[Retracted] New Operators of Cubic Picture Fuzzy Information with Applications

Tehreem, undefined; Gumaei, Abdu; Hussain, Amjad

doi:https://doi.org/10.1155/2021/9938181

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Research Article | Open Access

Volume 2021 | Article ID 9938181 | https://doi.org/10.1155/2021/9938181

[Retracted] New Operators of Cubic Picture Fuzzy Information with Applications

Tehreem,¹Abdu Gumaei,²and Amjad Hussain¹

Academic Editor: Naeem Jan

Received13 Mar 2021

Revised12 Apr 2021

Accepted20 Apr 2021

Published20 May 2021

Abstract

The researcher has been facing problems while handling imprecise and vague information, i.e., the problems of networking, decision-making, etc. For encountering such complicated data, the notion of fuzzy sets (FS) has been considered an influential tool. The notion was extended to its generalizations by a number of researchers in different ways which helps to understand and assess even more complex issues. This article characterizes imprecision with four kinds of values of membership. In this work, we aim to define and examine cubic picture fuzzy sets and give an application on averaging aggregation operators. We first introduce the notion of a cubic picture fuzzy set, which is a pair of interval-valued picture fuzzy set and a picture fuzzy set by giving examples. Then, we define two kinds of ordering on these sets and also discuss some set-theoretical properties. Moreover, we introduce three kinds of averaging aggregation operators based on cubic picture fuzzy sets and, at the end, we illustrate the results with a decision-making problem by using one of the provided aggregation operators.

1. Introduction

In 1965, Zadeh generalized the classical set and perceived the idea of fuzzy sets [1] to deal with uncertainty. This idea allows creating some new dimensions in the field of research and has been applied in many fields such as decision-making, medical diagnosis, and pattern recognition [1–6]. But in fuzzy set only the membership degree is considered. The limitation of fuzzy sets is that the nonmembership degree cannot be defined independently. To overcome this limitation, several extensions have been made by many researchers such as interval-valued fuzzy sets [7], intuitionistic fuzzy sets (IFSs) [8], cubic sets [9], and neutrosophic sets [10]. Among these various extensions of fuzzy sets, cubic set is one of the most prominent extensions. Jun [9] presented the idea of cubic sets in terms of interval-valued fuzzy set and fuzzy set in 2012. The very basic properties of cubic sets were studied, and some useful operations were defined successfully in his paper. Khir et al. [11] presented the idea of fuzzy sets and fuzzy logic and their application. Later on, the idea of cubic sets was applied to various fields by many authors (see [12–17]).

In recent years, the notion of fuzzy sets was further generalized by Coung et al. and they proposed the concept of picture fuzzy sets [18, 19], and this idea gained more and more attention from the researchers. Several similarity measures, correlation coefficients, and entropy measures for picture fuzzy sets were defined by many authors and they applied these sets in various fields (see [20–28]). Recently, Coung et al. [29] have extended the picture fuzzy sets to the interval-valued picture fuzzy sets. For some works on picture fuzzy sets and several types of aggregation operators, we refer the reader to [24, 25, 30–35].

Inspiring from the above study, we propose the concept of cubic picture fuzzy sets, which is an extension of cubic sets, picture fuzzy sets, and interval-valued picture fuzzy sets.

The rest of the paper is organized as follows. In Section 2, some basic definitions and results which are necessary for the main sections are discussed. In Section 3, the concept of cubic picture fuzzy sets which is a mixture of an interval-valued picture fuzzy set and a picture fuzzy set is introduced, and some basic operations on these sets were defined by giving several examples. Then the related theorems are studied. In Section 4, three types of aggregation operators in the environment of cubic picture fuzzy sets are discussed and, finally, one of them is applied in decision-making problem in the last section.

2. Preliminaries

Definition 1. (see [1]). Let be a nonempty set. Then is called a fuzzy set, where is a membership function that maps each element of in. Here we say that is a fuzzy subset of .

