# Correlation Measures for Cubic m-Polar Fuzzy Sets with Applications

Authors: Harish Garg; Muhammad Riaz; Muhammad Abdullah Khokhar; Maryam Saba
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A cubicm-polar fuzzy set (CmPFS) is a new hybrid extension of m-polar fuzzy set and cubic set. A CmPFS is a robust model to express multipolar information in terms of m fuzzy intervals representing membership grades and m fuzzy numbers representing nonmembership grades. In this article, we explore some new operational laws of CmPFSs, produce some related results, and discuss their consequences. We propose relative informational coefficients and relative noninformational coefficients for CmPFSs. These coefficients are analyzed to investigate further properties of CmPFSs. Based on these coefficients, we introduce new correlation measures and their weighted versions for CmPFSs. The value of proposed correlation measures is symmetrical and lies between −1 and 1. Moreover, the applications of the proposed correlation in pattern recognition and medical diagnosis are developed. The feasibility and efficiency of suggested correlation measures is determined by respective illustrative examples.


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## Body

## 1. Introduction

Modern systems and logic are the schematic study of the rules that can lead to the acceptance of a certain proposition on the basis of preassumed hypothesis. The development of the modern systems owes to the advancement of set theory and logic. To solve uncertain problems of real-life, the researchers have been established various sets and models. In 1965, Zadeh [1] extended crisp set to fuzzy set (FS) by assigning a real number in interval 0,1 which dictates the membership value of an element/object in the crisp set. Zadeh’s pioneer work has a revolutionary importance during the last decades as it laid a foundation to the modern “fuzzy set theory (FST) and its applications.” This theory, upon being uplifted to a certain degree, turned to fuzzy logics. Fuzzy logic and fuzzy sets have a large number of applications in computational intelligence, robotics, neural networks, data sciences, and pattern recognition. The idea of fuzzification proved to be an open end for the scientists. A lot of extensions of FST have been introduced so far. The notions of bipolar fuzzy set (BPFS) [2] and intuitionistic fuzzy set (IFS) [3] are both robust extensions of FS. An IFS assigns a pair of membership value and a nonmembership value which lie between 0 and 1. A BFS is a direct extension of FS which gives information which is bipolar in nature. Its positive membership value μ+ lies in the interval 0,1 and negative membership value μ− lies in the interval −1,0. The positive scores depict the consent of a certain property, and the negative scores assigned reflect the consent of a counter property. However, an important point to be noted here is that which type of necessity served as a mother of this extension, that is, which limitation led to this work? The innovative idea of BFS correctly addresses this problem. It should be acknowledged that the individual viewpoint or reasoning is not limited to bipolarity. There can be more than two independent view points to a certain crisp item. For example, in an electoral system, a vote being casted may have exact degree from 0,1m. These degrees may be assigned on the basis of personal mindset, honesty of the candidate, leadership qualities, education, and future plans. Therefore, bipolar fuzzy theory cannot answer this sort of problem and there is a need to invent a new extension. To address this limitation, m-polar fuzzy set (mPFS) theory has been proposed by Chen et al. [4]. An m-polar fuzzy set assigns m independent values/degrees to each crisp element/object.Jun et al. [5] introduced the concept of cubic set (CS) as an extension of intuitionistic fuzzy set (IFS). A CS assigns a membership grade which is a fuzzy interval and nonmembership which is fuzzy value to each crisp element/object. Besides, both types of grades are condition free. However, CS allows each of its alternatives to follow one attribute only. Multiattributed information is denied in CS theory. This is the main reason and answer to the important question “why cubic m-polar fuzzy set (CmPFS) theory has been developed.” The CmPFS (initiated by Riaz and Hashmi [6]) is a new hybrid structure of CS and mPFS, and it manipulates not only multiattributed but also cubic information as well. Therefore, CS and mPFS are the special cases of CmPFS.Some of the important extensions of fuzzy set are Pythagorean fuzzy set (PFS) [7], q-rung orthopair fuzzy set (q-ROFS) [8], neutrosophic set [9], single-valued neutrosophic set [10], spherical fuzzy set [11–13], and picture fuzzy sets (PicFSs) [14]. Soft set [15] is a robust independent extension of crisp sets for modeling uncertainty problems. Wang et al. [16] introduced Pythagorean fuzzy interactive Hamacher power aggregation operators for assessment of express service quality with entropy weight. Liu et al. [17] suggested Banzhaf–Choquet copula-based aggregation operators for managing q-ROF information. Akram et al. [18] established complex spherical fuzzy prioritized weighted aggregation operators. Sitara et al. [19] proposed novel decision-making analysis based on q-rung picture fuzzy graph structures. Garg [20, 21] developed MAGDM process based on operators for intuitionistic multiplicative set and interval-valued picture uncertain linguistic generalized Hamacher aggregation operators. Liu et al. [22] proposed group decision-making using complex q-ROF Bonferroni mean. Liu and Wang [23] established a new MAGDM method based on IF Einstein interactive operations.Recently, Riaz et al. [24, 25] introduced some new extensions of fuzzy sets named as linear Diophantine fuzzy set (LDFS) and spherical linear Diophantine sets. Kamaci [26] introduced algebraic structure on LDFS with an interesting application to coding theory which is based on LDFS codes. Ayub et al. [27] introduced LDF-relations and their algebraic properties with decision-making application. Almagrabi et al. [28] presented a new approach to the q-LDF emergency decision support system for COVID-19.Correlation and similarity measures are significant mathematical tools to deal with real-life problems. The development of fuzzy set theories give rise to new extension of correlation and similarity measures. The researchers have been used these measures to develop new techniques for decision-making, pattern recognition, and cluster analysis. Ganie and Singh [29] used similarity measures for MCDM in picture fuzzy environment. Mahmood and Ali [30] used correlation measures to propose the extended TOPSIS method for Pythagorean fuzzy environment. Zulqarnain et al. [31] proposed correlated TOPSIS for the mask selection.Correlation is a statistic that assesses the extent to which two objects move in relation to each other whereas similarity measure is a function which measures the closeness of two objects. Correlation gives some important applications to statistical analysis. Statistics has effectively used probability methods for dealing with many real space issues focused on random processing of data. Similarly, some statistical problems in engineering sciences have also been addressed by the same probabilistic approaches. However, these approaches have some drawbacks, since these methods are based on random sample selection to obtain a certain degree of trust. When the data are collected on a large-scale, massive fuzziness is involved therein which makes it difficult to optimize the idealistic probability. The main question is that how to overcome such obstacles to obtain idealistic results? The answer to this question is somewhat complicated. The first thing to be noted is that in every real-world problem, no matter it is an engineering, medical, or administrative issue, uncertainty always finds its place. In statistics, correlation is one of the most important concepts we encounter with and is digested by many other subjects such as economics, finance, engineering, and management sciences. However, the problem is that this type of correlation can cope with the crisp data only and we have been managing many real-life problems with the help of this without taking into account uncertainty. Since the birth of fuzzy set, many uncertain settings have been introduced to tackle variability. We have seen above that a lot of ambiguous sets have been defined so far to handle imprecise information. Therefore, there is a need to introduce correlation measures in every environment because a real problem may be diverse in its uncertain nature. To address this issue, Ejegwa et al. [32], Gerstenkorn and Mańko [33], and Szmidt and Kacprzyk [34] proposed some correlation measures for IFSs. Garg et al. [35] presented the correlation coefficient for cubic intuitionistic fuzzy set and hence presented a decision-making algorithm. Correlation measures are used in PFSs. Lin et al. [36] and Thao [37] also discussed limitations of the previously defined correlations and put forward new directional correlations for PFSs.The main objectives of the manuscript are listed as follows:(1)
A cubicm-polar fuzzy set (CmPFS) is a robust fusion of cubic sets and m-polar fuzzy set to solve more complicated decision-making problems in the multipolar and cubic environment. This model is more suitable to express vague information efficiently, and the decision-makers find more flexibility to give their assessment towards multicriterion, multiattribute, multisensor, multisystems, and multirelations.(2)
To define new operations on CmPFSs and discuss related results.(3)
To develop relative informational coefficients and relative noninformational coefficients for CmPFSs.(4)
To analyze these coefficients to investigate further properties of CmPFSs.(5)
To define new correlation measures and their weighted versions for CmPFSs.(6)
To construct new algorithms with the help of new correlation measures for solving the complicated problems of pattern recognition and medical diagnosis which cannot be tackled by existing techniques. The feasibility and efficiency of suggested correlation measures is determined by respective illustrative examples.Figure1 demonstrates novel hybrid extension of cubic set and m-polar fuzzy set named as cubic m-polar fuzzy set.Figure 1
Cubicm-polar fuzzy set: a hybrid model.This paper is organized as follows. In Section2, we first recall few basic concepts of fuzzy sets, cubic sets, and m-polar fuzzy sets. Then, we review definition and operations of CmPFSs. In Section 3, we discuss some novel features of CmPFSs depending on new operations. We also define support, core, height, level sets, and binary relations. In Section 4, we define correlation measures for CmPFSs. Their weighted versions have also been addressed. We prove that all types of correlation measures are symmetrical and lie in the interval −1,1, the correlation of two CmPFSs is 1 only if either both sets are equal or one of the sets is a positive linear multiple of the other, and the correlation of the two sets is −1 only if one set is a positive linear multiple of the complement of the other set. In Section 5, new algorithms are proposed using proposed correlation measures under CmPF environment for solving the complicated problems of pattern recognition and medical diagnosis. The feasibility and efficiency of suggested correlation measures is determined by respective illustrative examples. The conclusion of this research studies is presented in Section 6.


