maximum of the real part (also for the imaginary part) of the primary grade (also for the secondary grade) cannot exceed the unit interval. The aims of this manuscript are to discover the novel approach of CDT-2HFS and its operational laws. These operational laws are also justified with the help of an example. Additionally, based on a novel CDT-2HFS, we explored the correlation coefficient (CC) and entropy measures (EMs), and their special cases are discussed. TOPSIS method based on CDT-2HFS is also explored. Then, we applied our explored measures based on CDT-2HFSs in the environment of the TOPSIS method, medical diagnosis, pattern recognition, and clustering algorithm to cope with awkward and complicated information in realistic decision issues. Finally, four numerical examples are resolved to examine the proficiency and validity of the explored measures. Comparative analysis, advantages, and graphical interpretation of the explored measures with some other existing measures are also discussed.The aims of this manuscript are summarized as follows: in Section2, we review some basic notions like FSs, T-2FSs, HFSs, DHFSs, CFSs, and their basic laws. In Section 3, the theory of CDT-2HFS, which is a mixture of two different modifications of FS, that is, CFS and DT-2HFS, is presented. CDT-2HFS is a proficient technique to cope with unpredictable and awkward information in realistic decision problems. CDT-2HFS is composed of the grade of truth and the grade of falsity, and the grade truth (also for falsity grade) contains the grade of primary and secondary parts in the form of polar coordinates with the condition that the sum of the maximum of the real part (also for the imaginary part) of the primary grade (also for secondary grade) cannot exceed the unit interval. The aims of this manuscript were to discover the novel approach of CDT-2HFS and its operational laws. These operational laws are also justified with the help of examples. In Sections 4 and 5, based on a novel CDT-2HFS, we explored the correlation coefficient (CC) and entropy measures (EMs), and their special cases are discussed. In Section 6, TOPSIS method based on CDT-2HFS is also explored. Then, we applied our explored measures based on CDT-2HFSs in the environment of TOPSIS method, medical diagnosis, pattern recognition, and clustering algorithm to cope with awkward and complicated information in realistic decision issues. Finally, four numerical examples are resolved to examine the proficiency and validity of the explored measures. Comparative analysis, advantages, and graphical interpretation of the explored measures with some other existing measures are also discussed. The conclusion of this paper is discussed in Section 7.


## 2. Preliminaries

Basic notions of FSs, T-2FSs, HFSs, DHFSs, CFSs, and their operational laws are briefly reviewed in this study. Throughout this manuscript, the symbolXUNI denotes the fixed set.Definition 1 (see [4]).
A FS is an object of the form(1)QFS=x˜,MQFSx˜:x˜∈XUNI,where MQFS represents the grade of supporting with the condition that 0≤MQFS≤1.Definition 2 (see [5]).
A T-2FS is an object of the form(2)QT−2FS=x˜,x˜′,MQt−2FSx˜,x˜′:∀x˜∈XUNI,x˜′∈Jx˜⊆0,1,where MQt−2FSx˜,x˜′ represents the grade of type-2 supporting with the condition that 0≤MQt−2FSx˜,x˜′≤1.Definition 3 (see [9]).
A HFS is an object of the form(3)QHFS=x˜,MQHFSx˜:x˜∈XUNI,where MQHFS represents the grade of supporting in the form of the subset of the unit interval, with the condition that 0≤MaxMQHFS≤1, whenever MQHFS⊆0,1.Definition 4 (see [11]).
A DHFS is an object of the form(4)QDHFS=x˜,MQDHFSx˜,NQDHFSx˜:x˜∈XUNI,where MQDHFS and NQDHFS represent the grade of supporting and the grade of supporting against with the condition that 0≤maxMQDHFS+maxNQDHFS≤1, whenever MQDHFS,NQDHFS⊆0,1.
Additionally, we defined some operational laws based on DHFSs. For any two DHFSsQDHFS−1=MQDHFS−161x˜,NQDHFS−161x˜andQDHFS−1=MQDHFS−261x˜,NQDHFS−261x˜, we have(5)QDHFS−1∪QDHFS−2=maxMQDHFS−161x˜,MQDHFS−261x˜,minNQDHFS−161x˜,NQDHFS−261x˜,QDHFS−1∩QDHFS−2=minMQDHFS−161x˜,MQDHFS−261x˜,maxNQDHFS−161x˜,NQDHFS−261x˜.Definition 5 (see [13]).
A CFS is an object of the form(6)QCFS=x˜,MQCFSx˜:x˜∈XUNI,where MQCFS=MQCFRPx˜ei2πMQCEIPx˜ represents the grade of complex-valued supporting with the condition that 0≤MQCFRP,MQCFIP≤1.


## 3. Complex Dual Type-2 Hesitant Fuzzy Sets

Based on the existing drawbacks [30], in this study, we discovered the new theory of CDT-2HFSs and their operational laws. The presented operational laws are also justified with the help of some examples.Definition 6.
A CDT-2HFS is an object of the form(7)QCDTH=x˜,MQCDTHx˜,NQCDTHx˜:x˜∈XUNI.where MQCDTHx˜=MQCDTHRP−P6jx˜⋅ei2πMQCDTHRP−P6jx˜,MQCDTHRP−S6jx˜⋅ei2πMQCDTHRP−S6jx˜:j=1,2,3,…,n and NQCDTHx˜=NQCDTHRP−P6kx˜⋅ei2πMQCDTHRP−P6kx˜,NQCDTHRP−S6kx˜⋅ei2πMQCDTHRP−S6kx˜:k=1,2,3,…,m represent the grade of complex-valued supporting and the grade of complex-valued supporting against in the form of complex type-2 hesitant fuzzy elements (CT-2HFEs) with the following conditions: 0≤maxMQCDTHRP−Pбj+maxNQCDTHRP−Pбk≤1,0≤maxMQCDTHRP−Pбj+maxNQCDTHRP−Pбk≤1,0≤maxMQCDTHRP−Sбj+maxNQCDTHRP−Sбk≤1 and 0≤maxMQCDTHIP−Sбj+maxNQCDTHIP−Sбk≤1. The complex dual type-2 hesitant fuzzy set is expressed by(8)QCDTH=MQCDTHRP-Pбjx˜⋅ei2πMQCDTHIP-Pбjx˜,MQCDTHRP-Sбjx˜⋅ei2πMQCDTHIP-Sбjx˜,NQCDTHRP-Pбkx˜⋅ei2πNQCDTHIP-Pбkx˜,NQCDTHRP-Sбkx˜ei2πNQCDTHIP-Sбkx˜,j,k=1,2,3,…,n,m.
Additionally, we defined some operational laws based on CDT-2HFEs. For any two CDT-2HFEsQCDTH−1=MQCDTHRP−P−1б1x˜⋅ei2πMQCDTHIP−P−1б1x˜,MQCDTHRP−S−1б1x˜⋅ei2πMQCDTHIP−S−1б1x˜,NQCDTHRP−P−1б1x˜⋅ei2πNQCDTHIP−P−1б1x˜,NQCDTHRP−S−1б1x˜⋅ei2πMQCDTHIP−S−1б1x˜ and QCDTH−2=MQCDTHRP−P−2б1x˜⋅ei2πMQCDTHIP−P−1б1x˜,MQCDTHRP−S−1б1x˜⋅ei2πMQCDTHIP−S−1б1x˜,NQCDTHRP−P−2б1x˜⋅ei2πNQCDTHIP−P−2б1x˜,NQCDTHRP−S−2б1x˜⋅ei2πMQCDTHIP−S−2б1x˜, we have(9)QCDTH−1∪QCDTH−2=maxMQCDTHRP−P−1б1x˜,MQCDTHRP−P−2б1x˜ei2πmaxMQCDTHIP−P−1б1x˜,MQCDTHIP−P−2б1x˜,maxMQCDTHRP−S−1б1x˜,MQCDTHRP−S−2б1x˜ei2πmaxMQCDTHIP−S−2б1x˜,MQCDTHIP−S−2б1x˜,minNQCDTHRP−P−1б1x˜,NQCDTHRP−S−2б1x˜ei2πminNQCDTHIP−P−1б1x˜,NQCDHIP−S−2б1x˜,minNQCDTHRP−S−1б1x˜,NQCDTHRP−S−2б1x˜ei2πminNQCDTHRP−S−2б1x˜,NQCDTHIP−S−2б1x˜,(10)QCDTH−1∩QCDTH−2=minMQCDTHRP−P−1б1x˜,MQCDTHRP−P−2б1x˜ei2πminMQCDTHIP−P−1б1x˜,MQCDTHIP−P−2б1x˜,minMQCDTHRP−S−1б1x˜,MQCDTHRP−S−2б1x˜ei2πminMQCDTHIP−S−1б1x˜,MQCDTHIP−S−2б1x˜,maxNQCDTHRP−P−1б1x˜,NQCDTHRP−P−2б1x˜ei2πmaxNQCDTHIP−P−1б1x˜,NQCDTHIP−P−2б1x˜,maxNQCDTHRP−S−1б1x˜,NQCDTHRP−S−2б1x˜ei2πmaxNQCDTHIP−S−1б1x˜,NQCDTHIP−S−2б1x˜.Example 1.
For any two CDT-2HFSs, all their entries in the form of complex numbers are stated as follows:(11)QCDTH−1=0.1ei2π0.2,0.2ei2π0.3,0.2ei2π0.3,0.3ei2π0.4,0.3ei2π0.4,0.4ei2π0.5,0.01ei2π0.02,0.02ei2π0.03,0.03ei2π0.04,0.05ei2π0.06,QCDTH−1=0.7ei2π0.6,0.6ei2π0.5,0.6ei2π0.5,0.5ei2π0.4,0.5ei2π0.4,0.4ei2π0.3,0.07ei2π0.08,0.09ei2π0.01,0.01ei2π0.03,0.22ei2π0.03.
Then, by using equations (9) and (10), we get(12)QCDTH−1∪QCDTH−2=0.7ei2π0.6,0.6ei2π0.5,0.6ei2π0.5,0.5ei2π0.4,0.5ei2π0.4,0.4ei2π0.5,0.01ei2π0.02,0.02ei2π0.01,0.01ei2π0.03,0.05ei2π0.03,QCDTH−1∩QCDTH−2=0.1ei2π0.2,0.2ei2π0.3,0.2ei2π0.3,0.3ei2π0.4,0.3ei2π0.4,0.4ei2π0.3,0.07ei2π0.08,0.09ei2π0.03,0.03ei2π0.04,0.22ei2π0.06.
The explored notions, which are stated in the form of equations (7), (9), and (10), are more proficient and more modified than the existing drawbacks; for instance, if we choose the imaginary part of equations (7), (9), and (10) to be zero, then equations (7), (9), and (10) convert it for DT-2HFS [30].


