# Cubic Intuitionistic Fuzzy Topology with Application to Uncertain Supply Chain Management

Authors: Muhammad Riaz; Khadija Akmal; Yahya Almalki; Daud Ahmad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The concept of the cubic intuitionistic fuzzy set is an effective hybrid model for modeling uncertainties with an intuitionistic fuzzy set and an interval-valued intuitionistic fuzzy set, simultaneously. The primary objective of this study is to develop a topological structure on cubic intuitionistic fuzzy sets with P-order and R-order 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 decision-making 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.


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## Body

## 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 decision-makers. 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 q-rung orthopair fuzzy set (q-ROFS). 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 real-life complex challenges. Atanassov and Gargov [17] introduced interval-valued intuitionistic fuzzy sets. Cagman and Enginoglu [18] proposed decision-making applications based on soft-set theory. Karaaslan and Cagman [19] introduce the parameter trees based on soft set theory and their similarity measures. Chen [20] proposed m-polar fuzzy sets (mPFS) with m 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 m-polar fuzzy sets and cubic m-polar 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 interval-valued 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 m-polar fuzzy sets [33] have robust applications in decision-making. 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 decision-making 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 q-ROFNs and q-ROF aggregation operators. Ye [42] proposed MADM with new similarity measures based on the generalized distance of neutrosophic Z-number 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 fuzzy-assisted human resource management.Seikh et al. [47, 48] proposed the solution of matrix games with rough interval pay-offs and a defuzzification approach of type-2 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 price-dependent demand and a variable production rate in an interval environment. Ruidas et al. [50] suggested an interval environment with price revision using a single-period production inventory model. Ruidas et al. [51] introduced a production-repairing inventory model considering demand and the proportion of defective items as rough intervals. Seikh and Mandal [52] proposed q-rung orthopair fuzzy Frank aggregation operators and their application in MADM with unknown attribute weights. Seikh and Mandal [53] introduced the MADM method based on 3,4-quasirung fuzzy sets. Riaz and Farid [54] proposed the picture fuzzy aggregation approach and application to third-party logistic provider selection. Ashraf et al. [55] introduced the Maclaurin symmetric mean operator with an interval-valued 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 rough-MABAC-DoE-based 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 decision-making (MCGDM) is a branch of operation research in which the alternatives are evaluated by the group of decision-makers (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 decision-making by weighing a set of options against a set of criteria. The weighted product model and choice value method are well-known 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 P-order (P-CIFT) as well as R-order (R-CIFT) 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 interval-valued IFS, simultaneously.(2)
To examine various properties of the cubic intuitionistic fuzzy topology (CIFT) under P-order (R-order), 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 decision-making (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 Section2, 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 P-order is introduced in Section 3. We also investigated some key results on CIFSs in p-order. In Section 4, we discuss the major results of cubic intuitionistic fuzzy set topology with R-order. 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.Algorithm 1: Weighted product model (WPM).
Step 1. Obtain the decision matrix from the decision-makers, which indicates the alternative’sXj,j=1…m evaluation values on the basis of criterion ℂi,i=1,…n by Tji=Cji,σji, where Cji=ζjiL,ζjiU,ηjiL,ηjiU an IVIFN and σji=ζji,ηji is known as a cubic intuitionistic fuzzy number. The decision-makers provide the decision matrix M=Tjim×n of the form.X1X2Xmℂ1ℂ2……ℂnC11,σ11C12,σ12……C1n,σ1nC21,σ21C22,σ22……C2n,σ2n⋮⋮⋱⋱⋮Cm1,σm1Cm2,σm2……Cmn,σmn.Step 2. Then, the decision matrixM=Tjim×n is made normalized by a linear approach. Assume the criteria are categorized into benefit criteria B and cost criteria K. The normalization of every i∈B is defined asT¯ji=Tji/maxjTji,wheremaxjTji is defined asmaxjTji=maxζjiL,maxζjiU,minηjiL,minηjiU,minζji,maxηji.Similarly, the normalization of everyi∈K is defined asT¯ji=minjTji/Tji,whereminjTji is defined asminjTji=minζjiL,minζjiU,maxηjiL,maxηjiU,maxζji,minηji.The decision matrixM=Tjim×n is then transformed into normalized decision matrix M¯=T¯jim×n and is given asX1X2Xmℂ1ℂ2……ℂnC¯11,σ¯11C¯12,σ¯12……C¯1n,σ¯1nC¯21,σ¯21C22,σ22……C¯2n,σ¯2n⋮⋮⋱⋱⋮C¯m1,σ¯m1C¯m2,σ¯m2……C¯mn,σ¯mn.Step 3. According to CIFS-WPM, the relative importance ofj alternatives is denoted as Yj and is defined asYj=∏i=1nT¯jiwi,Here, we use the operation of power rule of CIFNs and also the product operation of CIFNs.Step 4. Find the score function of all valesYj.Step 5. Ranking of alternatives according to the score functions ofYj.Algorithm 2: Choice value method (CVM).
Step 1. Obtain the decision matrix from the decision-makers, with alternative’sXj evaluate on the basis of criterion ℂi by Tji=Cji,σji, where Cji=ζjiL,ζjiU,ηjiL,ηjiU an IVIFN and σji=ζji,ηji is known as cubic intuitionistic fuzzy number. The decision-makers provide the decision matrix M=Tjim×n of the form.X1X2Xmℂ1ℂ2……ℂnC11,σ11C12,σ12……C1n,σ1nC21,σ21C22,σ22……C2n,σ2n⋮⋮⋱⋱⋮Cm1,σm1Cm2,σm2……Cmn,σmn.Step 2. The decision-makers also give weightage to the criteria, with the condition that the sum of the weights must be equal to unity. We compute the multiplication of the decision matrix with criteria weights.C11,σ11C12,σ12……C1n,σ1nC21,σ21C22,σ22……C2n,σ2n⋮⋮⋱⋱⋮Cm1,σm1Cm2,σm2……Cmn,σmnW1W2Wn.Step 3. We find the score function of each value.Step 4. Compute the ranking of the alternatives according to their score function values.


## 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 k is described as(1)I=ℓ,ζℓ,ηℓ:0≤ζℓ+ηℓ≤1,ℓ∈k,where, ζ:k⟶0,1 represents the membership function, and the nonmembership function is denoted by η:k⟶0,1.Definition 2.
(see [34, 35]). Let I1=ζ1,η1 and I2=ζ2,η2 be two IFNs. Then, we have the following operations on IFNs.(i)
I1⊆I2 if ζ1≤ζ2 and η1≥η2 for all ℓ∈k(ii)
I1=I2 if I1⊆I2 and I2⊆I1(iii)
I1c=ℓ,η1ℓ,ζ1ℓ;ℓ∈k(iv)
I1∪I2=ℓ,∨ζ1,ζ2,∧η1,η2:ℓ∈k(v)
I1∩I2=ℓ,∧ζ1,ζ2,∨η1,η2:ℓ∈kIn 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 interval-valued intuitionistic fuzzy set (IVIFS).Definition 3.
(see [17]). Let k be a nonempty universal set. An interval-valued intuitionistic fuzzy set (IVIFS) on k is defined as(2)I=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ;ℓ∈k,where, ζLℓ,ζUℓ and ηLℓ,ηUℓ are the closed subintervals of 0,1 for every ℓ∈k. For simplicity, the pair I=ζLℓ,ζUℓ,ηLℓ,ηUℓ is called interval-valued 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 k is expressed as(3)∁=ℓ,Cℓ,σℓ:ℓ∈k,in which Cℓ is interval-valued fuzzy set and σℓ is fuzzy set on k. For use of ease, this pair is referred as ∁=C,σDefinition 5.
(see [21]). For any cubic fuzzy sets ∁i=Ci,σi, i∈Λ, we have(i)
P-union∪p∁i=∨i∈ΛCi,∨i∈Λσi(ii)
P-intersection∩p∁i=<∧i∈ΛCi,∧i∈Λσi>(iii)
R-union∪R∁i=∨i∈ΛCi,∨i∈Λσi(iv)
R-intersection∪p∁i=∨i∈ΛCi,∨i∈ΛσiDefinition 6.
(see [23, 24]). Let k be a universal set of discourse. A cubic intuitionistic fuzzy set (CIFS) on universal set k is described as(4)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,in which ζLℓ,ζUℓ,ηLℓ,ηUℓ is an IVIFS and ζ,η is an IFS in k. For ease of use, we denote these pairs as ℂI=C,σ, where C=ζL,ζU,ηL,ηU and σ=ζ,η is known as cubic intuitionistic fuzzy number (CIFN) with the condition that ζL,ζU,ηL,ηU⊆0,1, ζ,η∈0,1 and ζ+η≤1.
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 CIFSℂI=C,σ is defined as ℂIc=Cc,σc where Cc=ηLℓ,ηUℓ,ζLℓ,ζUℓ be the complement of the IVIFS, C=ζLℓ,ζUℓ,ηLℓ,ηUℓ and σc=η,ζ be the complement of the IFS, σ=ζ,η. Thus, the complement of CIFS is expressed as(5)ℂIc=ℓ,ηLℓ,ηUℓ,ζLℓ,ζUℓ,η,ζ;ℓ∈k.Definition 8.
Consider two CIFSs on a universal setk is given as follow:(6)ℂI1=ℓ,ζ1L,ζ1U,η1L,η1U,ζ1,η1;ℓ∈k,and(7)ℂI2=ℓ,ζ2L,ζ2U,η2L,η2U,ζ2,η2;ℓ∈k,we define(i)
(P-order)ℂI1⊆pℂI2 if ζ1L,ζ1U⊆ζ2L,ζ2U,η1L,η1U⊇η2L,η2U,ζ1≤ζ2 and η1≥η2(ii)
(R-order)ℂI1⊆RℂI2 if ζ1L,ζ1U⊆ζ2L,ζ2U,η1L,η1U⊇η2L,η2U,ζ1≥ζ2 and η1≤η2(iii)
(Equality)ℂI1=ℂI2 if ζ1L,ζ1U=ζ2L,ζ2U,η1L,η1U=η2L,η2U,ζ1=ζ2 and η1=η2Definition 9.
For any CIFSs(8)ℂIi=ζiL,ζiU,ηiL,ηiU,ζi,ηi;ℓ∈ki∈Λ,the operations listed have been defined as follows:(i)
(P-union)∪PℂIi=∨i∈ΛζiL,∨i∈ΛζiU,∧i∈ΛηiL,∧i∈ΛηiU,∨i∈Λζi,∧i∈Ληi(ii)
(P-intersection)∩PℂIi=∧i∈ΛζiL,∧i∈ΛζiU,∨i∈ΛηiL,∨i∈ΛηiU,∧i∈Λζi,∨i∈Ληi(iii)
(R-union)∪RℂIi=∨i∈ΛζiL,∨i∈ΛζiU,∧i∈ΛηiL,∧i∈ΛηiU,∧i∈Λζi,∨i∈Ληi(iv)
(R-intersection)∩RℂIi=∧i∈ΛζiL,∧i∈ΛζiU,∨i∈ΛηiL,∨i∈ΛηiU,∨i∈Λζi,∧i∈Ληi


### 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 CIFS(9)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=0,0,1,1 and ζ,η=1,0 for all ℓ∈k is denoted by ℂI0Definition 11.
A CIFS(10)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=1,1,0,0 and ζ,η=0,1 for all ℓ∈k is denoted by ℂI1Definition 12.
A CIFS(11)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=1,1,0,0 and ζ,η=0,1 for all ℓ∈k is denoted by ℂI0Definition 13.
A CIFS(12)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=1,1,0,0 and ζ,η=1,0 for all ℓ∈k is denoted by ℂI1Definition 14.
LetℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η be a CIFN. The score function SℂI and the accuracy function AℂI on for CIFNs are defined as
For P-order(13)SℂI=ζL+ζU−ηL−ηU2+ζ−η.
For R-order(14)SℂI=ζL+ζU−ηL−ηU2+η−ζ,AℂI=ζL+ζU+ηL+ηU2+ζ+η.The ranking of CIFNs in relation to the proposed scoring function and accuracy function is determined as.(i)
ℂI<ℂI1 if SℂI<SℂI1,(ii)
IfSℂI=SℂI1, then ℂI<ℂI1 if AℂI<AℂI1(iii)
IfSℂI=SℂI1 and AℂI=AℂI1, then ℂI=ℂI1Definition 15.
LetℂI=ℓ,ζL,ζU,ηL,ηU,ζ,η;ℓ∈k and(15)ℂIi=ℓ,ζiL,ζiU,ηiL,ηiU,ζi,ηi;ℓ∈k,i=1,2,be the CIFNs and let p>0 be any real number. The basic operations on CIFs are given as(i)
ℂI1+ℂI2=1−∏i=121−ζiL,1−∏i=121−ζiU,∏i=12ηiL,∏i=12ηiU,∏i=12ζi,1−∏i=12ηi(ii)
ℂI1×ℂI2=∏i=12ζiL,∏i=12ζiU,1−∏i=121−ηiL,1−∏i=121−1−ηiU)),1−∏i=12ζi,∏i=12ηi(iii)
pℂI=1−1−ζLp,1−1−ζUp,ηLp,ηUp,ζp,1−1−ηp(iv)
ℂIp=ζLp,ζUp,1−1−ηLp,1−1−ηUp,1−1−ζp,ηpDefinition 16.
Let(16)ℂIi=ℓ,ζiL,ζiU,ηiL,ηiU,ζi,ηi;ℓ∈k,i=1,2,be the CIFNs. Then, the division operator on CIFN is given as(17)ℂI1ℂI2=minζ1L,ζ2L,minζ1U,ζ2U,maxη1L,η2L,maxη1U,η2U,maxζ1,ζ2,minη1U,η2U.