## 2. Preliminaries

Some basic definitions and operations are discussed here in this section which can serve foundation for current research work. We shall useℵ as a universal space.Definition 1 (see [1]).
A fuzzy set (FS) on a universeℵ is a rule which assigns a number from the interval 0,1 to each element in ℵ. This number is referred to as membership grade of the crisp element. Mathematically, an FS is described by the membership function μ:ℵ⟶0,1.Definition 2 (see [4]).
Anm-polar fuzzy set (mPFS) on a universe ℵ is defined by(1)M=κ,μiκi=1m:κ∈ℵ=κ,μ1κ,μ2κ,…,μmκ:κ∈ℵ,where μi:0,1m⟶0,1;i=1,2,…,m, is the ith projection mapping.
Anm-polar fuzzy set is a direct extension of FS that describes the multipolarity of the objects. An mPFS assigns m fuzzy grades to each object/element of universe ℵ.Definition 3 (see [5]).
A cubic set (CS) is defined by a rule which assigns to each element/object of the discourseℵ a subinterval of 0,1 called membership value which is a fuzzy subinterval and a nonmembership value from 0,1. Mathematically, we can write(2)CSℵ=κ,Aκ,Bκ:κ∈ℵ,where Aκ is a fuzzy subinterval and Bκ is a fuzzy value assigned to the alternative κ. There is no restriction on Bκ to be a number from Aκ. If, however, Bκ∈Aκ, then CSℵ is called internal cubic set (ICS). Otherwise, it is called external cubic set (ECS). Thus, cubic information is of two categories, that is, internal and external. So, whenever we deal with cubic sets, we must ensure its behaviour.Definition 4 (see [6]).
A cubicm-polar fuzzy set (CmPFS) is a set which deals with multipolar information in cubic environment. It assigns to each member of the crisp discourse a cubic element allowing multipolarity. If we take ℵ as universe of discourse, the CmPFS can be viewed mathematically as follows:(3)CmPFSℵ=κ,ηi−κ,ηi+κ,ηiκi=1m.
For convenience, the termηj−,ηj+,ηjj=1m is referred to as cubic m-polar fuzzy number (CmPFN).


### 2.1. Operations on CmPFSs

LetAℵ1=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ and Aℵ2=κ,ζi−κ,ζi+κ,ζiκi=1m:κ∈ℵ be the CmPFSs. Then, the operations for these sets under P-ordering are defined as follows:(1)
Aℵ1=Aℵ2⇔ηi−κ=ζi−κ,ηi+κ=ζi+κandηiκ=ζiκ. For all i=1,2,…,m and κ∈ℵ(2)
Aℵ1c=κ,1−ηi+κ,1−ηi−κ,1−ηiκi=1m:κ∈ℵ(3)
Aℵ1∪PAℵ2=κ,ηi−κ∨ζi−κ,ηi+κ∨ζi+κ,ηiκ∨ζiκi=1m:κ∈ℵ(4)
Aℵ1∩PAℵ2=κ,ηi−κ∧ζi−κ,ηi+κ∧ζi+κ,ηiκ∧ζiκi=1m:κ∈ℵ(5)
Aℵ1⊆PAℵ2⇔ηi−κ≤ζi−κ,ηi+κ≤ζi+κandηiκ≤ζiκ. For all i=1,2,…,m and κ∈ℵ(6)
Aℵ1⊕PAℵ2=κ,ηi−κ+ζi−κ−ηi−κζi−κ,ηi+κ+ζi+κ−ηi+κζi+κ,ηiκ+ζiκ−ηiκζiκi=1m:κ∈ℵ(7)
Aℵ1⊗PAℵ2=κ,ηi−κζi−κ,ηi+κζi+κ,ηiκζiκi=1m:κ∈ℵ(8)
ςAℵ1=κ,1−1−ηi−κς,1−1−ηi+κς,1−1−ηiκςi=1m:κ∈ℵ(9)
Aℵ1ς=κ,ηi−κς,ηi+κς,ηiκςi=1m:κ∈ℵThe R-ordering operations for the abovementioned CmPFSs are given as follows:(Note: the equality and complement operations for these sets are universal; that is, they are independent or P/R-ordering.)(1)
Aℵ1∪RAℵ2=κ,ηi−κ∨ζi−κ,ηi+κ∨ζi+κ,ηiκ∧ζiκi=1m:κ∈ℵ(2)
Aℵ1∩RAℵ2=κ,ηi−κ∧ζi−κ,ηi+κ∧ζi+κ,ηiκ∨ζiκi=1m:κ∈ℵ(3)
Aℵ1⊆RAℵ2⇔ηi−κ≤ζi−κ,ηi+κ≤ζi+κandηiκ≥ζiκ. For all i=1,2,…,m and κ∈ℵ.(4)
Aℵ1⊕RAℵ2=κ,ηi−κ+ζi−κ−ηi−κζi−κ,ηi+κ+ζi+κ−ηi+κζi+κ,ηiκζiκi=1m:κ∈ℵ(5)
Aℵ1⊗RAℵ2=κ,ηi−κζi−κ,ηi+κζi+κ,ηiκ+ζiκ−ηiκζiκi=1m:κ∈ℵ(6)
ςAℵ1=κ,1−1−ηi−κς,1−1−ηi+κς,ηiκςi=1m:κ∈ℵ(7)
Aℵ1ς=κ,ηi−κς,ηi+κς,1−1−ηiκςi=1m:κ∈ℵNote: it is interesting to note that the collection of all CmPFSs on a universeℵ is closed under these operations.


## 2.1. Operations on CmPFSs

LetAℵ1=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ and Aℵ2=κ,ζi−κ,ζi+κ,ζiκi=1m:κ∈ℵ be the CmPFSs. Then, the operations for these sets under P-ordering are defined as follows:(1)
Aℵ1=Aℵ2⇔ηi−κ=ζi−κ,ηi+κ=ζi+κandηiκ=ζiκ. For all i=1,2,…,m and κ∈ℵ(2)
Aℵ1c=κ,1−ηi+κ,1−ηi−κ,1−ηiκi=1m:κ∈ℵ(3)
Aℵ1∪PAℵ2=κ,ηi−κ∨ζi−κ,ηi+κ∨ζi+κ,ηiκ∨ζiκi=1m:κ∈ℵ(4)
Aℵ1∩PAℵ2=κ,ηi−κ∧ζi−κ,ηi+κ∧ζi+κ,ηiκ∧ζiκi=1m:κ∈ℵ(5)
Aℵ1⊆PAℵ2⇔ηi−κ≤ζi−κ,ηi+κ≤ζi+κandηiκ≤ζiκ. For all i=1,2,…,m and κ∈ℵ(6)
Aℵ1⊕PAℵ2=κ,ηi−κ+ζi−κ−ηi−κζi−κ,ηi+κ+ζi+κ−ηi+κζi+κ,ηiκ+ζiκ−ηiκζiκi=1m:κ∈ℵ(7)
Aℵ1⊗PAℵ2=κ,ηi−κζi−κ,ηi+κζi+κ,ηiκζiκi=1m:κ∈ℵ(8)
ςAℵ1=κ,1−1−ηi−κς,1−1−ηi+κς,1−1−ηiκςi=1m:κ∈ℵ(9)
Aℵ1ς=κ,ηi−κς,ηi+κς,ηiκςi=1m:κ∈ℵThe R-ordering operations for the abovementioned CmPFSs are given as follows:(Note: the equality and complement operations for these sets are universal; that is, they are independent or P/R-ordering.)(1)
Aℵ1∪RAℵ2=κ,ηi−κ∨ζi−κ,ηi+κ∨ζi+κ,ηiκ∧ζiκi=1m:κ∈ℵ(2)
Aℵ1∩RAℵ2=κ,ηi−κ∧ζi−κ,ηi+κ∧ζi+κ,ηiκ∨ζiκi=1m:κ∈ℵ(3)
Aℵ1⊆RAℵ2⇔ηi−κ≤ζi−κ,ηi+κ≤ζi+κandηiκ≥ζiκ. For all i=1,2,…,m and κ∈ℵ.(4)
Aℵ1⊕RAℵ2=κ,ηi−κ+ζi−κ−ηi−κζi−κ,ηi+κ+ζi+κ−ηi+κζi+κ,ηiκζiκi=1m:κ∈ℵ(5)
Aℵ1⊗RAℵ2=κ,ηi−κζi−κ,ηi+κζi+κ,ηiκ+ζiκ−ηiκζiκi=1m:κ∈ℵ(6)
ςAℵ1=κ,1−1−ηi−κς,1−1−ηi+κς,ηiκςi=1m:κ∈ℵ(7)
Aℵ1ς=κ,ηi−κς,ηi+κς,1−1−ηiκςi=1m:κ∈ℵNote: it is interesting to note that the collection of all CmPFSs on a universeℵ is closed under these operations.