## 4. Correlation Coefficient for Complex Dual Type-2 Hesitant Fuzzy Sets

The aim of this study is to present the novel correlation, correlation coefficient (CC), maximum-based CC (MCC), weighted CC (WCC), and maximum-based WCC (MWCC). The special cases of the explored measures are also explored.Definition 7.
For any two CDT-2HFSs,QCDTH−1=MQCDTHRP−P−1бjxi˜⋅ei2πMQCDTHIP−P−1бjxi˜,MQCDTHRP−S−1бjxi˜⋅ei2πMQCDTHIP−S−1бjxi˜,NQCDTHRP−P−1бkxi˜⋅ei2πNQCDTHIP−P−1бkxi˜˜,MQCDTHRP−P−1бkxi˜⋅ei2πMQCDTHIP−S−1бkxi˜,j,k=1,2,3,…,n,m and QCDTH−2=MQCDTHRP−P−2бjxi˜⋅ei2πMQCDTHIP−P−2бjxi˜,MQCDTHIP−P−2бjxi˜⋅ei2πMQCDTHIP−S−2бjxi˜,NQCDTHRP−P−2бkxi˜⋅ei2πNQCDTHIP−P−2бkxi˜,NQCDTHRP−S−2бkxi˜⋅ei2πNQCDTHIP−S−2бkxi˜,j,k=1,2,3,…,l˜x˜i,l︷x˜i, the correlation is of the form(13)ÇCDTHF−cQCDTH−1,QCDTH−2=14n∑i=1n1l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjx˜iMQCDTHRP−P−2бjx˜i+MQCDTHRP−S−1бjx˜iMQCDTHRP−S−2бjx˜i+1l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjx˜iMQCDTHIP−P−2бjx˜i+MQCDTHIP−S−1бjx˜iMQCDTHIP−S−2бjx˜i+1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkx˜iNQCDTHRP−P−2бkx˜i+NQCDTHRP−P−1бkx˜iNQCDTHRP−P−2бkx˜i+1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkx˜iNQCDTHRP−P−2бkx˜i+NQCDTHRP−S−1бkx˜iNQCDTHRP−S−2бkx˜i.Definition 8.
For any two CDT-2HFSs,QCDTH−1=MQCDTHRP−P−1бjxi˜⋅ei2πMQCDTHIP−P−1бjxi˜,MQCDTHRP−S−1бjxi˜⋅ei2πMQCDTHIP−S−1бjxi˜,NQCDTHRP−P−1бkxi˜⋅ei2πMQCDTHIP−P−1бkxi˜,NQCDTHRP−S−1бkxi˜⋅ei2πNQCDTHIP−S−1бkxi˜,j,k=1,2,3,…,n,m and QCDTH−2MQCDTHRP−P−2бjxi˜⋅ei2πMQCDTHIP−P−2бjxi˜,MQCDTHRP−S−2бjxi˜⋅ei2πMQCDTHIP−S−2бjxi˜NQCDTHRP−P−2бkxi˜⋅ei2πNQCDTHIP−P−2бkxi˜,NQCDTHRP−S−2бkxi˜⋅ei2πNQCDTHIP−S−2бkxi˜,j,k=1,2,3,…,l˜x˜i,l︷x˜i, the correlation coefficient is of the form(14)ÇCDTHF−ccQCDTH−1,QCDTH−2=ÇCDTHF−cQCDTH−1,QCDTH−2ÇCDTHF−cQCDTH−1,QCDTH−11/2−ÇCDTHF−cQCDTH−1,QCDTH−21/2=∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜MQCDTHRP−P−2бjxi˜+MQCDTHRP−S−1бjxi˜MQCDTHRP−S−2бjxi˜+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜MQCDTHIP−P−2бjxi˜+MQCDTHIP−S−1бjxi˜MQCDTHIP−S−2бjxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+NQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бjxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜+NQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бjxi˜∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜2+MQCDTHRP−S−1бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜2+MQCDTHIP−S−1бjxi˜2+1/l˜x˜i∑j=1l˜x˜iNQCDTHRP−P−1бkxi˜2+NQCDTHRP−S−1бkxi˜2+1/l˜x˜i∑j=1l˜x˜iNQCDTHIP−P−1бkxi˜2+NQCDTHRP−S−1бjxi˜21/2×∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜2+MQCDTHRP−S−2бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜2+MQCDTHIP−S−2бjxi˜2+1/l˜x˜i∑j=1l˜x˜iNQCDTHRP−P−2бkxi˜2+NQCDTHRP−S−2бkxi˜2+1/l˜x˜i∑j=1l˜x˜iNQCDTHIP−P−2бkxi˜2+NQCDTHRP−S−2бjxi˜21/2.Proposition 1.
For any two CDT-2HFSs, QCDTH-1 and QCDTH-2, the CC among CDT-2HFSs satisfies the following axioms:(1)
0≤ÇCDTHF-ccQCDTH−1,QCDTH−2≤1(2)
ÇCDTHF−ccQCDTH−1,QCDTH−2=1⇔QCDTH−1=QCDTH−2(3)
ÇCDTHF-ccQCDTH−1,QCDTH−2=ÇCDTHF-ccQCDTH−2,QCDTH−1Proof.
We prove the three above conditions by using equation (14). By using the inequality, it is clear that 0≤ÇCDTHF-ccQCDTH−1,QCDTH−2; then we only prove that ÇCDTHF-ccQCDTH−1,QCDTH−2≤1. For this, we choose that(15)ÇCDTHF−cQCDTH−1,QCDTH−2=1l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜MQCDTHRP−P−2бjxi˜+MQCDTHRP−S−1бjxi˜MQCDTHRP−S−2бjxi˜+1l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜MQCDTHIP−P−2бjxi˜+MQCDTHIP−S−1бjxi˜MQCDTHIP−S−2бjxi˜+1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+NQCDTHRP−P−1бkxi˜NQCDTHRP−S−2бkxi˜+1l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜+NQCDTHIP−P−1бkxi˜NQCDTHIP−S−2бkxi˜=1l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjx1˜MQCDTHRP−P−2бjx1˜+MQCDTHRP−S−1бjx1˜MQCDTHRP−S−2бjx1˜+⋯+1l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxn˜MQCDTHRP−P−2бjxn˜+MQCDTHRP−S−1бjxn˜MQCDTHRP−S−2бjxn˜1l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjx1˜MQCDTHIP−P−2бjx1˜+MQCDTHIP−S−1бjx1˜MQCDTHIP−S−2бjx1˜+⋯+1l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxn˜MQCDTHIP−P−2бjxn˜+MQCDTHIP−S−1бjxn˜MQCDTHIP−S−2бjxn˜1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkx1˜NQCDTHRP−P−2бkx1˜+NQCDTHRP−P−1бkx1˜NQCDTHRP−S−2бkx1˜+⋯+1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxn˜NQCDTHRP−P−2бkxn˜+NQCDTHRP−P−1бkxn˜NQCDTHRP−S−2бkxn˜1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkx1˜NQCDTHRP−P−2бkx1˜+NQCDTHRP−P−1бkx1˜NQCDTHRP−S−2бkx1˜+⋯+1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxn˜NQCDTHRP−P−2бkxn˜+NQCDTHRP−P−1бkxn˜NQCDTHRP−S−2бkxn˜.
By using the Cauchy–Schwarz inequality,x1y1+x2y2+⋯+xnyn2x12+x22+⋯+xn2y12+y22+⋯+yn2, we have(16)ÇCDTHF−cQCDTH−1,QCDTH−22≤1l˜x˜1∑j=1l˜x˜iMQCDTHRP−P−1бjx1˜2+MQCDTHRP−S−1бjx1˜2+⋯+1l˜x˜n∑j=1l˜x˜iMQCDTHRP−P−1бjxn˜2+MQCDTHRP−S−1бjxn˜21l˜x˜1∑j=1l˜x˜iMQCDTHIP−P−1бj2x1˜+MQCDTHIP−S−1бj2x1˜+⋯+1l˜x˜n∑j=1l˜x˜iMQCDTHIP−P−1бj2xn˜+MQCDTHIP−S−1бj2xn˜1l˜x˜1∑k=1l˜x˜iNQCDTHRP−P−1бk2x1˜+NQCDTHRP−P−1бk2x1˜+⋯+1l˜x˜n∑k=1l˜x˜iNQCDTHRP−P−1бk2xn˜+NQCDTHRP−P−1бk2xn˜1l˜x˜1∑k=1l˜x˜iNQCDTHIP−P−1бk2x1˜+NQCDTHIP−P−1бk2x1˜+⋯+1l˜x˜n∑k=1l˜x˜iNQCDTHIP−P−1бk2xn˜+NQCDTHIP−P−1бk2xn˜×1l˜x˜1∑j=1l˜x˜1MQCDTHRP−P−2бj2x1˜+MQCDTHRP−S−2бj2x1˜⋯+1l˜x˜n∑j=1l˜x˜iMQCDTHRP−P−1бj2xn˜+MQCDTHRP−S−1бj2xn˜1l˜x˜1∑j=1l˜x˜iMQCDTHIP−P−1бj2x1˜+MQCDTHIP−S−1бj2x1˜+⋯+1l˜x˜n∑j=1l˜x˜iMQCDTHIP−P−1бj2xn˜+MQCDTHIP−S−1бj2xn˜1l˜x˜1∑k=1l˜x˜iNQCDTHRP−P−1бk2x1˜+NQCDTHRP−P−1бk2x1˜+⋯+1l˜x˜n∑k=1l˜x˜iNQCDTHRP−P−1бk2xn˜+NQCDTHRP−P−1бk2xn˜1l˜x˜1∑k=1l˜x˜iNQCDTHIP−P−1бk2x1˜+NQCDTHIP−P−1бk2x1˜+⋯+1l˜x˜n∑k=1l˜x˜iNQCDTHIP−P−1бk2xn˜+NQCDTHIP−P−1бk2xn˜=ÇCDTHF−cQCDTH−1,QCDTH−1.ÇCDTHF−cQCDTH−2,QCDTH−2.
ÇCDTHF−cQCDTH−1,QCDTH−2≤ÇCDTHF−cQCDTH−1,QCDTH−11/2ÇCDTHF−cQCDTH−2,QCDTH−21/2; thus 0≤ÇCDTHF−ccQCDTH−1,QCDTH−2≤1. Further, we prove the second part by using (13)fd14. By hypothesis, it is given that QCDTH−1=QCDTH−2, and then MQCDTHRP−P−1бjxi˜=MQCDTHRP−P−2бjxi˜,MQCDTHIP−P−1бjxi˜=MQCDTHIP−P−2бjxi˜,MQCDTHRP−S−1бjxi˜=MQCDTHRP−S−2бjxi˜,MQCDTHIP−S−1бjxi˜=MQCDTHRP−S−2бjxi˜,MQCDTHIP−S−1бjxi˜=MQCDTHIP−S−2бjxi˜,NQCDTHRP−P−1бk=NQCDTHRP−P−2бk,NQCDTHRP−P−1бkxi˜˜=NQCDTHIP−P−2бkxi˜˜, and NQCDTHRP−S−1бk=NQCDTHRP−S−2бk,NQCDTHIP−S−1бkxi˜˜=NQCDTHIP−S−2бkxi˜˜; then, by using (13)fd14, we get ÇCDTHF−ccQCDTH−1,QCDTH−2=1. Additionally, we prove the third condition such that(17)ÇCDTHF−ccQCDTH−1,QCDTH−2=∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜MQCDTHRP−P−2бjxi˜+MQCDTHRP−S−1бjxi˜MQCDTHRP−S−2бjxi˜+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜MQCDTHIP−P−2бjxi˜+MQCDTHIP−S−1бjxi˜MQCDTHIP−S−2бjxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+NQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜+NQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜2+MQCDTHRP−S−1бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜2+MQCDTHIP−S−1бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜2+NQCDTHRP−S−1бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜2+NQCDTHIP−S−1бkxi˜21/2×∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜2+MQCDTHRP−S−2бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜2+MQCDTHIP−S−2бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−2бkxi˜2+NQCDTHRP−S−2бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−2бkxi˜2+NQCDTHIP−S−2бkxi˜21/2=∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜MQCDTHRP−P−1бjxi˜+MQCDTHRP−S−2бjxi˜MQCDTHRP−S−1бjxi˜+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜MQCDTHIP−P−1бjxi˜+MQCDTHIP−S−2бjxi˜MQCDTHIP−S−1бjxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−2бkxi˜NQCDTHRP−P−1бkxi˜+NQCDTHRP−P−2бkxi˜NQCDTHRP−P−1бkxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−2бkxi˜NQCDTHIP−P−1бkxi˜+NQCDTHIP−P−2бkxi˜NQCDTHIP−P−1бkxi˜∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜2+MQCDTHRP−S−2бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜2+MQCDTHIP−S−2бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−2бkxi˜2+NQCDTHRP−S−2бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−2бkxi˜2+NQCDTHIP−S−2бkxi˜21/2×∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜2+MQCDTHRP−S−1бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜2+MQCDTHIP−S−1бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜2+NQCDTHRP−S−1бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜2+NQCDTHIP−S−1бkxi˜21/2=ÇCDTHF−ccQCDTH−1,QCDTH−2Definition 9.
For any two CDT-2HFSs,QCDTH−1=MQCDTHRP−P−1бjxi˜⋅ei2πMQCDTHIP−P−1бjxi˜,MQCDTHRP−S−1бjxi˜⋅ei2πMQCDTHIP−S−1бjxi˜,NQCDTHRP−P−1бkxi˜⋅ei2πNQCDTHIP−P−1бkxi˜,NQCDTHRP−S−1бkxi˜⋅ei2πNQCDTHIP−S−1бkxi˜,j,k=1,2,3,…,n,m and QCDTH−2=MQCDTHRP−P−2бjxi˜⋅ei2πMQCDTHIP−P−1бjxi˜,MQCDTHRP−S−2бjxi˜⋅ei2πMQCDTHIP−S−1бjxi˜,NQCDTHRP−P−2бkxi˜⋅ei2πNQCDTHIP−P−1бkxi˜,NQCDTHRP−S−2бkxi˜⋅ei2πNQCDTHIP−S−1бjxi˜,j,k=1,2,3,…,l˜x˜i,l︷x˜i, the maximum-based correlation coefficient (MCC) is of the form(18)ÇCDTHF−mccQCDTH−1,QCDTH−2=ÇCDTHF−cQCDTH−1,QCDTH−2maxÇCDTHF−cQCDTH−1,QCDTH−11/2,ÇCDTHF−cQCDTH−1,QCDTH−21/2=∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜MQCDTHRP−P−2бjxi˜+MQCDTHRP−S−1бjxi˜MQCDTHRP−S−2бjxi˜+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜MQCDTHIP−P−2бjxi˜+MQCDTHIP−S−1бjxi˜MQCDTHIP−S−2бjxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+NQCDTHRP−S−1бkxi˜NQCDTHRP−S−2бkxi˜+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜+NQCDTHIP−S−1бkxi˜NQCDTHIP−S−2бkxi˜max∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜2+MQCDTHRP−S−1бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜2+MQCDTHRP−S−1бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜2+NQCDTHRP−S−1бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜2+NQCDTHRP−S−1бkxi˜21/2,∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜2+MQCDTHRP−S−2бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜2+MQCDTHIP−S−2бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−2бkxi˜2+NQCDTHRP−S−2бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−2бkxi˜2+NQCDTHIP−S−2бkxi˜2.Proposition 2.
For any two CDT-2HFSs, QCDTH-1 and QCDTH-2, the MCC among CDT-2HFSs satisfies the following axioms:(1)
0≤ÇCDTH−mccQCDTH−1,QCDTH−2≤1(2)
ÇCDTHF−mccQCDTH−1,QCDTH−2=1⇔QCDTH−1=QCDTH−2(3)
ÇCDTHF−mccQCDTH−1,QCDTH−2=ÇCDTHF−mccQCDTH−2,QCDTH−1Proof.
We prove the three above conditions by using equation (18). By using the inequality, it is clear that 0≤ÇCDTH−mccQCDTH−1,QCDTH−2; then, we only prove that ÇCDTH−mccQCDTH−1,QCDTH−2≤1. For this, we choose that(19)ÇCDTHF−cQCDTH−1,QCDTH−2=1l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜MQCDTHRP−P−2бjxi˜+MQCDTHRP−S−1бjxi˜MQCDTHRP−S−2бjxi˜+1l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜MQCDTHIP−P−2бjxi˜+MQCDTHIP−S−1бjxi˜MQCDTHIP−S−2бjxi˜+1l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+NQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+1l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜+NQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜=1l˜x˜1∑j=1l˜x˜iMQCDTHRP−P−1бjx1˜MQCDTHRP−P−2бjx1˜+MQCDTHRP−P−2−S−1бjx1˜MQCDTHRP−P−2−S−2бjx1˜+⋯+1l˜x˜n∑j=1l˜x˜iMQCDTHRP−P−1бjxn˜MQCDTHRP−P−2−P−2бjxn˜+MQCDTHRP−P−2−S−1бjxn˜MQCDTHRP−P−2−S−2бjxn˜1l˜x˜1∑j=1l˜x˜iMQCDTHIP−P−1бjx1˜MQCDTHIP−P−2бjx1˜+MQCDTHIP−S−1бjx1˜MQCDTHIP−S−2бjx1˜+⋯+1l˜x˜n∑j=1l˜x˜iMQCDTHIP−P−1бjxn˜MQCDTHIP−P−2бjxn˜+MQCDTHIP−S−1бjxn˜MQCDTHIP−S−2бjxn˜1l˜x˜1∑k=1l˜x˜iNQCDTHRP−P−1бkx1˜NQCDTHRP−P−2бkx1˜+NQCDTHRP−P−1бkx1˜NQCDTHRP−P−2бkx1˜+⋯+1l˜x˜n∑k=1l˜x˜iNQCDTHRP−P−1бkxn˜NQCDTHRP−P−2бkxn˜+NQCDTHRP−P−1бkxn˜NQCDTHRP−P−2бkxn˜1l˜x˜1∑k=1l˜x˜iNQCDTHIP−P−1бkx1˜NQCDTHIP−P−2бkx1˜+NQCDTHIP−P−1бkx1˜NQCDTHIP−P−2бkx1˜+⋯+1l˜x˜n∑k=1l˜x˜iNQCDTHIP−P−1бkxn˜NQCDTHIP−P−2бkxn˜+NQCDTHIP−P−1бkxn˜NQCDTHIP−P−2бkxn˜.
By using the Cauchy–Schwarz inequality,x1y1+x2y2+⋯+xnyn2≤x12+x22+⋯+xn2y12+y22+⋯yn2, we have(20)ÇCDTHF-cQCDTH−1,QCDTH−22≤1l˜x˜1∑j=1lx˜i˜MQCDTHRP−P−1бj2x1˜+MQCDTHRP−S−1бj2x1˜+⋯+1l˜x˜n∑j=1lx˜i˜MQCDTHRP−P−1бj2xn˜+MQCDTHRP−S−1бj2xn˜1l˜x˜1∑j=1lx˜i˜MQCDTHIP−P−1бj2x1˜+MQCDTHIP−S−1бj2x1˜+⋯+1l˜x˜n∑j=1lx˜i˜MQCDTHIP−P−1бj2xn˜+MQCDTHIP−S−1бj2xn˜1l˜x˜1∑k=1lx˜i˜NQCDTHRP−P−1бk2x1˜+NQCDTHRP−P−1бk2x1˜+⋯+1l˜x˜n∑k=1lx˜i˜NQCDTHRP−P−1бk2xn˜+NQCDTHRP−P−1бk2xn˜1l˜x˜1∑k=1lx˜i˜NQCDTHIP−P−1бk2x1˜+NQCDTHIP−P−1бk2x1˜+⋯+1l˜x˜n∑k=1lx˜i˜NQCDTHIP−P−1бk2xn˜+NQCDTHIP−P−1бk2xn˜×1l˜x˜1∑j=1lx˜i˜MQCDTHRP−P−2бj2x1˜+MQCDTHRP−S−2бj2x1˜+⋯+1l˜x˜n∑j=1lx˜i˜MQCDTHRP−P−2бj2xn˜+MQCDTHRP−S−2бj2xn˜1l˜x˜1∑j=1lx˜i˜MQCDTHIP−P−2бj2x1˜+MQCDTHIP−S−2бj2x1˜+..+1l˜x˜n∑j=1lx˜i˜MQCDTHIP−P−2бj2xn˜+MQCDTHIP−S−2бj2xn˜1l˜x˜1∑k=1lx˜i˜NQCDTHRP−P−2бk2x1˜+NQCDTHRP−P−2бk2x1˜+⋯+1l˜x˜n∑k=1lx˜i˜NQCDTHRP−P−2бk2xn˜+NQCDTHRP−P−2бk2xn˜1l˜x˜1∑k=1lx˜i˜NQCDTHIP−P−2бk2x1˜+NQCDTHIP−P−2бk2x1˜+⋯+1l˜x˜n∑k=1lx˜i˜NQCDTHIP−P−2бk2xn˜+NQCDTHIP−P−2бk2xn˜=ÇCDTHF−cQCDTH−1,QCDTH−1.ÇCDTHF−cQCDTH−2,QCDTH−2.
ÇCDTHF−cQCDTH−1,QCDTH−2≤ÇCDTHF−cQCDTH−1,QCDTH−11/2ÇCDTHF−cQCDTH−2,QCDTH−21/2; thus 0≤ÇCDTHF−mccQCDTH−1,QCDTH−2≤1. Furthermore, we prove the second part by using equation (18). By hypothesis, it is given that QCDTH−1=QCDTH−2; then MQCDTHRP−P−1бjxi˜=MQCDTHRP−P−2бjxi˜,MQCDTHIP−P−1бjxi˜=MQCDTHIP−P−2бjxi˜,MQCDTHRP−S−1бjxi˜=MQCDTHRP−S−2бjxi˜,MQCDTHIP−S−1бjxi˜=MQCDTHIP−S−2бjxi˜,NQCDTHRP−P−1бk=NQCDTHRP−P−2бk,NQCDTHIP−P−2бkxi˜˜=NQCDTHIP−P−2бkxi˜˜, and NQCDTHRP−S−1бk=NQCDTHRP−S−2бk,NQCDTHIP−S−1бkxi˜˜=NQCDTHIP−S−2бkxi˜˜; then, by using equation (18), we get ÇCDTHF−mccQCDTH−1,QCDTH−2=1. Additionally, we prove the third condition such that(21)ÇCDTHF−mccQCDTH−1,QCDTH−2=∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜MQCDTHRP−P−2бjxi˜+MQCDTHRP−S−1бjxi˜MQCDTHRP−S−2бjxi˜1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜MQCDTHIP−P−2бjxi˜+MQCDTHIP−S−1бjxi˜MQCDTHIP−S−2бjxi˜1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜+NQCDTHRP−P−1бkxi˜NQCDTHRP−P−2бkxi˜1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜+NQCDTHIP−P−1бkxi˜NQCDTHIP−P−2бkxi˜max∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜2+MQCDTHRP−S−1бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜2+MQCDTHIP−S−1бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜2+NQCDTHRP−S−1бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜2+NQCDTHIP−S−1бkxi˜21/2∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜2+MQCDTHRP−S−2бjxi˜21/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜2+MQCDTHIP−S−2бjxi˜21/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−2бkxi˜2+NQCDTHRP−S−2бkxi˜21/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−2бkxi˜2+NQCDTHIP−S−2бkxi˜21/2=∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜MQCDTHRP−P−1бjxi˜+MQCDTHRP−S−2бjxi˜MQCDTHRP−S−1бjxi˜1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜MQCDTHIP−P−1бjxi˜+MQCDTHIP−S−2бjxi˜MQCDTHIP−S−1бjxi˜1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−2бkxi˜NQCDTHRP−P−1бkxi˜+NQCDTHRP−P−2бkxi˜NQCDTHRP−P−1бkxi˜1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−2бkxi˜NQCDTHIP−P−1бkxi˜+NQCDTHIP−P−2бkxi˜NQCDTHIP−P−1бkxi˜max∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−2бjxi˜2+MQCDTHRP−S−2бjxi˜2+1/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−2бjxi˜2+MQCDTHIP−S−2бjxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−2бkxi˜2+NQCDTHRP−S−2бkxi˜2+1/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−2бkxi˜2+NQCDTHIP−S−2бkxi˜21/2∑i=1n1/l˜x˜i∑j=1l˜x˜iMQCDTHRP−P−1бjxi˜2+MQCDTHRP−S−1бjxi˜21/l˜x˜i∑j=1l˜x˜iMQCDTHIP−P−1бjxi˜2+MQCDTHIP−S−1бjxi˜21/l˜x˜i∑k=1l˜x˜iNQCDTHRP−P−1бkxi˜2+NQCDTHRP−S−1бkxi˜21/l˜x˜i∑k=1l˜x˜iNQCDTHIP−P−1бkxi˜2+NQCDTHIP−S−1бkxi˜21/2=ÇCDTHF−mccQCDTH−1,QCDTH−2.Definition 10.
For any two CDT-2HFSs,QCDTH−1=MQCDTHRP−P−1бjxi˜⋅ei2πMQCDTHIP−P−1бjxi˜,MQCDTHRP−S−1бjxi˜.ei2πMQCDTHIP−S−1бjxi˜,NQCDTHRP−P−1бkxi˜⋅ei2πNQCDTHIP−P−1бkxi˜˜,NQCDTHRP−S−1бkxi˜⋅ei2πNQCDTHIP−S−1бkxi˜,j,k=1,2,3,…,n,m and QCDTH−2=MQCDTHRP−P−2бjxi˜⋅ei2πMQCDTHIP−P−2бjxi˜,MQCDTHRP−S−2бjxi˜⋅ei2πMQCDTHIP−S−2бjxi˜,NQCDTHRP−P−2бkxi˜⋅ei2πNQCDTHIP−P−2бkxi˜,NQCDTHRP−S−2бkxi˜⋅ei2πNQCDTHIP−S−2бkxi˜,j,k=1,2,3,…,l˜x˜i,l︷x˜i, the weighted correlation coefficient is of the form(22)ÇCDTHF−wccQCDTH−1,QCDTH−2=ÇCDTHF−wcQCDTH−1−QCDTH−2ÇCDTHF−wcQCDTH−1,QCDTH−11/2×ÇCDTHF−wcQCDTH−1,QCDTH−21/2=∑i=1nΩi1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜MQCDTHRP−P−2бjxι˜+MQCDTHRP−S−1бjxι˜MQCDTHRP−S−2бjxι˜+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜MQCDTHIP−P−2бjxι˜+MQCDTHIP−S−1бjxι˜MQCDTHIP−S−2бjxι˜+1/ι˜x˜i∑j=1ι˜x˜iNQCDTHRP−P−1бkxι˜NQCDTHRP−P−2бkxι˜+NQCDTHRP−S−1бkxι˜NQCDTHRP−S−2бkxι˜+1/ι˜x˜i∑j=1ι˜x˜iNQCDTHIP−P−1бkxι˜NQCDTHIP−P−2бkxι˜+NQCDTHIP−S−1бkxι˜NQCDTHIP−S−2бkxι˜∑i=1nΩi/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜2+MQCDTHRP−S−1бjxι˜2+Ωi/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜2+MQCDTHIP−S−1бjxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−1бkxι˜2+NQCDTHRP−S−1бkxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−1бkxι˜2+NQCDTHIP−S−1бkxι˜2+1/2×∑i=1nΩi/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−2бjxι˜2+MQCDTHRP−S−2бjxι˜2+Ωi/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−2бjxι˜2+MQCDTHIP−S−2бjxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−2бkxι˜2+NQCDTHRP−S−1бkxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−2бkxι˜2+NQCDTHIP−S−1бkxι˜2+1/2,where Ω=Ω1,Ω2,…,ΩnT represents weight vector with the condition that ∑i=1nΩi=1,Ωi∈0,1.Proposition 3.
For any two CDT-2HFSs, QCDTH-1 and QCDTH-2, the CC among CDT-2HFSs satisfies the following axioms:(1)
0≤ÇCDTHF−wccQCDTH−1,QCDTH−2≤1(2)
ÇCDTHF−wccQCDTH−1,QCDTH−2=1⇔QCDTH−1=QCDTH−2(3)
ÇCDTHF−wccQCDTH−1,QCDTH−2=ÇCDTHF−wccQCDTH−2,QCDTH−1Proof.
The proof is straightforward.Definition 11.
For any two CDT-2HFSs,QCDTH−1=MQCDTHRP−P−1бjxι˜⋅ei2πMQCDTHIP−P−1бjxι˜,MQCDTHRP−S−1бjxι˜⋅ei2πMQCDTHIP−S−1бjxι˜,NQCDTHRP−P−1бkxι˜˜⋅ei2πMQCDTHIP−P−1бkxι˜,NQCDTHRP−S−1бkxι˜⋅ei2πMQCDTHIP−S−1бkxι˜,j,k=1,2,3,…,n,m and QCDTH−2=MQCDTHRP−P−2бjxι˜⋅ei2πMQCDTHIP−P−2бjxι˜,MQCDTHRP−S−2бjxι˜⋅ei2πMQCDTHIP−S−2бjxι˜,NQCDTHRP−P−2бjxι˜⋅ei2πMQCDTHIP−P−2бjxι˜,NQCDTHRP−S−2бjxι˜⋅ei2πMQCDTHIP−S−2бjxι˜,j,k=1,2,3,…,ι˜x˜i,ι˜x˜i, the maximum-based weighted correlation coefficient (MCC) is of the form(23)ÇCDTHF−mwccQCDTH−1,QCDTH−2=ÇCDTHF−wcQCDTH−1−QCDTH−2maxÇCDTHF−wcQCDTH−1,QCDTH−11/2×ÇCDTHF−wcQCDTH−1,QCDTH−21/2=∑i=1nΩi1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜MQCDTHRP−P−2бjxι˜+MQCDTHRP−S−1бjxι˜MQCDTHRP−S−2бjxι˜+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜MQCDTHIP−P−2бjxι˜+MQCDTHIP−S−1бjxι˜MQCDTHIP−S−2бjxι˜+1/ι˜x˜i∑j=1ι˜x˜iNQCDTHRP−P−1бkxι˜NQCDTHRP−P−2бkxι˜+NQCDTHRP−S−1бkxι˜NQCDTHRP−S−2бkxι˜+1/ι˜x˜i∑j=1ι˜x˜iNQCDTHIP−P−1бkxι˜NQCDTHIP−P−2бkxι˜+NQCDTHIP−S−1бkxι˜NQCDTHIP−S−2бkxι˜max∑i=1nΩi/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜2+MQCDTHRP−S−1бjxι˜2+Ωi/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜2+MQCDTHIP−S−1бjxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−1бkxι˜2+NQCDTHRP−S−1бkxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−1бkxι˜2+NQCDTHIP−S−1бkxι˜2+1/2×∑i=1nΩi/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−2бjxι˜2+MQCDTHRP−S−2бjxι˜2+Ωi/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−2бjxι˜2+MQCDTHIP−S−2бjxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−2бkxι˜2+NQCDTHRP−S−1бkxι˜2+Ωi/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−2бkxι˜2+NQCDTHIP−S−1бkxι˜2+1/2.where Ω=Ω1,Ω2,…,ΩnT represents weight vector with the condition that ∑i=1nΩi=1,Ωi∈0,1.Proposition 4.
For any two CDT-2HFSs, QCDTH-1 and QCDTH-2, the MCC among CDT-2HFSs satisfies the following axioms:(1)
0≤ÇCDTHF−mwccQCDTH−1,QCDTH−2≤1(2)
ÇCDTHF−mwccQCDTH−1,QCDTH−2=1⇔QCDTH−1=QCDTH−2(3)
ÇCDTHF−mwccQCDTH−1,QCDTH−2=ÇCDTHF−mwccQCDTH−2,QCDTH−1Proof.
The proof is straightforward.
The explored notions, which are stated in the form of equations (13)–(23), are more proficient and more modified than the existing drawbacks; for instance, if we choose the imaginary part of equations (13)–(23) to be zero, then equations (13)–(23) convert it for DT-2HFS.


