Abstract
The concept of the cubic intuitionistic fuzzy set is an effective hybrid model for modeling uncertainties with an intuitionistic fuzzy set and an intervalvalued intuitionistic fuzzy set, simultaneously. The primary objective of this study is to develop a topological structure on cubic intuitionistic fuzzy sets with Porder and Rorder as well as to define some fundamental characteristics and significant results with illustrations. Taking advantage of topological data analysis with cubic intuitionistic information, novel multicriteria group decisionmaking methods are developed for an uncertain supply chain management. Algorithms 1 and 2 are proposed for extensions of the weighted product model and the choice value method towards a cubic intuitionistic fuzzy environment, respectively. A comparative analysis is also given to discuss the validity and advantages of the proposed techniques.
1. Introduction
Topological data analysis (TDA) methods are rapidly growing approaches to inferring persistent key features from possibly complex data [1]. We deal with complex issues in our daily lives due to vague and uncertain information, and if we do not use the proper modeling techniques for them, we eventually wind up with vague and unclear reasoning. For this reason, making rational and logical conclusions in the face of such imprecise and inexplicit facts is a difficult task for decisionmakers. As a result, dealing with vagueness and uncertainty is a necessary part of dealing with such challenges and difficulties. Zadeh [2] initiated the notion of fuzzy set (FS) theory, which is an instantaneous extension of a crisp set. Various sets of theories and models have been developed by researchers to manage the complexity of daily life problems that include vague and uncertain information. Atanassov [3] presented the idea of an intuitionistic fuzzy set (IFS), and Atanassov [4] further initiated the notion of circular intuitionistic fuzzy sets. Yager [5, 6] introduced the concept of a Pythagorean fuzzy set (PFS), and further Yager [7] developed the notion of a qrung orthopair fuzzy set (qROFS). Molodtsov [8] was the first who proposed the idea of a soft set (SS), and Zhang [9, 10] originally introduced the notion of a bipolar fuzzy set (BFS) to address bipolarity and bipolar information. Smarandache [11, 12] initiated the concept of a neutrosophic set (NS). Cuong [13] introduced the idea of a picture fuzzy set (PiFS). Gundogdu and Kahraman [14], Mahmood et al. [15], and Ashraf et al. [16] independently introduced the notion of a spherical fuzzy set (SFS). These models have a strong foothold when it comes to modeling uncertainty in a reallife complex challenges. Atanassov and Gargov [17] introduced intervalvalued intuitionistic fuzzy sets. Cagman and Enginoglu [18] proposed decisionmaking applications based on softset theory. Karaaslan and Cagman [19] introduce the parameter trees based on soft set theory and their similarity measures. Chen [20] proposed mpolar fuzzy sets (mPFS) with membership values to address the multipolarity of objects.
Jun et al. [21] developed the cubic set (CS) and its internal and external environment. But CS has some limitations, as it does not convert membership degree grades into nonmembership grades. Riaz and Hashmi [22] proposed cubic mpolar fuzzy sets and cubic mpolar fuzzy averaging aggregation operators for MAGDM. So, for this, Kaur and Garg [23, 24] presented the concept of a cubic intuitionistic fuzzy set by combining the concepts of IFSs, CFSs, and IVIFSs. So, CIFS, rather than IFSs or IVIFSs, is a handy technique to address information more precisely throughout the DMP. Young et al. [25] proposed cubic intervalvalued intuitionistic fuzzy sets. Senapati et al. [26] introduced a cubic intuitionistic WASPAS technique. Garg and Kaur [27] suggested cubic intuitionistic fuzzy Bonferroni mean operators. Garg and Kaur [28] proposed cubic intuitionistic fuzzy TOPSIS for nonlinear programming.
Classical topology derives its inspiration from classical analysis and has a wide range of scientific applications. In 1968, Chang [29] proposed the concept of fuzzy topology. Coker [30] pioneered intuitionistic fuzzy topology. Olgun [31] expanded on this concept by introducing Pythagorean fuzzy topology. Topological structures on fuzzy soft sets [32] and cubic mpolar fuzzy sets [33] have robust applications in decisionmaking. Xu and Yager [34] and Xu [35] originated the notion of an intuitionistic fuzzy number (IFN) and their aggregation operators. Zhang and Xu [36] developed an extension of TOPSIS for Pythagorean fuzzy numbers (PyFNs). They also suggested a domestic airline MCDM application to examine the service quality of airlines. Feng et al. [37] proposed the MADM application by using a new score function for ranking alternatives with generalized orthopair fuzzy membership grades. Akram [38] initiated the concept of BFS graphs, and Akram et al. [39] suggested a hybrid decisionmaking framework by using aggregation operators under a complex spherical fuzzy prioritization approach. Alghamdi et al. [40] proposed some MCDM methods for bipolar fuzzy environments. Liu and Wang [41] proposed some basic operational laws of qROFNs and qROF aggregation operators. Ye [42] proposed MADM with new similarity measures based on the generalized distance of neutrosophic Znumber sets. Senapati and Yager [43] proposed WPM for Fermatean fuzzy numbers. Kahraman and Alkan [44] developed the TOPSIS method for circular intuitionistic fuzzy sets. Sinha and Sarmah [45] developed supply chain coordination using fuzzy set theory. Alshurideh et al. [46] proposed supply chain management with fuzzyassisted human resource management.