## 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 CIFSℂI=C,σ is defined as ℂIc=Cc,σc where Cc=ηLℓ,ηUℓ,ζLℓ,ζUℓ be the complement of the IVIFS, C=ζLℓ,ζUℓ,ηLℓ,ηUℓ and σc=η,ζ be the complement of the IFS, σ=ζ,η. Thus, the complement of CIFS is expressed as(5)ℂIc=ℓ,ηLℓ,ηUℓ,ζLℓ,ζUℓ,η,ζ;ℓ∈k.Definition 8.
Consider two CIFSs on a universal setk is given as follow:(6)ℂI1=ℓ,ζ1L,ζ1U,η1L,η1U,ζ1,η1;ℓ∈k,and(7)ℂI2=ℓ,ζ2L,ζ2U,η2L,η2U,ζ2,η2;ℓ∈k,we define(i)
(P-order)ℂI1⊆pℂI2 if ζ1L,ζ1U⊆ζ2L,ζ2U,η1L,η1U⊇η2L,η2U,ζ1≤ζ2 and η1≥η2(ii)
(R-order)ℂI1⊆RℂI2 if ζ1L,ζ1U⊆ζ2L,ζ2U,η1L,η1U⊇η2L,η2U,ζ1≥ζ2 and η1≤η2(iii)
(Equality)ℂI1=ℂI2 if ζ1L,ζ1U=ζ2L,ζ2U,η1L,η1U=η2L,η2U,ζ1=ζ2 and η1=η2Definition 9.
For any CIFSs(8)ℂIi=ζiL,ζiU,ηiL,ηiU,ζi,ηi;ℓ∈ki∈Λ,the operations listed have been defined as follows:(i)
(P-union)∪PℂIi=∨i∈ΛζiL,∨i∈ΛζiU,∧i∈ΛηiL,∧i∈ΛηiU,∨i∈Λζi,∧i∈Ληi(ii)
(P-intersection)∩PℂIi=∧i∈ΛζiL,∧i∈ΛζiU,∨i∈ΛηiL,∨i∈ΛηiU,∧i∈Λζi,∨i∈Ληi(iii)
(R-union)∪RℂIi=∨i∈ΛζiL,∨i∈ΛζiU,∧i∈ΛηiL,∧i∈ΛηiU,∧i∈Λζi,∨i∈Ληi(iv)
(R-intersection)∩RℂIi=∧i∈ΛζiL,∧i∈ΛζiU,∨i∈ΛηiL,∨i∈ΛηiU,∨i∈Λζi,∧i∈Ληi


## 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 CIFS(9)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=0,0,1,1 and ζ,η=1,0 for all ℓ∈k is denoted by ℂI0Definition 11.
A CIFS(10)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=1,1,0,0 and ζ,η=0,1 for all ℓ∈k is denoted by ℂI1Definition 12.
A CIFS(11)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=1,1,0,0 and ζ,η=0,1 for all ℓ∈k is denoted by ℂI0Definition 13.
A CIFS(12)ℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η;ℓ∈k,for which ζLℓ,ζUℓ,ηLℓ,ηUℓ=1,1,0,0 and ζ,η=1,0 for all ℓ∈k is denoted by ℂI1Definition 14.
LetℂI=ℓ,ζLℓ,ζUℓ,ηLℓ,ηUℓ,ζ,η be a CIFN. The score function SℂI and the accuracy function AℂI on for CIFNs are defined as
For P-order(13)SℂI=ζL+ζU−ηL−ηU2+ζ−η.
For R-order(14)SℂI=ζL+ζU−ηL−ηU2+η−ζ,AℂI=ζL+ζU+ηL+ηU2+ζ+η.The ranking of CIFNs in relation to the proposed scoring function and accuracy function is determined as.(i)
ℂI<ℂI1 if SℂI<SℂI1,(ii)
IfSℂI=SℂI1, then ℂI<ℂI1 if AℂI<AℂI1(iii)
IfSℂI=SℂI1 and AℂI=AℂI1, then ℂI=ℂI1Definition 15.
LetℂI=ℓ,ζL,ζU,ηL,ηU,ζ,η;ℓ∈k and(15)ℂIi=ℓ,ζiL,ζiU,ηiL,ηiU,ζi,ηi;ℓ∈k,i=1,2,be the CIFNs and let p>0 be any real number. The basic operations on CIFs are given as(i)
ℂI1+ℂI2=1−∏i=121−ζiL,1−∏i=121−ζiU,∏i=12ηiL,∏i=12ηiU,∏i=12ζi,1−∏i=12ηi(ii)
ℂI1×ℂI2=∏i=12ζiL,∏i=12ζiU,1−∏i=121−ηiL,1−∏i=121−1−ηiU)),1−∏i=12ζi,∏i=12ηi(iii)
pℂI=1−1−ζLp,1−1−ζUp,ηLp,ηUp,ζp,1−1−ηp(iv)
ℂIp=ζLp,ζUp,1−1−ηLp,1−1−ηUp,1−1−ζp,ηpDefinition 16.
Let(16)ℂIi=ℓ,ζiL,ζiU,ηiL,ηiU,ζi,ηi;ℓ∈k,i=1,2,be the CIFNs. Then, the division operator on CIFN is given as(17)ℂI1ℂI2=minζ1L,ζ2L,minζ1U,ζ2U,maxη1L,η2L,maxη1U,η2U,maxζ1,ζ2,minη1U,η2U.


## 3. Cubic Intuitionistic Topology under P-Order

In this section, we introduce the concept of a P-cubic intuitionistic fuzzy topology (P-CIFT) or a cubic intuitionistic fuzzy topology with P-order.Definition 17.
Considerk to be a nonempty universal set, and let cik to be the accumulation of all CIFSs in k. If the collection TℂIP containing CIFSs satisfies the following conditions, it is termed as a cubic intuitionistic fuzzy topology with a P-order (P-CIFT).(1)
ℂI0, ℂI1, ℂI0¯ and ℂI0¯∈TℂIP(2)
IfℂIPi∈TℂIP∀i∈Λ then ∪pℂIPi∈TℂIP(3)
IfℂIP1,ℂIP2∈TℂIP then ℂIP1∩pℂIP2∈TℂIP
Then, the pairk,TℂIP is called cubic intuitionistic fuzzy topological space with a P-order (P-CIFT).Example 1.
Letk be a universal set. Then, cik be the assemblage of all P-cubic intuitionistic fuzzy sets PCIFSs in k. Consider P-order fuzzy subsets of cik given as(18)ℂIP1=0.20,0.31,0.41,0.52,0.32,0.44,ℂIP2=0.20,0.31,0.41,0.52,1,0,ℂIP3=0.20,0.31,0.41,0.52,0,1,ℂIP4=1,1,0,0,0.32,0.44,ℂIP5=0,0,1,1,0.32,0.44.
The union and intersection with a P-order for the above CIFSs are given in Tables1 and 2,, respectively.
Clearly,(19)TℂIP1=ℂI0,ℂI1,ℂI1¯,ℂI1¯,and(20)TℂIP2=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,are cubic intuitionistic topology with a P-order.Table 1
Union under P-order.
∪pℂI0ℂI1ℂI0¯ℂI1¯ℂIP1ℂIP2ℂIP3ℂIP4ℂIP5ℂI0ℂI0ℂI1¯ℂI0ℂI1¯ℂIP2ℂIP2ℂIP2ℂI1¯ℂI01ℂI1ℂI¯ℂI1ℂI1ℂI1¯ℂIP4ℂI1¯ℂI1ℂIP4ℂIP40ℂIℂI0ℂI10ℂI¯1ℂI¯ℂIP1ℂIP2ℂIP3ℂIP4ℂIP51ℂI¯1ℂI¯1ℂI¯1ℂI¯1ℂI¯1ℂI¯1ℂI¯1ℂI¯1ℂI¯1ℂI¯ℂIP1ℂIP2ℂIP4ℂIP11ℂI¯ℂIP1ℂIP2ℂIP1ℂIP4ℂIP1ℂIP2ℂIP21ℂI¯ℂIP21ℂI¯ℂIP2ℂIP2ℂIP21ℂI¯ℂIP2ℂIP3ℂIP21ℂIℂIP31ℂI¯ℂIP1ℂIP2ℂIP3ℂIP4ℂIP1ℂIP41ℂI¯ℂIP4ℂIP41ℂI¯ℂIP41ℂI¯ℂIP4ℂIP4ℂIP4ℂIP50ℂIℂIP4ℂIP51ℂI¯ℂIP1ℂIP2ℂIP1ℂIP4ℂIP5Table 2
Intersection under P-order.
∩p0ℂI1ℂI0ℂI¯1ℂI¯ℂIP1ℂIP2ℂIP3ℂIP4ℂIP50ℂI0ℂI0ℂI¯0ℂI¯0ℂIℂIP50ℂI0ℂIℂIP5ℂIP51ℂI0ℂI¯1ℂI0ℂI¯1ℂIℂIP3ℂIP3ℂIP31ℂI0ℂI¯0ℂI¯0ℂI¯0ℂI¯0ℂI¯0ℂI¯0ℂI¯0ℂI¯0ℂI¯0ℂI¯0ℂI¯1ℂI¯0ℂI1ℂI0ℂI¯1ℂI¯ℂIP1ℂIP2ℂIP3ℂIP4ℂIP5ℂIP1ℂIP5ℂIP30ℂI¯ℂIP1ℂIP1ℂIP1ℂIP3ℂIP1ℂIP5ℂIP20ℂIℂIP30ℂI¯ℂIP2ℂIP1ℂIP2ℂIP3ℂIP1ℂIP5ℂIP30ℂI¯ℂIP30ℂI¯ℂIP3ℂIP3ℂIP3ℂIP3ℂIP30ℂI¯ℂIP4ℂIP51ℂI0ℂI¯ℂIP4ℂIP1ℂIP1ℂIP3ℂIP4ℂIP5ℂIP5ℂIP50ℂI¯0ℂI¯ℂIP5ℂIP5ℂIP50ℂI¯ℂIP5ℂIP5Definition 18.
Letk be a nonempty set and TℂIP=ℂIPk where ℂIPk represent the cubic intuitionistic fuzzy subsets of universal set k. Then, TℂIP is termed as a P-cubic intuitionistic fuzzy topology on k and it is the largest P-cubic intuitionistic fuzzy topology on k and is entitled as P-discrete cubic intuitionistic fuzzy topology.Definition 19.
Letk be a universal set and TℂIP=ℂI0,ℂI1,ℂI0¯,ℂI1¯ be the assemblage of cubic intuitionistic fuzzy sets. Then, TℂIP is termed as a P-cubic intuitionistic fuzzy topology on universal set k and is the smallest P-cubic intuitionistic fuzzy topology on k and is entitled as P-indiscrete cubic intuitionistic fuzzy topology.Definition 20.
The elements of a P-cubic intuitionistic fuzzy topologyTℂIP is termed as P-cubic intuitionistic fuzzy open sets PCIFOS in k,TℂIP.Theorem 1.
Ifk,TℂIP is any P-cubic intuitionistic fuzzy topological space. Then,(1)
ℂI0,ℂI1,ℂI0¯ and 1ℂI¯ are PCIFOSs(2)
The P-union of any number of PCIFOSs is PCIFOS(3)
The P-intersection of finite PCIFOSs is PCIFOSProof.
(1)
By the Definition 4.2 of a P-cubic intuitionistic fuzzy topology (P-CIFT),ℂI0,ℂI1,ℂI0¯ and ℂI1¯∈TℂIP. Hence, ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are PCIFOSs.(2)
LetℂIPi|i∈Λ be PCIFOSs. Then, ℂIPi∈TℂIP. From the definition of P-CIFT(21)∪pℂIPi∈TℂIP.Hence,∪pℂIPi∈TℂIP is PCIFOSs.(3)
LetℂIP1,ℂIP2,…,ℂIPn be PCIOSs. Then, from definition of P-CIFT(22)∩pℂIPi∈TℂIP.Hence,∩pℂIPi is PCIFOSs.Definition 21.
The complement of elements of P-cubic intuitionistic fuzzy open sets is termed as P-cubic intuitionistic fuzzy closed sets PCIFCSs ink,TℂIP.Theorem 2.
Ifk,TℂIP is any P-cubic intuitionistic fuzzy topological space. Then,(1)
ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are PCIFCSs(2)
The P-intersection of any number of PCIFCSs is PCIFCS(3)
The P-union of finite PCIFCSs is PCIFCSProof.
(1)
ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are PCIFOSs. From the definition of P-CIFT(23)ℂI0,ℂI1,ℂI0¯,ℂI1¯∈TℂIP.Since the complement ofℂI0=ℂI1, ℂI1=ℂI0, ℂI0¯=ℂI1¯ and ℂI1¯=ℂI0¯. So, ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are PCIFCSs.(2)
LetℂIPi|i∈Λ be PCIFCSs. Then,(24)ℂIPic∈TℂIP.From the definition of P-CIFT,(25)∪pℂIPic∈TℂIP.Hence,∪pℂIPic is PCIFOSs, but(26)∪pℂIPic=∩pℂIPic.So,∩pℂIPi is PCIFCSs.(3)
LetℂIP1,ℂIP2,…,ℂIPn be PCmPCSs. Then, ℂIP1c,ℂIP2c,…,ℂIPnc are PCIFOSs. So,(27)ℂIP1c,ℂIP2c,…,ℂIPnc∈TℂIP.From the definition of P-CIFT,(28)∩pℂIPic∈TℂIP.This gives∩pℂIPic∈TℂIP is PCIFOSs, but(29)∩pℂIPic=∪pℂIPic.Hence,∪pℂIPi is PCIFCSs.Definition 22.
The P-cubic intuitionistic fuzzy sets PCIFSs, which are PCIFOSs and PCIFCSs, are entitled as P-cubic intuitionistic fuzzy clopen sets ink,TℂIP.Proposition 1.
(1)
For everyTℂIP, ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are P-cubic intuitionistic fuzzy clopen sets(2)
For discrete P-order cubic intuitionistic fuzzy topology, all the cubic intuitionistic subsets ofk are P-cubic intuitionistic fuzzy clopen sets(3)
For in-discrete P-order cubic intuitionistic fuzzy topology,ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are only P-cubic intuitionistic fuzzy clopen setsDefinition 23.
Letk,TℂIP1 and k,TℂIP2 be two P-CIFTs in k. Two P-CIFTs are called comparable if(30)TℂIP1⊆PTℂIP2,or(31)TℂIP2⊆PTℂIP1.
IfTℂIP1⊆PTℂIP2 then, TℂIP1 is called P-cubic intuitionistic fuzzy coarser than TℂIP2 and TℂIP2 is called P-cubic intuitionistic fuzzy finer than. TℂIP1Example 2.
Letk be a nonempty set and from Example 1(32)TℂIP1=ℂI0,ℂI1,ℂI0¯,ℂI1¯,and(33)TℂIP2=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,are P-cubic intuitionistic fuzzy topologies on universal set. Then, TℂIP1⊆PTℂIP2. Hence, TℂIP1 is called a P-cubic intuitionistic fuzzy coarser then, TℂIP2.