Definition 2. (see [8]). Consider closed subinterval of where is called an interval number, where . The set of all interval numbers is denoted by . A function is said to be an interval-valued fuzzy (IVF) set of . The set of all IVF sets of is denoted by . For each and, is called the degree of membership of an element to ; in this case and are fuzzy subsets of ; these sets are known as lower fuzzy set and upper fuzzy subset of , respectively.

Definition 3. (see [9]). The cubic set of a nonempty set is defined as follows: , where is an interval-valued fuzzy (IVF) set of and is a fuzzy subset of . A cubic set is simply denoted by .

Definition 4. (see [9]). A cubic set is known to be(1)an internal cubic set (briefly, ICS) if (2)an external cubic set (briefly, ECS) if

Example 1. If is an IVF set of , then and , where are cubic sets of .

Example 2. Let be a cubic set of and and , for each . Then is an ICS. If and , for each , then is an ECS. If , for each , then is neither an ICS nor an ECS.

Definition 5. (see [9]). Let be a nonempty set and let and be two cubic sets of . Then the orderings are defined in the following way:(1)(Equality) and (2)(P-Order) and (3)(R-Order) and

Definition 6. (see [9]). For arbitrary indexed family of cubic sets , where , we define the P-union, P-intersection, R-union, and R-intersection as follows:(1)(2)(3)(4)The complement of is also a cubic set that is defined by Obviously, for any indexed family of cubic sets :(1) and (2) and

Definition 7. (see [18, 19]). A picture fuzzy set (briefly, PFS) of a universe is an object in the form of , where are fuzzy sets that satisfy for each . Then the values are called the degree of positive membership of in , the degree of neutral membership of in , and the degree of negative membership of in , respectively. Now could be called the degree of refusal membership of in. Let represent the set of all picture fuzzy sets of a universe.

Definition 8. (see [18, 19]). Let and be the PFSs. Then the set of operations are defined as follows:(1)(2)(3)(4)Now, a generalization of interval-valued fuzzy set is proposed. Here int stands for the set of all closed subintervals of .

Definition 9. (see [29]). An interval-valued picture fuzzy set (briefly, IVPFS) of a universe is an object in the following form: , whereThe following condition is satisfied: . The denotes the set of all interval-valued picture fuzzy sets of .

3. Cubic Picture Fuzzy Sets

In this section, we propose the notion of a cubic picture fuzzy set and investigate its set-theoretical operations and some basic properties by giving illustrative examples.

Definition 10. A cubic picture fuzzy set (briefly, CPFS) of is denoted and defined by , where is an interval-valued picture fuzzy set and is a picture fuzzy set of . A CPFS is simply denoted by .

Definition 11. A cubic picture fuzzy set of is said to be(1)positive internal CPFS if , where are the lower and the upper positive degrees in, respectively(2)negative internal CPFS if , where are the lower and the upper negative degrees in, respectively(3)indeterminacy internal CPFS if , where are the lower and the upper indeterminacy degrees in, respectivelyWhen conditions (1), (2), and (3) hold, then it is called an internal cubic picture fuzzy set (ICPFS) in .

Definition 12. A cubic picture fuzzy set is said to be an external cubic picture fuzzy set (ECPFS) if for all .

Example 3. Let be given. Then, the CPFS,is an internal cubic picture fuzzy set of .

Example 4. Let be given; then the CPFS,is an external cubic picture fuzzy set of .

Theorem 1. If is a cubic picture fuzzy set, which is not an ECPFS, then there exists such that ,, and .

Proof. The proof is straightforward and therefore is omitted.

Theorem 2. If is both ICPFS and ECPFS, then the following is satisfied for each : , , and .

Proof. Assume that is both ICPFS and ECPFS. Then, by using the definitions of ICPFS and ECPFS, we have and and and for all . Thus or , implying that or implying that or implying that .

Definition 13. If and are the cubic picture fuzzy sets, then equality, P-order, and R-order are defined as follows:(1)(Equality) and (2)(P-order) and (3)(R-order) and

Definition 14. For any indexed family of CPFSs we define the following:(1) (P-union)(2) (P-intersection)(3) (R-union)(4) (R-intersection)The complement of is also a cubic picture fuzzy set which is defined by .