Seikh et al. [47, 48] proposed the solution of matrix games with rough interval payoffs and a defuzzification approach of type2 fuzzy variables to solving matrix games. They developed applications of matrix games to the telecom market share problem and the plastic ban problem. Ruidas et al. [49] developed an EPQ model with stock and selling pricedependent demand and a variable production rate in an interval environment. Ruidas et al. [50] suggested an interval environment with price revision using a singleperiod production inventory model. Ruidas et al. [51] introduced a productionrepairing inventory model considering demand and the proportion of defective items as rough intervals. Seikh and Mandal [52] proposed qrung orthopair fuzzy Frank aggregation operators and their application in MADM with unknown attribute weights. Seikh and Mandal [53] introduced the MADM method based on quasirung fuzzy sets. Riaz and Farid [54] proposed the picture fuzzy aggregation approach and application to thirdparty logistic provider selection. Ashraf et al. [55] introduced the Maclaurin symmetric mean operator with an intervalvalued picture fuzzy model. Baig et al. [56] developed new methods for enhancing resilience in developing countries for oil supply chains. Chattopadhyay et al. [57] proposed the idea of the development of a roughMABACDoEbased metamodel for iron and steel supplier selection. Karamasa et al. [58] studied weighting the factors affecting logistics outsourcing. Bairagi [59] developed a novel MCDM model for warehouse location selection in supply chain management. Recently, some applications of fuzzy modeling have been developed, such as uncertain supply chains [60], medical tourism supply chains [61], sustainable plastic recycling processes [62], and pattern recognition [63].
Multicriteria group decisionmaking (MCGDM) is a branch of operation research in which the alternatives are evaluated by the group of decisionmakers (DMs) under multiple criteria to find a ranking of alternatives and an optimal decision. It is an important aspect of MCGDM to evaluate alternatives based on their characteristics. It is extremely difficult for an individual to choose an option in a variety of situations due to inconsistencies in the data caused by human errors or a lack of knowledge. Dealing with vagueness and uncertainties in MCGDM problems is very crucial to dealing with daily life problems. For this purpose, a variety of strategies have been utilized to evaluate the stability of human decisionmaking by weighing a set of options against a set of criteria. The weighted product model and choice value method are wellknown methods and are often utilized to rank the alternatives according to certain criteria.
The main objectives of this research work are given as follows:(1)To develop a topological structure on cubic intuitionistic fuzzy sets (CIFSs) with Porder (PCIFT) as well as Rorder (RCIFT) and to validate some significant results and fundamental characteristics with examples. The concept of the CIFS is a strong hybrid model for modeling uncertainties with an IFS and an intervalvalued IFS, simultaneously.(2)To examine various properties of the cubic intuitionistic fuzzy topology (CIFT) under Porder (Rorder), such as open sets of CIFT, closed sets of CIFT, interior in CIFT, closure in CIFT, subspace of CIFT, exterior in CIFT, a frontier in CIFT, and a basis of CIFT.(3)Taking advantage of topological data analysis with cubic intuitionistic fuzzy information, we proposed two multicriteria group decisionmaking (MCGDM) methods.(4)To develop Algorithm 1 for a weighted product model (WPM) and Algorithm 2 for a choice value method (CVM). An application of the proposed techniques is also designed for the uncertain supply chain management.(5)ranking of feasible alternatives is computed, and a comparative analysis of proposed methods with existing methods is also given to discuss the validity and advantage of the proposed techniques.
The remaining sections of this paper are organized as follows. In Section 2, we reviewed some fundamental concepts such as IFS, IVIFS, cubic sets, CIFS, operations on CIFSs, and some essential results on CIFSs. The idea of cubic intuitionistic fuzzy set topology with Porder is introduced in Section 3. We also investigated some key results on CIFSs in porder. In Section 4, we discuss the major results of cubic intuitionistic fuzzy set topology with Rorder. In Section 5, we discuss a useful application that employs the weighted product model and choice value method. The conclusion of the paper is given in Section 6.