### 3.1. Subspace ofℂIFTp

Definition 24.
Letk,TℂIPk be a ℂIFTp. Let Y⊆k and TℂIPY is a ℂIFTp on Y and whose PCIFOSs are(34)ℂIPY=TℂIPk∩pY,where ℂIPk are PCIFOSs of TℂIPk, TℂIPY are PCIFOSs of TℂIPY and Y˜ is any P-cubic subset of PCIFS on Y. Then, TℂIPY is called a P-cubic intuitionistic fuzzy subspace of TℂIPk, i.e.,(35)TℂIPY=ℂIPY:ℂIPY=ℂIPk∩pY,ℂIPk∈TℂIPk.Example 3.
Letk be a nonempty set. From Example 1,(36)TℂIP=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,is a P-cubic intuitionistic fuzzy topology on k.
Now, consider any P-cubic fuzzy subset onk such that Y⊆k is(37)Y=0.98,0.23,0.46,0.61,0.27,0.49.
Since,(38)Y∩p0ℂI=0,0,1,1,0.27,0.49=ℂIP⟶,Y∩p1ℂI=0.98,0.23,0.46,0.61,0,1=ℂ˜IP,Y∩p0ℂI¯=0,0,1,1,0,1=ℂIP′,Y∩p1ℂI¯=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP1=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP2=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP3=0.98,0.23,0.46,0.61,0,1=ℂ˜IP,Y∩pℂIP4=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP5=0,0,1,1,0.27,0.49=ℂIP⟶.
Then,(39)TℂIPY=ℂIP⟶,ℂIP˜,ℂIP′,Y,is a P-cubic intuitionistic fuzzy relative topology of TℂIPk


### 3.2. Interior, Closure, Frontier and Exterior of PCIFSs

Definition 25.
letk,TℂIP be ℂIFTp and ℂIP∈cik, the interior of ℂIP is expressed as ℂIP0 and is described as union of all P-cubic intuitionistic fuzzy open subsets contained in ℂIP. It is the greatest P-cubic intuitionistic fuzzy open set contained in ℂIP.Example 4.
Consider a P-cubic intuitionistic fuzzy topological space as constructed in Example1. Let ℂIP6∈cik given as(40)ℂIP6=0.23,0.39,0.37,0.48,0.46,0.33.
Then,(41)ℂIP60=ℂI0¯∪pℂIP1∪pℂIP3∪pℂIP5=ℂIP1.Theorem 3.
Letk,TℂIP be ℂIFT p and ℂIP∈cik. Then, ℂIP is open CIFS iff ℂIP0=ℂIP.Proof.
IfℂIP is open CIFS, then we say that the greatest open CIFS contained in ℂIP is ℂIP itself. Thus,(42)ℂIP0=ℂIP.Conversely, ifℂIP0=ℂIP, then ℂIP0 is open CIFS. This implies ℂIP is open CIFS.Theorem 4.
Letk,TℂIP be ℂIFTp and ℂIP1,ℂIP2∈cik. Then,(i)
ℂIP00=ℂIP0(ii)
ℂIP1⊆pℂIP2⇒ℂIP10⊆pℂIP20(iii)
ℂIP1∩pℂIP20=ℂIP10⊆pℂIP20(iv)
ℂIP1∪pℂIP20⊇pℂIP10∪pℂIP20Proof.
Proof is trivial.Definition 26.
letk,TℂIP be ℂIFTp and ℂIP∈cik, the closure of ℂIP¯ is expressed as ℂIP¯ and is described as the intersection of all the P-cubic intuitionistic fuzzy closed supersets of ℂIP. It is the smallest P-cubic intuitionistic fuzzy closed superset of ℂIP.Example 5.
Let us consider a P-cubic intuitionistic topological space as constructed in Example1. Then, the closed CIFSs are given as(43)ℂI0c=1,1,0,0,0,1,ℂI1c=0,0,1,1,1,0,ℂI0¯c=1,1,0,0,1,0,ℂI1¯c=0,0,1,1,0,1,ℂIP1c=0.41,0.52,0.20,0.31,0.44,0.32,ℂIP2c=0.41,0.52,0.20,0.31,0,1,ℂIP3c=0.41,0.52,0.20,0.31,1,0,ℂIP4c=0,0,1,1,0.44,0.32,ℂIP5c=1,1,0,0,0.44,0.32.
LetℂIP7∈cik given as(44)ℂIP7=0.34,0.50,0.27,0.38,0.33,0.41.
Then,(45)ℂIP7¯=ℂI0¯c∩pℂIP1c∩pℂIP3c∩pℂIP5c=ℂIP1c.Theorem 5.
Letk,TℂIP be ℂIFT p and ℂIP∈cik. Then ℂIP¯ is closed CIFS iff ℂIP¯=ℂIP.Proof.
IfℂIP is closed CIFS, then we can say that the smallest closed CIFS superset of ℂIP is ℂIP itself. Thus,(46)ℂIP¯=ℂIP.
Conversely, ifℂIP¯=ℂIP, then ℂIP¯ is closed CIFS. This implies ℂIP is closed CIFS. □Definition 27.
LetℂIP be a P-cubic intuitionistic fuzzy subset of k,TℂIP, then its boundary or frontier is defined as(47)FrℂIP=ℂIP¯∩pℂIPc¯.Definition 28.
LetℂIP be a P-cubic intuitionistic fuzzy subset of k,TℂIP, then the exterior is defined as(48)ExtℂIP=ℂIP¯c=ℂIPc0.Example 6.
Consider a P-cubic intuitionistic topological space as constructed in Example1 and ℂIP6 and ℂIP7 from Examples 4 and 5. Then,(49)ℂIP60=ℂIP1,ℂIP6¯=ℂIP3c,FrℂIP6=ℂIP1c,ExtℂIP6=ℂIP3,ℂIP70=ℂIP1,ℂIP7¯=ℂIP1c,FrℂIP7=ℂIP1c,ExtℂIP7=ℂIP1.Theorem 6.
Letk,TℂIP be ℂIFTp and ℂIP∈cik. Then,(1)
ℂIP0c=ℂIPc¯(2)
ℂIP¯c=ℂIPc0(3)
ExtℂIPc=ℂIP0(4)
ExtℂIP=ℂIPc0(5)
ExtℂIP∪pFrℂIP∪pℂIP0≠1ℂIP(6)
FrℂIP=FrℂIPc(7)
FrℂIP∩pℂIP0≠0ℂIPProof.
(1)
The proof is obvious.(2)
The proof is obvious.(3)
ExtℂIPc=ℂIPc¯c.ExtℂIPc=ℂIPcc0.ExtℂIPc=ℂIP0.(4)
ExtℂIP=ℂIP¯c.ExtℂIP=ℂIPc0.(5)
ExtℂIP∪pFrℂIP∪pℂIP0≠1ℂIP. By Example 13, we can see that ExtℂIP6∪pFrℂIP6∪pℂIP0≠1ℂI.(6)
FrℂIPc=ℂIPc¯∩pℂIPcc¯FrℂIPc=ℂIPc¯∩pℂIP¯=FrℂIP.(7)
FrℂIP∩pℂIP0≠ℂIP. From Example 13, we can see that FrℂIP6∩pℂIP0≠0ℂI.


### 3.3. P-Cubic Intuitionistic Fuzzy Basis

Definition 29.
Letk,TℂIP be ℂIFTp. Then, B⊆TℂIP is called P-cubic intuitionistic fuzzy basis for TℂIP if for every ℂIP∈TℂIP, ∃B∈B such that(50)ℂIP=∪pB.Example 7.
From Example1,(51)TℂIP=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,is a P-cubic intuitionistic fuzzy topology of k. Then,(52)B=ℂI0,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,Is a P-cubic intuitionistic fuzzy basis for TℂIP.