Proposition 1. For any CPFS , , , and , we have the following: (1)If (2)If (3)If (4)If (5)If (6)If (7)If (8)If

Proof. The proof is straightforward and therefore is omitted.

Remark 1. Te following are noted:(1)If (2)If

Example 5. Letand then .
Since we obtain .

Example 6. Letand then .
Since we have .

Theorem 3. Let be a cubic picture fuzzy set. If is an ICPFS (resp., ECPFS), then is an ICPFS (resp., ECPFS).

Proof. The proof is straightforward and therefore is omitted.

Theorem 4. P-union and P-intersection of arbitrary indexed family of ICPFSs are ICPFSs.

Proof. As is an ICPFSs,for each . This implies thatand, likewise,Hence, P-union and P-intersection of are CPFSs.

Remark 2. P-union and P-intersection of ECPFSs need not be an ECPFS.

Example 7. Let and be the ECPFSs of , wherefor all .(1)We know that and for all . Hence, is not an ECPFS.(2)We know that and for all . Hence is not an ECPFS.The following example shows that the R-union and R-intersection of CPFSs need not be an CPFS.

Example 8. Let and be CPFSs in , where and for all.(1)We know that and for all. Hence, is not a CPFS.(2)We know that and for all . Hence, is not a CPFS.The following example shows that “R-union” and “R-intersection” of ECPFS need not be an ECPFS.

Example 9. Let and be ECPFSs of in which, and, and for all .(1)We know that ; clearly . Hence, is not an ECPFS in .(2)We know that ; clearly for all . Hence, is not an ECPFS.

Theorem 5. Let and be the CPFSs, such that for each . Then the “R-union” of and is a CPFS.

Proof. Let and be two CPFSs, which satisfy the conditions given in Theorem 5; then we have , and .
These imply that . It follows from the assumption that where is a CPFS. For two ECPFSs and of , two CPFSs and derived from the given sets need not be CPFSs.

Example 10. Let and be ECPFSs of , in which for all .
It is seen that and are not CPFSs, because in and in .
The following example shows that the “P-union” of two ECPFSs need not be a CPFS.

Example 11. Let and be ECPFSs of in which. and for all ; clearly, . Hence is not a CPFS of .

Theorem 6. Let and be the CPFSs in satisfying the following inequalities:for all . Then the “R-intersection” of and is a CPFS.

Proof. The proof is straightforward and therefore is omitted.

Theorem 7. Let and be the ECPFSs. if and are CPFSs, then P-union of and is a CPFS.

Proof. The proof is straightforward and therefore is omitted.

Theorem 8. Let and be the ECPFSs. If and are CPFSs, then P-intersection of and is a CPFS.

Proof. The proof is straightforward and therefore is omitted.

Remark 3. For two ECPFSs and of , the derived CPFSs and need not be ECPFSs.

Example 12. Let and be ECPFSs of in whichfor all . Now, from the above, we observe that and are not ECPFSs, because, in , and, in .

Theorem 9. Let and be two ECPFSs. If and are ECPFSs, then P-union of and is an ECPFS.

Proof. Let and be ECPFSs, such that and are ECPFSs. Then we obtain that ,, and , and , , and .
Hence,This implies thatHence, is an ECPFS.

Theorem 10. Let and be the ECPFSs of such thatfor all .
Then, the P-intersection of and is an ECPFS.

Proof. The proof is straightforward by the definitions in [11, 13].

Theorem 11. Let and be the CPFSs, such that the following implications are valid:for all . Then the P-intersection of and is both an ECPFS and a CPFS.

Proof. It is straightforward by the definitions in [11, 13].
The following example shows that the P-union of two ECPFSs needs not be an ECPFS.

Example 13. Let and be two ECPFSs of defined as follows:for all .
Since , clearly . Hence is not an ECPFS of .