## 5. Entropy Measures for Complex Dual Type-2 Hesitant Fuzzy Sets

The aim of this study is to present the novel of two types of entropy measures (EMs). The special cases of the explored measures are also explored.Definition 12.
For any two CDT-2HFSsQCDTH−1=MQCDTHRP−P−1бjxι˜⋅ei2πMQCDTHIP−P−1бjxι˜,MQCDTHRP−S−1бjxι˜⋅ei2πMQCDTHIP−S−1бjxι˜,NQCDTHRP−P−1бkxι˜⋅ei2πMQCDTHIP−P−1бkxι˜˜,NQCDTHRP−S−1бkxι˜⋅ei2πMQCDTHIP−S−1бkxι˜,j,k=1,2,3,…,n,m, the two EMs are defined by(24)J1QCDTH−1=1n∑i=1nsin2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜+MQCDTHRP−S−1бjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−1бkxι˜+NQCDTHRP−S−1бkxι˜+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜+MQCDTHIP−S−1бjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−1бkxι˜+NQCDTHIP−S−1бkxι˜8sinπ×2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜+MQCDTHRP−S−1бjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−1бkxι˜+NQCDTHRP−S−1бkxι˜−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜+MQCDTHIP−S−1бjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−1бkxι˜+NQCDTHIP−S−1бkxι˜8−1,(25)J1QCDTH−1=1n∑i=1ncosπ×2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜+MQCDTHRP−S−1бjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−1бkxι˜+NQCDTHRP−S−1бkxι˜−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜+MQCDTHIP−S−1бjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−1бkxι˜+NQCDTHIP−S−1бkxι˜8cosπ×2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−1бjxι˜+MQCDTHRP−S−1бjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−1бkxι˜+NQCDTHRP−S−1бkxι˜−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−1бjxι˜+MQCDTHIP−S−1бjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−1бjkxι˜+NQCDTHIP−S−1бkxι˜8−1.
The two EMs based on CDT-2HFSs has the following properties:(1)
IfMQCDTHRP−P−1бj=MQCDTHRP−S−1бj=1,NQCDTHRP−P−1бk=NQCDTHRP−S−1бk=0,MQCDTHIP−P−1бj=MQCDTHIP−S−1бj=1,NQCDTHIP−P−1бk=NQCDTHIP−S−1бk=0 or MQCDTHRP−P−2бj=MQCDTHRP−S−2бj=1,NQCDTHRP−P−2бk=NQCDTHRP−S−2бk=0,MQCDTHIP−P−2бj=MQCDTHIP−S−2бj=1,NQCDTHIP−P−2бj=NQCDTHIP−S−2бj=0(2)
IfMQCDTHRP−P−1бj,MQCDTHRP−S−1бj≤MQCDTHRP−P−2бj,MQCDTHRP−S−2бj,NQCDTHRP−P−1бk,NQCDTHRP−S−1бk≥NQCDTHRP−P−2бk,NQCDTHRP−S−2бk,MQCDTHIP−P−1бj,MQCDTHIP−S−1бj≤MQCDTHIP−P−1бj,MQCDTHIP−S−1бj,NQCDTHIP−P−1бk,NQCDTHIP−S−1бk≥NQCDTHIP−P−2бk,NQCDTHIP−S−2бk then, J1QCDTH−1≤J1QCDTH−2, and if we change ≤ into ≥, then J1QCDTH−1≥J1QCDTH−2(3)
J1QCDTH−1=J1QCDTH−1c and J2QCDTH−1=J2QCDTH−1c.
The explored notions, which are stated in the form of (23) and (24)fd25, are more proficient and more modified than the existing drawbacks; for instance, if we choose the imaginary part of (23) and (24)fd25 to be zero, then (23) and (24)fd25 convert it for DT-2HFS.