## 3.1. Subspace ofℂIFTp

Definition 24.
Letk,TℂIPk be a ℂIFTp. Let Y⊆k and TℂIPY is a ℂIFTp on Y and whose PCIFOSs are(34)ℂIPY=TℂIPk∩pY,where ℂIPk are PCIFOSs of TℂIPk, TℂIPY are PCIFOSs of TℂIPY and Y˜ is any P-cubic subset of PCIFS on Y. Then, TℂIPY is called a P-cubic intuitionistic fuzzy subspace of TℂIPk, i.e.,(35)TℂIPY=ℂIPY:ℂIPY=ℂIPk∩pY,ℂIPk∈TℂIPk.Example 3.
Letk be a nonempty set. From Example 1,(36)TℂIP=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,is a P-cubic intuitionistic fuzzy topology on k.
Now, consider any P-cubic fuzzy subset onk such that Y⊆k is(37)Y=0.98,0.23,0.46,0.61,0.27,0.49.
Since,(38)Y∩p0ℂI=0,0,1,1,0.27,0.49=ℂIP⟶,Y∩p1ℂI=0.98,0.23,0.46,0.61,0,1=ℂ˜IP,Y∩p0ℂI¯=0,0,1,1,0,1=ℂIP′,Y∩p1ℂI¯=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP1=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP2=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP3=0.98,0.23,0.46,0.61,0,1=ℂ˜IP,Y∩pℂIP4=0.98,0.23,0.46,0.61,0.27,0.49=Y,Y∩pℂIP5=0,0,1,1,0.27,0.49=ℂIP⟶.
Then,(39)TℂIPY=ℂIP⟶,ℂIP˜,ℂIP′,Y,is a P-cubic intuitionistic fuzzy relative topology of TℂIPk


## 3.2. Interior, Closure, Frontier and Exterior of PCIFSs

Definition 25.
letk,TℂIP be ℂIFTp and ℂIP∈cik, the interior of ℂIP is expressed as ℂIP0 and is described as union of all P-cubic intuitionistic fuzzy open subsets contained in ℂIP. It is the greatest P-cubic intuitionistic fuzzy open set contained in ℂIP.Example 4.
Consider a P-cubic intuitionistic fuzzy topological space as constructed in Example1. Let ℂIP6∈cik given as(40)ℂIP6=0.23,0.39,0.37,0.48,0.46,0.33.
Then,(41)ℂIP60=ℂI0¯∪pℂIP1∪pℂIP3∪pℂIP5=ℂIP1.Theorem 3.
Letk,TℂIP be ℂIFT p and ℂIP∈cik. Then, ℂIP is open CIFS iff ℂIP0=ℂIP.Proof.
IfℂIP is open CIFS, then we say that the greatest open CIFS contained in ℂIP is ℂIP itself. Thus,(42)ℂIP0=ℂIP.Conversely, ifℂIP0=ℂIP, then ℂIP0 is open CIFS. This implies ℂIP is open CIFS.Theorem 4.
Letk,TℂIP be ℂIFTp and ℂIP1,ℂIP2∈cik. Then,(i)
ℂIP00=ℂIP0(ii)
ℂIP1⊆pℂIP2⇒ℂIP10⊆pℂIP20(iii)
ℂIP1∩pℂIP20=ℂIP10⊆pℂIP20(iv)
ℂIP1∪pℂIP20⊇pℂIP10∪pℂIP20Proof.
Proof is trivial.Definition 26.
letk,TℂIP be ℂIFTp and ℂIP∈cik, the closure of ℂIP¯ is expressed as ℂIP¯ and is described as the intersection of all the P-cubic intuitionistic fuzzy closed supersets of ℂIP. It is the smallest P-cubic intuitionistic fuzzy closed superset of ℂIP.Example 5.
Let us consider a P-cubic intuitionistic topological space as constructed in Example1. Then, the closed CIFSs are given as(43)ℂI0c=1,1,0,0,0,1,ℂI1c=0,0,1,1,1,0,ℂI0¯c=1,1,0,0,1,0,ℂI1¯c=0,0,1,1,0,1,ℂIP1c=0.41,0.52,0.20,0.31,0.44,0.32,ℂIP2c=0.41,0.52,0.20,0.31,0,1,ℂIP3c=0.41,0.52,0.20,0.31,1,0,ℂIP4c=0,0,1,1,0.44,0.32,ℂIP5c=1,1,0,0,0.44,0.32.
LetℂIP7∈cik given as(44)ℂIP7=0.34,0.50,0.27,0.38,0.33,0.41.
Then,(45)ℂIP7¯=ℂI0¯c∩pℂIP1c∩pℂIP3c∩pℂIP5c=ℂIP1c.Theorem 5.
Letk,TℂIP be ℂIFT p and ℂIP∈cik. Then ℂIP¯ is closed CIFS iff ℂIP¯=ℂIP.Proof.
IfℂIP is closed CIFS, then we can say that the smallest closed CIFS superset of ℂIP is ℂIP itself. Thus,(46)ℂIP¯=ℂIP.
Conversely, ifℂIP¯=ℂIP, then ℂIP¯ is closed CIFS. This implies ℂIP is closed CIFS. □Definition 27.
LetℂIP be a P-cubic intuitionistic fuzzy subset of k,TℂIP, then its boundary or frontier is defined as(47)FrℂIP=ℂIP¯∩pℂIPc¯.Definition 28.
LetℂIP be a P-cubic intuitionistic fuzzy subset of k,TℂIP, then the exterior is defined as(48)ExtℂIP=ℂIP¯c=ℂIPc0.Example 6.
Consider a P-cubic intuitionistic topological space as constructed in Example1 and ℂIP6 and ℂIP7 from Examples 4 and 5. Then,(49)ℂIP60=ℂIP1,ℂIP6¯=ℂIP3c,FrℂIP6=ℂIP1c,ExtℂIP6=ℂIP3,ℂIP70=ℂIP1,ℂIP7¯=ℂIP1c,FrℂIP7=ℂIP1c,ExtℂIP7=ℂIP1.Theorem 6.
Letk,TℂIP be ℂIFTp and ℂIP∈cik. Then,(1)
ℂIP0c=ℂIPc¯(2)
ℂIP¯c=ℂIPc0(3)
ExtℂIPc=ℂIP0(4)
ExtℂIP=ℂIPc0(5)
ExtℂIP∪pFrℂIP∪pℂIP0≠1ℂIP(6)
FrℂIP=FrℂIPc(7)
FrℂIP∩pℂIP0≠0ℂIPProof.
(1)
The proof is obvious.(2)
The proof is obvious.(3)
ExtℂIPc=ℂIPc¯c.ExtℂIPc=ℂIPcc0.ExtℂIPc=ℂIP0.(4)
ExtℂIP=ℂIP¯c.ExtℂIP=ℂIPc0.(5)
ExtℂIP∪pFrℂIP∪pℂIP0≠1ℂIP. By Example 13, we can see that ExtℂIP6∪pFrℂIP6∪pℂIP0≠1ℂI.(6)
FrℂIPc=ℂIPc¯∩pℂIPcc¯FrℂIPc=ℂIPc¯∩pℂIP¯=FrℂIP.(7)
FrℂIP∩pℂIP0≠ℂIP. From Example 13, we can see that FrℂIP6∩pℂIP0≠0ℂI.


## 3.3. P-Cubic Intuitionistic Fuzzy Basis

Definition 29.
Letk,TℂIP be ℂIFTp. Then, B⊆TℂIP is called P-cubic intuitionistic fuzzy basis for TℂIP if for every ℂIP∈TℂIP, ∃B∈B such that(50)ℂIP=∪pB.Example 7.
From Example1,(51)TℂIP=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,is a P-cubic intuitionistic fuzzy topology of k. Then,(52)B=ℂI0,ℂI1¯,ℂIP1,ℂIP2,ℂIP3,ℂIP4,ℂIP5,Is a P-cubic intuitionistic fuzzy basis for TℂIP.


## 4. Cubic Intuitionistic Topology under R-Order

In this section, we introduce the concept of an R-cubic intuitionistic fuzzy topology (R-CIFT) or a cubic intuitionistic fuzzy topology with an R-order.Definition 30.
Considerk to be a nonempty universal set, and let cik to be the collection of all CIFSs in k. If the collection TℂIℝ containing CIFSs satisfies the following conditions, it is termed a cubic intuitionistic fuzzy topology with an R-order (R-CIFT).(1)
ℂI0, ℂI1, ℂI0¯ and ℂI0¯∈TℂIℝ(2)
IfℂIℝi∈TℂIℝ∀i∈Λ then ∪RℂIℝi∈TℂIℝ(3)
IfℂIℝ1,ℂIℝ2∈TℂIℝ then ℂIℝ1∩RℂIℝ2∈TℂIℝ
Then, the pairk,TℂIℝ is termed as a cubic intuitionistic fuzzy topological space with an R-order (R-CIFT).Example 8.
Letk be a nonempty universal set. Then, cik be the accumulation of all R-cubic intuitionistic fuzzy sets RCIFSs in k. Consider the R-order fuzzy subsets of cik given as(53)ℂIℝ1=0.31,0.42,0.47,0.56,0.29,0.39,ℂIℝ2=0.31,0.42,0.47,0.56,0,1,ℂIℝ3=1,1,0,0,0.29,0.39,ℂIℝ4=0,0,1,1,0.29,0.39,ℂIℝ5=0.31,0.42,0.47,0.56,1,0.
The union and intersection with a P-order for the above CIFSs are given in Tables3 and 4,, respectively.
Clearly,(54)TℂIℝ1=ℂI0,ℂI1,ℂI0¯,ℂI1¯,and(55)TℂIℝ2=0ℂI,1ℂI,0ℂI¯,1ℂI¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,are the cubic intuitionistic topology with an R-order.Table 3
Union under R-order.
∪RℂI01ℂI0ℂI¯1ℂI¯ℂIℝ1ℂIℝ2ℂIℝ3ℂIℝ4ℂIℝ50ℂI0ℂI1ℂI0ℂI¯ℂI1¯ℂIℝ1ℂIℝ2ℂIℝ3ℂIℝ4ℂIℝ5ℂI11ℂI1ℂI1ℂI1ℂI1ℂI1ℂI1ℂI1ℂI1ℂIℂI0¯0ℂI¯1ℂI0ℂI¯1ℂIℂIℝ2ℂIℝ21ℂI0ℂI¯0ℂI¯ℂI1¯1ℂI¯1ℂI1ℂI1ℂI¯ℂIℝ31ℂIℂIℝ3ℂIℝ31ℂI¯ℂIℝ1ℂIℝ11ℂIℂIℝ2ℂIℝ3ℂIℝ1ℂIℝ2ℂIℝ3ℂIℝ1ℂIℝ1ℂIℝ2ℂIℝ21ℂIℂIℝ21ℂIℂIℝ2ℂIℝ21ℂIℂIℝ2ℂIℝ2ℂIℝ3ℂIℝ31ℂI1ℂIℂIℝ3ℂIℝ31ℂIℂIℝ3ℂIℝ3ℂIℝ3ℂIℝ4ℂIℝ4ℂI10ℂI¯ℂIℝ3ℂIℝ1ℂIℝ2ℂIℝ3ℂIℝ4ℂIℝ1ℂIℝ5ℂIℝ5ℂI10ℂI¯1ℂI¯ℂIℝ1ℂIℝ2ℂIℝ3ℂIℝ1ℂIℝ5Table 4
Intersection under R-order.
∩R0ℂI1ℂI0ℂI¯1ℂI¯ℂIℝ1ℂIℝ2ℂIℝ3ℂIℝ4ℂIℝ5ℂI00ℂI0ℂI0ℂI0ℂI0ℂI0ℂI0ℂI0ℂI0ℂI1ℂI0ℂI1ℂI0ℂI¯1ℂI¯ℂIℝ1ℂIℝ2ℂIℝ3ℂIℝ4ℂIℝ50ℂI¯0ℂI0ℂI¯0ℂI¯0ℂIℂIℝ40ℂI¯ℂIℝ4ℂIℝ40ℂI1ℂI¯0ℂI1ℂI¯0ℂI1ℂI¯ℂIℝ5ℂIℝ51ℂI¯0ℂIℂIℝ5ℂIℝ10ℂIℂIℝ1ℂIℝ4ℂIℝ5ℂIℝ1ℂIℝ1ℂIℝ4ℂIℝ5ℂIℝ5ℂIℝ20ℂIℂIℝ20ℂI¯ℂIℝ5ℂIℝ1ℂIℝ2ℂIℝ1ℂIℝ4ℂIℝ5ℂIℝ30ℂIℂIℝ3ℂIℝ41ℂI¯ℂIℝ1ℂIℝ1ℂIℝ3ℂIℝ4ℂIℝ5ℂIℝ40ℂIℂIℝ4ℂIℝ40ℂIℂIℝ4ℂIℝ4ℂIℝ4ℂIℝ40ℂIℂIℝ50ℂIℂIℝ50ℂIℂIℝ5ℂIℝ5ℂIℝ5ℂIℝ50ℂI0ℂIDefinition 31.
Letk be a nonempty set, and TℂIℝ=ℂIℝk where ℂIℝk represent the cubic intuitionistic fuzzy subsets of the universal set k. Then, TℂIℝ is termed as an R-cubic intuitionistic fuzzy topology on k and it is the largest R-cubic intuitionistic fuzzy topology on k and is entitled as an R-discrete cubic intuitionistic fuzzy topology.Definition 32.
Letk be a universal set and TℂIℝ=0ℂI,1ℂI,0ℂI¯,1ℂI¯ be the assemblage of cubic intuitionistic fuzzy sets. Then, TℂIℝ is termed as an R-cubic intuitionistic fuzzy topology on the universal set k and is the smallest R-cubic intuitionistic fuzzy topology on k and is entitled as an R-indiscrete cubic intuitionistic fuzzy topology.Definition 33.
The elements of an R-cubic intuitionistic fuzzy topologyTℂIℝ is termed as the R-cubic intuitionistic fuzzy open sets RCIFOS in k,TℂIℝ.Theorem 7.
Ifk,TℂIℝ is any R-cubic intuitionistic fuzzy topological space. Then,(1)
ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are RCIFOSs.(2)
The R-union of any number of RCIFOSs is RCIFOS.(3)
The R-intersection of finite RCIFOSs is RCIFOS.Proof.
The proof is trivial.Definition 34.
The complement of elements of an R-cubic intuitionistic fuzzy open sets is termed as the R-cubic intuitionistic fuzzy closed sets RCIFCSs ink,TℂIℝ.Theorem 8.
Ifk,TℂIℝ is any R-cubic intuitionistic fuzzy topological space. Then,(1)
ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are RCIFCSs.(2)
The R-intersection of any number of RCIFCSs is RCIFCS.(3)
The R-union of finite RCIFCSs is RCIFCS.Proof.
The proof is trivial.Definition 35.
The R-cubic intuitionistic fuzzy sets RCIFSs, which are RCIFOSs and RCIFCSs, are entitled as the R-cubic intuitionistic fuzzy clopen sets ink,TℂIℝ.Proposition 2.
(1)
For everyTℂIℝ, ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are R-cubic intuitionistic fuzzy clopen sets.(2)
For the discrete R-order cubic intuitionistic fuzzy topology, all the cubic intuitionistic subsets ofk are R-cubic intuitionistic fuzzy clopen sets.(3)
For the in-discrete R-order cubic intuitionistic fuzzy topology,ℂI0,ℂI1,ℂI0¯ and ℂI1¯ are only R-cubic intuitionistic fuzzy clopen sets.Definition 36.
Letk,TℂIℝ1 and k,TℂIℝ2 be two R-CIFTs in k. Two R-CIFTs are called comparable if(56)TℂIℝ1⊆RTℂIℝ2,or(57)TℂIℝ2⊆RTℂIℝ1.
IfTℂIℝ2⊆RTℂIℝ2 then, TℂIℝ1 is called R-cubic intuitionistic fuzzy coarser than TℂIℝ2 and TℂIℝ2 is called the R-cubic intuitionistic fuzzy finer than. TℂIℝ1Example 9.
Letk be a nonempty set and from Example 8,(58)TℂIℝ1=ℂI0,ℂI1,ℂI0¯,ℂI1¯,and(59)TℂIℝ2=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,are R-cubic intuitionistic fuzzy topologies on the universal set. Then, TℂIℝ1⊆RTℂIℝ2. Hence, TℂIℝ1 is called the R-cubic intuitionistic fuzzy coarser then, TℂIℝ2.