Theorem 12. Let and be two ECPFSs, such that the following are satisfied:for all . Then, P-union of and is an ECPFS.

Proof. The proof is straightforward and therefore is omitted.

Theorem 13. Let and be two ECPFSs, which satisfy the followings conditions:for all . Then R-union of and is an ECPFS.

Proof. The proof is straightforward.

Theorem 14. Let and be the ECPFSs, such that the following are satisfied:for all . Then R-intersection of and is an ECPFS.

Proof. The proof is straightforward.

Remark 4. Let and be two ECPFSs, such that the following are satisfied:for all . Then R-intersection of and may not be an ECPFS.

Theorem 15. Let and be two ECPFSs, such that the following are satisfied: for all . Then R-intersection of and is both an ECPFS and a CPFS.

Proof. The proof is straightforward.

Theorem 16. Let and be two CPFSs. If the implications are satisfied, for all ,then the R-union of and is an EPCFS.

Proof. The proof is straightforward.

Theorem 17. Let and be two CPFSs. If the following implications are satisfied for all ,, and, then R-intersection of and is an ECPFS.

Proof. The proof is straightforward.

Theorem 18. Let and be two ECPFSs, such that the following implications hold:for all . Then, R-union of and is a CPFS.

Proof. The proof is straightforward.

4. Averaging Aggregation Operators

In this section, we present three types of new aggregation operators called cubic picture fuzzy weighted averaging, cubic picture fuzzy ordered weighted averaging, and cubic picture fuzzy hybrid weighted averaging operators based on cubic picture fuzzy sets. Let denote the collection of all CPFSs.

Definition 15. (see [11]). A function T: [0, 1] × [0, 1] −⟶ [0, 1] is said to be a t-norm which satisfies the following: (1)Boundary: T (0, 0) = 0; T (x, 1) = T (1, x) = x for all x ∈ [0, 1](2)Monotonicity: If and , then (3)Commutativity: (4)Associativity: A function defined by is called t-co-norm. A decreasing function generates a t-norm as such that and function generates the t-co-norm as , where . Based on these norms’ generators, and will be used in the next theorems.

Definition 16. Let , and be three CPFSs. Then the operations are defined as follows: (1)(2)(3)(4)

Theorem 19. Let , and be three CPFNs and , and . Then, we have the following:(1)(2)(3)(4)(5)(6)

Proof. It is easily obtained by the above definition.

4.1. Cubic Picture Fuzzy Weighted Averaging (CPFWA) Operators

Definition 17. Let be a collection of CPFSs. Then the is defined as follows:where is the weighted vector of , s.t. and .

Theorem 20. f is the collection of CPFSs, then the averaging value by using CPFWA operator is still CPFS and is given by

Proof. We shall prove the result by using the principle of mathematical induction on “.” Step 1. For , we have ; thus, by the operations of CPFSs, we get Hence, by additive properties of CPFSs, we get Then, the results hold for . Step 2. If equation (30) holds for , then, for , we haveSince the results hold for , hence, by the principle of mathematical induction, the result given in equation (30) holds for all positive integers .

Remark 5. If is taken to be , then by equation (30) we have that which is called cubic picture fuzzy Archimedean weighted averaging operator.

4.2. Cubic Picture Fuzzy Ordered Weighted Averaging (CPFOWA) Operator

In this section, we intend to take the idea of OWA into CPFWA operator and propose a new operator which is defined as follows.

Definition 18. Let be a collection of CPFSs. Then the is defined in the following way:where is the weighted vector of , such that and . Here is the permutation of , such that , and is the ith largest of CPFSs .

Theorem 21. Let be a collection of CPFSs. Then, based on the CPFOWA operator, the aggregated CPFSs can be expressed as follows: In particular, if for all , then equation (36) reduces towhich becomes cubic intuitionistic OWA operator.

Proof. The proof follows from Theorem 19.

4.3. Cubic Picture Fuzzy Hybrid Averaging (CPFHA) Operator

CPFWA operator weighs the CPFSs only, while CPFOWA weighs the ordered positions of it. However, in order to combine these two aspects in one, we introduce CPFHA operator.