## 6. TOPSIS Method Based on CDT-2HFSs

Basically, a novel TOPSIS method using CC and EM is provided to handle the MADM problems based on Cq-ROFS. Previously, TOPSIS method was proposed based on sample SMs, but in our proposed work we considered the CC and EM. The DM cannot accurately examine the proximity of each alternative to ideal solution in some particular cases. So, we replace the TOPSIS method with the CC instead of DM to check the efficacy and effectiveness of the proposed work.


### 6.1. Problem Description

Consider an MADM problem, whosem alternatives and n attributes are denoted by C=c1,c2,c3,…,cm and U=u1,u2,u3…,un, respectively, with respect to weight vectors represented by Ω=Ω1,Ω2,…,ΩnT with the conditions that Ω ∈ 0, 1 and ∑i=1nΩi=1. Each attribute of each alternative is simplified using CDT-2HFSs QCDTH−z=MQCDTHRP−P−zбkxι˜⋅ei2πMQCDTHIP−P−zбkxι˜,MQCDTHRP−S−zбkxι˜⋅ei2πMQCDTHIP−S−zбkxι˜,NQCDTHRP−P−zбkxι˜⋅ei2πNQCDTHIP−P−zбkxι˜,NQCDTHRP−S−zбkxι˜⋅ei2πNQCDTHIP−S−zбkxι˜,j,k=1,2,3,…,n,m1,2,3,…,n,m,z=1,2,3,…,n, satisfying the following conditions: 0≤maxMQCDTHRP−Pбj+maxNQCDTHRP−Pбk≤1,0≤maxMQCDTHIP−Pбj+maxNQCDTHIP−Pбk≤1,0≤maxMQCDTHRP−Sбj+maxNQCDTHRP−Sбk≤1and0≤maxMQCDTHIP−Sбj+maxNQCDTHIP−Pбk≤1. All the attributes values of the alternatives are in the form of CDT-2HF decision matrix (CDT-2HFDM); that is, QCDTH−yzm×n=MQCDTHRP−P−yzбkxι˜⋅ei2πMQCDTHIP−P−yzбkxι˜,MQCDTHRP−S−yzбkxι˜⋅ei2πMQCDTHIP−S−yzбkxι˜,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜,j,k=1,2,3,…,n,m1,2,3,…,n,m,y,z=1,2,3,…,m,n.Due to tension, time limitations of the decision-makers (DMs), and complication of problems, it is awkward to give the weight information of attribute in advance. To handle such type of issue, we compute the weights of attributes and consider the proposed EM examining the weight of each attribute as(26)Ωj=1−ℋjn−∑i=1nℋj,where ℋj∈0,1,j=1,2,3,…,n, is defined as(27)ℋj=1n∑i=1nsinπ×2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8+sinπ×2−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8−1,(28)ℋj=1n∑i=1ncosπ×2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8+cosπ×2−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8−1When we consider that the imaginary part of (26) and (27) will be zero, (26) and (27) will be converted for DT-2HFS.Procedure for MADM problem based on the above analysis by considering the proposed CDT-2HF TOPSIS method using CC and EM is explained below.


### 6.2. Application

The steps of the CDT-2HF TOPSIS method using WCC are as follows:(i)
Step 1: some decision-making problems also contain benefits and cost types of informations, so for this, we normalized the decision matrix by considering the following formula. We have(29)ryz=MQCDTHRP−P−yzбjxι˜⋅ei2πMQCDTHIP−P−yzбjxι˜,MQCDTHRP−S−yzбjxι˜⋅ei2πMQCDTHIP−S−yzбjxι˜,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜=MQCDTHRP−P−yzбjxι˜⋅ei2πMQCDTHIP−P−yzбjxι˜,MQCDTHRP−S−yzбjxι˜⋅ei2πMQCDTHIP−S−yzбjxι˜,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜forbenefittypes ofattributes,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜,MQCDTHRP−P−yzбjxι˜⋅ei2πMQCDTHIP−P−yzбjxι˜,MQCDTHRP−S−yzбjxι˜⋅ei2πMQCDTHIP−S−yzбjxι˜forcosttypesofattributes.Step 2: by using (25), we examine the weight vector of the attributes.Step 3: by using (30) and (31), we examine the PIS and NIS among the alternatives.(30)R+=ri1+,ri2+,ri3+,…,rin+,rij+maxMQCDTHRP−P−yzбjx˜⋅ei2πMQCDTHIP−P−yzбjx˜,maxMQCDTHRP−S−yzбjx˜⋅ei2πMQCDTHIP−S−yzбjx˜,minNQCDTHRP−P−yzбkx˜⋅ei2πNQCDTHIP−P−yzбkx˜,minNQCDTHRP−S−yzбkx˜⋅ei2πNQCDTHIP−S−yzбkx˜,(31)R−=ri1−,ri2−,ri3−,…,rin−,rij−minMQCDTHRP−P−yzбjx˜⋅ei2πMQCDTHIP−P−yzбjx˜,minMQCDTHRP−S−yzбjx˜⋅ei2πMQCDTHIP−S−yzбjx˜,maxNQCDTHRP−P−yzбkx˜⋅ei2πNQCDTHIP−P−yzбkx˜,maxNQCDTHRP−S−yzбkx˜⋅ei2πNQCDTHIP−S−yzбkx˜.Step 4: by using (21), we examine the CDT-2HF PIS, and we have(32)ÇCDTHF−wcc−yryz,R+.We also examine complex CDT-2HF NIS by using (21), and we have(33)ÇCDTHF−wcc−yryz,R−.Step 5: by using (34), we examine the closeness of each of the alternatives, and we have(34)Pi=ÇCDTHF−wcc−yryz,R+ÇCDTHF−wcc−yryz,R++ÇCDTHF−wcc−yryz,R−⟶.Step 6: we rank all alternatives and examine the best optimal one.Step 7: the end.Example 2.
The company of intranet is usually attacked by malicious intrusions. To enhance the security of the intranet, the company plans to purchase the firewall production and put it between the intranet and extranet for blocking illegal access. Basically, there are four types of firewall productions given to be considered, which are detailed as follows:C=c1,c2,c3,c4. When choosing the firewall production, the company pays attention to the factors detailed as follows: u1⟶the promotion, u2⟶configuration simplicity, u3⟶Security level, and u4⟶maintenance sever level, the weight vector of which is denoted and defined by Ωj∈0,1, ∑j=1nΩj=1, and Ω=Ω1,Ω2,Ω3,Ω4T. To examine the firewall production with respect to their factors, we consider the following matrix, and the decision matrix is given in the form of Table 1.
The steps of the proposed complex dual type-2 hesitant fuzzy TOPSIS method are as follows:(i)
Step 1: some decision-making problems also contain benefits and cost types of informations, so for this, we normalized the decision matrix by considering (29), but the considered information cannot be normalized. So, we have used the information available in Table 1 and go to step 2.(ii)
Step 2: by using (26), we examine the weight vector of the attributes.(35)Ω=0.342,0.155,0.067,0.2,0.236T.(iii)
Step 3: by using (30) and (31), we examine the PIS and NIS among the alternatives.(36)R+=0.62ei2π0.62,0.12ei2π0.12,0.22ei2π0.22,0.24ei2π0.24,0.21ei2π0.21,0.25ei2π0.25,0.23ei2π0.23,0.23ei2π0.23,0.1ei2π0.1,0.23ei2π0.23,0.5ei2π0.5,0.04ei2π0.04,0.31ei2π0.31,0.33ei2π0.33,0.04ei2π0.04,0.43ei2π0.43,0.12ei2π0.120.1ei2π0.1,0.01ei2π0.01,0.01ei2π0.010.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.43ei2π0.34,0.4ei2π0.43,0.5ei2π0.5,0.1ei2π0.1,0.12ei2π0.12,0.2ei2π0.2,0.3ei2π0.3,0.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.4ei2π0.4,0.24ei2π0.6,0.23ei2π0.23,0.2ei2π0.2,0.1ei2π0.1,R−=0.36ei2π0.36,0.12ei2π0.12,0.12ei2π0.12,0.14ei2π0.24,0.1ei2π0.1,0.15ei2π0.15,0.33ei2π0.33,0.33ei2π0.33,0.34ei2π0.34,0.31ei2π0.31,0.15ei2π0.15,0.01ei2π0.01,0.31ei2π0.31,0.13ei2π0.13,0.04ei2π0.04,0.14ei2π0.14,0.32ei2π0.320.31i2π0.31,0.04ei2π0.04,0.4ei2π0.40.12ei2π0.12,0.31ei2π0.31,0.13ei2π0.13,0.14ei2π0.14,0.14ei2π0.4,0.25ei2π0.25,0.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.2,0.33ei2π0.33,0.1ei2π0.1,0.2ei2π0.2,0.2ei2π0.2,0.3ei2π0.3,0.3ei2π0.3,0.4ei2π0.4,0.41ei2π0.41,0.33ei2π0.33,0.32ei2π0.32,0.31ei2π0.31.(iv)
Step 4: by using (22), we examine the complex dual type-2 hesitant fuzzy PIS, and we have(37)K1r1j,R+=0.7782,K2r2j,R+=0.6645,K3r3j,R+=0.6612,K4r4j,R+=0.5534.We also examine complex dual type-2 hesitant NIS by using equation (34), and we have(38)K1r1j,R−=0.7146,K2r2j,R−=0.5627,K3r3j,R−=0.5537,K4r4j,R−=0.5718.(v)
Step 5: by using (32), we examine the closeness of each of the alternatives, and we have(39)P1=0.5213,P2=0.5415,P3=0.5442,P4=0.4918.(v)
Step 6: we rank all alternatives and examine the best optimal one.(40)P3≥P2≥P1≥P4.(vii)
Step 7: the end. The comparative analysis of the explored measure with existing measures [30] is discussed in the form of Table 2.Table 1
Original decision matrix, all items of which are in the form of complex numbers.
Symbolsu1u2c10.6ei2π0.60.1ei2π0.1,0.2ei2π0.2,0.4ei2π0.4,0.1ei2π0.10.5ei2π0.5,0.3ei2π0.30.3ei2π0.3,0.1ei2π0.1,0.3ei2π0.30.5ei2π0.50.01ei2π0.01,0.1ei2π0.10.3ei2π0.3,0.01ei2π0.010.4ei2π0.4,0.2ei2π0.20.1ei2π0.1,0.01ei2π0.01,0.03ei2π0.03c20.61ei2π0.610.11ei2π0.11,0.12ei2π0.120.14ei2π0.14,0.11ei2π0.11,0.15ei2π0.51,0.13ei2π0.130.31ei2π0.13,0.11ei2π0.11,0.13ei2π0.130.15ei2π0.150.02ei2π0.02,0.11ei2π0.110.13ei2π0.13,0.02ei2π0.02,0.14ei2π0.14,0.12ei2π0.120.11ei2π0.11,0.02ei2π0.02,0.01ei2π0.01c30.62ei2π0.62,0.12ei2π0.12,0.22ei2π0.220.24ei2π0.24,0.21ei2π0.21,0.25ei2π0.25,0.23ei2π0.230.23ei2π0.23,0.21ei2π0.21,0.23ei2π0.230.25ei2π0.250.03ei2π0.03,0.21ei2π0.210.23ei2π0.23,0.03ei2π0.3,0.24ei2π0.24,0.22ei2π0.220.21ei2π0.21,0.02ei2π0.2,0.02ei2π0.02c40.36ei2π0.36,0.31ei2π0.13,0.32ei2π0.32,0.34ei2π0.34,0.31ei2π0.31,0.35ei2π0.35,0.33ei2π0.33,0.33ei2π0.33,0.31ei2π0.31,0.33ei2π0.330.35ei2π0.35,0.04ei2π0.04,0.31ei2π0.31,0.33ei2π0.33,0.04ei2π0.04,0.43ei2π0.34;0.32ei2π0.32,0.31ei2π0.31,0.04ei2π0.04,0.04ei2π0.04u3u4c10.2ei2π0.2,0.3ei2π0.3,0.3ei2π0.3,0.4ei2π0.4,0.4ei2π0.4,0.5ei2π0.5,0.01ei2π0.1,0.2ei2π0.2,0.2ei2π0.2,0.3ei2π0.30.1ei2π0.1,0.2ei2π0.6,0.2ei2π0.6,0.3ei2π0.3,0.3ei2π0.3,0.4ei2π0.4,0.4ei2π0.4,0.3ei2π0.3,0.2ei2π0.2,0.1ei2π0.1c20.12ei2π0.12,0.31ei2π0.31,0.13ei2π0.13,0.14ei2π0.14,0.14ei2π0.41,0.51ei2π0.51,0.11ei2π0.11,0.12ei2π0.12,0.21ei2π0.12,0.31ei2π0.310.11ei2π0.11,0.21ei2π0.21,0.21ei2π0.210.31ei2π0.31,0.31ei2π0.31,0.41ei2π0.41,0.41ei2π0.41,0.31ei2π0.31,0.21ei2π0.21,0.11ei2π0.11c30.22ei2π0.22,0.32ei2π0.23,0.23ei2π0.230.24ei2π0.24,0.24ei2π0.42,0.25ei2π0.25,0.21ei2π0.21,0.22ei2π02.2,0.22ei2π0.22,0.23ei2π0.230.21ei2π0.21,0.22ei2π0.22,0.22ei2π0.22,0.23ei2π0.23,0.23ei2π0.23,0.24ei2π0.24,0.24ei2π0.24,0.23ei2π0.23,0.22ei2π0.22,0.12ei2π0.21c40.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.43ei2π0.34,0.34ei2π0.43,0.35ei2π0.35,0.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.32,0.33ei2π0.330.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.34ei2π0.34,0.34ei2π0.34,0.33ei2π0.33,0.32ei2π0.32,0.31ei2π0.31Table 2
Comparison of the explored work with existing work.
MethodScore valuesRankingKaraaslan et al. [30]P1=0.5249,P2=0.5417,P3=0.5439,P4=0.4923P1≥P2≥P1≥P4Explored approachP1=0.5213,P2=0.5415,P3=0.5442,P4=0.4918P3≥P2≥P1≥P4The geometrical representation of the explored measures, which are discussed in Table2, is described with the help of Figure 1.Figure 1
Graphical representation for the information of Table2.