## 3. Some Novel Features of CmPFSs

In this section, we give some properties and related results for CmPFSs.Theorem 1.
LetAℵ1, Aℵ2, and Aℵ3 be CmPFSs. Then,(1)
Aℵ1∪PAℵ2=Aℵ2∪PAℵ1(2)
Aℵ1∩PAℵ2=Aℵ2∩PAℵ1(3)
Aℵ1⊆PAℵ2 and Aℵ2⊆PAℵ3 implies Aℵ1⊆PAℵ3(4)
Aℵ1∪PAℵ2∪PAℵ3=Aℵ1∪PAℵ2∪PAℵ3(5)
Aℵ1∩PAℵ2∩PAℵ3=Aℵ1∩PAℵ2∩PAℵ3(6)
Aℵ1∪PAℵ2∩PAℵ3=Aℵ1∪PAℵ2∩PAℵ1∪PAℵ3(7)
Aℵ1∩PAℵ2∪PAℵ3=Aℵ1∩PAℵ2∪PAℵ1∩PAℵ3Proof.
LetAℵ1=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ, Aℵ2=κ,ζi−κ,ζi+κ,ζiκi=1m:κ∈ℵ, and Aℵ3=κ,θi−κ,θi+κ,θiκi=1m:κ∈ℵ be the three CmPFSs on ℵ.
Proofs (1) and (2) are trivial.(3)Aℵ1⊆PAℵ2impliesηi−κ≤ζi−κ,ηi+κ≤ζi+κandηiκ≤ζiκ.SimilarlyAℵ2⊆PAℵ3impliesζi−κ≤θi−κ,ζi+κ≤θi+κandζiκ≤θiκ.Combining the two facts, we obtainηi−κ≤θi−κ,ηi+κ≤θi+κandηiκ≤θiκ for all i=1,2,…,m and κ∈ℵ. This implies that Aℵ1⊆PAℵ3.(4) ConsiderAℵ1∪PAℵ2∪PAℵ3=Aℵ1∪Pκ,ζi−κ∨θi−κ,ζi+κ∨θi+κ,ζiκ∨θiκi=1m:κ∈ℵ=κ,ηi−κ∨ζi−κ∨θi−κ,ηi+κ∨ζi+κ∨θi+κ,ηi−κ∨ζiκ∨θiκi=1m:κ∈ℵ=κ,ηi−κ∨ζi−κ∨θi−κ,ηi+κ∨ζi+κ∨θi+κ,ηi−κ∨ζiκ∨θiκi=1m:κ∈ℵ=Aℵ1∪PAℵ2∪PAℵ3.(5) Similar to (4).(6)Aℵ1∪PAℵ2∩PAℵ3=Aℵ1∪Pκ,ζi−κ∧θi−κ,ζi+κ∧θi+κ,ζiκ∧θiκi=1m:κ∈ℵ=κ,ηi−κ∨ζi−κ∧θi−κ,ηi+κ∨ζi+κ∧θi+κ,ηiκ∨ζiκ∧θiκi=1m:κ∈ℵ=〈κ,ηi−κ∨ζi−κ∧ηi−κ∨θi−κ,ηi+κ∨ζi+κ∧ηi+κ∨θi+κ,ηiκ∨ζiκ∧ηiκ∨θiκ〉i=1m:κ∈ℵ=Aℵ1∪PAℵ2∩PAℵ1∪PAℵ3.(7) Similar to (6).Remark 1.
In rest of the section, we will takeAℵ1=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ, Aℵ2=κ,ζi−κ,ζi+κ,ζiκi=1m:κ∈ℵ, and Aℵ3=κ,θi−κ,θi+κ,θiκi=1m:κ∈ℵ as CmPFSs.Proposition 1.
LetAℵ1 and Aℵ2. Then,(1)
Aℵ1∪PAℵ2c=Aℵ1c∩PAℵ2c(2)
Aℵ1∩PAℵ2c=Aℵ1c∪PAℵ2cProof.
LetAℵ1=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵandAℵ2=κ,ζi−κ,ζi+κ,ζiκi=1m:κ∈ℵ.(1)
Aℵ1∪PAℵ2c=κ,ηi−κ∨ζi−κ,ηi+κ∨ζi+κ,ηiκ∨ζiκi=1m:κ∈ℵc=κ,1−ηi−κ∨ζi−κ,1−ηi+κ∨ζi+κ,1−ηiκ∨ζiκi=1m:κ∈ℵ=κ,1−ηi−κ∧1−ζi−κ,1−ηi+κ∧1−ζi+κ,1−ηiκ∧1−ζiκi=1m:κ∈ℵ=Aℵ1c∩PAℵ2c.(2)
Similar to (1).Definition 5.
LetAℵ=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ be a CmPFS with discourse ℵ. Then, the support of Aℵ is the collection of all those crisp elements from ℵ which contribute to its cubic m-polar behaviour. That is,(4)SuppAℵ=κ∈ℵ:ηi−κ,ηi+κ≠0morηiκ≠1for at least onei,where 0m is the null CmPFN.Example 1.
Letℵ=κ1,κ2,κ3,κ4 be the universe, then a CmPFS Aℵ on ℵ is given by(5)Aℵ=κ1,0.1,0.2,0.8,0.9,0,0,0.9,1,0.3κ2,0,0,0,0,0,0,1,1,1κ3,0,0,0,0,0,0.7,1,1,0.5κ4,1,1,1,1,1,1,0,0,0.
The CmPFSAℵ can be expressed in the tabular form as given by Table 1. Then, SuppAℵ=κ1,κ3,κ4.Table 1
CmPFSAℵ.
κ10.1,0.2,0.8,0.9,0,0,0.9,1,0.3κ20,0,0,0,0,0,1,1,1κ30,0,0,0,0,0.7,1,1,0.5κ41,1,1,1,1,1,0,0,0Definition 6.
The core of a CmPFSAℵ=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ is the collection of crisp elements κ from ℵ for which ηi−κ,ηi+κ=1m and ηiκ=0m for all i. Mathematically,(6)CorAℵ=κ∈ℵ:ηi−κ,ηi+κ=1mandηiκ=0mfor alli,where 1m is the absolute CmPFN.Example 2.
The core of the CmPFS in Example1 is given by(7)CorAℵ=κ4.Definition 7.
The height of a CmPFS is the maximum membership value that a certain alternative can attain. IfAℵ=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ is a CmPFS, then the height is given by(8)ht¯Aℵ=maxκ∈ℵ,iηi+κ.
If the height of a CmPFS finds its value to be 1, then it is said to be a normal CmPFS:(9)Aℵis normal⇔ht¯Aℵ=1.Example 3.
The CmPFS given in Example1 can be referred as a normal one because its height is 1, that is, ht¯Aℵ=1.Remark 2.
IfAℵ1 and Aℵ2 are CmPFSs over a crisp domain ℵ, then(10)Aℵ1⊆PAℵ2⇒ht¯Aℵ1≤ht¯Aℵ2,Aℵ1⊆RAℵ2⇒ht¯Aℵ1≤ht¯Aℵ2.
However, the converse does not hold in general. The counter example is given as follows.Example 4.
Consider the two CmPFSs, namely,Aℵ1 and Aℵ2 as given by(11)Aℵ1=κ1,0.10,0.12,0.30,0.36,0.26,0.40,0.71,0.68,0.90κ2,0.40,0.47,0.50,0.60,0.47,0.57,0.80,0.80,0.76κ3,0.50,0.55,0.60,0.66,0.23,0.34,0.56,0.69,0.44κ4,0.89,0.93,0.10,0.36,0.56,0.60,0.70,0.20,0.47,Aℵ2=κ1,0.88,0.10,0.64,0.70,0.60,0.74,0.29,0.32,0.10κ2,0.53,0.60,0.40,0.50,0.43,0.53,0.20,0.20,0.24κ3,0.45,0.50,0.34,0.40,0.66,0.77,0.44,0.31,0.56κ4,0.07,0.11,0.64,0.90,0.40,0.44,0.30,0.80,0.53.
Clearlyht¯Aℵ1=0.93≤1=ht¯Aℵ2. However, Aℵ1⊈P,RAℵ2.Theorem 2.
For a CmPFSAℵ, following results are straightforward:(1)
Aℵ=Aℵ∪PAℵ, Aℵ∩PAℵ=Aℵ(2)
Aℵ=Aℵ∪RAℵ, Aℵ∩RAℵ=Aℵ(3)
Aℵ⊕PAℵ=Aℵ⇔Aℵ=1mor1m(4)
Aℵ⊗PAℵ=Aℵ⇔Aℵ=0mor1m(5)
Aℵ⊕RAℵ=Aℵ⇔Aℵ=0mor1m(6)
Aℵ⊗RAℵ=Aℵ⇔Aℵ=0mor1m(7)
ςAℵ=AℵandAℵς=Aℵ⇔ς=1Theorem 3.
LetAℵ, Bℵ, Aℵ1, and Aℵ2 be the CmPFS with usual notations. Then,(1)
IfAℵ⊆PAℵ1andAℵ⊆PAℵ2,thenAℵ⊆PAℵ1∩pAℵ2(2)
IfAℵ1⊆PAℵandAℵ2⊆PAℵ,thenAℵ1∪pAℵ2⊆PAℵ(3)
IfAℵ1⊆PAℵandAℵ2⊆PBℵ,thenAℵ1∪pAℵ2⊆PAℵ∪PBℵandAℵ1∩pAℵ2⊆PAℵ∩pBℵTheorem 4.
LetAℵ, Aℵ1, and Aℵ2 be the usual CmPFS. Then,(1)
Aℵcc=Aℵ(2)
Aℵ1⊆PAℵ2⇔Aℵ2c⊆PAℵ1c(3)
Aℵ1=Aℵ2⇔Aℵ1c=Aℵ2c(4)
Aℵ1⊕PAℵ2c=Aℵ1c⊗PAℵ2c(5)
Aℵ1⊕PAℵ2c=Aℵ1c⊗PAℵ2c(6)
Aℵςc=ςAℵc(7)
ςAℵcc=Aℵς(8)
ςAℵc=AℵcςProof.
(1) and (3) are straightforward.(2)
LetAℵ1⊆PAℵ2⇔ηi−κ≤ζi−κ,ηi+κ≤ζi+κandηiκ≤ζiκ⇔1−ζi−κ≤1−ηi−κ,1−ζi+κ≤1−ηi+κand1−ζiκ≤1−ηiκ⇔Aℵ2c⊆PAℵ11.(4)
Aℵ1⊕PAℵ2c=〈κ,ηi−κ+ζi−κ−ηi−κζi−κ,ηi+κ+ζi+κ−ηi+κζi+κ,ηiκ+ζiκ−ηiκζiκ〉i=1m:κ∈ℵc=〈κ,1−ηi+κ−ζi+κ+ηi+κζi+κ,1−ηi−κ−ζi−κ+ηi−κζi−κ,1−ηiκ−ζiκ+ηiκζiκ〉i=1m:κ∈ℵc=κ,1−ηi+κ,1−ηi−κ,1−ηiκi=1m:κ∈ℵ⊗Pκ,1−ζi+κ,1−ζi−κ,1−ζiκi=1m:κ∈ℵ=Aℵ1c⊗PAℵ2c.(5)
Aℵ1⊗PAℵ2c=κ,ηi−κζi−κ,ηi+κζi+κ,ηiκζiκi=1m:κ∈ℵc=κ,1−ηi+κζi+κ,1−ηi−κζi−κ,1−ηiκζiκi=1m:κ∈ℵ=κ,1−ηi+κ,1−ηi−κ,1−ηiκi=1m:κ∈ℵ⊕Pκ,1−ζi+κ,1−ζi−κ,1−ζiκi=1m:κ∈ℵ=Aℵ1c⊕PAℵ2c.(6)
Aℵςc=κ,ηi−κς,ηi+κς,ηiκςi=1m:κ∈ℵc=κ,1−ηi+κς,1−ηi−κς,1−ηiκςi=1m:κ∈ℵ=κ,1−1−1−ηi+κς,1−1−1−ηi−κς,1−1−1−ηiκςi=1m:κ∈ℵ=ςκ,1−ηi+κ,1−ηi−κ,1−ηiκi=1m:κ∈ℵ=ςAℵ1c.(7)
ςAℵcc=ςκ,1−ηi+κ,1−ηi−κ,1−ηiκi=1m:κ∈ℵc=κ,1−1−1−ηi+κς,1−1−1−ηi−κς,1−1−1−ηiκςi=1m:κ∈ℵc=κ,1−ηi+κς,1−ηi−κς,1−ηiκςi=1m:κ∈ℵc=κ,ηi−κς,ηi+κς,ηiκςi=1m:κ∈ℵ=Aℵς.(8)