### 4.1. Subspace ofℂIFTr

Definition 37.
Letk,TℂIℝk be a ℂIFTr. Let Y⊆k and TℂIℝY is a ℂIFTr on Y and whose RCIFOSs are(60)ℂIℝY=TℂIℝk∩RY,where ℂIℝk are RCIFOSs of TℂIℝk, TℂIℝY are RCIFOSs of TℂIℝY and Y˜ is any R-cubic subset of RCIFS on Y. Then, TℂIℝY is called the R-cubic intuitionistic fuzzy subspace of TℂIℝk i.e.,(61)TℂIℝY=ℂIℝY:ℂIℝY=ℂIℝk∩RY,ℂIℝk∈TℂIℝk.Example 10.
Letk be a nonempty set. From Example 8,(62)TℂIℝ=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,is an R-cubic intuitionistic fuzzy topology on k.
Now, consider any R-cubic fuzzy subset onk such that Y⊆k is(63)Y=0.27,0.38,0.52,0.67,0.34,0.28.
Also,(64)Y∩R0ℂI=0,0,1,1,1,0=ℂIℝ⟶,Y∩R1ℂI=0.27,0.38,0.52,0.67,1,0=ℂIℝ˜,Y∩R0ℂI¯=0,0,1,1,0.34,0.28=ℂIℝ′,Y∩R1ℂI¯=0.27,0.38,0.52,0.67,1,0=ℂIℝ˜,Y∩RℂIℝ1=0.27,0.38,0.52,0.67,0.34,0.28=Y,Y∩RℂIℝ2=0.27,0.38,0.52,0.67,0.34,0.28=Y,Y∩RℂIℝ3=0.27,0.38,0.52,0.67,0.34,0.28=Y,Y∩RℂIℝ4=0,0,1,1,0.34,0.28=ℂIℝ′,Y∩RℂIℝ5=0.27,0.38,0.52,0.67,1,0=ℂIℝ˜.
Then,(65)TℂIℝY=ℂIℝ⟶,ℂIℝ˜,ℂIℝ′,Y,is an R-cubic intuitionistic fuzzy relative topology of TℂIℝk


### 4.2. Interior, Closure, Frontier, and Exterior of RCIFSs

Definition 38.
letk,TℂIℝ be ℂIFTr and ℂIℝ∈cik, the interior of ℂIℝ is expressed as ℂIℝ0 and is described as a union of all the R-cubic intuitionistic fuzzy open subsets contained in ℂIℝ. It is the greatest R-cubic intuitionistic fuzzy open set contained in ℂIℝ.Example 11.
Consider an R-cubic intuitionistic fuzzy topological space as constructed in Example8. Let ℂIℝ6∈cik given as(66)ℂIℝ6=0.38,0.46,0.45,0.51,0.26,0.40.
Then,(67)ℂIℝ60=0ℂI∪RℂIℝ1∪RℂIP4∪RℂIℝ5=ℂIℝ1.Theorem 9.
Letk,TℂIℝ be ℂIFT r and ℂIℝ∈cik. Then, ℂIℝ is open CIFS iff ℂIℝ0=ℂIℝ.Proof.
The proof is trivial.Theorem 10.
Letk,TℂIP be ℂIFTp and ℂIP1,ℂIP2∈cik. Then,(i)
ℂIℝ00=ℂIℝ0(ii)
ℂIℝ1⊆RℂIℝ2⇒ℂIℝ10⊆RℂIℝ20(iii)
ℂIℝ1∩RℂIℝ20=ℂIℝ10⊆RℂIℝ20(iv)
ℂIℝ1∪RℂIℝ20⊇RℂIℝ10∪RℂIℝ20Proof.
Proof is trivial.Definition 39.
letk,TℂIℝ be ℂIFTr and ℂIℝ∈cik, the closure of ℂIℝ is expressed as ℂIℝ¯ and is described as the intersection of all the R-cubic intuitionistic fuzzy closed supersets of ℂIℝ. It is the smallest R-cubic intuitionistic fuzzy closed superset of ℂIℝ.Example 12.
Let us consider an R-cubic intuitionistic topological space as constructed in Example8. Then, the closed CIFSs are given as(68)ℂI0c=1,1,0,0,0,1,ℂI1c=0,0,1,1,1,0,ℂI0¯c=1,1,0,0,1,0,ℂI1¯c=0,0,1,1,0,1,ℂIP1c=0.47,0.56,0.31,0.42,0.39,0.29,ℂIP2c=0.47,0.56,0.31,0.42,1,0,ℂIP3c=0,0,1,1,0.39,0.29,ℂIP4c=1,1,0,0,0.39,0.29,ℂIP5c=0.47,0.56,0.31,0.42,0,1.
LetℂIℝ7∈cik given as(69)ℂIℝ7=0.30,0.37,0.48,0.61,0.38,0.24.
Then,(70)ℂIℝ7¯=ℂI0¯c∩RℂIℝ5c=ℂIℝ5c.Theorem 11.
Letk,TℂIℝ be ℂIFT r and ℂIℝ∈cik. Then, ℂIℝ is closed CIFS iff ℂIℝ¯=ℂIℝ..Proof.
The proof is trivial.Definition 40.
LetℂIℝ be an R-cubic intuitionistic fuzzy subset of k,TℂIℝ, then its boundary or frontier is defined as(71)FrℂIℝ=ℂIℝ¯∩RℂIℝc¯.Definition 41.
LetℂIℝ be an R-cubic intuitionistic fuzzy subset of k,TℂIℝ, then the exterior is defined as(72)ExtℂIℝ=ℂIℝ¯c=ℂIℝc0.Example 4.13.
Consider an R-cubic intuitionistic topological space as constructed in Example8 and ℂIℝ6 and ℂIℝ7 from Examples 11 and 12. Then,(73)ℂIℝ60=ℂIℝ1,ℂIP6¯=ℂIℝ5c,FrℂIℝ6=ℂIℝ5c,ExtℂIℝ6=ℂIℝ5,ℂIℝ70=ℂIℝ4,ℂIℝ7¯=ℂIℝ5c,FrℂIℝ7=ℂIℝ5c,ExtℂIℝ7=ℂIℝ5.Theorem 12.
Letk,TℂIℝ be ℂIFTr and ℂIℝ∈cik. Then,(1)
ℂIℝ0c=ℂIℝc¯(2)
ℂIℝ¯c=ℂIℝc0(3)
ExtℂIℝc=ℂIℝ0(4)
ExtℂIℝ=ℂIℝc0(5)
ExtℂIℝ∪RFrℂIℝ∪RℂIℝ0≠1ℂIℝ(6)
FrℂIℝ=FrℂIℝc(7)
FrℂIℝ∩RℂIℝ0≠0ℂIℝProof.
The proof is trivial.


### 4.3. R-Cubic Intuitionistic Fuzzy Basis

Definition 42.
Letk,TℂIℝ be ℂIFTr. Then B⊆TℂIℝ is called an R-cubic intuitionistic fuzzy basis for TℂIℝ if for every ℂIℝ∈TℂIℝ, ∃B∈B such that(74)ℂIℝ=∪RB.Example 14.
From Example8,(75)TℂIℝ=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,is an R-cubic intuitionistic fuzzy topology of k. Then,(76)B=ℂI1,ℂI1¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,is an R-cubic intuitionistic fuzzy basis for TℂIℝ.


## 4.1. Subspace ofℂIFTr

Definition 37.
Letk,TℂIℝk be a ℂIFTr. Let Y⊆k and TℂIℝY is a ℂIFTr on Y and whose RCIFOSs are(60)ℂIℝY=TℂIℝk∩RY,where ℂIℝk are RCIFOSs of TℂIℝk, TℂIℝY are RCIFOSs of TℂIℝY and Y˜ is any R-cubic subset of RCIFS on Y. Then, TℂIℝY is called the R-cubic intuitionistic fuzzy subspace of TℂIℝk i.e.,(61)TℂIℝY=ℂIℝY:ℂIℝY=ℂIℝk∩RY,ℂIℝk∈TℂIℝk.Example 10.
Letk be a nonempty set. From Example 8,(62)TℂIℝ=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,is an R-cubic intuitionistic fuzzy topology on k.
Now, consider any R-cubic fuzzy subset onk such that Y⊆k is(63)Y=0.27,0.38,0.52,0.67,0.34,0.28.
Also,(64)Y∩R0ℂI=0,0,1,1,1,0=ℂIℝ⟶,Y∩R1ℂI=0.27,0.38,0.52,0.67,1,0=ℂIℝ˜,Y∩R0ℂI¯=0,0,1,1,0.34,0.28=ℂIℝ′,Y∩R1ℂI¯=0.27,0.38,0.52,0.67,1,0=ℂIℝ˜,Y∩RℂIℝ1=0.27,0.38,0.52,0.67,0.34,0.28=Y,Y∩RℂIℝ2=0.27,0.38,0.52,0.67,0.34,0.28=Y,Y∩RℂIℝ3=0.27,0.38,0.52,0.67,0.34,0.28=Y,Y∩RℂIℝ4=0,0,1,1,0.34,0.28=ℂIℝ′,Y∩RℂIℝ5=0.27,0.38,0.52,0.67,1,0=ℂIℝ˜.
Then,(65)TℂIℝY=ℂIℝ⟶,ℂIℝ˜,ℂIℝ′,Y,is an R-cubic intuitionistic fuzzy relative topology of TℂIℝk