### 6.3. Medical Diagnosis

In this study, we explored the idea of the algorithm which is taken from [31] which is very effective and meaningful for explored works. The new algorithm utilizes the complex dual type-2 hesitant fuzzy similarity and entropy measures which obtain excellent results in application.Problem statement: suppose that five patients, namely, Lil, Jones, Deby, Ramot, and Inas, visit a given laboratory for medical diagnosis. They are observed to have the following symptoms: heart pain, temperature, cough, liver pain, and kidney pain. That is, the set of patients QCDTH is as follows:(41)QCDTH=viz, Lil, Jones, Deby, Ramot, Inas,and the set of symptoms X is as follows:(42)X=x1heart pain,x2temperature,x3cough,x4liver pain,x5kidney pain.Then we will find which patient has which kind of disease. The information related to this problem is given in Example3, which is discussed below.Example 3.
Suppose a set of diagnosesQCDTH=QCDTH−1Heart problem,QCDTH−2Fever,QCDTH−3Flu,QCDTH−4Liver problem,QCDTH−5Kidney problem and a set of symptoms X=x1heart problem,x2temperature,x3cough,x4liver pain,x5kidney pain. Suppose a sick person according to all symptoms is represented by the CDT-2HFSs given as follows:(43)QCDTH=0.91ei2π0.91,0.92ei2π0.91,0.93ei2π0.93,0.94ei2π0.94,0.95ei2π0.95,0.86ei2π0.850.002ei2π0.003,0.004ei2π0.003,0.003ei2π0.003,0.001ei2π0.001.
All diagnosesQCDTH−ii=1,2,3,4,5 that can be represented as CDT-2HFSs according to all symptoms are given below:(44)QCDTH−1heart problem=0.81ei2π0.71,0.76ei2π0.57,0.82ei2π0.72,0.77ei2π0.57,0.85ei2π0.75,0.8ei2π0.61,0.07ei2π0.06,0.06ei2π0.07,0.02ei2π0.02,0.07ei2π0.08,QCDTH−1fever=0.82ei2π0.72,0.77ei2π0.58,0.83ei2π0.73,0.78ei2π0.59,0.86ei2π0.76,0.81ei2π0.62,0.08ei2π0.07,0.07ei2π0.08,0.03ei2π0.03,0.08ei2π0.05,QCDTH−4flu=0.83ei2π0.73,,0.78ei2π0.59,0.837ei2π0.74,0.79ei2π0.6,0.87ei2π0.77,0.82ei2π0.63,0.01ei2π0.01,,0.03ei2π0.06,0.04ei2π0.01,,0.00ei2π0.00,QCDTH−4liver problem=0.83ei2π0.73,,0.78ei2π0.59,0.84ei2π0.74,0.79ei2π0.6,0.87ei2π0.77,0.82ei2π0.63,0.03ei2π0.04,,0.05ei2π0.06,0.06ei2π0.03,,0.02ei2π0.08,QCDTH−4kidney problem=0.84ei2π0.74,,0.77ei2π0.6,0.84ei2π0.75,0.8ei2π0.61,0.88ei2π0.78,0.83ei2π0.64,0.04ei2π0.03,,0.05ei2π0.05,0.06ei2π0.02,,0.03ei2π0.07.
The aim of this work is to examine the best alternative from the family of alternatives by using the measures. The information of the resultant values of the explored measure and some existing measures is stated in Table3. The, ÇCDTH-mcc-iQCDTH−1,QCDTH−2=Pi,i=1,2,3,4,5, are follow as.Table 3
Comparison of the explored work with existing work.
MethodMeasuresRankingKaraaslan et al. [30]P1=0.803,P2=0.853,P3=0.889,P4=0.902,P5=0.943P5≥P4≥P3≥P2≥P1Explored approach, equation (18)P1=0.834,P2=0.851,P3=0.888,P4=0.916,P5=0.951P5≥P4≥P3≥P2≥P1The geometrical representation of the explored measures, which is discussed in Table3, is described with the help of Figure 2.


### 6.4. Pattern Recognition

The instruments of likeness measures have applications in design classiﬁcation. In such a marvel, the class of an obscure example or item is discovered utilizing some likeness estimating devices and a few inclinations of leaders. In this segment, the likeness estimates that grew so far in Section3 are applied to an example acknowledgment (building design acknowledgment) issue, where the class of an obscure structure material should have been assessed. The outcomes got utilizing the similitude proportions of CHFSs are then examined for portrayal of the benefits of proposed work and the constraints of existing work. To clarify the marvel, an illustrative model adjusted from [31] is discussed.Figure 2
Graphical representation for the information of Table3.To evaluate the proficiency of the explored measures, we adopt the pattern recognition model form [31]. The purpose of this application is to find the reliability and skill of the presented measures; we solve a numerical example that contains the CDT-2HFNs and utilized it in the environment of pattern recognition.Example 4.
We consider five knowns with their class labels being represented as follows:PCQ−1,PCQ−2,PCQ−3,PCQ−4,andPCQ−5 and QCDTH−1,QCDTH−2,QCDTH−3,QCDTH−4,andQCDTH−5. The information of the above patterns is in the form of CDT-2HFNs for universal set XUNI=x1,x2,x3,x4,x5, which is stated as follows:(45)QCDTH−1=0.8ei2π0.7,0.75ei2π0.56,0.81ei2π0.71,0.76ei2π0.57,0.84ei2π0.74,0.79ei2π0.60,0.08ei2π0.07,0.07ei2π0.08,0.03ei2π0.03,0.08ei2π0.09,QCDTH−2=0.81ei2π0.71,0.76ei2π0.57,0.82ei2π0.72,0.77ei2π0.58,0.85ei2π0.75,0.8ei2π0.61,0.09ei2π0.08,0.08ei2π0.09,0.04ei2π0.04,0.09ei2π0.06,QCDTH−3=0.82ei2π0.72,0.77ei2π0.58,0.83ei2π0.73,0.78ei2π0.59,0.86ei2π0.76,0.81ei2π0.62,0.02ei2π0.03,0.04ei2π0.05,0.05ei2π0.02,0.01ei2π0.07,QCDTH−4=0.83ei2π0.73,0.78ei2π0.59,0.84ei2π0.74,0.79ei2π0.6,0.87ei2π0.77,0.82ei2π0.63,0.03ei2π0.03,0.05ei2π0.06,0.06ei2π0.03,0.02ei2π0.08,QCDTH−5=0.84ei2π0.74,0.77i2π0.6,0.84ei2π0.75,0.8i2π0.61,0.88ei2π0.78,0.83i2π0.64,0.04ei2π0.03,0.05i2π0.05,0.06ei2π0.02,0.03i2π0.07.
Their unknown pattern is stated as follows:(46)QCDTH=0.9ei2π0.9,0.91ei2π0.91,0.92ei2π0.92,0.93ei2π0.930.94ei2π0.94,0.85ei2π0.850.02ei2π0.03,0.04ei2π0.03,0.03ei2π0.03,0.01ei2π0.01.
The aim of this work is to examine the best alternative from the family of alternatives by using the measures. The information of the resultant values of the explored measure and some existing measures is stated in Table4. The, ÇCDTHF-mcc-iQCCDTH−1,QCDTH−2=Pi,i=1,2,3,4,5, are follow as.
The geometrical representation of the explored measures, which is discussed in Table4, is described with the help of Figure 3.Table 4
Comparison of the explored work with existing work.
MethodMeasuresRankingKaraaslan and Ozlu [30]P1=0.814,P2=0.862,P3=0.892,P4=0.911,P5=0951P5≥P4≥P3≥P2≥P1Explored approach equation (18)P1=0.845,P2=0.866,P3=0.897,P4=0.913,P5=0.947P5≥P4≥P3≥P2≥P1Figure 3
Graphical representation for the information of Table4.