ςAℵc=κ,1−1−ηi−κς,1−1−ηi+κς,1−1−ηiκςi=1m:κ∈ℵc=κ,1−ηi+κς,1−ηi−κς,1−ηiκςi=1m:κ∈ℵ=κ,1−ηi+κ,1−ηi−κ,1−ηiκi=1m:κ∈ℵς=Aℵcς.Theorem 5.
The two famous laws of crisp set theory, law of contradiction and excluded middle, do not hold in CmPF P-environment as well as R-environment. That is,(1)
Aℵ∩PAℵc≠0m(2)
Aℵ∪PAℵc≠1m(3)
Aℵ∩RAℵc≠1m(4)
Aℵ∪RAℵc≠1m
The counter example for CmPFSs is given as follows.Example 5.
Let(12)Aℵ=κ1κ2κ3κ40.10,0.12,0.30,0.36,0.26,0.40,0.71,0.68,0.900.40,0.47,0.50,0.60,0.47,0.57,0.80,0.80,0.760.50,0.55,0.60,0.66,0.23,0.34,0.56,0.69,0.440.89,0.93,0.10,0.36,0.56,0.60,0.70,0.20,0.47be a CmPFS. Its complement is given by(13)Aℵc=κ1κ2κ3κ40.88,0.90,0.64,0.70,0.60,0.74,0.29,0.32,0.100.53,0.60,0.40,0.50,0.43,0.53,0.20,0.20,0.240.45,0.50,0.34,0.40,0.66,0.77,0.44,0.31,0.560.07,0.11,0.64,0.90,0.40,0.44,0.30,0.80,0.53.
Then,(14)Aℵ∩PAℵc=κ1κ2κ3κ40.10,0.12,0.30,0.36,0.26,0.40,0.29,0.32,0.100.40,0.47,0.40,0.50,0.43,0.53,0.20,0.20,0.240.45,0.50,0.34,0.40,0.23,0.34,0.44,0.31,0.440.07,0.11,0.10,0.36,0.40,0.44,0.30,0.20,0.47≠0m,Aℵ∩RAℵc=κ1κ2κ3κ40.10,0.12,0.30,0.36,0.26,0.40,0.71,0.68,0.900.40,0.47,0.40,0.50,0.43,0.53,0.80,0.80,0.760.45,0.50,0.34,0.40,0.23,0.34,0.56,0.69,0.560.07,0.11,0.10,0.36,0.40,0.44,0.70,0.80,0.53≠0m.
Similarly, other cases can be addressed.Theorem 6.
LetAℵ, Aℵ1, and Aℵ2 be the usual CmPFS. Then, (1)
Aℵ1⊆PAℵ2⇔ςAℵ1⊆PςAℵ2(2)
Aℵ1=Aℵ2⇔ςAℵ1=ςAℵ2(3)
ςAℵ1⊕PAℵ2=Aℵ1⊕PAℵ2cςc(4)
ςAℵ1⊗PAℵ2=Aℵ1⊗PAℵ2cςc(5)
Aℵ1⊗PAℵ2ς=Aℵ1ς⊗PAℵ2ς(6)
ςAℵ1⊗PςAℵ2=Aℵ1cς⊕PAℵ2cςc(7)
ςAℵ1⊕PAℵ2=ςAℵ1⊕PςAℵ2Proof.
We discuss proof for (1), (3), (6), and (7). Rest of the parts have similar proof.(1)Aℵ1⊆PAℵ2⇔ηi−κ≤ζi−κ,ηi+κ≤ζi+κandηiκ≤ζiκ,for alliandκ∈ℵ⇔1−ηi−κ≤1−ζi−κ,1−ηi+κ≤1−ζi+κand1−ηiκ≤1−ζiκ⇔1−ηi−κς≤1−ζi−κς,1−ηi+κς≤1−ζi+κςand1−ηiκς≤1−ζiκς⇔1−1−ηi−κς≤1−1−ζi−κς,1−1−ηi+κς≤1−1−ζi+κςand1−1−ηiκς≤1−1−ζiκ⇔ςAℵ1⊆PςAℵ2ς.(3)ςAℵ1⊕PAℵ2=ς〈κ,ηi−κ+ζi−κ−ηi−κζi−κ,ηi+κ+ζi+κ−ηi+κζi+κ,ηiκ+ζiκ−ηiκζiκ〉i=1m:κ∈ℵ=〈κ,1−1−ηi−κ+ζi−κ−ηi−κζi−κς,1−1−ηi+κ+ζi+κ−ηi+κζi+κς,1−1−ηiκ+ζiκ−ηiκζiκς〉i=1m:κ∈ℵ=〈κ,1−ηi+κ−ζi+κ+ηi+κζi+κς,1−ηi−κ−ζi−κ+ηi−κζi−κς,1−ηiκ−ζiκ+ηiκζiκς〉i=1m:κ∈ℵc=〈κ,1−ηi+κ−ζi+κ+ηi+κζi+κ,1−ηi−κ−ζi−κ+ηi−κζi−κ,1−ηiκ−ζiκ+ηiκζiκ〉i=1m:κ∈ℵςc=Aℵ1⊕PAℵ2cςc.(6)ςAℵ1⊗PςAℵ2=κ,1−1−ηi−κς,1−1−ηi+κς,1−1−ηiκςi=1m:κ∈ℵ⊗P〈κ,1−1−ζi−κς,1−1−ζi+κς,1−1−ζiκς〉i=1m:κ∈ℵ=〈κ,1−1−ηi−κς−1−ζi−κς+1−ηi−κ−ζi−κ+ηi−κζi−κς,1−1−ηi+κς−1−ζi+κς+1−ηi+κ−ζi+κ+ηi+κζi+κς,1−1−ηiκς−1−ζiκς+1−ηiκ−ζiκ+ηiκζiκς〉i=1m:κ∈ℵ=〈κ,1−ηi−κς+1−ζi−κς−1−ηi−κ−ζi−κ+ηi−κζi−κς,1−ηi+κς+1−ζi+κς−1−ηi+κ−ζi+κ+ηi+κζi+κς,1−ηiκς+1−ζiκς−1−ηiκ−ζiκ+ηiκζiκς〉i=1m:κ∈ℵ=κ,1−ηi−κς,1−ηi+κς,1−ηiκςi=1m:κ∈ℵ⊕P〈κ,1−ζi−κς,1−ζi+κς,1−ζiκς〉i=1m:κ∈ℵc=Aℵ1cς⊕PAℵ2cςc.(7)ςAℵ1⊕PςAℵ2=κ,1−1−ηi−κς,1−1−ηi+κς,1−1−ηiκςi=1m:κ∈ℵ⊕Pκ,1−1−ζi−κς,1−1−ζi+κς,1−1−ζiκςi=1m:κ∈ℵ=〈κ,1−1−ηi−κ−ζi−κ+ηi−κζi−κς,1−1−ηi+κ−ζi+κ+ηi+κζi+κς,1−1−ηiκ−ζiκ+ηiκζiκς〉i=1m:κ∈ℵ=ς〈κ,ηi−κ+ζi−κ−ηi−κζi−κ,ηi+κ+ζi+κ−ηi+κζi+κ,ηiκ+ζiκ−ηiκζiκ〉i=1m:κ∈ℵ=ςAℵ1⊕PAℵ2.Definition 8.
Letg,h∈0,1. The upper g-level set of a CmPFS,(15)Aℵ=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ,is denoted by UAℵ,g and is defined as(16)UAℵ,g=κ∈ℵ:ηi−κ≥g.
The lower h-level set of the setAℵ is denoted by LAℵ,h and is defined as(17)LAℵ,h=κ∈ℵ:ηiκ≤h.
The strong upper g-level set and the strong lower h-level set ofAℵ can be, respectively, defined as(18)UsAℵ,g=κ∈ℵ:ηi−κ>g,LsAℵ,h=κ∈ℵ:ηiκ<h.
Moreover, if we put the restrictiong+h≤1, then we can define (g, h)-level set of Aℵ as(19)Aℵg,h=κ∈ℵ:ηi−κ≥g,ηiκ≤h.
Similarly, strong (g, h)-level set ofAℵ can be defined as(20)Aℵg,hs=κ∈ℵ:ηi−κ>g,ηiκ<h.Example 6.
In Example4, the level sets are given as follows:(21)UAℵ,0.23=κ2,κ3⋅UsAℵ,0.23=κ2,LAℵ,0.8=κ2,κ3,κ4⋅LsAℵ,0.68=κ3,κ4,Aℵ0.23,0.8=κ2,κ3⋅Aℵ0.23,0.8s=.Remark 3.
Aℵg,h=κ∈ℵ:ηi−κ≥g,ηiκ≤hκ∈ℵ:ηi−κ≥g∩κ∈ℵ:ηiκ≤h=UAℵ,g∩LAℵ,h.
This shows that (g,h)-level set ofAℵ is actually an intersection of upper g-level and lower h-level sets.Definition 9.
Two CmPFSs are given by(22)Aℵ1=κ1,ηi−κ1,ηi+κ1,ηiκ1i=1m:κ1∈ℵ1,Aℵ2=κ2,ζi−κ2,ζi+κ2,ζiκ2i=1m:κ2∈ℵ2.
The four versions of Cartesian products of the two CmPFSs are given by(23)Aℵ1×PAℵ2=κ1,κ2,ηi−κ1ζi−κ2,ηi+κ1ζi+κ2,ηiκ1ζiκ2i=1m:κ2∈ℵ2,κ1∈ℵ1,Aℵ1×RAℵ2=κ1,κ2,ηi−κ1ζi−κ2,ηi+κ1ζi+κ2,ηiκ1+ζiκ2−ηiκ1ζiκ2i=1m:κ2∈ℵ2,κ1∈ℵ1,Aℵ1×PAℵ2=κ1,κ2,ηi−κ1∧ζi−κ2,ηi+κ1∧ζi+κ2,ηiκ1∧ζiκ2i=1m:κ2∈ℵ2,κ1∈ℵ1,Aℵ1×RAℵ2=κ1,κ2,ηi−κ1∧ζi−κ2,ηi+κ1∧ζi+κ2,ηiκ1∨ζiκ2i=1m:κ2∈ℵ2,κ1∈ℵ1.
Interestingly, the cubic Cartesian product of the CmPFSs is again a CmPFS.Theorem 7.
LetAℵ1, Aℵ2, and Aℵ3 be CmPFSs. Then, (1)
Aℵ1∪RAℵ2=Aℵ2∪RAℵ1(2)
Aℵ1∩RAℵ2=Aℵ2∩RAℵ1(3)
Aℵ1⊆RAℵ2 and Aℵ2⊆RAℵ3 implies Aℵ1⊆RAℵ3(4)
Aℵ1∪RAℵ2∪RAℵ3=Aℵ1∪RAℵ2∪RAℵ3(5)
Aℵ1∩RAℵ2∩RAℵ63=Aℵ1∩RAℵ2∩RAℵ3(6)
Aℵ1∪RAℵ2∩RAℵ3=Aℵ1∪RAℵ2∩RAℵ1∪RAℵ3(7)
Aℵ1∩RAℵ2∪RAℵ3=Aℵ1∩RAℵ2∪RAℵ1∩RAℵ3Theorem 8.
letAℵ, Φm, and Um be the CmPFS, and null and absolute CmPFSs, respectively. Then, (1)
Aℵ∪RΦm=Aℵ(2)
Aℵ∩RΦm=Φm(3)
Aℵ=Aℵ∪RAℵ(4)
Aℵ∩RAℵ=Aℵ(5)
Aℵ∪RUm=Um(6)
Aℵ∩RUm=AℵProposition 2.
LetAℵ1 and Aℵ2. Then,(1)
Aℵ1∪RAℵ2c=Aℵ1c∩RAℵ2c(2)
Aℵ1∩RAℵ2c=Aℵ1c∪RAℵ2cTheorem 9.
LetAℵ, Bℵ, Aℵ1, and Aℵ2 be the CmPFS with usual notations. Then,(1)
IfAℵ1⊆RAℵ2andAℵ2⊆RAℵ,thenAℵ1⊆RAℵ(2)
IfAℵ⊆RAℵ1andAℵ⊆RAℵ2,thenAℵ⊆RAℵ1∩RAℵ2(3)
IfAℵ1⊆RAℵandAℵ2⊆RAℵ,thenAℵ1∪RAℵ2⊆RAℵ(4)
IfAℵ1⊆RAℵandAℵ2⊆RBℵ,thenAℵ1∪RAℵ2⊆RAℵ∪RBℵandAℵ1∩RAℵ2⊆RAℵ∩RBℵTheorem 10.
LetAℵ, Aℵ1, and Aℵ2 be the usual CmPFS. Then,(1)
Aℵ1⊆RAℵ2⇔Aℵ2c⊆RAℵ1c(2)
Aℵ1⊕RAℵ2c=Aℵ1c⊗RAℵ2c(3)
Aℵ1⊗RAℵ2c=Aℵ1c⊕RAℵ2cTheorem 11.
LetAℵ, Aℵ1, and Aℵ2 be the usual CmPFS. Then,(1)
Aℵ1⊆RAℵ2⇔ςAℵ1⊆RςAℵ2(2)
ςAℵ1⊕RAℵ2=Aℵ1⊕RAℵ2cςc(3)
ςAℵ1⊗RAℵ2=Aℵ1⊗RAℵ2cςc(4)
Aℵ1⊗RAℵ2ς=Aℵ1ς⊗RAℵ2ς(5)
ςAℵ1⊗RςAℵ2=Aℵ1cς⊕RAℵ2cςc(6)
ςAℵ1⊕RAℵ2=ςAℵ1⊕RςAℵ2Proof.
Straightforward.