2. Preliminaries
In this section, we study some basic concepts of IFSs, IVIFSs, CSs, and CIFSs. We also review some fundamental properties of CIFSs that are necessary to understand the topological structures of CIFSs.
Definition 1. (see [3]). An intuitionistic fuzzy set (IFS) in a set is described aswhere, represents the membership function, and the nonmembership function is denoted by .
Definition 2. (see [34, 35]). Let and be two IFNs. Then, we have the following operations on IFNs.(i) if and for all (ii) if and (iii)(iv)(v)
In reality, it is difficult to determine the exact membership and nonmembership degrees of an element in a set. In this situation, a range of values may be a better measurement to accommodate the uncertainty. For this, Atanassov and Gargov [17] introduce the idea of an intervalvalued intuitionistic fuzzy set (IVIFS).
Definition 3. (see [17]). Let be a nonempty universal set. An intervalvalued intuitionistic fuzzy set (IVIFS) on is defined aswhere, and are the closed subintervals of for every . For simplicity, the pair is called intervalvalued intuitionistic fuzzy number (IVIFN).
By fusing the concept of IFS and IVIFS, Jun et al. [21] defined the cubic intuitionistic set as follows:
Definition 4. (see [21]). A cubic set on a universal set is expressed asin which is intervalvalued fuzzy set and is fuzzy set on . For use of ease, this pair is referred as
Definition 5. (see [21]). For any cubic fuzzy sets , , we have(i)Punion (ii)Pintersection (iii)Runion (iv)Rintersection
Definition 6. (see [23, 24]). Let be a universal set of discourse. A cubic intuitionistic fuzzy set (CIFS) on universal set is described asin which is an IVIFS and is an IFS in . For ease of use, we denote these pairs as , where and is known as cubic intuitionistic fuzzy number (CIFN) with the condition that , and .
That is why the CIFS has the advantage of being capable to contain a lot more data to represent both the IVIFN and the IFN at the same time.
2.1. Operations on CIFSs
Now we review some fundamental operations of CIFSs, which have been explored in [23, 24].
Definition 7. The complement of the CIFS is defined as where be the complement of the IVIFS, and be the complement of the IFS, . Thus, the complement of CIFS is expressed as
Definition 8. Consider two CIFSs on a universal set is given as follow:andwe define(i)(Porder) if and (ii)(Rorder) if and (iii)(Equality) if and
Definition 9. For any CIFSsthe operations listed have been defined as follows:(i)(Punion) (ii)(Pintersection) (iii)(Runion) (iv)(Rintersection)
2.2. Some Results on CIFSs
Now we review some essential properties and results of CIFSs, which have been explored in [23, 24].
Definition 10. A CIFSfor which and for all is denoted by
Definition 11. A CIFSfor which and for all is denoted by
Definition 12. A CIFSfor which and for all is denoted by
Definition 13. A CIFSfor which and for all is denoted by
Definition 14. Let be a CIFN. The score function and the accuracy function on for CIFNs are defined as
For PorderFor Rorder
The ranking of CIFNs in relation to the proposed scoring function and accuracy function is determined as.(i) if ,(ii)If , then if (iii)If and , then
Definition 15. Let andbe the CIFNs and let be any real number. The basic operations on CIFs are given as(i)(ii)(iii)(iv)
Definition 16. Letbe the CIFNs. Then, the division operator on CIFN is given as
3. Cubic Intuitionistic Topology under POrder
In this section, we introduce the concept of a Pcubic intuitionistic fuzzy topology (PCIFT) or a cubic intuitionistic fuzzy topology with Porder.
Definition 17. Consider to be a nonempty universal set, and let to be the accumulation of all CIFSs in . If the collection containing CIFSs satisfies the following conditions, it is termed as a cubic intuitionistic fuzzy topology with a Porder (PCIFT).(1), , and (2)If then (3)If then Then, the pair is called cubic intuitionistic fuzzy topological space with a Porder (PCIFT).
Example 1. Let be a universal set. Then, be the assemblage of all Pcubic intuitionistic fuzzy sets PCIFSs in . Consider Porder fuzzy subsets of given asThe union and intersection with a Porder for the above CIFSs are given in Tables 1 and 2,, respectively.
Clearly,andare cubic intuitionistic topology with a Porder.
Definition 18. Let be a nonempty set and where represent the cubic intuitionistic fuzzy subsets of universal set . Then, is termed as a Pcubic intuitionistic fuzzy topology on and it is the largest Pcubic intuitionistic fuzzy topology on and is entitled as Pdiscrete cubic intuitionistic fuzzy topology.