### 6.5. Clustering Algorithm Based on CDT-2HFSs

The aim of this study is to present the clustering algorithm based on the novel approach of CDT-2HFSs to examine the reliability and proficiency of the explored approach. For this, we choose the set of alternatives and their attributes with weight vectors, whose expressions are in the form ofAAL=AAL−1,AAL−2,…,AAL−m,CAT=CAT−1,CAT−2,…,CAT−n, and Ω=Ω1,Ω2,…,ΩnT and Ωj∈0,1,∑j=1nΩj=1. The technique of the clustering algorithm is summarized as follows:Step 1: construct the decision matrix, all entities of which are in the form of CDT-2HFSs.Step 2: construct the correlation matrix by using (12). The correlation matrix is expressed by ÇCDTHF−cc=ÇCDTHF−cc−yzQCDTH−y,QCDTH−z where ÇCDTHF−ccm×m=C.Step 3: we checked whether the correlation matrixC satisfies C2⊆C, where C=C∘C=ÇCDTHF−cc−yzm×m,ÇCDTHF−cc−yz=maxzminÇCDTHF−cc−yx,ÇCDTHF−cc−xz. If C does not satisfy condition C2⊆C, then the equivalent correlation matrix C2k will be formed: C⟶C2⟶C4⟶⋯⟶C2k⟶⋯untilC2k+1.Step 4: furthermore, we examine theλ-cutting matrix by using(47)Cλ=λÇCDTHF−cc−yzm×m=0,ÇCDTHF−cc−yz<λ,1,ÇCDTHF−cc−yz≥λ,whereλ∈0,1 denotes the confidence level.Step 5: we build up all possible classifications based onλ-cutting matrix. If all elements of the yth line (column) in Cλ are the same as the corresponding elements of the zth line (column) in Cλ, then the CDT-2HFS Dy and Dz are of the same type. For simplicity, we draw the graphical shape of the explored clustering algorithm, which is stated with the help of Table 5.Example 5.
(see [30]). Consider a speculation organization that needs to put a total cash in the most ideal choice and along these lines organization officials decide five choices by considering different standards to recognize the best choice to put away the cash: (a) AAL−5 is a structure organization; (b) AAL−2 is an airplane organization; (c) AAL−3 is a food organization; (d) AAL−4 is an electronic things organization; (e) AAL−5 is a cowhide organization; (f) AAL−6 is a vehicle organization; (g) AAL−7 is a correspondence organization; (h) AAL−8 is a product organization; (I) AAL−9 is a paper creation organization; (j) AAL−10 is a plastic creation organization. The speculation organization must make a choice as indicated by the five rules: (a) x1 is the transportation; (b) x2 is the work; (c) x3 is an ecological effect; (d) x4 is the vicinity to crude material; (e) x5 is the experience. The loads of standards x1,x2,x3,x4andx5 are given by Ω1=0.2,Ω2=0.25,Ω3=0.1,Ω4=0.3,Ω5=0.15, individually. The 10 choices are assessed under the standards by etymological evaluations yielded in Table 6 and given by decision‐makers. The technique of the clustering algorithm is summarized as follows:Step 1: we construct the decision matrix, all entities of which are in the form of CDT-2HFSs; see Table5.Step 2: we construct the correlation matrix by using equation (22). The correlation matrix is expressed by ÇCDTHF-cc=ÇCDTHF-cc-yzQCDTH−y,QCDTH−z, where ÇCDTHF−ccm×m=C, such that(48)ÇCDTHF−cc2=10.790.850.810.80.720.750.660.670.70.7910.830.810.750.720.650.650.640.690.850.8310.890.890.880.820.750.740.780.810.810.8910.870.80.760.720.690.790.80.750.890.8710.850.830.730.750.80.720.720.880.80.8510.870.790.830.820.750.650.820.760.830.8710.840.830.850.660.650.750.750.730.790.8410.920.890.670.640.740.690.750.830.830.9210.860.70.690.780.790.80.820.850.890.861.Step 3: we checked whether the correlation matrixC satisfies C2⊆C, where C2=C∘C=ÇCDTHF−cc−yzm×m,ÇCDTHF−cc−yz=maxzminÇCDTHF−cc−yx,ÇCDTHF−cc−xz.(49)ÇCDTHF−cc2=10.830.850.850.850.850.820.750.750.80.8310.830.830.830.830.820.750.750.790.850.8310.890.890.880.870.820.830.820.850.830.8910.890.880.830.790.80.80.850.830.890.8910.880.850.830.830.830.850.830.880.880.8810.870.840.830.850.820.820.870.830.850.8710.850.850.850.750.750.820.790.830.840.8510.920.890.750.750.830.80.830.830.850.9210.890.80.790.820.80.830.850.850.890.891,whereC does not satisfy the condition C2⊆C; then the equivalent correlation matrix C2k will be formed: C⟶C2⟶C4⟶⋯⟶C2k⟶⋯untilC2k=C2k+1. Then(50)ÇCDTHF−cc4=10.830.850.850.850.850.850.840.830.850.8310.830.830.830.830.830.830.830.830.850.8310.890.890.880.870.850.850.850.850.830.8910.890.880.870.840.830.850.850.830.890.8910.880.870.850.850.850.850.830.880.880.8810.870.850.850.850.850.830.870.870.870.8710.850.850.850.840.830.850.840.850.850.8510.920.890.830.830.850.830.850.850.850.9210.890.80.790.820.80.830.850.850.890.891,ÇCDTHF−cc8=10.830.850.850.850.850.850.850.850.850.8310.830.830.830.830.830.830.830.830.850.8310.890.890.880.870.850.850.850.850.830.8910.890.880.870.840.830.850.850.830.890.8910.880.870.850.850.850.850.830.880.880.8810.870.850.850.850.850.830.870.870.870.8710.850.850.850.850.830.850.850.850.850.8510.920.890.850.830.850.850.850.850.850.9210.890.850.830.850.850.850.850.850.890.891,ÇCDTHF−cc16=10.830.850.850.850.850.850.850.850.850.8310.830.830.830.830.830.830.830.830.850.8310.890.890.880.870.850.850.850.850.830.8910.890.880.870.850.850.850.850.830.890.8910.880.870.850.850.850.850.830.880.880.8810.870.850.850.850.850.830.870.870.870.8710.850.850.850.850.830.850.850.850.850.8510.920.890.850.830.850.850.850.850.850.9210.890.850.830.850.850.850.850.850.890.891=ÇCDTHF−cc8.Therefore, we getÇCDTHF−cc16=ÇCDTHF−cc8.Step 4: furthermore, we examine theλ-cutting matrix by using(51)Cλ=λÇCDTHF−cc−yzm×m=0,ÇCDTHF−cc−yz<λ,1,ÇCDTHF−cc−yz≥λ,whereλ∈0,1 denotes the confidence level, such that(52)C0<λ≤0.83=1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111,C0.83<λ≤0.85=1011111111010000000010111111111011111111101111111110111111111011111111101111111110111111111011111111,C0.85<λ≤0.87=1000000000010000000000111110000011111000001111100000111110000011111000000000011100000001110000000111,C0.87<λ≤0.88=1000000000010000000000111110000011111000001111100000111110000000001000000000011100000001110000000111,C0.88<λ≤0.89=1000000000010000000000111000000011100000001110000000000100000000001000000000011100000001110000000111,C0.89<λ≤0.92=1000000000010000000000100000000001000000000010000000000100000000001000000000011100000001110000000111,C0.92<λ≤1=1000000000010000000000100000000001000000000010000000000100000000001000000000010000000000100000000001.Step 5: we build up all possible classifications based onλ-cutting matrix. If all elements of the yth line (column) in Cλ are the same as the corresponding elements of the zth line (column) in Cλ, then the CDT-2HFS Dy and Dz are of the same type. For simplicity, we draw the graphical shape of the explored clustering algorithm, which is stated with the help of Table 5.
Therefore, from the above analysis, we get the result that the explored notions and their measures are more powerful and more proficient than exiting measures. CDT-2HFS is a proficient technique to cope with unpredictable and awkward information in realistic decision problems. CDT-2HFS is composed of the grade of truth and the grade of falsity, and the grade of truth (also for falsity grade) contains the grade of primary and secondary parts in the form of polar coordinates with the condition that the sum of the maximum of the real part (also for imaginary part) of the primary grade (also for secondary grade) cannot exceed the unit interval.Table 5
All possible classifications based onλ-cutting matrices.
ClassificationsRepresentationsLimitationsAAL−i of the same characteristicAAL−1,AAL−2,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−8,AAL−9,AAL−100<λ≤0.83AAL−i of the two characteristicsAAL−1,AAL−2,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−8,AAL−9,AAL−10,AAL−20.83<λ≤0.85AAL−i of the three characteristicsAAL−1,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−2,AAL−8,AAL−9,AAL−100.85<λ≤0.87AAL−i of the five characteristicsAAL−1,AAL−3,AAL−4,AAL−5,AAL−6,AAL−2,AAL−7AAL−8,AAL−9,AAL−100.87<λ≤0.88AAL−i of the six characteristicsAAL−1,AAL−3,AAL−4,AAL−5,,AAL−2,AAL−6,AAL−7AAL−8,AAL−9,AAL−100.88<λ≤0.89AAL−i of the nine characteristicsAAL−1,AAL−3AAL−4,AAL−5AAL−2,AAL−6,AAL−7AAL−8,AAL−9,AAL−100.89<λ≤0.92AAL−i of the ten characteristicsAAL−1,AAL−2,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−8,AAL−9,AAL−100.92<λ≤1Table 6
Common evaluations of alternatives performed by decision-makers.
x1x2x3AAL−10.6,0.1,0.2,0.8,0.1,0.5,0.3,0.6,0.1,0.30.59,0.1,0.4,0.72,0.3,0.2,0.4,0.5,0.3,0.10.7,0.2,0.6,0.1,0.2,0.4,0.3,0.3,0.2,0.0AAL−20.35,0.4,0.2,0.9,0.1,0.1,0.3,0.5,0.2,0.80.4,0.5,0.1,0.9,0.1,0.62,0.5,0.3,0.4,0.30.7,0.2,0.6,0.1,0.2,0.4,0.3,0.3,0.2,0.0AAL−30.65,0.12,0.21,0.7,0.2,0.3,0.35,0.1,0.34,0.0650.7,0.,0.35,0.71,0.3,0.2,0.3,0.1,0.2,0.50.7,0.6,0.6,0.1,0.2,0.4,0.3,0.3,0.15,0.0AAL−40.85,0.12,0.21,0.7,0.2,0.3,0.15,0.1,0.14,0.650.3,0.2,0.15,0.71,0.15,0.1,0.3,0.1,0.2,0.50.5,0.5,0.2,0.4,0.2,0.1,0.3,0.3,0.15,0.0AAL−50.25,0.12,0.21,0.7,0.1,0.3,0.75,0.1,0.14,0.650.6,0.2,0.45,0.71,0.35,0.1,0.3,0.1,0.2,0.50.5,0.8,0.3,0.1,0.14,0.4,0.45,0.0,0.3,0.3AAL−60.35,0.12,0.11,0.9,0.1,0.8,0.55,0.1,0.14,0.650.6,0.3,0.35,0.71,0.25,0.1,0.4,0.1,0.1,0.90.5,0.8,0.3,0.1,0.14,0.4,0.35,0.1,0.3,0.3AAL−70.45,0.12,0.11,0.75,0.1,0.15,0.45,0.1,0.14,0.650.9,0.3,0.25,0.71,0.25,0.1,0.1,0.1,0.0,0.90.5,0.8,0.3,0.9,0.3,0.3,0.35,0.1,0.2,0.9AAL−80.95,0.82,0.11,0.75,0.1,0.55,0.05,0.10,0.14,0.650.9,0.3,0.65,0.1,0.25,0.71,0.1,0.7,0.0,0.90.4,0.8,0.3,0.6,0.2,0.3,0.35,0.1,0.2,0.9AAL−90.55,0.22,0.21,0.75,0.0,0.95,0.45,0.1,0.24,0.150.67,0.1,0.5,0.3,0.25,0.71,0.3,0.9,0.23,0.700.4,0.8,0.3,0.6,0.2,0.3,0.35,0.1,0.2,0.9AAL−100.83,0.82,0.21,0.75,0.1,0.65,0.17,0.1,0.04,0.550.9,0.2,0.25,0.71,0.25,0.1,0.1,0.7,0.0,0.90.7,0.8,0.35,0.65,0.22,0.23,0.25,0.1,0.2,0.9x4x5AAL−10.3,0.1,0.25,0.65,0.2,0.2,0.7,0.3,0.66,0.600.8,0.1,0.27,0.89,0.14,0.5,0.2,0.6,0.1,0.34AAL−20.35,0.45,0.3,0.1,0.1,0.3,0.55,0.1,0.3,0.20.45,0.45,0.2,0.1,0.1,0.4,0.55,0.1,0.5,0.2AAL−30.4,0.1,0.25,0.75,0.2,0.2,0.6,0.3,0.5,0.60.35,0.1,0.35,0.1,0.25,0.15,0.6,0.3,0.3,0.6AAL−40.4,0.1,0.25,0.75,0.2,0.2,0.5,0.6,0.45,0.30.5,0.4,0.48,0.1,0.15,0.75,0.5,0.6,0.35,0.3AAL−50.57,0.1,0.25,0.75,0.23,0.2,0.45,0.6,0.35,0.30.37,0.1,0.25,0.15,0.23,0.25,0.35,0.9,0.33,0.6AAL−60.57,0.1,0.25,0.75,0.23,0.2,0.35,0.3,0.13,0.90.15,0.65,0.15,0.2,0.15,0.1,0.85,0.3,0.13,0.9AAL−70.85,0.1,0.35,0.75,0.25,0.1,0.15,0.3,0.13,0.90.75,0.1,0.25,0.75,0.25,0.1,0.25,0.3,0.23,0.9AAL−80.55,0.1,0.45,0.2,0.05,0.75,0.15,0.3,0.13,0.90.65,0.2,0.55,0.1,0.4,0.45,0.25,0.3,0.03,0.9AAL−90.55,0.1,0.45,0.2,0.05,0.75,0.15,0.3,0.13,0.90.65,0.55,0.35,0.2,0.35,0.1,0.25,0.3,0.23,0.9AAL−100.65,0.25,0.45,0.2,0.45,0.1,0.15,0.35,0.13,0.50.45,0.25,0.35,0.1,0.25,0.2,0.43,0.7,0.11,0.35


## 6.1. Problem Description

Consider an MADM problem, whosem alternatives and n attributes are denoted by C=c1,c2,c3,…,cm and U=u1,u2,u3…,un, respectively, with respect to weight vectors represented by Ω=Ω1,Ω2,…,ΩnT with the conditions that Ω ∈ 0, 1 and ∑i=1nΩi=1. Each attribute of each alternative is simplified using CDT-2HFSs QCDTH−z=MQCDTHRP−P−zбkxι˜⋅ei2πMQCDTHIP−P−zбkxι˜,MQCDTHRP−S−zбkxι˜⋅ei2πMQCDTHIP−S−zбkxι˜,NQCDTHRP−P−zбkxι˜⋅ei2πNQCDTHIP−P−zбkxι˜,NQCDTHRP−S−zбkxι˜⋅ei2πNQCDTHIP−S−zбkxι˜,j,k=1,2,3,…,n,m1,2,3,…,n,m,z=1,2,3,…,n, satisfying the following conditions: 0≤maxMQCDTHRP−Pбj+maxNQCDTHRP−Pбk≤1,0≤maxMQCDTHIP−Pбj+maxNQCDTHIP−Pбk≤1,0≤maxMQCDTHRP−Sбj+maxNQCDTHRP−Sбk≤1and0≤maxMQCDTHIP−Sбj+maxNQCDTHIP−Pбk≤1. All the attributes values of the alternatives are in the form of CDT-2HF decision matrix (CDT-2HFDM); that is, QCDTH−yzm×n=MQCDTHRP−P−yzбkxι˜⋅ei2πMQCDTHIP−P−yzбkxι˜,MQCDTHRP−S−yzбkxι˜⋅ei2πMQCDTHIP−S−yzбkxι˜,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜,j,k=1,2,3,…,n,m1,2,3,…,n,m,y,z=1,2,3,…,m,n.Due to tension, time limitations of the decision-makers (DMs), and complication of problems, it is awkward to give the weight information of attribute in advance. To handle such type of issue, we compute the weights of attributes and consider the proposed EM examining the weight of each attribute as(26)Ωj=1−ℋjn−∑i=1nℋj,where ℋj∈0,1,j=1,2,3,…,n, is defined as(27)ℋj=1n∑i=1nsinπ×2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8+sinπ×2−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8−1,(28)ℋj=1n∑i=1ncosπ×2+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜+1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜−1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8+cosπ×2−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHRP−P−iбjxι˜+MQCDTHRP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHRP−P−iбkxι˜+NQCDTHRP−S−iбkxι˜−1/ι˜x˜i∑j=1ι˜x˜iMQCDTHIP−P−iбjxι˜+MQCDTHIP−S−iбjxι˜+1/ι˜x˜i∑k=1ι˜x˜iNQCDTHIP−P−iбkxι˜+NQCDTHIP−S−iбkxι˜8−1When we consider that the imaginary part of (26) and (27) will be zero, (26) and (27) will be converted for DT-2HFS.Procedure for MADM problem based on the above analysis by considering the proposed CDT-2HF TOPSIS method using CC and EM is explained below.