### 3.1. Cubicm-Polar Fuzzy Relations

Now, we define cubicm-polar fuzzy relations (CmPFRs) and give their different types.Definition 10.
LetXc and Yc be the two crisp universes. A CmPFR Xc⟶Yc is an entity given by(24)RXc⟶Yc=u,v,ηi−,ηi+,ηii=1m:u∈Xc,v∈Yc,where 0≤ηi−,ηi+,ηi≤1, and ηi−≤ηi+ for all i=1,2,…,m.Definition 11.
The different types of the CmPFRs are given as follows:(i)
A CmPFRRXc⟶Yc is said to be a reflexive if ηi−,ηi+=1 and ηi=0 for all i and u∈Xc,v∈Yc(ii)
RXc⟶Yc is said to be CmPFR of symmetric type if RXc⟶Ycu,v=RXc⟶Ycv,u for all u∈Xc,v∈Yc(iii)
RXc⟶Yc is said to be CmPFR of transitive type if RXc⟶Ycu,w≥P∨vRXc⟶Ycu,v∧RXc⟶Ycv,w


## 3.1. Cubicm-Polar Fuzzy Relations

Now, we define cubicm-polar fuzzy relations (CmPFRs) and give their different types.Definition 10.
LetXc and Yc be the two crisp universes. A CmPFR Xc⟶Yc is an entity given by(24)RXc⟶Yc=u,v,ηi−,ηi+,ηii=1m:u∈Xc,v∈Yc,where 0≤ηi−,ηi+,ηi≤1, and ηi−≤ηi+ for all i=1,2,…,m.Definition 11.
The different types of the CmPFRs are given as follows:(i)
A CmPFRRXc⟶Yc is said to be a reflexive if ηi−,ηi+=1 and ηi=0 for all i and u∈Xc,v∈Yc(ii)
RXc⟶Yc is said to be CmPFR of symmetric type if RXc⟶Ycu,v=RXc⟶Ycv,u for all u∈Xc,v∈Yc(iii)
RXc⟶Yc is said to be CmPFR of transitive type if RXc⟶Ycu,w≥P∨vRXc⟶Ycu,v∧RXc⟶Ycv,w