Definition 19. Let be a universal set and be the assemblage of cubic intuitionistic fuzzy sets. Then, is termed as a Pcubic intuitionistic fuzzy topology on universal set and is the smallest Pcubic intuitionistic fuzzy topology on and is entitled as Pindiscrete cubic intuitionistic fuzzy topology.
Definition 20. The elements of a Pcubic intuitionistic fuzzy topology is termed as Pcubic intuitionistic fuzzy open sets PCIFOS in .
Theorem 1. If is any Pcubic intuitionistic fuzzy topological space. Then,(1) and are PCIFOSs(2)The Punion of any number of PCIFOSs is PCIFOS(3)The Pintersection of finite PCIFOSs is PCIFOS
Proof. (1)By the Definition 4.2 of a Pcubic intuitionistic fuzzy topology (PCIFT), and . Hence, and are PCIFOSs.(2)Let be PCIFOSs. Then, . From the definition of PCIFT Hence, is PCIFOSs.(3)Let be PCIOSs. Then, from definition of PCIFT Hence, is PCIFOSs.
Definition 21. The complement of elements of Pcubic intuitionistic fuzzy open sets is termed as Pcubic intuitionistic fuzzy closed sets PCIFCSs in .
Theorem 2. If is any Pcubic intuitionistic fuzzy topological space. Then,(1) and are PCIFCSs(2)The Pintersection of any number of PCIFCSs is PCIFCS(3)The Punion of finite PCIFCSs is PCIFCS
Proof. (1) and are PCIFOSs. From the definition of PCIFT Since the complement of , , and . So, and are PCIFCSs.(2)Let be PCIFCSs. Then, From the definition of PCIFT, Hence, is PCIFOSs, but So, is PCIFCSs.(3)Let be PCmPCSs. Then, are PCIFOSs. So, From the definition of PCIFT, This gives is PCIFOSs, but Hence, is PCIFCSs.
Definition 22. The Pcubic intuitionistic fuzzy sets PCIFSs, which are PCIFOSs and PCIFCSs, are entitled as Pcubic intuitionistic fuzzy clopen sets in .
Proposition 1. (1)For every , and are Pcubic intuitionistic fuzzy clopen sets(2)For discrete Porder cubic intuitionistic fuzzy topology, all the cubic intuitionistic subsets of are Pcubic intuitionistic fuzzy clopen sets(3)For indiscrete Porder cubic intuitionistic fuzzy topology, and are only Pcubic intuitionistic fuzzy clopen sets
Definition 23. Let and be two PCIFTs in . Two PCIFTs are called comparable iforIf then, is called Pcubic intuitionistic fuzzy coarser than and is called Pcubic intuitionistic fuzzy finer than.
Example 2. Let be a nonempty set and from Example 1andare Pcubic intuitionistic fuzzy topologies on universal set. Then, . Hence, is called a Pcubic intuitionistic fuzzy coarser then, .
3.1. Subspace of p
Definition 24. Let be a p. Let and is a p on and whose PCIFOSs arewhere are PCIFOSs of , are PCIFOSs of and is any Pcubic subset of PCIFS on . Then, is called a Pcubic intuitionistic fuzzy subspace of , i.e.,
Example 3. Let be a nonempty set. From Example 1,is a Pcubic intuitionistic fuzzy topology on .
Now, consider any Pcubic fuzzy subset on such that isSince,Then,is a Pcubic intuitionistic fuzzy relative topology of
3.2. Interior, Closure, Frontier and Exterior of PCIFSs
Definition 25. let be p and , the interior of is expressed as and is described as union of all Pcubic intuitionistic fuzzy open subsets contained in . It is the greatest Pcubic intuitionistic fuzzy open set contained in .
Example 4. Consider a Pcubic intuitionistic fuzzy topological space as constructed in Example 1. Let given asThen,
Theorem 3. Let be p and . Then, is open CIFS iff .
Proof. If is open CIFS, then we say that the greatest open CIFS contained in is itself. Thus,
Conversely, if , then is open CIFS. This implies is open CIFS.
Theorem 4. Let be p and . Then,(i)(ii)(iii)(iv)
Proof. Proof is trivial.
Definition 26. let be p and , the closure of is expressed as and is described as the intersection of all the Pcubic intuitionistic fuzzy closed supersets of . It is the smallest Pcubic intuitionistic fuzzy closed superset of .
Example 5. Let us consider a Pcubic intuitionistic topological space as constructed in Example 1. Then, the closed CIFSs are given asLet given asThen,
Theorem 5. Let be p and . Then is closed CIFS iff .
Proof. If is closed CIFS, then we can say that the smallest closed CIFS superset of is itself. Thus,Conversely, if , then