## 4.2. Interior, Closure, Frontier, and Exterior of RCIFSs

Definition 38.
letk,TℂIℝ be ℂIFTr and ℂIℝ∈cik, the interior of ℂIℝ is expressed as ℂIℝ0 and is described as a union of all the R-cubic intuitionistic fuzzy open subsets contained in ℂIℝ. It is the greatest R-cubic intuitionistic fuzzy open set contained in ℂIℝ.Example 11.
Consider an R-cubic intuitionistic fuzzy topological space as constructed in Example8. Let ℂIℝ6∈cik given as(66)ℂIℝ6=0.38,0.46,0.45,0.51,0.26,0.40.
Then,(67)ℂIℝ60=0ℂI∪RℂIℝ1∪RℂIP4∪RℂIℝ5=ℂIℝ1.Theorem 9.
Letk,TℂIℝ be ℂIFT r and ℂIℝ∈cik. Then, ℂIℝ is open CIFS iff ℂIℝ0=ℂIℝ.Proof.
The proof is trivial.Theorem 10.
Letk,TℂIP be ℂIFTp and ℂIP1,ℂIP2∈cik. Then,(i)
ℂIℝ00=ℂIℝ0(ii)
ℂIℝ1⊆RℂIℝ2⇒ℂIℝ10⊆RℂIℝ20(iii)
ℂIℝ1∩RℂIℝ20=ℂIℝ10⊆RℂIℝ20(iv)
ℂIℝ1∪RℂIℝ20⊇RℂIℝ10∪RℂIℝ20Proof.
Proof is trivial.Definition 39.
letk,TℂIℝ be ℂIFTr and ℂIℝ∈cik, the closure of ℂIℝ is expressed as ℂIℝ¯ and is described as the intersection of all the R-cubic intuitionistic fuzzy closed supersets of ℂIℝ. It is the smallest R-cubic intuitionistic fuzzy closed superset of ℂIℝ.Example 12.
Let us consider an R-cubic intuitionistic topological space as constructed in Example8. Then, the closed CIFSs are given as(68)ℂI0c=1,1,0,0,0,1,ℂI1c=0,0,1,1,1,0,ℂI0¯c=1,1,0,0,1,0,ℂI1¯c=0,0,1,1,0,1,ℂIP1c=0.47,0.56,0.31,0.42,0.39,0.29,ℂIP2c=0.47,0.56,0.31,0.42,1,0,ℂIP3c=0,0,1,1,0.39,0.29,ℂIP4c=1,1,0,0,0.39,0.29,ℂIP5c=0.47,0.56,0.31,0.42,0,1.
LetℂIℝ7∈cik given as(69)ℂIℝ7=0.30,0.37,0.48,0.61,0.38,0.24.
Then,(70)ℂIℝ7¯=ℂI0¯c∩RℂIℝ5c=ℂIℝ5c.Theorem 11.
Letk,TℂIℝ be ℂIFT r and ℂIℝ∈cik. Then, ℂIℝ is closed CIFS iff ℂIℝ¯=ℂIℝ..Proof.
The proof is trivial.Definition 40.
LetℂIℝ be an R-cubic intuitionistic fuzzy subset of k,TℂIℝ, then its boundary or frontier is defined as(71)FrℂIℝ=ℂIℝ¯∩RℂIℝc¯.Definition 41.
LetℂIℝ be an R-cubic intuitionistic fuzzy subset of k,TℂIℝ, then the exterior is defined as(72)ExtℂIℝ=ℂIℝ¯c=ℂIℝc0.Example 4.13.
Consider an R-cubic intuitionistic topological space as constructed in Example8 and ℂIℝ6 and ℂIℝ7 from Examples 11 and 12. Then,(73)ℂIℝ60=ℂIℝ1,ℂIP6¯=ℂIℝ5c,FrℂIℝ6=ℂIℝ5c,ExtℂIℝ6=ℂIℝ5,ℂIℝ70=ℂIℝ4,ℂIℝ7¯=ℂIℝ5c,FrℂIℝ7=ℂIℝ5c,ExtℂIℝ7=ℂIℝ5.Theorem 12.
Letk,TℂIℝ be ℂIFTr and ℂIℝ∈cik. Then,(1)
ℂIℝ0c=ℂIℝc¯(2)
ℂIℝ¯c=ℂIℝc0(3)
ExtℂIℝc=ℂIℝ0(4)
ExtℂIℝ=ℂIℝc0(5)
ExtℂIℝ∪RFrℂIℝ∪RℂIℝ0≠1ℂIℝ(6)
FrℂIℝ=FrℂIℝc(7)
FrℂIℝ∩RℂIℝ0≠0ℂIℝProof.
The proof is trivial.


## 4.3. R-Cubic Intuitionistic Fuzzy Basis

Definition 42.
Letk,TℂIℝ be ℂIFTr. Then B⊆TℂIℝ is called an R-cubic intuitionistic fuzzy basis for TℂIℝ if for every ℂIℝ∈TℂIℝ, ∃B∈B such that(74)ℂIℝ=∪RB.Example 14.
From Example8,(75)TℂIℝ=ℂI0,ℂI1,ℂI0¯,ℂI1¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,is an R-cubic intuitionistic fuzzy topology of k. Then,(76)B=ℂI1,ℂI1¯,ℂIℝ1,ℂIℝ2,ℂIℝ3,ℂIℝ4,ℂIℝ5,is an R-cubic intuitionistic fuzzy basis for TℂIℝ.


## 5. Multicriteria Group Decision-Making

The weighted product model (WPM) is a renowned and widely used MCGDM approach for evaluating a set of choices using a set of criteria. Each choice is contrasted to the others by calculating a number of ratios, one per choice criterion. Every ratio is multiplied by the proportional weight of the criterion in consideration. For the selection of one or more options from the set of alternatives based on a number of criteria is a fundamental task in MCGDM problems. Let us considerm alternatives, n criteria with weighted vectors, with the condition that the sum of the weights will be one, for an MCGDM problem in a cubic intuitionistic fuzzy set domain.Figure1 shows the flow chart of WPM.Figure 1
Flow chart of CIF WPM.


### 5.1. Application to Uncertain Supply Chain Management

Communication and information technologies are affecting every area of the industrial sector at a rapid pace. In reality, it would be difficult to pinpoint an organization that does not use or is not touched by information and communications technologies in some way. In many cases, if technology is not employed appropriately, the firm’s survival is jeopardized. Companies nowadays use technology to boost productivity, streamline operations, and form electronic conglomerates. Advanced technologies and electronic systems are radically altering how businesses operate and stay competitive. Many businesses are making strategic technology investments to obtain and maintain a competitive advantage in their industry. Management teams must use technology throughout the organization to enhance information flow, reduce cost, streamline operations, provide product variety, formulate connections with suppliers, and reduce response times to customers’ needs to gain a competitive advantage through the use of information and communications technology.Administrators and top executives should be associated with the development of enterprise-wide information systems (EIS), which should take into account such matters as computer hardware and software and infrastructure facilities, online systems, digital applications, electronic commerce, and alterations to current processes and practices. Managers can integrate data and telecommunications technologies throughout the corporation and connect all business areas by developing an enterprise broad information systems plan. Enterprise-wide integration of technology enables firms to allow consumers to get timely access to the information they need to make informed decisions. Recent research has looked at information systems as useful tools for integrating systems like enterprise resource planning, knowledge management, e-commerce, electronic markets, and supply chain management (SCM) to enhance organizational profit and efficiency.Companies must analyze both internal and external processes for the production and exchange of products and services to be more efficient and competitive. The managers will be able to evaluate the value of actions for each process to determine how to boost the value among these operations that form a supply chain from supplier to business to dealer to customer through the evaluation of these processes. The level of integration among suppliers, business associates, and buyers, independent of their geographical location, determines the value chain’s or supply chain’s effectiveness.The construction of an integrated organizational system capable of information sharing, resources, and services in the supply chain is central to the digital supply chain management paradigm. To gain and maintain competitive advantages, companies use digital information and communications networks to standardize manufacturing processes, reduce cycle time, increase the effectiveness of procurement procedures and logistical support, reduce production costs, and increase customer satisfaction, among other things. Supply chain management based on the Internet allows a company to streamline its supply chain, increase speed, reduce costs, and be more adaptable. It can also increase consumer and supplier communications as well as smooth the ongoing flow of goods along the supply chain.Supplier selection is highly essential in supply chain management. The objective is to locate a supplier who can offer the best products and services for the lowest price. Proper supplier selection delivers a high profit and quality level. In this strategic collaboration, the supplier is viewed as a significant element of the business. Because of the increasing focus on sustainability, identifying these providers has become more challenging. Environmental studies, often known as sustainability studies, have become increasingly popular around the world. Identifying these suppliers has become increasingly difficult as a result of the rapidly increasing emphasis on sustainability. Many methodologies for sustainable supply chain selection have been developed.To determine the most suitable supplier selection, MCGDM techniques can be used successfully. In this section, the suggested model is used to determine the selection of appropriate suppliers for fast-moving consumer products, with the goal of selecting the best supplier among various possibilities. Several criteria have been established based on expert opinions to evaluate supplier choices. In this study,X1, X2, X3, X4, X5 and X6 are examined as possible fast-moving customers goods suppliers using the four criteria established.


### 5.2. CIF Weighted Product Model

The proposed method is used to choose the best supplier among six alternatives. These alternatives are weighed against four criteriaℂ1 = price, ℂ2 = quality, ℂ3 = performance, and ℂ4 = delivery, derived from thorough expert opinions. A group of decision-makers has been assembled to assess the suppliers using the recommended methodology. Six decision-makers D1, D2, D3, D4, D5, and D6 were chosen, consisting of supplier experts and expert academics on multicriteria decision-making in a fuzzy environment:,,,Step 1. Consider the decision matrix M=Tjim×n given by the decision-makers in the form of CIFNs on the basis of the cubic intuitionistic fuzzy linguistic scale to evaluate suppliers in accordance with established objectives and criteria is given in Table 5.Step 2. With the help of the linear approach, we normalize the matrix M=Tjim×n. We divide the criteria into two subsets, benefit criteria B and cost criteria K. Here, X3 and X6 belong to benefit criteria B, and the remaining others belong to cost criteria K. For this, first, we find maxjTji and minjTji, which are given in Table 5. We normalized the decision matrix by utilizing the 1st and 2nd equations in Algorithm 1 and this is given in Table 6.Normalized decision matrixes from DMs are expressed in Table7.Steps 3 and 4. We find the relative importance of all alternatives by utilizing the 3rd equation in Algorithm 1, and then, we calculate their score function as given in Table 8.Step 5. Rank the alternatives according to the score function, and the final ranking is(77)X3≻X5≻X1≻X6≻X2≻X4.As we can see thatX3 is the most appropriate supplier among the six alternatives with the best of qualities of all criteria.Table 5
Cubic intuitionistic decision matrix from DMs.
Criteriaℂ1ℂ2ℂ3ℂ4X10.17,0.24,0.36,0.43,0.56,0.320.20,0.28,0.29,0.31,0.27,0.200.18,0.21,0.21,0.32,0.39,0.220.20,0.37,0.21,0.43,0.54,0.23X20.19,0.22,0.39,0.42,0.59,0.400.27,0.34,0.33,0.40,0.43,0.210.24,0.30,0.30,0.39,0.50,0.200.32,0.40,0.19,0.51,0.52,0.30X30.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.72,0.170.31,0.39,0.18,0.40,0.67,0.140.14,0.63,0.24,0.50,0.70,0.18X40.31,0.37,0.36,0.49,0.50,0.360.18,0.33,0.28,0.52,0.40,0.320.23,0.40,0.24,0.51,0.50,0.330.08,0.74,0.32,0.40,0.46,0.42X50.40,0.48,0.51,0.60,0.52,0.300.29,0.41,0.30,0.39,0.48,0.400.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.53,0.37X60.29,0.38,0.27,0.42,0.60,0.270.40,0.51,0.41,0.50,0.47,0.380.32,0.38,0.24,0.72,0.43,0.320.42,0.50,0.37,0.53,0.53,0.27Table 6
maxjTji and minjTji values.
maxjTjiminjTjiX10.20,0.37,0.21,0.31,0.50,0.200.17,0.21,0.36,0.43,0.27,0.32X20.32,0.40,0.19,0.39,0.59,0.200.19,0.22,0.39,0.51,0.43,0.40X30.31,0.63,0.18,0.40,0.81,0.130.14,0.29,0.42,0.50,0.67,0.18X40.31,0.74,0.24,0.40,0.50,0.320.08,0.33,0.36,0.52,0.40,0.42X50.40,0.64,0.30,0.39,0.56,0.290.13,0.41,0.51,0.60,0.48,0.40X60.42,0.51,0.24,0.42,0.60,0.270.29,0.38,0.41,0.72,0.43,0.38Table 7
Normalized decision matrix from DMs.
Criteriaℂ1ℂ2ℂ3ℂ4X10.17,0.21,0.36,0.43,0.56,0.320.17,0.21,0.36,0.43,0.27,0.200.17,0.21,0.36,0.43,0.39,0.220.17,0.21,0.36,0.43,0.54,0.23X20.19,0.22,0.39,0.51,0.59,0.400.19,0.22,0.39,0.51,0.43,0.210.19,0.22,0.39,0.51,0.50,0.200.19,0.22,0.39,0.51,0.52,0.30X30.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.81,0.130.31,0.39,0.18,0.40,0.81,0.130.14,0.63,0.24,0.50,0.81,0.13X40.08,0.33,0.36,0.52,0.50,0.360.08,0.33,0.36,0.52,0.40,0.320.08,0.33,0.36,0.52,0.50,0.330.08,0.33,0.36,0.52,0.46,0.42X50.40,0.48,0.51,0.60,0.56,0.290.29,0.41,0.30,0.39,0.56,0.290.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.56,0.29X60.29,0.38,0.41,0.72,0.60,0.270.29,0.38,0.41,0.72,0.47,0.380.29,0.38,0.41,0.72,0.43,0.320.29,0.38,0.41,0.72,0.53,0.27Table 8
Relative importance and score function.
AlternativesCIFN-WPM valuesScore valuesX10.16,0.20,0.34,0.41,0.9980,0.000010.8029X20.18,0.21,0.38,0.49,0.9993,0.000020.7592X30.21,0.45,0.29,0.46,0.9999,0.00000070.9548X40.07,0.31,0.34,0.50,0.9990,0.000070.6294X50.25,0.49,0.39,0.50,0.9996,0.000020.9245X60.28,0.34,0.39,0.71,0.9994,0.000040.7593