## 4. Correlation Measures for CmPFSs

In this section, we define correlation measure for two cubicm-polar fuzzy sets (CmPFSs) which can describe the informational settings for CmPFSs to vary or tend to vary together. If two settings tend to vary in some direction, simultaneously, they are said to be correlated. If both of them increase/decrease, simultaneously, they are positively correlated, otherwise negatively correlated. Correlation actually measures the closeness of the two variables. We establish certain properties of proposed correlation measures and also discuss novelty and effectiveness of these measures.Definition 12.
LetAℵ=κ,ηi−κ,ηi+κ,ηiκi=1m:κ∈ℵ be a CmPFS over the discourse ℵ. The average or mean of Aℵ can be defined as follows:(25)Aℵ¯=ηi−¯,ηi+¯,ηi¯i=1m=1ℵ∑κηi−κ,1ℵ∑κηi+κ,1ℵ∑κηiκi=1m.
If we take the universeℵ=κ1,κ2,κ3,κ4 and CmPFS(26)Aℵ=κ1κ2κ3κ4η11−,η11+,η12−,η12+,η13−,η13+,η14,η15,η16η21−,η21+,η22−,η22+,η23−,η23+,η24,η25,η26η31−,η31+,η32−,η32+,η33−,η33+,η34,η35,η36η41−,η41+,η42−,η42+,η43−,η43+,η44,η45,η46,then the average of Aℵ is given by Aℵ¯=η1−¯,η1+¯,η2−¯,η2+¯,η3−¯,η3+¯,η4¯,η5¯,η6¯, where(27)η1−¯=14∑j=14ηj1−,η1+¯=14∑j=14ηj1+,η2−¯=14∑j=14ηj2−,η2+¯=14∑j=14ηj2+,η3−¯=14∑j=14ηj3−,η3+¯=14∑j=14ηj3+,η4¯=14∑j=14ηj4,η5¯=14∑j=14ηj5,η6¯=14∑j=14ηj6.Definition 13.
Let(28)Aℵ=κ1κ2⋮κnη11−,η11+,η12−,η12+,…,η1m−,η1m+,η11,η12,…,η1mη21−,η21+,η22−,η22+,…,η2m−,η2m+,η21,η22,…,η2m⋯ηn1−,ηn1+,ηn2−,ηn2+,…,ηnm−,ηnm+,ηn1,ηn2,…,ηnm,Bℵ=κ1κ2⋮κnζ11−,ζ11+,ζ12−,ζ12+,…,ζ1m−,ζ1m+,ζ11,ζ12,…,ζ1mζ21−,ζ21+,ζ22−,ζ22+,…,ζ2m−,ζ2m+,ζ21,ζ22,…,ζ2m⋯ζn1−,ζn1+,ζn2−,ζn2+,…,ζnm−,ζnm+,ζn1,ζn2,…,ζnmbe the two CmPFSs over ℵ=κ1,κ2,κ3,…,κn. The left and right relative informational coefficients (RIC) of Aℵ over Bℵ are, respectively, defined by(29)IjAℵ⟶Bℵ−=∑i=1nηij−−ηj−¯ζij−−ζj−¯,IjAℵ⟶Bℵ+=∑i=1nηij+−ηj+¯ζij+−ζj+¯,j=1,2,…,m,and relative noninformational coefficients (RNICs) by(30)IjAℵ⟶Bℵ=∑i=1nηij−ηj¯ζij−ζj¯,j=1,2,…,m.
Similarly, the left and right informational coefficients ofAℵ are, respectively, given by(31)ℂj−Aℵ=∑i=1nηij−−ηj−¯2,ℂj+Aℵ=∑i=1nηij+−ηj+¯2,and noninformational coefficient by(32)ℂjAℵ=∑i=1nηij−ηj¯2,j=1,2,…,m.Proposition 3.
The RICs/RNICs and the ICs/NICs for the CmPFSsAℵ and Bℵ follow the following properties:(a)
IjAℵ⟶Aℵ−=ℂj−Aℵ, IjAℵ⟶Aℵ+=ℂj+Aℵ, IjAℵ⟶Aℵ=ℂjAℵ(b)
IjAℵ⟶Bℵ−=IjBℵ⟶Aℵ−, IjAℵ⟶Bℵ+=IjBℵ⟶Aℵ+, IjAℵ⟶Bℵ=IjBℵ⟶Aℵ(c)
IjAℵ⟶Bℵ−≤ℂj−Aℵℂj−Bℵ, IjAℵ⟶Bℵ+≤ℂj+Aℵℂj+Bℵ, IjAℵ⟶Bℵ≤ℂjAℵℂjBℵProof.
(a) and (b) are easy to address.(c) The well-known Cauchy–Schwarz inequality states that∑q=1kuqvq2≤∑q=1kuq2∑q=1kvq2 for uq,vq∈R. We will use the above inequality to prove our result.
Consider(33)IjAℵ⟶Bℵ−2=∑i=1nηij−−ηj−¯ζij−−ζj−¯2≤∑i=1nηij−−ηj−¯2∑i=1nζij−−ζj−¯2=ℂj−Aℵℂj−Bℵ⇒IjAℵ⟶Bℵ−≤ℂj−Aℵℂj−Bℵ,forallj.
The other two results are similar.Definition 14.
IfAℵ and Bℵ are the two CmPFSs with the usual notations, then their relative correlation coefficient (RCC) is given by(34)∂Aℵ,Bℵ=13m∑j=1m∂j−Aℵ,Bℵ+∑j=1m∂j+Aℵ,Bℵ+∑j=1m∂jAℵ,Bℵ,where(35)∂j−Aℵ,Bℵ=∑i=1nηij−−ηj−¯ζij−−ζj−¯∑i=1nηij−−ηj−¯2∑i=1nζij−−ζj−¯2,∂j+Aℵ,Bℵ=∑i=1nηij+−ηj+¯ζij+−ζj+¯∑i=1nηij+−ηj+¯2∑i=1nζij+−ζj+¯2,∂jAℵ,Bℵ=∑i=1nηij−ηj¯ζij−ζj¯∑i=1nηij−ηj¯2∑i=1nζij−ζj¯2,j=1,2,…,m.Theorem 12.
For the CmPFSsAℵ and Bℵ, we have(a)
The correlation∂ is symmetrical, that is, ∂Aℵ,Bℵ=∂Bℵ,Aℵ(b)
The correlation∂ finds the interval −1,1, that is, −1≤∂Aℵ,Bℵ≤1(c)
∂Aℵ,Bℵ=1 if Aℵ=Bℵ(d)
∂Aℵ,Bℵ=1 if Aℵ=λBℵ, λ>0(here,λBℵ means λζij−,λζij+,λζijj=1m)(e)
∂Aℵ,Bℵ=−1 if Aℵ=λBℵc, λ>0Proof.
(a)
Straightforward.(b)
From Proposition3, we have(36)IjAℵ⟶Bℵ−≤ℂj−Aℵℂj−Bℵ⇒−ℂj−Aℵℂj−Bℵ≤IjAℵ⟶Bℵ−≤ℂj−Aℵℂj−Bℵ⇒−1≤IjAℵ⟶Bℵ−ℂj−Aℵℂj−Bℵ≤1.Hence,−1≤∂j−Aℵ,Bℵ≤1.Similarly,−1≤∂j+Aℵ,Bℵ≤1 and −1≤∂jAℵ,Bℵ≤1, for all j=1,2,…,m.Consequently,−1≤∂Aℵ,Bℵ≤1, being the mean of the quantities between -1 and 1.(c)
IfAℵ=Bℵ, then ηij−=ζij−,ηij+=ζij+,ηij=ζij.Therefore,∂j−Aℵ,Bℵ=∑i=1nηij−−ηj−¯ζij−−ζj−¯/∑i=1nηij−−ηj−¯2∑i=1nζij−−ζj−¯2=∑i=1nηij−−ηj−¯ηij−−ηj−¯/∑i=1nηij−−ηj−¯2∑i=1nηij−−ηj−¯2=1.Similarly,∂j+Aℵ,Bℵ=1 and ∂jAℵ,Bℵ=1, and hence, ∂Aℵ,Bℵ=1.(d)
IfAℵ=λBℵ, λ>0, then ηij−=λζij−, ηij+=λζij+, and ηij=λζij. Therefore,(37)∂j−Aℵ,Bℵ=∑i=1nηij−−ηj−¯ζij−−ζj−¯∑i=1nηij−−ηj−¯2∑i=1nζij−−ζj−¯2=∑i=1nλζij−−λζij−¯ζij−−ζj−¯∑i=1nλζij−−λζij−¯2∑i=1nζij−−ζj−¯2=1.Similarly,∂j+Aℵ,Bℵ=1, ∂jAℵ,Bℵ=1, and hence, ∂Aℵ,Bℵ=1.(e)
IfAℵ=λBℵc, λ>0, then ηij−=λ1−ζij−, ηij+=λ1−ζij+, and ηij=λ1−ζij.
Therefore,ηj−¯=1−ζj−¯ and(38)∂j−Aℵ,Bℵ=∑i=1nηij−−ηj−¯ζij−−ζj−¯∑i=1nηij−−ηj−¯2∑i=1nζij−−ζj−¯2=∑i=1nλ1−ζij−−λ1−ζij−¯ζij−−ζj−¯∑i=1nλ1−ζij−−λ1−ζij−¯2∑i=1nζij−−ζj−¯2=−∑i=1nζij−−ζij−¯ζij−−ζj−¯∑i=1nζij−−ζj−¯2∑i=1nζij−−ζj−¯2=−∑i=1nζij−−ζij−¯2∑i=1nζij−−ζj−¯2∑i=1nζij−−ζj−¯2=−1.
Similarly,∂j+Aℵ,Bℵ=−1, ∂jAℵ,Bℵ=−1, and hence, ∂Aℵ,Bℵ=−1.Definition 15.
For the CmPFSsAℵ and Bℵ, the RCC may also be defined as(39)σAℵ,Bℵ=13m∑j=1mσj−Aℵ,Bℵ+∑j=1mσj+Aℵ,Bℵ+∑j=1mσjAℵ,Bℵ,where(40)σj−Aℵ,Bℵ=∑i=1nηij−−ηj−¯ζij−−ζj−¯max∑i=1nηij−−ηj−¯2,∑i=1nζij−−ζj−¯2,σj+Aℵ,Bℵ=∑i=1nηij+−ηj+¯ζij+−ζj+¯max∑i=1nηij+−ηj+¯2,∑i=1nζij+−ζj+¯2,σjAℵ,Bℵ=∑i=1nηij−ηj¯ζij−ζj¯max∑i=1nηij−ηj¯2,∑i=1nζij−ζj¯2,j=1,2,…,m.Theorem 13.
For the CmPFSsAℵ and Bℵ, we have(a)
The correlationσ is symmetrical, that is, σAℵ,Bℵ=σBℵ,Aℵ(b)
The correlationσ finds the interval −1,1, that is, −1≤σAℵ,Bℵ≤1(c)
σAℵ,Bℵ=1 if Aℵ=BℵProof.
(a) and (b) are straightforward.(c) SupposeAℵ=Bℵ. Then, ηij−=ζij−, ηij+=ζij+, and ηij=ζij. In this case,(41)σj−Aℵ,Bℵ=∑i=1nηij−−ηj−¯ζij−−ζj−¯max∑i=1nηij−−ηj−¯2,∑i=1nζij−−ζj−¯2=∑i=1nζij−−ζj−¯ζij−−ζj−¯max∑i=1nζij−−ζj−¯2,∑i=1nζij−−ζj−¯2=∑i=1nζij−−ζj−¯2max∑i=1nζij−−ζj−¯2,∑i=1nζij−−ζj−¯2=1.
Similarly,σj+Aℵ,Bℵ=1 and σjAℵ,Bℵ=1. Hence, σAℵ,Bℵ=1.In most real-world problems, two major aspects should be addressed, that is, uncertainty and weights. While dealing with such problems, uncertainty is a key factor which can affect our decisions and analysis. Besides, most of the uncertain mathematical models are effected by weights. To deal with such problems, we define weighted CCs.Definition 16.
LetAℵ and Bℵ be the two CmPFSs (usual notation followed), and the alternatives κ1,κ2,…,κn are affected by the weights ω1,ω2,…,ωn with the condition ∑i=1nωi=1 and 0≤ωi≤1. We define relative informational weighted coefficients (RIWCs) of Aℵ over Bℵ by(42)LeftRIWCωIjAℵ⟶Bℵ−=∑i=1nωiηij−−ηj−¯ζij−−ζj−¯,RightRIWCωIjAℵ⟶Bℵ+=∑i=1nωiηij+−ηj+¯ζij+−ζj+¯,j=1,2,…,m,and the relative noninformational weighted coefficients (RNIWCs) by(43)IωjAℵ⟶Bℵ=∑i=1nωiηij−ηj¯ζij−ζj¯,j=1,2,…,m.
Similarly, the informational (left and right) and noninformational weighted coefficients ofAℵ are, respectively, given by(44)ℂωj−Aℵ=∑i=1nωiηij−−ηj−¯2,ℂωj+Aℵ=∑i=1nωiηij+−ηj+¯2,ℂωjAℵ=∑i=1nωiηij−ηj¯2,j=1,2,…,m.Proposition 4.
The RIWCs/RNIWCs and the IWCs/NIWCs for the CmPFSsAℵ and Bℵ follow the following properties:(a)
IωjAℵ⟶Aℵ−=ℂωj−Aℵ, IωjAℵ⟶Aℵ+=ℂωj+Aℵ, IωjAℵ⟶Aℵ=ℂωjAℵ(b)
IωjAℵ⟶Bℵ−=IωjBℵ⟶Zℵ−, IωjAℵ⟶Bℵ+=IωjBℵ⟶Zℵ+, IωjAℵ⟶Bℵ=IωjBℵ⟶Aℵ(c)
IωjAℵ⟶Bℵ−≤ℂωj−Aℵℂωj−Bℵ, IωjAℵ⟶Bℵ+≤ℂωj+Aℵℂωj+Bℵ, IωjAℵ⟶Bℵ≤ℂωjAℵℂωjBℵProof.
(a) and (b) are easy to address.(c) We can prove the results in mirror of Cauchy–Schwarz inequality∑q=1kuqvq2≤∑q=1kuq2∑q=1kvq2 for uq,vq∈R.
Consider(45)IωjAℵ⟶Bℵ−2=∑i=1nωiηij−−ηj−¯ζij−−ζj−¯2=∑i=1nωiηij−−ηj−¯ωiζij−−ζj−¯2≤∑i=1nωiηij−−ηj−¯2∑i=1nωiζij−−ζj−¯2=ℂωj−Aℵℂωj−Bℵ⇒IωjAℵ⟶Bℵ−≤ℂωj−Aℵℂωj−Bℵ,for allj.
The other two results are similar.Definition 17.
IfAℵ and Bℵ are the two CmPFSs, then their weighted correlation coefficient (WCC) is given by(46)∂ωAℵ,Bℵ=13m∑j=1m∂ωj−Aℵ,Bℵ+∑j=1m∂ωj+Aℵ,Bℵ+∑j=1m∂ωjAℵ,Bℵ,where(47)∂ωj−Aℵ,Bℵ=∑i=1nωiηij−−ηj−¯ζij−−ζj−¯∑i=1nωiηij−−ηj−¯2∑i=1nωiζij−−ζj−¯2,∂ωj+Aℵ,Bℵ=∑i=1nωiηij+−ηj+¯ζij+−ζj+¯∑i=1nωiηij+−ηj+¯2∑i=1nωiζij+−ζj+¯2,∂ωjAℵ,Bℵ=∑i=1nωiηij−ηj¯ζij−ζj¯∑i=1nωiηij−ηj¯2∑i=1nωiζij−ζj¯2,j=1,2,…,m.Theorem 14.
IfAℵ and Bℵ are CmPFSs with the preassigned weights ω1,ω2,…,ωn, then(a)
the correlation∂ω is symmetrical, that is, ∂ωAℵ,Bℵ=∂ωBℵ,Aℵ(b)
the correlation∂ω finds the interval −1,1, that is, −1≤∂ωAℵ,Bℵ≤1(c)
∂ωAℵ,Bℵ=1 if Aℵ=λBℵ, λ>0(here,λBℵ means λζij−,λζij+,λζijj=1m)(d)
∂ωAℵ,Bℵ=1 if Aℵ=BℵProof.
(a)
Obvious.(b)
By Proposition4, we have(48)IωjAℵ⟶Bℵ−≤ℂωj−Aℵℂωj−Bℵ⇒−ℂωj−Aℵℂωj−Bℵ≤IωjAℵ⟶Bℵ−≤ℂωj−Aℵℂωj−Bℵ⇒−1≤IωjAℵ⟶Bℵ−ℂωj−Aℵℂωj−Bℵ≤1⇒−1≤∑i=1nωiηij−−ηj−¯ζij−−ζj−¯∑i=1nωiηij−−ηj−¯2∑i=1nωiζij−−ζj−¯2≤1⇒−1≤∂ωj−Aℵ,Bℵ≤1.(c)
SupposeAℵ=λBℵ. Then, ηij−=λζij−, ηij+=λζij+, and ηij=λζij. In this case,(49)∂ωj−Aℵ,Bℵ=∑i=1nωiηij−−ηj−¯ζij−−ζj−¯∑i=1nωiηij−−ηj−¯2∑i=1nωiζij−−ζj−¯2=∑i=1nωiλζij−−λζj−¯ζij−−ζj−¯∑i=1nωiλζij−−λζj−¯2∑i=1nωiζij−−ζj−¯2=∑i=1nωiζij−−ζj−¯2∑i=1nωiζij−−ζj−¯2∑i=1nωiζij−−ζj−¯2=1.Similarly,∂ωj+Aℵ,Bℵ=1, ∂ωjAℵ,Bℵ=1, and hence, ∂ωAℵ,Bℵ=1, being the average of 1′s.(d)
is similar to (c).Definition 18.
For the CmPFSsAℵ and Bℵ, the WCC may also be defined as(50)σωAℵ,Bℵ=13m∑j=1mσωj−Aℵ,Bℵ+∑j=1mσωj+Aℵ,Bℵ+∑j=1mσωjAℵ,Bℵ,where(51)σωj−Aℵ,Bℵ=∑i=1nωiηij−−ηj−¯ζij−−ζj−¯max∑i=1nωiηij−−ηj−¯2,∑i=1nωiζij−−ζj−¯2,σωj+Aℵ,Bℵ=∑i=1nωiηij+−ηj+¯ζij+−ζj+¯max∑i=1nωiηij+−ηj+¯2,∑i=1nωiζij+−ζj+¯2,σωjAℵ,Bℵ=∑i=1nωiηij−ηj¯ζij−ζj¯max∑i=1nωiηij−ηj¯2,∑i=1nωiζij−ζj¯2,j=1,2,…,m.Theorem 15.
IfAℵ and Bℵ are CmPFSs which are dictated by the weights, then(a)
the correlationσω is symmetrical, that is, σωAℵ,Bℵ=σωBℵ,Aℵ(b)
the correlationσω finds the interval −1,1, that is, −1≤σωAℵ,Bℵ≤1(c)
σωAℵ,Bℵ=1 if Aℵ=BℵProof.
The proof is obvious.The two types of the weighted correlation coefficients are equally effective no matter what types of weights are assigned to the alternatives. However, the weights should follow the mentioned conditions. If there are “n” alternatives and are affected by the weights 1/n,1/n,…,1/n, then we see that(52)∂ωj−Aℵ,Bℵ=∑i=1nωiηij−−ηj−¯ζij−−ζj−¯∑i=1nωiηij−−ηj−¯2∑i=1nωiζij−−ζj−¯2=∑i=1n1/nηij−−ηj−¯ζij−−ζj−¯∑i=1n1/nηij−−ηj−¯2∑i=1n1/nζij−−ζj−¯2=∑i=1nηij−−ηj−¯ζij−−ζj−¯∑i=1nηij−−ηj−¯2∑i=1nζij−−ζj−¯2=∂j−Aℵ,Bℵ,σωj−Aℵ,Bℵ=∑i=1nωiηij−−ηj−¯ζij−−ζj−¯max∑i=1nωiηij−−ηj−¯2,∑i=1nωiζij−−ζj−¯2=∑i=1n1/nηij−−ηj−¯ζij−−ζj−¯max∑i=1n1/nηij−−ηj−¯2,∑i=1n1/nζij−−ζj−¯2=∑i=1nηij−−ηj−¯ζij−−ζj−¯max∑i=1nηij−−ηj−¯2,∑i=1nζij−−ζj−¯2=σj−Aℵ,Bℵ,and similarly others. This shows that the above defined correlations are invariant under the weights 1/n,1/n,…,1/n. So, whenever we are encountered by such weights, we can simply ignore them and proceed in an unweighted manner. That is, the weighted alternatives can be treated as unweighted ones.