Definition 19. Let be a collection of CPFSs. Then the is defined as follows:where is the standard weight vector of , such that and is the ith largest of the weighted CPFSs is the number of CPFSs. Then CPFHA is called cubic picture fuzzy hybrid averaging operator.

Theorem 22. Let be a collection of CPFSs; then, based on CPFHA operator, the aggregated CPFSs can be expressed as

Proof. The proof is similar to Theorem 20, so it is omitted here.

5. MCDM Based on the Proposed Operation

In this section, we need the previous aggregation operators in a decision-making for CPFSs with illustrative example for evaluating the approach.

Let a set of alternatives denoted by be found by the decision-maker under the set of the unlikely criteria whose weight vector is such that and .

Suppose that the ranking of an alternative on the criteria is assessed by the decision-maker in the form of CPFSs , where is the degree of VPFS and is the degree of PFS that the alternative does not satisfy the attribute . So we develop an approach for evaluating the best alternative based on the proposed operators for MCGDM problem whose steps are as follows: Step 1. Construct the decision matrix of CPFSs. , where are VPFNs and are the PFSs towards the alternative and hence construct a cubic picture fuzzy decision matrix . Step 2. Normalized decision matrix, namely, cost and benefits , so we normalize where is the complement of . Step 3. Aggregated assessment of alternative, based on the decision matrix, as taken from step 2, all the aggregated values of the alternatives under the different criteria are obtained by using either CPFWA or CPFOWA or CPFHA operator and we collect the value of for each alternative . Step 4. We compute the score values of . Step 5. At last, we find that the rank of the alternatives according to the descending value of the score value are most valuable.

6. Illustrative Example

In this section, we illustrate with the mathematical example for the decision-making studied as follows.

Suppose few companies design their financial strategy for the next fiscal year, and according to their plan of strategy, they are picking three alternatives defined as follows: : to invest in the “Chinese markets”; : to invest in the “Indian markets”; and : to invest in “USA markets.” These proceed for finding the aspect as follows: : “the increases analysis,” : “the decreases analysis,” and : “the neutral analysis,” whose weight vector .

6.1. Example by the CPFWA Operator

The example is applied in CPFWA operator to calculate the best one. Step 1. These three alternatives are to be solved by an expert under the three aspects by using cubic picture fuzzy decision matrix for . Step 2. Since the criteria and are the porches criteria while are losses criteria , equation (40) is used as follows: Step 3. By following the CPFWA given in equation (30) with generator, we obtain the overall rating value of each alternative as Step 4. The definitions of the score functions of are , , and . Step 5. Since , we have . Hence, the gorgeous financial strategy is , that is, to invest in the Chinese markets.

7. Conclusion

The article is based on a novel approach to CPFSs as a generalization of two new strong concepts of CSs and PFSs. The basic operations for CPFSs are developed and exemplified. Some related results based on proposed operations are discussed. Several aggregation operators are defined for CPFSs and their properties are investigated. The proposed aggregation operators are subjected to a decision-making problem and the results are discussed. Furthermore, we developed multicriteria decision-making (MCDM) to prove the effectiveness and validity of the proposed methodology. A numerical example showed that the proposed operators can resolve decision-making more accurately. We compared these with predefined operators to show the validity and effectiveness of the proposed methodology.

In the future, some similarity measures for CPFS can be developed and can be applied in pattern recognition problems. We will define other methods with CPFS such as Dombi aggregation operators and introduce the idea of cubic picture fuzzy Dombi weighted average (CPFDWA), cubic picture fuzzy Dombi ordered weighted average (CPFDOWA), cubic picture fuzzy Dombi weighted geometric (CPFDWG), cubic picture fuzzy Dombi ordered weighted geometric (CPFDOWG), and generalized operators in multicriteria decision-making.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The authors are grateful to the Deanship of Scientific Research, King Saud University, for funding through Vice Deanship of Scientific Research Chairs.

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