### 5.3. CIF Choice Value Method

The choice value method is a renowned and widely used MCGDM basis for evaluating a set of choices using a set of criteria. Each choice is contrasted to the others by calculating a number of ratios, one per choice criterion. Every ratio is multiplied by the proportional weight of the criterion in consideration. A fundamental task in MCGDM problems is the selection of one or more options from the set of alternatives based on a number of criteria. Let us considerm alternatives, n criteria with weighted vectors, with the condition that the sum of weights will be one, for an MCGDM problem in a cubic intuitionistic fuzzy set domain.,Table 9
Cubic intuitionistic decision matrix from DMs.
Criteriaℂ1ℂ2ℂ3ℂ4X10.17,0.24,0.36,0.43,0.56,0.320.20,0.28,0.29,0.31,0.27,0.200.18,0.21,0.21,0.32,0.39,0.220.20,0.37,0.21,0.43,0.54,0.23X20.19,0.22,0.39,0.42,0.59,0.400.27,0.34,0.33,0.40,0.43,0.210.24,0.30,0.30,0.39,0.50,0.200.32,0.40,0.19,0.51,0.52,0.30X30.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.72,0.170.31,0.39,0.18,0.40,0.67,0.140.14,0.63,0.24,0.50,0.70,0.18X40.31,0.37,0.36,0.49,0.50,0.360.18,0.33,0.28,0.52,0.40,0.320.23,0.40,0.24,0.51,0.50,0.330.08,0.74,0.32,0.40,0.46,0.42X50.40,0.48,0.51,0.60,0.52,0.300.29,0.41,0.30,0.39,0.48,0.400.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.53,0.37X60.29,0.38,0.27,0.42,0.60,0.270.40,0.51,0.41,0.50,0.47,0.380.32,0.38,0.24,0.72,0.43,0.320.42,0.50,0.37,0.53,0.53,0.27Table 10
Score values.
AlternativesScore valuesX10.1125X20.1705X30.5490X40.0725X50.1435X60.204


### 5.4. MCDGM Application

Step 1. Consider the decision matrix M=Tjim×n given by the decision-makers in the form of CIFNs given in Table 9.Step 2. Decision-makers gives the weights to the four criteria as W1=0.18, W2=0.24, W3=0.26, and W4=0.32 with ∑Wi=1(78)0.17,0.24,0.36,0.43,0.56,0.320.20,0.28,0.29,0.31,0.27,0.200.18,0.21,0.21,0.32,0.39,0.220.20,0.37,0.21,0.43,0.54,0.230.19,0.22,0.39,0.42,0.59,0.400.27,0.34,0.33,0.40,0.43,0.210.24,0.30,0.30,0.39,0.50,0.200.32,0.40,0.19,0.51,0.52,0.300.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.72,0.170.31,0.39,0.18,0.40,0.67,0.140.14,0.63,0.24,0.50,0.70,0.180.31,0.37,0.36,0.49,0.50,0.360.18,0.33,0.28,0.52,0.40,0.320.23,0.40,0.24,0.51,0.50,0.330.08,0.74,0.32,0.40,0.46,0.420.40,0.48,0.51,0.60,0.52,0.300.29,0.41,0.30,0.39,0.48,0.400.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.53,0.370.29,0.38,0.27,0.42,0.60,0.270.40,0.51,0.41,0.50,0.47,0.380.32,0.38,0.24,0.72,0.43,0.320.42,0.50,0.37,0.53,0.53,0.270.180.240.260.320.032,0.048,0.832,0.859,0.900,0.067+0.052,0.075,0.742,0.754,0.730,0.052+0.050,0.059,0.666,0.743,0.782,0.062+0.068,0.137,0.606,0.763,0.821,0.0800.037,0.043,0.844,0.855,0.909,0.087+0.072,0.094,0.766,0.802,0.816,0.055+0.068,0.088,0.731,0.782,0.835,0.056+0.116,0.150,0.587,0.806,0.811,0.1070.039,0.059,0.847,0.885,0.962,0.024+0.085,0.161,0.812,0.846,0.924,0.043+0.091,0.120,0.640,0.788,0.901,0.038+0.047,0.272,0.633,0.801,0.892,0.0610.064,0.079,0.832,0.879,0.882,0.077+0.046,0.091,0.736,0.854,0.802,0.088+0.065,0.124,0.690,0.839,0.835,0.098+0.026,0.350,0.694,0.745,0.779,0.1590.087,0.111,0.885,0.912,0.888,0.062+0.078,0.118,0.749,0.797,0.888,0.115+0.124,0.152,0.777,0.875,0.860,0.085+0.043,0.278,0.745,0.785,0.816,0.1370.059,0.082,0.790,0.855,0.912,0.055+0.115,0.157,0.807,0.846,0.834,0.108+0.095,0.116,0.690,0.918,0.802,0.095+0.159,0.198,0.727,0.816,0.840,0.0950.187,0.284,0.249,0.367,0.421,0.2360.263,0.327,0.277,0.432,0.502,0.2720.238,0.494,0.278,0.472,0.714,0.1560.186,0.523,0.293,0.469,0.460,0.3610.294,0.519,0.383,0.499,0.522,0.3440.366,0.451,0.278,0.537,0.512,0.309.Step 3. We compute the score values for each alternative. The score values are expressed in Table 10.Step 4. Rank the alternatives according to their score values.(79)X3≻X6≻X2≻X5≻X1≻X4.As a result,X3 is best supplier among six alternatives with qualities of all criteria.


### 5.5. Comparative Analysis

This paper describes techniques for dealing with the cubic intuitionistic situation. We compare our two cubic intuitionistic strategies that are already in use. If we use CIF-WPM to assemble the alternatives, they are ranked as(80)X3≻X5≻X1≻X6≻X2≻X4.On the other side, when we use the technique of choice value method, the ranking of alternatives is(81)X3≻X6≻X2≻X5≻X1≻X4.Based on these findings, it seemed that the ranking of theX3 alternative was the same as that produced by the suggested cubic intuitionistic procedures. The rest of the alternatives have been altered, as can be seen. As a result, we concluded that in the case of only IVIFSs, the best choice matches with the indicated one; however, the other alternatives are altered, resulting in numerous decisions. As a result, this CIS condition improves the application range of the membership and nonmembership intervals by considering IFS membership values in line with it.Figure2 shows the bar chart of ranking of feasible alternatives by using the WPM and CVM methods.Figure 2
Ranking of feasible alternatives.The comparison analysis of the proposed CIF-WPM and CIF-CVM with other existing techniques is expressed in Table11.Table 11
Comparative analysis and ranking of alternatives.
MethodsRanking of alternativesTop alternativeCIF-TOPSIS (Garg and Kaur [23])X3≻X6≻X2≻X5≻X1≻X4X3CIF-WASPAS (Senapati et al. [26])X3≻X6≻X1≻X2≻X5≻X4X3Frank AO (Seikh and Mandal [52])X3≻X6≻X2≻X5≻X1≻X4X3CIF-WPM (Algorithm 1)X3≻X6≻X2≻X5≻X1≻X4X3CIF-CVM (Algorithm 2)X3≻X5≻X1≻X6≻X2≻X4X3


## 5.1. Application to Uncertain Supply Chain Management

Communication and information technologies are affecting every area of the industrial sector at a rapid pace. In reality, it would be difficult to pinpoint an organization that does not use or is not touched by information and communications technologies in some way. In many cases, if technology is not employed appropriately, the firm’s survival is jeopardized. Companies nowadays use technology to boost productivity, streamline operations, and form electronic conglomerates. Advanced technologies and electronic systems are radically altering how businesses operate and stay competitive. Many businesses are making strategic technology investments to obtain and maintain a competitive advantage in their industry. Management teams must use technology throughout the organization to enhance information flow, reduce cost, streamline operations, provide product variety, formulate connections with suppliers, and reduce response times to customers’ needs to gain a competitive advantage through the use of information and communications technology.Administrators and top executives should be associated with the development of enterprise-wide information systems (EIS), which should take into account such matters as computer hardware and software and infrastructure facilities, online systems, digital applications, electronic commerce, and alterations to current processes and practices. Managers can integrate data and telecommunications technologies throughout the corporation and connect all business areas by developing an enterprise broad information systems plan. Enterprise-wide integration of technology enables firms to allow consumers to get timely access to the information they need to make informed decisions. Recent research has looked at information systems as useful tools for integrating systems like enterprise resource planning, knowledge management, e-commerce, electronic markets, and supply chain management (SCM) to enhance organizational profit and efficiency.Companies must analyze both internal and external processes for the production and exchange of products and services to be more efficient and competitive. The managers will be able to evaluate the value of actions for each process to determine how to boost the value among these operations that form a supply chain from supplier to business to dealer to customer through the evaluation of these processes. The level of integration among suppliers, business associates, and buyers, independent of their geographical location, determines the value chain’s or supply chain’s effectiveness.The construction of an integrated organizational system capable of information sharing, resources, and services in the supply chain is central to the digital supply chain management paradigm. To gain and maintain competitive advantages, companies use digital information and communications networks to standardize manufacturing processes, reduce cycle time, increase the effectiveness of procurement procedures and logistical support, reduce production costs, and increase customer satisfaction, among other things. Supply chain management based on the Internet allows a company to streamline its supply chain, increase speed, reduce costs, and be more adaptable. It can also increase consumer and supplier communications as well as smooth the ongoing flow of goods along the supply chain.Supplier selection is highly essential in supply chain management. The objective is to locate a supplier who can offer the best products and services for the lowest price. Proper supplier selection delivers a high profit and quality level. In this strategic collaboration, the supplier is viewed as a significant element of the business. Because of the increasing focus on sustainability, identifying these providers has become more challenging. Environmental studies, often known as sustainability studies, have become increasingly popular around the world. Identifying these suppliers has become increasingly difficult as a result of the rapidly increasing emphasis on sustainability. Many methodologies for sustainable supply chain selection have been developed.To determine the most suitable supplier selection, MCGDM techniques can be used successfully. In this section, the suggested model is used to determine the selection of appropriate suppliers for fast-moving consumer products, with the goal of selecting the best supplier among various possibilities. Several criteria have been established based on expert opinions to evaluate supplier choices. In this study,X1, X2, X3, X4, X5 and X6 are examined as possible fast-moving customers goods suppliers using the four criteria established.