## 5. Applications of the Proposed Correlation Measures

Correlation actions are common indicators and effective approaches to address MCDM problems. In different uncertain environments, various scholars have used correlations to solve such problems efficiently and effectively. The correlations defined in this article are used here in this section to accurately recognize the patterns and medical diagnosis. Pattern recognition (PR) finds its roots in machine learning and statistics. Due to the availability of big data, the recognition problems are subjected to automatization. Bishop [38] gave a modern definition of PR as, “The field of pattern recognition is handled with the automated design of swings in data through the use of computer algorithms and with the use of these rhythms to take operations such as ordering the data into separate classes [38].” PR algorithms commonly endeavor to present a sensible answer for all attainable inputs and to achieve “most likely” matching of the inputs, using the statement of their statistical variation. They are classified according to the variety of learning procedures used to produce the output value.


### 5.1. Illustrative Examples

In the following example, we use our proposed correlations to categorize an unknown pattern in CmPF environment (Algorithm1).Algorithm 1: Steps of the proposed approach.
Step 1: construct the patternP˜ which is to be categorized.Step 2: assemble the known patternsP˜1,P˜2,…,P˜t, over a discourse ℵ, among which the recognition is to be performed.Step 3 (a): find the correlation coefficients of the patternP˜ from the known patterns P˜1,P˜2,…,P˜t using formula (34) or (39).Step 3 (b): if the alternatives from the discourseℵ follow some weights, then find the respective weighted correlation coefficients using formula (46) or (50).Step 4: if the patternP˜j finds the greatest correlation with P˜, then the pattern P˜ belongs to P˜j.The flowchart for the proposed algorithm is demonstrated in Figure2.Figure 2
Flowchart for the proposed algorithm.Example 7.
In this example, we give a solution to a pattern recognition problem. Suppose a mining firm discovers a new type of carbon while working on a site. The experts analyze it completely keeping in view its structure, properties, and smell and give their opinion about its pattern. According to the expert’s point of view, suppose its pattern comes out to be a cubic 3-polar fuzzy set over the discourseℵ=κ1,κ2,κ3, given by(53)Pκ1κ2κ30.10,0.12,0.30,0.36,0.26,0.4,0.71,0.68,0.900.40,0.47,0.50,0.60,0.47,0.57,0.80,0.80,0.760.50,0.55,0.60,0.66,0.23,0.34,0.56,0.69,0.44.
The experts are aware that there are three patterns (allotropes) of carbon, that is, diamond, graphite, and fullerenes, which are already known. Now, the problem is to identify the discovered pattern among the three already known ones. For this purpose, the three known allotropes are reanalyzed and the same type of CmPF patterns is assigned to them:(54)AℵdiamondP˜1κ1κ2κ30.20,0.40,0.29,0.48,0.32,0.55,0.57,0.66,0.420.32,0.58,0.25,0.39,0.48,0.56,0.60,0.50,0.400.49,0.62,0.31,0.43,0.36,0.48,0.20,0.35,0.50,graphiteP˜2κ1κ2κ30.13,0.69,0.33,0.42,0.46,0.50,0.39,0.47,0.680.17,0.48,0.25,0.36,0.17,0.35,0.42,0.39,0.720.42,0.53,0.30,0.42,0.22,0.37,0.48,0.56,0.75,fullerenesP˜3κ1κ2κ30.39,0.48,0.40,0.60,0.32,0.55,0.16,0.53,0.650.31,0.67,0.39,0.51,0.23,0.49,0.50,0.66,0.770.44,0.55,0.42,0.69,0.28,0.48,0.39,0.49,0.70.
We find the correlations ofP from P˜1,P˜2,P˜3 (using formulae (34) or (39), respectively) as follows:(55)∂P,P˜1=0.33,∂P,P˜2=−0.49,∂P,P˜3=0.16,σP,P˜1=0.24,σP,P˜2=−0.20,σP,P˜3=0.10.
This shows thatP finds the greatest correlations with P˜1, so the pattern P belongs to P˜1.
However, if the alternativesκ1,κ2,κ3 follow the weights ω=0.29,0.33,0.38, then we find the respective weighted correlations (using formulae (46) and (50), respectively) as follows:(56)∂ωP,P˜1=0.35,∂ωP,P˜2=−0.47,∂ωP,P˜3=0.18,∂ωP,P˜1=0.24,∂ωP,P˜2=−0.19,∂ωP,P˜3=0.11.
The same result is obtained using the weighted correlations. Hence, our proposed correlations are consistent with each other.Example 8.
Suppose a patient P is to be diagnosed in a medical center. A medical team examines him completely, and the following symptoms are observed in the patient:(57)S=κ1,headache,κ2,cough,κ3,chest pain,κ4,shortness of breath.
The set of diagnoses which the patient can observe is given byϒ1corona,ϒ2diarrhea,ϒ3asthma,ϒ4sore throat. Suppose the expert’s point of views about the patient, taking these symptoms into account, comes out to be a CmPFS (Table 2).
Therefore, each diagnosis must also be viewed as CmPFS, given by Table3.
Our aim is to find that to which disease the patient belongs, that is, to examine that from which disease the patient is suffering from. For this purpose, we find the following correlation coefficients (using formulae (34) or (39), respectively):(58)∂P,ϒ1=0.42,∂P,ϒ2=0.33,∂P,ϒ3=−0.50,∂P,ϒ4=0.28,σP,ϒ1=0.35,σP,ϒ2=0.27,σP,ϒ3=−0.31,σP,ϒ4=0.21.
However, if the symptoms bear some credibility weightsω=0.17,0.18,0.25,0.40, then we find the respective weighted correlations (using formulae (46) and (50), respectively) as follows:(59)∂ωP,ϒ1=0.42,∂ωP,ϒ2=0.35,∂ωP,ϒ3=−0.43,∂ωP,ϒ4=0.29,∂ωP,ϒ1=0.37,∂ωP,ϒ2=0.28,∂ωP,ϒ3=−0.24,∂ωP,ϒ4=0.22.
We have used all of the four proposed correlations, and we foundϒ1 as optimal solution. Hence, the patient P is suffering from corona.Table 2
Rating values of the patient in the CmPFS environment.
PPatientκ10.20,0.40,0.18,0.39,0.15,0.27,0.69,0.72,0.88κ20.42,0.60,0.30,0.50,0.19,0.33,0.40,0.60,0.80κ30.55,0.62,0.42,0.55,0.18,0.25,0.12,0.27,0.39κ40.49,0.69,0.52,0.64,0.42,0.49,0.43,0.55,0.67Table 3
Rating values of each diagnosis in the CmPFS environment.
ϒ1Coronaκ10.32,0.44,0.22,0.27,0.09,0.27,0.42,0.53,0.64κ20.14,0.26,0.30,0.39,0.42,0.51,0.48,0.59,0.68κ30.16,0.43,0.33,0.40,0.32,0.47,0.20,0.30,0.40κ40.21,0.33,0.49,0.56,0.40,0.50,0.17,0.39,0.44ϒ2Diarrheaκ10.16,0.24,0.23,0.36,0.27,0.39,0.39,0.47,0.55κ20.04,0.17,0.14,0.24,0.37,0.47,0.40,0.45,0.60κ30.24,0.36,0.39,0.47,0.45,0.57,0.37,0.49,0.54κ40.32,0.48,0.42,0.54,0.40,0.51,0.25,0.27,0.33ϒ3Asthmaκ10.31,0.43,0.40,0.55,0.60,0.71,0.41,0.39,0.55κ20.24,0.36,0.26,0.40,0.42,0.49,0.47,0.56,0.63κ30.13,0.28,0.09,0.13,0.37,0.47,0.44,0.53,0.72κ40.15,0.33,0.29,0.37,0.36,0.56,0.80,0.90,0.70ϒ4Sore throatκ10.41,0.54,0.29,0.47,0.68,0.73,0.39,0.48,0.57κ20.38,0.59,0.39,0.47,0.40,0.51,0.41,0.47,0.56κ30.62,0.80,0.55,0.65,0.32,0.46,0.62,0.55,0.39κ40.40,0.70,0.38,0.42,0.62,0.70,0.60,0.72,0.54