## 6.2. Application

The steps of the CDT-2HF TOPSIS method using WCC are as follows:(i)
Step 1: some decision-making problems also contain benefits and cost types of informations, so for this, we normalized the decision matrix by considering the following formula. We have(29)ryz=MQCDTHRP−P−yzбjxι˜⋅ei2πMQCDTHIP−P−yzбjxι˜,MQCDTHRP−S−yzбjxι˜⋅ei2πMQCDTHIP−S−yzбjxι˜,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜=MQCDTHRP−P−yzбjxι˜⋅ei2πMQCDTHIP−P−yzбjxι˜,MQCDTHRP−S−yzбjxι˜⋅ei2πMQCDTHIP−S−yzбjxι˜,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜forbenefittypes ofattributes,NQCDTHRP−P−yzбkxι˜⋅ei2πNQCDTHIP−P−yzбkxι˜,NQCDTHRP−S−yzбkxι˜⋅ei2πNQCDTHIP−S−yzбkxι˜,MQCDTHRP−P−yzбjxι˜⋅ei2πMQCDTHIP−P−yzбjxι˜,MQCDTHRP−S−yzбjxι˜⋅ei2πMQCDTHIP−S−yzбjxι˜forcosttypesofattributes.Step 2: by using (25), we examine the weight vector of the attributes.Step 3: by using (30) and (31), we examine the PIS and NIS among the alternatives.(30)R+=ri1+,ri2+,ri3+,…,rin+,rij+maxMQCDTHRP−P−yzбjx˜⋅ei2πMQCDTHIP−P−yzбjx˜,maxMQCDTHRP−S−yzбjx˜⋅ei2πMQCDTHIP−S−yzбjx˜,minNQCDTHRP−P−yzбkx˜⋅ei2πNQCDTHIP−P−yzбkx˜,minNQCDTHRP−S−yzбkx˜⋅ei2πNQCDTHIP−S−yzбkx˜,(31)R−=ri1−,ri2−,ri3−,…,rin−,rij−minMQCDTHRP−P−yzбjx˜⋅ei2πMQCDTHIP−P−yzбjx˜,minMQCDTHRP−S−yzбjx˜⋅ei2πMQCDTHIP−S−yzбjx˜,maxNQCDTHRP−P−yzбkx˜⋅ei2πNQCDTHIP−P−yzбkx˜,maxNQCDTHRP−S−yzбkx˜⋅ei2πNQCDTHIP−S−yzбkx˜.Step 4: by using (21), we examine the CDT-2HF PIS, and we have(32)ÇCDTHF−wcc−yryz,R+.We also examine complex CDT-2HF NIS by using (21), and we have(33)ÇCDTHF−wcc−yryz,R−.Step 5: by using (34), we examine the closeness of each of the alternatives, and we have(34)Pi=ÇCDTHF−wcc−yryz,R+ÇCDTHF−wcc−yryz,R++ÇCDTHF−wcc−yryz,R−⟶.Step 6: we rank all alternatives and examine the best optimal one.Step 7: the end.Example 2.
The company of intranet is usually attacked by malicious intrusions. To enhance the security of the intranet, the company plans to purchase the firewall production and put it between the intranet and extranet for blocking illegal access. Basically, there are four types of firewall productions given to be considered, which are detailed as follows:C=c1,c2,c3,c4. When choosing the firewall production, the company pays attention to the factors detailed as follows: u1⟶the promotion, u2⟶configuration simplicity, u3⟶Security level, and u4⟶maintenance sever level, the weight vector of which is denoted and defined by Ωj∈0,1, ∑j=1nΩj=1, and Ω=Ω1,Ω2,Ω3,Ω4T. To examine the firewall production with respect to their factors, we consider the following matrix, and the decision matrix is given in the form of Table 1.
The steps of the proposed complex dual type-2 hesitant fuzzy TOPSIS method are as follows:(i)
Step 1: some decision-making problems also contain benefits and cost types of informations, so for this, we normalized the decision matrix by considering (29), but the considered information cannot be normalized. So, we have used the information available in Table 1 and go to step 2.(ii)
Step 2: by using (26), we examine the weight vector of the attributes.(35)Ω=0.342,0.155,0.067,0.2,0.236T.(iii)
Step 3: by using (30) and (31), we examine the PIS and NIS among the alternatives.(36)R+=0.62ei2π0.62,0.12ei2π0.12,0.22ei2π0.22,0.24ei2π0.24,0.21ei2π0.21,0.25ei2π0.25,0.23ei2π0.23,0.23ei2π0.23,0.1ei2π0.1,0.23ei2π0.23,0.5ei2π0.5,0.04ei2π0.04,0.31ei2π0.31,0.33ei2π0.33,0.04ei2π0.04,0.43ei2π0.43,0.12ei2π0.120.1ei2π0.1,0.01ei2π0.01,0.01ei2π0.010.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.43ei2π0.34,0.4ei2π0.43,0.5ei2π0.5,0.1ei2π0.1,0.12ei2π0.12,0.2ei2π0.2,0.3ei2π0.3,0.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.4ei2π0.4,0.24ei2π0.6,0.23ei2π0.23,0.2ei2π0.2,0.1ei2π0.1,R−=0.36ei2π0.36,0.12ei2π0.12,0.12ei2π0.12,0.14ei2π0.24,0.1ei2π0.1,0.15ei2π0.15,0.33ei2π0.33,0.33ei2π0.33,0.34ei2π0.34,0.31ei2π0.31,0.15ei2π0.15,0.01ei2π0.01,0.31ei2π0.31,0.13ei2π0.13,0.04ei2π0.04,0.14ei2π0.14,0.32ei2π0.320.31i2π0.31,0.04ei2π0.04,0.4ei2π0.40.12ei2π0.12,0.31ei2π0.31,0.13ei2π0.13,0.14ei2π0.14,0.14ei2π0.4,0.25ei2π0.25,0.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.2,0.33ei2π0.33,0.1ei2π0.1,0.2ei2π0.2,0.2ei2π0.2,0.3ei2π0.3,0.3ei2π0.3,0.4ei2π0.4,0.41ei2π0.41,0.33ei2π0.33,0.32ei2π0.32,0.31ei2π0.31.(iv)
Step 4: by using (22), we examine the complex dual type-2 hesitant fuzzy PIS, and we have(37)K1r1j,R+=0.7782,K2r2j,R+=0.6645,K3r3j,R+=0.6612,K4r4j,R+=0.5534.We also examine complex dual type-2 hesitant NIS by using equation (34), and we have(38)K1r1j,R−=0.7146,K2r2j,R−=0.5627,K3r3j,R−=0.5537,K4r4j,R−=0.5718.(v)
Step 5: by using (32), we examine the closeness of each of the alternatives, and we have(39)P1=0.5213,P2=0.5415,P3=0.5442,P4=0.4918.(v)
Step 6: we rank all alternatives and examine the best optimal one.(40)P3≥P2≥P1≥P4.(vii)
Step 7: the end. The comparative analysis of the explored measure with existing measures [30] is discussed in the form of Table 2.Table 1
Original decision matrix, all items of which are in the form of complex numbers.
Symbolsu1u2c10.6ei2π0.60.1ei2π0.1,0.2ei2π0.2,0.4ei2π0.4,0.1ei2π0.10.5ei2π0.5,0.3ei2π0.30.3ei2π0.3,0.1ei2π0.1,0.3ei2π0.30.5ei2π0.50.01ei2π0.01,0.1ei2π0.10.3ei2π0.3,0.01ei2π0.010.4ei2π0.4,0.2ei2π0.20.1ei2π0.1,0.01ei2π0.01,0.03ei2π0.03c20.61ei2π0.610.11ei2π0.11,0.12ei2π0.120.14ei2π0.14,0.11ei2π0.11,0.15ei2π0.51,0.13ei2π0.130.31ei2π0.13,0.11ei2π0.11,0.13ei2π0.130.15ei2π0.150.02ei2π0.02,0.11ei2π0.110.13ei2π0.13,0.02ei2π0.02,0.14ei2π0.14,0.12ei2π0.120.11ei2π0.11,0.02ei2π0.02,0.01ei2π0.01c30.62ei2π0.62,0.12ei2π0.12,0.22ei2π0.220.24ei2π0.24,0.21ei2π0.21,0.25ei2π0.25,0.23ei2π0.230.23ei2π0.23,0.21ei2π0.21,0.23ei2π0.230.25ei2π0.250.03ei2π0.03,0.21ei2π0.210.23ei2π0.23,0.03ei2π0.3,0.24ei2π0.24,0.22ei2π0.220.21ei2π0.21,0.02ei2π0.2,0.02ei2π0.02c40.36ei2π0.36,0.31ei2π0.13,0.32ei2π0.32,0.34ei2π0.34,0.31ei2π0.31,0.35ei2π0.35,0.33ei2π0.33,0.33ei2π0.33,0.31ei2π0.31,0.33ei2π0.330.35ei2π0.35,0.04ei2π0.04,0.31ei2π0.31,0.33ei2π0.33,0.04ei2π0.04,0.43ei2π0.34;0.32ei2π0.32,0.31ei2π0.31,0.04ei2π0.04,0.04ei2π0.04u3u4c10.2ei2π0.2,0.3ei2π0.3,0.3ei2π0.3,0.4ei2π0.4,0.4ei2π0.4,0.5ei2π0.5,0.01ei2π0.1,0.2ei2π0.2,0.2ei2π0.2,0.3ei2π0.30.1ei2π0.1,0.2ei2π0.6,0.2ei2π0.6,0.3ei2π0.3,0.3ei2π0.3,0.4ei2π0.4,0.4ei2π0.4,0.3ei2π0.3,0.2ei2π0.2,0.1ei2π0.1c20.12ei2π0.12,0.31ei2π0.31,0.13ei2π0.13,0.14ei2π0.14,0.14ei2π0.41,0.51ei2π0.51,0.11ei2π0.11,0.12ei2π0.12,0.21ei2π0.12,0.31ei2π0.310.11ei2π0.11,0.21ei2π0.21,0.21ei2π0.210.31ei2π0.31,0.31ei2π0.31,0.41ei2π0.41,0.41ei2π0.41,0.31ei2π0.31,0.21ei2π0.21,0.11ei2π0.11c30.22ei2π0.22,0.32ei2π0.23,0.23ei2π0.230.24ei2π0.24,0.24ei2π0.42,0.25ei2π0.25,0.21ei2π0.21,0.22ei2π02.2,0.22ei2π0.22,0.23ei2π0.230.21ei2π0.21,0.22ei2π0.22,0.22ei2π0.22,0.23ei2π0.23,0.23ei2π0.23,0.24ei2π0.24,0.24ei2π0.24,0.23ei2π0.23,0.22ei2π0.22,0.12ei2π0.21c40.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.43ei2π0.34,0.34ei2π0.43,0.35ei2π0.35,0.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.32,0.33ei2π0.330.31ei2π0.31,0.32ei2π0.32,0.32ei2π0.32,0.33ei2π0.33,0.33ei2π0.33,0.34ei2π0.34,0.34ei2π0.34,0.33ei2π0.33,0.32ei2π0.32,0.31ei2π0.31Table 2
Comparison of the explored work with existing work.
MethodScore valuesRankingKaraaslan et al. [30]P1=0.5249,P2=0.5417,P3=0.5439,P4=0.4923P1≥P2≥P1≥P4Explored approachP1=0.5213,P2=0.5415,P3=0.5442,P4=0.4918P3≥P2≥P1≥P4The geometrical representation of the explored measures, which are discussed in Table2, is described with the help of Figure 1.Figure 1
Graphical representation for the information of Table2.


## 6.3. Medical Diagnosis

In this study, we explored the idea of the algorithm which is taken from [31] which is very effective and meaningful for explored works. The new algorithm utilizes the complex dual type-2 hesitant fuzzy similarity and entropy measures which obtain excellent results in application.Problem statement: suppose that five patients, namely, Lil, Jones, Deby, Ramot, and Inas, visit a given laboratory for medical diagnosis. They are observed to have the following symptoms: heart pain, temperature, cough, liver pain, and kidney pain. That is, the set of patients QCDTH is as follows:(41)QCDTH=viz, Lil, Jones, Deby, Ramot, Inas,and the set of symptoms X is as follows:(42)X=x1heart pain,x2temperature,x3cough,x4liver pain,x5kidney pain.Then we will find which patient has which kind of disease. The information related to this problem is given in Example3, which is discussed below.Example 3.
Suppose a set of diagnosesQCDTH=QCDTH−1Heart problem,QCDTH−2Fever,QCDTH−3Flu,QCDTH−4Liver problem,QCDTH−5Kidney problem and a set of symptoms X=x1heart problem,x2temperature,x3cough,x4liver pain,x5kidney pain. Suppose a sick person according to all symptoms is represented by the CDT-2HFSs given as follows:(43)QCDTH=0.91ei2π0.91,0.92ei2π0.91,0.93ei2π0.93,0.94ei2π0.94,0.95ei2π0.95,0.86ei2π0.850.002ei2π0.003,0.004ei2π0.003,0.003ei2π0.003,0.001ei2π0.001.
All diagnosesQCDTH−ii=1,2,3,4,5 that can be represented as CDT-2HFSs according to all symptoms are given below:(44)QCDTH−1heart problem=0.81ei2π0.71,0.76ei2π0.57,0.82ei2π0.72,0.77ei2π0.57,0.85ei2π0.75,0.8ei2π0.61,0.07ei2π0.06,0.06ei2π0.07,0.02ei2π0.02,0.07ei2π0.08,QCDTH−1fever=0.82ei2π0.72,0.77ei2π0.58,0.83ei2π0.73,0.78ei2π0.59,0.86ei2π0.76,0.81ei2π0.62,0.08ei2π0.07,0.07ei2π0.08,0.03ei2π0.03,0.08ei2π0.05,QCDTH−4flu=0.83ei2π0.73,,0.78ei2π0.59,0.837ei2π0.74,0.79ei2π0.6,0.87ei2π0.77,0.82ei2π0.63,0.01ei2π0.01,,0.03ei2π0.06,0.04ei2π0.01,,0.00ei2π0.00,QCDTH−4liver problem=0.83ei2π0.73,,0.78ei2π0.59,0.84ei2π0.74,0.79ei2π0.6,0.87ei2π0.77,0.82ei2π0.63,0.03ei2π0.04,,0.05ei2π0.06,0.06ei2π0.03,,0.02ei2π0.08,QCDTH−4kidney problem=0.84ei2π0.74,,0.77ei2π0.6,0.84ei2π0.75,0.8ei2π0.61,0.88ei2π0.78,0.83ei2π0.64,0.04ei2π0.03,,0.05ei2π0.05,0.06ei2π0.02,,0.03ei2π0.07.
The aim of this work is to examine the best alternative from the family of alternatives by using the measures. The information of the resultant values of the explored measure and some existing measures is stated in Table3. The, ÇCDTH-mcc-iQCDTH−1,QCDTH−2=Pi,i=1,2,3,4,5, are follow as.Table 3
Comparison of the explored work with existing work.
MethodMeasuresRankingKaraaslan et al. [30]P1=0.803,P2=0.853,P3=0.889,P4=0.902,P5=0.943P5≥P4≥P3≥P2≥P1Explored approach, equation (18)P1=0.834,P2=0.851,P3=0.888,P4=0.916,P5=0.951P5≥P4≥P3≥P2≥P1The geometrical representation of the explored measures, which is discussed in Table3, is described with the help of Figure 2.


## 6.4. Pattern Recognition

The instruments of likeness measures have applications in design classiﬁcation. In such a marvel, the class of an obscure example or item is discovered utilizing some likeness estimating devices and a few inclinations of leaders. In this segment, the likeness estimates that grew so far in Section3 are applied to an example acknowledgment (building design acknowledgment) issue, where the class of an obscure structure material should have been assessed. The outcomes got utilizing the similitude proportions of CHFSs are then examined for portrayal of the benefits of proposed work and the constraints of existing work. To clarify the marvel, an illustrative model adjusted from [31] is discussed.Figure 2
Graphical representation for the information of Table3.To evaluate the proficiency of the explored measures, we adopt the pattern recognition model form [31]. The purpose of this application is to find the reliability and skill of the presented measures; we solve a numerical example that contains the CDT-2HFNs and utilized it in the environment of pattern recognition.Example 4.
We consider five knowns with their class labels being represented as follows:PCQ−1,PCQ−2,PCQ−3,PCQ−4,andPCQ−5 and QCDTH−1,QCDTH−2,QCDTH−3,QCDTH−4,andQCDTH−5. The information of the above patterns is in the form of CDT-2HFNs for universal set XUNI=x1,x2,x3,x4,x5, which is stated as follows:(45)QCDTH−1=0.8ei2π0.7,0.75ei2π0.56,0.81ei2π0.71,0.76ei2π0.57,0.84ei2π0.74,0.79ei2π0.60,0.08ei2π0.07,0.07ei2π0.08,0.03ei2π0.03,0.08ei2π0.09,QCDTH−2=0.81ei2π0.71,0.76ei2π0.57,0.82ei2π0.72,0.77ei2π0.58,0.85ei2π0.75,0.8ei2π0.61,0.09ei2π0.08,0.08ei2π0.09,0.04ei2π0.04,0.09ei2π0.06,QCDTH−3=0.82ei2π0.72,0.77ei2π0.58,0.83ei2π0.73,0.78ei2π0.59,0.86ei2π0.76,0.81ei2π0.62,0.02ei2π0.03,0.04ei2π0.05,0.05ei2π0.02,0.01ei2π0.07,QCDTH−4=0.83ei2π0.73,0.78ei2π0.59,0.84ei2π0.74,0.79ei2π0.6,0.87ei2π0.77,0.82ei2π0.63,0.03ei2π0.03,0.05ei2π0.06,0.06ei2π0.03,0.02ei2π0.08,QCDTH−5=0.84ei2π0.74,0.77i2π0.6,0.84ei2π0.75,0.8i2π0.61,0.88ei2π0.78,0.83i2π0.64,0.04ei2π0.03,0.05i2π0.05,0.06ei2π0.02,0.03i2π0.07.
Their unknown pattern is stated as follows:(46)QCDTH=0.9ei2π0.9,0.91ei2π0.91,0.92ei2π0.92,0.93ei2π0.930.94ei2π0.94,0.85ei2π0.850.02ei2π0.03,0.04ei2π0.03,0.03ei2π0.03,0.01ei2π0.01.
The aim of this work is to examine the best alternative from the family of alternatives by using the measures. The information of the resultant values of the explored measure and some existing measures is stated in Table4. The, ÇCDTHF-mcc-iQCCDTH−1,QCDTH−2=Pi,i=1,2,3,4,5, are follow as.
The geometrical representation of the explored measures, which is discussed in Table4, is described with the help of Figure 3.Table 4
Comparison of the explored work with existing work.
MethodMeasuresRankingKaraaslan and Ozlu [30]P1=0.814,P2=0.862,P3=0.892,P4=0.911,P5=0951P5≥P4≥P3≥P2≥P1Explored approach equation (18)P1=0.845,P2=0.866,P3=0.897,P4=0.913,P5=0.947P5≥P4≥P3≥P2≥P1Figure 3
Graphical representation for the information of Table4.