## 5.2. CIF Weighted Product Model

The proposed method is used to choose the best supplier among six alternatives. These alternatives are weighed against four criteriaℂ1 = price, ℂ2 = quality, ℂ3 = performance, and ℂ4 = delivery, derived from thorough expert opinions. A group of decision-makers has been assembled to assess the suppliers using the recommended methodology. Six decision-makers D1, D2, D3, D4, D5, and D6 were chosen, consisting of supplier experts and expert academics on multicriteria decision-making in a fuzzy environment:,,,Step 1. Consider the decision matrix M=Tjim×n given by the decision-makers in the form of CIFNs on the basis of the cubic intuitionistic fuzzy linguistic scale to evaluate suppliers in accordance with established objectives and criteria is given in Table 5.Step 2. With the help of the linear approach, we normalize the matrix M=Tjim×n. We divide the criteria into two subsets, benefit criteria B and cost criteria K. Here, X3 and X6 belong to benefit criteria B, and the remaining others belong to cost criteria K. For this, first, we find maxjTji and minjTji, which are given in Table 5. We normalized the decision matrix by utilizing the 1st and 2nd equations in Algorithm 1 and this is given in Table 6.Normalized decision matrixes from DMs are expressed in Table7.Steps 3 and 4. We find the relative importance of all alternatives by utilizing the 3rd equation in Algorithm 1, and then, we calculate their score function as given in Table 8.Step 5. Rank the alternatives according to the score function, and the final ranking is(77)X3≻X5≻X1≻X6≻X2≻X4.As we can see thatX3 is the most appropriate supplier among the six alternatives with the best of qualities of all criteria.Table 5
Cubic intuitionistic decision matrix from DMs.
Criteriaℂ1ℂ2ℂ3ℂ4X10.17,0.24,0.36,0.43,0.56,0.320.20,0.28,0.29,0.31,0.27,0.200.18,0.21,0.21,0.32,0.39,0.220.20,0.37,0.21,0.43,0.54,0.23X20.19,0.22,0.39,0.42,0.59,0.400.27,0.34,0.33,0.40,0.43,0.210.24,0.30,0.30,0.39,0.50,0.200.32,0.40,0.19,0.51,0.52,0.30X30.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.72,0.170.31,0.39,0.18,0.40,0.67,0.140.14,0.63,0.24,0.50,0.70,0.18X40.31,0.37,0.36,0.49,0.50,0.360.18,0.33,0.28,0.52,0.40,0.320.23,0.40,0.24,0.51,0.50,0.330.08,0.74,0.32,0.40,0.46,0.42X50.40,0.48,0.51,0.60,0.52,0.300.29,0.41,0.30,0.39,0.48,0.400.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.53,0.37X60.29,0.38,0.27,0.42,0.60,0.270.40,0.51,0.41,0.50,0.47,0.380.32,0.38,0.24,0.72,0.43,0.320.42,0.50,0.37,0.53,0.53,0.27Table 6
maxjTji and minjTji values.
maxjTjiminjTjiX10.20,0.37,0.21,0.31,0.50,0.200.17,0.21,0.36,0.43,0.27,0.32X20.32,0.40,0.19,0.39,0.59,0.200.19,0.22,0.39,0.51,0.43,0.40X30.31,0.63,0.18,0.40,0.81,0.130.14,0.29,0.42,0.50,0.67,0.18X40.31,0.74,0.24,0.40,0.50,0.320.08,0.33,0.36,0.52,0.40,0.42X50.40,0.64,0.30,0.39,0.56,0.290.13,0.41,0.51,0.60,0.48,0.40X60.42,0.51,0.24,0.42,0.60,0.270.29,0.38,0.41,0.72,0.43,0.38Table 7
Normalized decision matrix from DMs.
Criteriaℂ1ℂ2ℂ3ℂ4X10.17,0.21,0.36,0.43,0.56,0.320.17,0.21,0.36,0.43,0.27,0.200.17,0.21,0.36,0.43,0.39,0.220.17,0.21,0.36,0.43,0.54,0.23X20.19,0.22,0.39,0.51,0.59,0.400.19,0.22,0.39,0.51,0.43,0.210.19,0.22,0.39,0.51,0.50,0.200.19,0.22,0.39,0.51,0.52,0.30X30.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.81,0.130.31,0.39,0.18,0.40,0.81,0.130.14,0.63,0.24,0.50,0.81,0.13X40.08,0.33,0.36,0.52,0.50,0.360.08,0.33,0.36,0.52,0.40,0.320.08,0.33,0.36,0.52,0.50,0.330.08,0.33,0.36,0.52,0.46,0.42X50.40,0.48,0.51,0.60,0.56,0.290.29,0.41,0.30,0.39,0.56,0.290.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.56,0.29X60.29,0.38,0.41,0.72,0.60,0.270.29,0.38,0.41,0.72,0.47,0.380.29,0.38,0.41,0.72,0.43,0.320.29,0.38,0.41,0.72,0.53,0.27Table 8
Relative importance and score function.
AlternativesCIFN-WPM valuesScore valuesX10.16,0.20,0.34,0.41,0.9980,0.000010.8029X20.18,0.21,0.38,0.49,0.9993,0.000020.7592X30.21,0.45,0.29,0.46,0.9999,0.00000070.9548X40.07,0.31,0.34,0.50,0.9990,0.000070.6294X50.25,0.49,0.39,0.50,0.9996,0.000020.9245X60.28,0.34,0.39,0.71,0.9994,0.000040.7593


## 5.3. CIF Choice Value Method

The choice value method is a renowned and widely used MCGDM basis for evaluating a set of choices using a set of criteria. Each choice is contrasted to the others by calculating a number of ratios, one per choice criterion. Every ratio is multiplied by the proportional weight of the criterion in consideration. A fundamental task in MCGDM problems is the selection of one or more options from the set of alternatives based on a number of criteria. Let us considerm alternatives, n criteria with weighted vectors, with the condition that the sum of weights will be one, for an MCGDM problem in a cubic intuitionistic fuzzy set domain.,Table 9
Cubic intuitionistic decision matrix from DMs.
Criteriaℂ1ℂ2ℂ3ℂ4X10.17,0.24,0.36,0.43,0.56,0.320.20,0.28,0.29,0.31,0.27,0.200.18,0.21,0.21,0.32,0.39,0.220.20,0.37,0.21,0.43,0.54,0.23X20.19,0.22,0.39,0.42,0.59,0.400.27,0.34,0.33,0.40,0.43,0.210.24,0.30,0.30,0.39,0.50,0.200.32,0.40,0.19,0.51,0.52,0.30X30.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.72,0.170.31,0.39,0.18,0.40,0.67,0.140.14,0.63,0.24,0.50,0.70,0.18X40.31,0.37,0.36,0.49,0.50,0.360.18,0.33,0.28,0.52,0.40,0.320.23,0.40,0.24,0.51,0.50,0.330.08,0.74,0.32,0.40,0.46,0.42X50.40,0.48,0.51,0.60,0.52,0.300.29,0.41,0.30,0.39,0.48,0.400.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.53,0.37X60.29,0.38,0.27,0.42,0.60,0.270.40,0.51,0.41,0.50,0.47,0.380.32,0.38,0.24,0.72,0.43,0.320.42,0.50,0.37,0.53,0.53,0.27Table 10
Score values.
AlternativesScore valuesX10.1125X20.1705X30.5490X40.0725X50.1435X60.204


## 5.4. MCDGM Application

Step 1. Consider the decision matrix M=Tjim×n given by the decision-makers in the form of CIFNs given in Table 9.Step 2. Decision-makers gives the weights to the four criteria as W1=0.18, W2=0.24, W3=0.26, and W4=0.32 with ∑Wi=1(78)0.17,0.24,0.36,0.43,0.56,0.320.20,0.28,0.29,0.31,0.27,0.200.18,0.21,0.21,0.32,0.39,0.220.20,0.37,0.21,0.43,0.54,0.230.19,0.22,0.39,0.42,0.59,0.400.27,0.34,0.33,0.40,0.43,0.210.24,0.30,0.30,0.39,0.50,0.200.32,0.40,0.19,0.51,0.52,0.300.20,0.29,0.40,0.51,0.81,0.130.31,0.52,0.42,0.50,0.72,0.170.31,0.39,0.18,0.40,0.67,0.140.14,0.63,0.24,0.50,0.70,0.180.31,0.37,0.36,0.49,0.50,0.360.18,0.33,0.28,0.52,0.40,0.320.23,0.40,0.24,0.51,0.50,0.330.08,0.74,0.32,0.40,0.46,0.420.40,0.48,0.51,0.60,0.52,0.300.29,0.41,0.30,0.39,0.48,0.400.40,0.47,0.38,0.60,0.56,0.290.13,0.64,0.40,0.47,0.53,0.370.29,0.38,0.27,0.42,0.60,0.270.40,0.51,0.41,0.50,0.47,0.380.32,0.38,0.24,0.72,0.43,0.320.42,0.50,0.37,0.53,0.53,0.270.180.240.260.320.032,0.048,0.832,0.859,0.900,0.067+0.052,0.075,0.742,0.754,0.730,0.052+0.050,0.059,0.666,0.743,0.782,0.062+0.068,0.137,0.606,0.763,0.821,0.0800.037,0.043,0.844,0.855,0.909,0.087+0.072,0.094,0.766,0.802,0.816,0.055+0.068,0.088,0.731,0.782,0.835,0.056+0.116,0.150,0.587,0.806,0.811,0.1070.039,0.059,0.847,0.885,0.962,0.024+0.085,0.161,0.812,0.846,0.924,0.043+0.091,0.120,0.640,0.788,0.901,0.038+0.047,0.272,0.633,0.801,0.892,0.0610.064,0.079,0.832,0.879,0.882,0.077+0.046,0.091,0.736,0.854,0.802,0.088+0.065,0.124,0.690,0.839,0.835,0.098+0.026,0.350,0.694,0.745,0.779,0.1590.087,0.111,0.885,0.912,0.888,0.062+0.078,0.118,0.749,0.797,0.888,0.115+0.124,0.152,0.777,0.875,0.860,0.085+0.043,0.278,0.745,0.785,0.816,0.1370.059,0.082,0.790,0.855,0.912,0.055+0.115,0.157,0.807,0.846,0.834,0.108+0.095,0.116,0.690,0.918,0.802,0.095+0.159,0.198,0.727,0.816,0.840,0.0950.187,0.284,0.249,0.367,0.421,0.2360.263,0.327,0.277,0.432,0.502,0.2720.238,0.494,0.278,0.472,0.714,0.1560.186,0.523,0.293,0.469,0.460,0.3610.294,0.519,0.383,0.499,0.522,0.3440.366,0.451,0.278,0.537,0.512,0.309.Step 3. We compute the score values for each alternative. The score values are expressed in Table 10.Step 4. Rank the alternatives according to their score values.(79)X3≻X6≻X2≻X5≻X1≻X4.As a result,X3 is best supplier among six alternatives with qualities of all criteria.


## 5.5. Comparative Analysis

This paper describes techniques for dealing with the cubic intuitionistic situation. We compare our two cubic intuitionistic strategies that are already in use. If we use CIF-WPM to assemble the alternatives, they are ranked as(80)X3≻X5≻X1≻X6≻X2≻X4.On the other side, when we use the technique of choice value method, the ranking of alternatives is(81)X3≻X6≻X2≻X5≻X1≻X4.Based on these findings, it seemed that the ranking of theX3 alternative was the same as that produced by the suggested cubic intuitionistic procedures. The rest of the alternatives have been altered, as can be seen. As a result, we concluded that in the case of only IVIFSs, the best choice matches with the indicated one; however, the other alternatives are altered, resulting in numerous decisions. As a result, this CIS condition improves the application range of the membership and nonmembership intervals by considering IFS membership values in line with it.Figure2 shows the bar chart of ranking of feasible alternatives by using the WPM and CVM methods.Figure 2
Ranking of feasible alternatives.The comparison analysis of the proposed CIF-WPM and CIF-CVM with other existing techniques is expressed in Table11.Table 11
Comparative analysis and ranking of alternatives.
MethodsRanking of alternativesTop alternativeCIF-TOPSIS (Garg and Kaur [23])X3≻X6≻X2≻X5≻X1≻X4X3CIF-WASPAS (Senapati et al. [26])X3≻X6≻X1≻X2≻X5≻X4X3Frank AO (Seikh and Mandal [52])X3≻X6≻X2≻X5≻X1≻X4X3CIF-WPM (Algorithm 1)X3≻X6≻X2≻X5≻X1≻X4X3CIF-CVM (Algorithm 2)X3≻X5≻X1≻X6≻X2≻X4X3


## 6. Conclusion

A cubic intuitionistic fuzzy set is an effective method for dealing with various uncertainties in multicriteria group decision-making (MCGDM) settings. A cubic set is a two-component system that would be used to describe data with a fuzzy interval and a fuzzy number. The notion of cubic intuitionistic fuzzy sets (CIFS) is a strong hybrid model of IFSs and IVIFSs. A CIFS has two components, one indicating the IVIFS and the other indicating the IFS. A CIFS is a new fuzzy model for data analysis, computational intelligence, neural computing, soft computing, and others. The idea of cubic hesitant fuzzy topology defined on CIFS can be utilized to seek solutions to various problems of information analysis, information fusion, big data, and decision analysis.Main findings in this manuscript are as follows:(1)
We introduced the concepts of “P-cubic intuitionistic fuzzy topology” as well as “R-cubic intuitionistic fuzzy topology.” Topological structures provide robust approaches for data analysis and decision analysis under an uncertain environment.(2)
Certain properties of CIF topology under P(R)-order are explored, and their related results are elaborated with illustrations.(3)
The notions of CIF-open set, CIF-closed set, CIF-closure, CIF-interior, CIF-exterior, as well as CIF-frontier, CIF-dense set, and CIF-basis are investigated with a corresponding example.(4)
Algorithms 1 and 2 are proposed for extension of the weighted product model and the choice value method, respectively.(5)
The symmetry of optimal decisions is analyzed by computations with Algorithms 1 and 2. The numerical values of alternatives are very close by using Algorithm 1. However, the numerical values of alternatives have a clear difference when using Algorithm 2.(6)
An application of proposed methods named CIF-WPM and CIF-CVM towards uncertain supply chain management is presented.(7)
To discuss the advantages, flexibility, and validity of proposed methods, a comparison analysis is also expressed.For forthcoming analysis, due to the flexibility of CIF topology towards data analysis and information analysis, one can extend this work to develop new MCDM techniques with CIF-VIKOR, CIF-AHP, and CIF-aggregation operators.



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