## 5.1. Illustrative Examples

In the following example, we use our proposed correlations to categorize an unknown pattern in CmPF environment (Algorithm1).Algorithm 1: Steps of the proposed approach.
Step 1: construct the patternP˜ which is to be categorized.Step 2: assemble the known patternsP˜1,P˜2,…,P˜t, over a discourse ℵ, among which the recognition is to be performed.Step 3 (a): find the correlation coefficients of the patternP˜ from the known patterns P˜1,P˜2,…,P˜t using formula (34) or (39).Step 3 (b): if the alternatives from the discourseℵ follow some weights, then find the respective weighted correlation coefficients using formula (46) or (50).Step 4: if the patternP˜j finds the greatest correlation with P˜, then the pattern P˜ belongs to P˜j.The flowchart for the proposed algorithm is demonstrated in Figure2.Figure 2
Flowchart for the proposed algorithm.Example 7.
In this example, we give a solution to a pattern recognition problem. Suppose a mining firm discovers a new type of carbon while working on a site. The experts analyze it completely keeping in view its structure, properties, and smell and give their opinion about its pattern. According to the expert’s point of view, suppose its pattern comes out to be a cubic 3-polar fuzzy set over the discourseℵ=κ1,κ2,κ3, given by(53)Pκ1κ2κ30.10,0.12,0.30,0.36,0.26,0.4,0.71,0.68,0.900.40,0.47,0.50,0.60,0.47,0.57,0.80,0.80,0.760.50,0.55,0.60,0.66,0.23,0.34,0.56,0.69,0.44.
The experts are aware that there are three patterns (allotropes) of carbon, that is, diamond, graphite, and fullerenes, which are already known. Now, the problem is to identify the discovered pattern among the three already known ones. For this purpose, the three known allotropes are reanalyzed and the same type of CmPF patterns is assigned to them:(54)AℵdiamondP˜1κ1κ2κ30.20,0.40,0.29,0.48,0.32,0.55,0.57,0.66,0.420.32,0.58,0.25,0.39,0.48,0.56,0.60,0.50,0.400.49,0.62,0.31,0.43,0.36,0.48,0.20,0.35,0.50,graphiteP˜2κ1κ2κ30.13,0.69,0.33,0.42,0.46,0.50,0.39,0.47,0.680.17,0.48,0.25,0.36,0.17,0.35,0.42,0.39,0.720.42,0.53,0.30,0.42,0.22,0.37,0.48,0.56,0.75,fullerenesP˜3κ1κ2κ30.39,0.48,0.40,0.60,0.32,0.55,0.16,0.53,0.650.31,0.67,0.39,0.51,0.23,0.49,0.50,0.66,0.770.44,0.55,0.42,0.69,0.28,0.48,0.39,0.49,0.70.
We find the correlations ofP from P˜1,P˜2,P˜3 (using formulae (34) or (39), respectively) as follows:(55)∂P,P˜1=0.33,∂P,P˜2=−0.49,∂P,P˜3=0.16,σP,P˜1=0.24,σP,P˜2=−0.20,σP,P˜3=0.10.
This shows thatP finds the greatest correlations with P˜1, so the pattern P belongs to P˜1.
However, if the alternativesκ1,κ2,κ3 follow the weights ω=0.29,0.33,0.38, then we find the respective weighted correlations (using formulae (46) and (50), respectively) as follows:(56)∂ωP,P˜1=0.35,∂ωP,P˜2=−0.47,∂ωP,P˜3=0.18,∂ωP,P˜1=0.24,∂ωP,P˜2=−0.19,∂ωP,P˜3=0.11.
The same result is obtained using the weighted correlations. Hence, our proposed correlations are consistent with each other.Example 8.
Suppose a patient P is to be diagnosed in a medical center. A medical team examines him completely, and the following symptoms are observed in the patient:(57)S=κ1,headache,κ2,cough,κ3,chest pain,κ4,shortness of breath.
The set of diagnoses which the patient can observe is given byϒ1corona,ϒ2diarrhea,ϒ3asthma,ϒ4sore throat. Suppose the expert’s point of views about the patient, taking these symptoms into account, comes out to be a CmPFS (Table 2).
Therefore, each diagnosis must also be viewed as CmPFS, given by Table3.
Our aim is to find that to which disease the patient belongs, that is, to examine that from which disease the patient is suffering from. For this purpose, we find the following correlation coefficients (using formulae (34) or (39), respectively):(58)∂P,ϒ1=0.42,∂P,ϒ2=0.33,∂P,ϒ3=−0.50,∂P,ϒ4=0.28,σP,ϒ1=0.35,σP,ϒ2=0.27,σP,ϒ3=−0.31,σP,ϒ4=0.21.
However, if the symptoms bear some credibility weightsω=0.17,0.18,0.25,0.40, then we find the respective weighted correlations (using formulae (46) and (50), respectively) as follows:(59)∂ωP,ϒ1=0.42,∂ωP,ϒ2=0.35,∂ωP,ϒ3=−0.43,∂ωP,ϒ4=0.29,∂ωP,ϒ1=0.37,∂ωP,ϒ2=0.28,∂ωP,ϒ3=−0.24,∂ωP,ϒ4=0.22.
We have used all of the four proposed correlations, and we foundϒ1 as optimal solution. Hence, the patient P is suffering from corona.Table 2
Rating values of the patient in the CmPFS environment.
PPatientκ10.20,0.40,0.18,0.39,0.15,0.27,0.69,0.72,0.88κ20.42,0.60,0.30,0.50,0.19,0.33,0.40,0.60,0.80κ30.55,0.62,0.42,0.55,0.18,0.25,0.12,0.27,0.39κ40.49,0.69,0.52,0.64,0.42,0.49,0.43,0.55,0.67Table 3
Rating values of each diagnosis in the CmPFS environment.
ϒ1Coronaκ10.32,0.44,0.22,0.27,0.09,0.27,0.42,0.53,0.64κ20.14,0.26,0.30,0.39,0.42,0.51,0.48,0.59,0.68κ30.16,0.43,0.33,0.40,0.32,0.47,0.20,0.30,0.40κ40.21,0.33,0.49,0.56,0.40,0.50,0.17,0.39,0.44ϒ2Diarrheaκ10.16,0.24,0.23,0.36,0.27,0.39,0.39,0.47,0.55κ20.04,0.17,0.14,0.24,0.37,0.47,0.40,0.45,0.60κ30.24,0.36,0.39,0.47,0.45,0.57,0.37,0.49,0.54κ40.32,0.48,0.42,0.54,0.40,0.51,0.25,0.27,0.33ϒ3Asthmaκ10.31,0.43,0.40,0.55,0.60,0.71,0.41,0.39,0.55κ20.24,0.36,0.26,0.40,0.42,0.49,0.47,0.56,0.63κ30.13,0.28,0.09,0.13,0.37,0.47,0.44,0.53,0.72κ40.15,0.33,0.29,0.37,0.36,0.56,0.80,0.90,0.70ϒ4Sore throatκ10.41,0.54,0.29,0.47,0.68,0.73,0.39,0.48,0.57κ20.38,0.59,0.39,0.47,0.40,0.51,0.41,0.47,0.56κ30.62,0.80,0.55,0.65,0.32,0.46,0.62,0.55,0.39κ40.40,0.70,0.38,0.42,0.62,0.70,0.60,0.72,0.54


## 6. Conclusion

The paper aims to present the novel correlation coefficient between the pairs of the cubicm-polar fuzzy sets. The considered CmPFS is a more integrated set to handle the uncertainties in the data than the existing m-polar fuzzy set and the cubic set. The advantage of the CmPFS over these existing sets is that it represents the information about the element by using m fuzzy intervals as well as m fuzzy numbers. Keeping these advantages, we investigate some operational laws about the CmPFS and hence investigate their properties. Furthermore, we elaborate on the information energy and hence define the correlation coefficient between the pairs of CmPFS. The major advantage of the proposed measure is that it considered the range of correlation measures as [−1, 1] instead of [0, 1] only. Therefore, the proposed measure has a stronger measure of the negative relationship between the variables also. Furthermore, the applicability of the proposed measures has been demonstrated through a case study of pattern recognition as well as medical diagnosis problems. The results of the proposed method are compared with several existing approaches which reveal the effectiveness of the presented work. In future studies, we shall study applications of the proposed methodology under the different fuzzy environment and solve some more practical problems [39–45].



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