## 6.5. Clustering Algorithm Based on CDT-2HFSs

The aim of this study is to present the clustering algorithm based on the novel approach of CDT-2HFSs to examine the reliability and proficiency of the explored approach. For this, we choose the set of alternatives and their attributes with weight vectors, whose expressions are in the form ofAAL=AAL−1,AAL−2,…,AAL−m,CAT=CAT−1,CAT−2,…,CAT−n, and Ω=Ω1,Ω2,…,ΩnT and Ωj∈0,1,∑j=1nΩj=1. The technique of the clustering algorithm is summarized as follows:Step 1: construct the decision matrix, all entities of which are in the form of CDT-2HFSs.Step 2: construct the correlation matrix by using (12). The correlation matrix is expressed by ÇCDTHF−cc=ÇCDTHF−cc−yzQCDTH−y,QCDTH−z where ÇCDTHF−ccm×m=C.Step 3: we checked whether the correlation matrixC satisfies C2⊆C, where C=C∘C=ÇCDTHF−cc−yzm×m,ÇCDTHF−cc−yz=maxzminÇCDTHF−cc−yx,ÇCDTHF−cc−xz. If C does not satisfy condition C2⊆C, then the equivalent correlation matrix C2k will be formed: C⟶C2⟶C4⟶⋯⟶C2k⟶⋯untilC2k+1.Step 4: furthermore, we examine theλ-cutting matrix by using(47)Cλ=λÇCDTHF−cc−yzm×m=0,ÇCDTHF−cc−yz<λ,1,ÇCDTHF−cc−yz≥λ,whereλ∈0,1 denotes the confidence level.Step 5: we build up all possible classifications based onλ-cutting matrix. If all elements of the yth line (column) in Cλ are the same as the corresponding elements of the zth line (column) in Cλ, then the CDT-2HFS Dy and Dz are of the same type. For simplicity, we draw the graphical shape of the explored clustering algorithm, which is stated with the help of Table 5.Example 5.
(see [30]). Consider a speculation organization that needs to put a total cash in the most ideal choice and along these lines organization officials decide five choices by considering different standards to recognize the best choice to put away the cash: (a) AAL−5 is a structure organization; (b) AAL−2 is an airplane organization; (c) AAL−3 is a food organization; (d) AAL−4 is an electronic things organization; (e) AAL−5 is a cowhide organization; (f) AAL−6 is a vehicle organization; (g) AAL−7 is a correspondence organization; (h) AAL−8 is a product organization; (I) AAL−9 is a paper creation organization; (j) AAL−10 is a plastic creation organization. The speculation organization must make a choice as indicated by the five rules: (a) x1 is the transportation; (b) x2 is the work; (c) x3 is an ecological effect; (d) x4 is the vicinity to crude material; (e) x5 is the experience. The loads of standards x1,x2,x3,x4andx5 are given by Ω1=0.2,Ω2=0.25,Ω3=0.1,Ω4=0.3,Ω5=0.15, individually. The 10 choices are assessed under the standards by etymological evaluations yielded in Table 6 and given by decision‐makers. The technique of the clustering algorithm is summarized as follows:Step 1: we construct the decision matrix, all entities of which are in the form of CDT-2HFSs; see Table5.Step 2: we construct the correlation matrix by using equation (22). The correlation matrix is expressed by ÇCDTHF-cc=ÇCDTHF-cc-yzQCDTH−y,QCDTH−z, where ÇCDTHF−ccm×m=C, such that(48)ÇCDTHF−cc2=10.790.850.810.80.720.750.660.670.70.7910.830.810.750.720.650.650.640.690.850.8310.890.890.880.820.750.740.780.810.810.8910.870.80.760.720.690.790.80.750.890.8710.850.830.730.750.80.720.720.880.80.8510.870.790.830.820.750.650.820.760.830.8710.840.830.850.660.650.750.750.730.790.8410.920.890.670.640.740.690.750.830.830.9210.860.70.690.780.790.80.820.850.890.861.Step 3: we checked whether the correlation matrixC satisfies C2⊆C, where C2=C∘C=ÇCDTHF−cc−yzm×m,ÇCDTHF−cc−yz=maxzminÇCDTHF−cc−yx,ÇCDTHF−cc−xz.(49)ÇCDTHF−cc2=10.830.850.850.850.850.820.750.750.80.8310.830.830.830.830.820.750.750.790.850.8310.890.890.880.870.820.830.820.850.830.8910.890.880.830.790.80.80.850.830.890.8910.880.850.830.830.830.850.830.880.880.8810.870.840.830.850.820.820.870.830.850.8710.850.850.850.750.750.820.790.830.840.8510.920.890.750.750.830.80.830.830.850.9210.890.80.790.820.80.830.850.850.890.891,whereC does not satisfy the condition C2⊆C; then the equivalent correlation matrix C2k will be formed: C⟶C2⟶C4⟶⋯⟶C2k⟶⋯untilC2k=C2k+1. Then(50)ÇCDTHF−cc4=10.830.850.850.850.850.850.840.830.850.8310.830.830.830.830.830.830.830.830.850.8310.890.890.880.870.850.850.850.850.830.8910.890.880.870.840.830.850.850.830.890.8910.880.870.850.850.850.850.830.880.880.8810.870.850.850.850.850.830.870.870.870.8710.850.850.850.840.830.850.840.850.850.8510.920.890.830.830.850.830.850.850.850.9210.890.80.790.820.80.830.850.850.890.891,ÇCDTHF−cc8=10.830.850.850.850.850.850.850.850.850.8310.830.830.830.830.830.830.830.830.850.8310.890.890.880.870.850.850.850.850.830.8910.890.880.870.840.830.850.850.830.890.8910.880.870.850.850.850.850.830.880.880.8810.870.850.850.850.850.830.870.870.870.8710.850.850.850.850.830.850.850.850.850.8510.920.890.850.830.850.850.850.850.850.9210.890.850.830.850.850.850.850.850.890.891,ÇCDTHF−cc16=10.830.850.850.850.850.850.850.850.850.8310.830.830.830.830.830.830.830.830.850.8310.890.890.880.870.850.850.850.850.830.8910.890.880.870.850.850.850.850.830.890.8910.880.870.850.850.850.850.830.880.880.8810.870.850.850.850.850.830.870.870.870.8710.850.850.850.850.830.850.850.850.850.8510.920.890.850.830.850.850.850.850.850.9210.890.850.830.850.850.850.850.850.890.891=ÇCDTHF−cc8.Therefore, we getÇCDTHF−cc16=ÇCDTHF−cc8.Step 4: furthermore, we examine theλ-cutting matrix by using(51)Cλ=λÇCDTHF−cc−yzm×m=0,ÇCDTHF−cc−yz<λ,1,ÇCDTHF−cc−yz≥λ,whereλ∈0,1 denotes the confidence level, such that(52)C0<λ≤0.83=1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111,C0.83<λ≤0.85=1011111111010000000010111111111011111111101111111110111111111011111111101111111110111111111011111111,C0.85<λ≤0.87=1000000000010000000000111110000011111000001111100000111110000011111000000000011100000001110000000111,C0.87<λ≤0.88=1000000000010000000000111110000011111000001111100000111110000000001000000000011100000001110000000111,C0.88<λ≤0.89=1000000000010000000000111000000011100000001110000000000100000000001000000000011100000001110000000111,C0.89<λ≤0.92=1000000000010000000000100000000001000000000010000000000100000000001000000000011100000001110000000111,C0.92<λ≤1=1000000000010000000000100000000001000000000010000000000100000000001000000000010000000000100000000001.Step 5: we build up all possible classifications based onλ-cutting matrix. If all elements of the yth line (column) in Cλ are the same as the corresponding elements of the zth line (column) in Cλ, then the CDT-2HFS Dy and Dz are of the same type. For simplicity, we draw the graphical shape of the explored clustering algorithm, which is stated with the help of Table 5.
Therefore, from the above analysis, we get the result that the explored notions and their measures are more powerful and more proficient than exiting measures. CDT-2HFS is a proficient technique to cope with unpredictable and awkward information in realistic decision problems. CDT-2HFS is composed of the grade of truth and the grade of falsity, and the grade of truth (also for falsity grade) contains the grade of primary and secondary parts in the form of polar coordinates with the condition that the sum of the maximum of the real part (also for imaginary part) of the primary grade (also for secondary grade) cannot exceed the unit interval.Table 5
All possible classifications based onλ-cutting matrices.
ClassificationsRepresentationsLimitationsAAL−i of the same characteristicAAL−1,AAL−2,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−8,AAL−9,AAL−100<λ≤0.83AAL−i of the two characteristicsAAL−1,AAL−2,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−8,AAL−9,AAL−10,AAL−20.83<λ≤0.85AAL−i of the three characteristicsAAL−1,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−2,AAL−8,AAL−9,AAL−100.85<λ≤0.87AAL−i of the five characteristicsAAL−1,AAL−3,AAL−4,AAL−5,AAL−6,AAL−2,AAL−7AAL−8,AAL−9,AAL−100.87<λ≤0.88AAL−i of the six characteristicsAAL−1,AAL−3,AAL−4,AAL−5,,AAL−2,AAL−6,AAL−7AAL−8,AAL−9,AAL−100.88<λ≤0.89AAL−i of the nine characteristicsAAL−1,AAL−3AAL−4,AAL−5AAL−2,AAL−6,AAL−7AAL−8,AAL−9,AAL−100.89<λ≤0.92AAL−i of the ten characteristicsAAL−1,AAL−2,AAL−3,AAL−4,AAL−5,AAL−6,AAL−7,AAL−8,AAL−9,AAL−100.92<λ≤1Table 6
Common evaluations of alternatives performed by decision-makers.
x1x2x3AAL−10.6,0.1,0.2,0.8,0.1,0.5,0.3,0.6,0.1,0.30.59,0.1,0.4,0.72,0.3,0.2,0.4,0.5,0.3,0.10.7,0.2,0.6,0.1,0.2,0.4,0.3,0.3,0.2,0.0AAL−20.35,0.4,0.2,0.9,0.1,0.1,0.3,0.5,0.2,0.80.4,0.5,0.1,0.9,0.1,0.62,0.5,0.3,0.4,0.30.7,0.2,0.6,0.1,0.2,0.4,0.3,0.3,0.2,0.0AAL−30.65,0.12,0.21,0.7,0.2,0.3,0.35,0.1,0.34,0.0650.7,0.,0.35,0.71,0.3,0.2,0.3,0.1,0.2,0.50.7,0.6,0.6,0.1,0.2,0.4,0.3,0.3,0.15,0.0AAL−40.85,0.12,0.21,0.7,0.2,0.3,0.15,0.1,0.14,0.650.3,0.2,0.15,0.71,0.15,0.1,0.3,0.1,0.2,0.50.5,0.5,0.2,0.4,0.2,0.1,0.3,0.3,0.15,0.0AAL−50.25,0.12,0.21,0.7,0.1,0.3,0.75,0.1,0.14,0.650.6,0.2,0.45,0.71,0.35,0.1,0.3,0.1,0.2,0.50.5,0.8,0.3,0.1,0.14,0.4,0.45,0.0,0.3,0.3AAL−60.35,0.12,0.11,0.9,0.1,0.8,0.55,0.1,0.14,0.650.6,0.3,0.35,0.71,0.25,0.1,0.4,0.1,0.1,0.90.5,0.8,0.3,0.1,0.14,0.4,0.35,0.1,0.3,0.3AAL−70.45,0.12,0.11,0.75,0.1,0.15,0.45,0.1,0.14,0.650.9,0.3,0.25,0.71,0.25,0.1,0.1,0.1,0.0,0.90.5,0.8,0.3,0.9,0.3,0.3,0.35,0.1,0.2,0.9AAL−80.95,0.82,0.11,0.75,0.1,0.55,0.05,0.10,0.14,0.650.9,0.3,0.65,0.1,0.25,0.71,0.1,0.7,0.0,0.90.4,0.8,0.3,0.6,0.2,0.3,0.35,0.1,0.2,0.9AAL−90.55,0.22,0.21,0.75,0.0,0.95,0.45,0.1,0.24,0.150.67,0.1,0.5,0.3,0.25,0.71,0.3,0.9,0.23,0.700.4,0.8,0.3,0.6,0.2,0.3,0.35,0.1,0.2,0.9AAL−100.83,0.82,0.21,0.75,0.1,0.65,0.17,0.1,0.04,0.550.9,0.2,0.25,0.71,0.25,0.1,0.1,0.7,0.0,0.90.7,0.8,0.35,0.65,0.22,0.23,0.25,0.1,0.2,0.9x4x5AAL−10.3,0.1,0.25,0.65,0.2,0.2,0.7,0.3,0.66,0.600.8,0.1,0.27,0.89,0.14,0.5,0.2,0.6,0.1,0.34AAL−20.35,0.45,0.3,0.1,0.1,0.3,0.55,0.1,0.3,0.20.45,0.45,0.2,0.1,0.1,0.4,0.55,0.1,0.5,0.2AAL−30.4,0.1,0.25,0.75,0.2,0.2,0.6,0.3,0.5,0.60.35,0.1,0.35,0.1,0.25,0.15,0.6,0.3,0.3,0.6AAL−40.4,0.1,0.25,0.75,0.2,0.2,0.5,0.6,0.45,0.30.5,0.4,0.48,0.1,0.15,0.75,0.5,0.6,0.35,0.3AAL−50.57,0.1,0.25,0.75,0.23,0.2,0.45,0.6,0.35,0.30.37,0.1,0.25,0.15,0.23,0.25,0.35,0.9,0.33,0.6AAL−60.57,0.1,0.25,0.75,0.23,0.2,0.35,0.3,0.13,0.90.15,0.65,0.15,0.2,0.15,0.1,0.85,0.3,0.13,0.9AAL−70.85,0.1,0.35,0.75,0.25,0.1,0.15,0.3,0.13,0.90.75,0.1,0.25,0.75,0.25,0.1,0.25,0.3,0.23,0.9AAL−80.55,0.1,0.45,0.2,0.05,0.75,0.15,0.3,0.13,0.90.65,0.2,0.55,0.1,0.4,0.45,0.25,0.3,0.03,0.9AAL−90.55,0.1,0.45,0.2,0.05,0.75,0.15,0.3,0.13,0.90.65,0.55,0.35,0.2,0.35,0.1,0.25,0.3,0.23,0.9AAL−100.65,0.25,0.45,0.2,0.45,0.1,0.15,0.35,0.13,0.50.45,0.25,0.35,0.1,0.25,0.2,0.43,0.7,0.11,0.35


## 7. Conclusion

The theory of CDT-2HFS is a mixture of two different modifications of FS, called CFS and DT-2HFS. CDT-2HFS is a proficient technique to cope with unpredictable and awkward information in realistic decision problems. The intention of this manuscript is to determine the novel methodology of CDT-2HFSs and to discuss their operational laws. These operational laws are also defensible with the assistance of examples. Furthermore, based on novel CDT-2HFS, we reconnoitered the CC and EMs, and their special cases are also discussed. TOPSIS method based on CDT-2HFS is also explored. Then, we applied our explored measures based on CDT-2HFSs in the environment of TOPSIS method, medical diagnosis, pattern recognition, and clustering algorithm to cope with awkward and complicated information in realistic decision issues. Finally, some numerical examples are given and discussed to examine the proficiency and validity of the explored measures. Comparative analysis, advantages, and graphical interpretation of the explored measures with some other existing measures are also discussed.In the future, the concept of complex dual type-2 hesitant fuzzy sets can be applied to group MADM problems. Moreover, the problems discussed in this manuscript can be discussed in the environment of complex q-rung orthopair fuzzy sets [32–39], T-spherical fuzzy sets [40, 41], and some others [42–45].



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