Topological Data Analysis with Cubic Hesitant Fuzzy TOPSIS Approach

: A hesitant fuzzy set (HFS) and a cubic set (CS) are two independent approaches to deal with hesitancy and vagueness simultaneously. An HFS assigns an essential hesitant grade to each object in the universe, whereas a CS deals with uncertain information in terms of fuzzy sets as well as interval-valued fuzzy sets. A cubic hesitant fuzzy set (CHFS) is a new computational intelligence approach that combines CS and HFS. The primary objective of this paper is to deﬁne topological structure of CHFSs under P(R)-order as well as to develop a new topological data analysis technique. For these objectives, we propose the concept of “cubic hesitant fuzzy topology (CHF topology)”, which is based on CHFSs with both P(R)-order. The idea of CHF points gives rise to the study of several properties of CHF topology, such as CHF closure, CHF exterior, CHF interior, CHF frontier, etc. We also deﬁne the notion of CHF subspace and CHF base in CHF topology and related results. We proposed two algorithms for extended cubic hesitant fuzzy TOPSIS and CHF topology method, respectively. The symmetry of optimal decision is analyzed by computations with both algorithms. A numerical analysis is illustrated to discuss similar medical diagnoses. We also discuss a case study of heart failure diagnosis based on CHF information and the modiﬁed TOPSIS approach.


Preliminaries
Topological data analysis (TDA) methods are rapidly growing approaches to infer persistent key features for possibly complex data [1]. Machine learning techniques have several limitations due to repetition in data collection. TDA presents a classifier for such problems during sampling from the data space and defines a data topology based on network graph [2]. TDA can be used independently or in combination with other data analysis and statistical learning techniques. Topological approaches have been used in contemporary data science to examine the structural characteristics of big data, leading to further information analysis. Researchers have developed a lot of TDA methods for information aggregation. However, the classical methods cannot deal with vague and uncertain information. To address these issues, Zadeh [3] established the notion of "fuzzy set (FS) theory" as an extension of classical set theory. In fact, this an extension of characteristic functions, with codomain {0, 1} towards membership function (MF) having codomain [0, 1]. The idea of MF in FS theory is a robust innovation for computational intelligence and many other fields. An FS model characterized by a membership function assigns a membership grade under a specific criterion to each object in {0, 1}. Atanassov [4] highlighted the significance of non-membership function (NMF) in addition to MF of FS theory and proposed "intuitionistic fuzzy set (IFS) theory" to describe the positive and negative aspects of an object under a specific criteria. Pythagorean fuzzy set (PFS) is an extension of IFS [5][6][7], and q-rung orthopair fuzzy set (q-ROFS) is an extension of PFS [8].
Molodtsov [9] proposed "soft set (SS) theory" to study uncertainty in complex problems by means of parameterizations. Neutrosophic sets [10], spherical fuzzy sets [11,12] and picture fuzzy sets [13] have been successfully applied to numerous problems of computational intelligence and decision-making problems.
Jun et al. [14] proposed the perspective of "cubic sets (CSs) theory" that includes several novel concepts such as internal CSs, external CSs, P(R)-union and P(R)-intersection, etc. Torra [15] suggested the concept of "hesitant fuzzy sets" (HFSs) as a generalization of fuzzy sets to better describe this circumstance, which allows a collection of possible values in the closed interval [0,1] to be aided by the membership degree. HFSs were used to manage scenarios in which experts had to choose between diverse feasible membership values in order to evaluate an alternative. HFS theory has various applications in different fields such as decision support systems, clustering and pattern recognition as well as various features of algebraic structures and topological structures. Some rudimentary concepts of hesitant fuzzy sets are given in Table 1.

HFS Models Applications
HFSs (Torra [15,16]) Operational laws of HFSs Interval-valued hesitant fuzzy set (Chen et al. [17]) IVHF clustring algorithm Information measures for HFSs and IVHFSs (Farhadinia [18]) Entropy, similarity measures, clustering HF information aggregation (Xia and Xu [19]) Development of large project (strategy initiatives) Generalized HFSs (Qian et al. [20]) Adopting an information system to stimulate university work productivity Chang [21] laid the foundation of "fuzzy topology", which is defined on a collection of fuzzy sets. Coker [22] proposed "intuitionistic fuzzy topology" which is defined on the collection of IFSs, and Olgun et al. [23] defined "Pythagorean fuzzy topology" on PFSs. The conceptualization of fuzzy soft topology [24] suggested modified topological structures. Lee and Hur [25] introduced "hesitant fuzzy topology" based on HFSs and defined the notions of HF product space and HF continuous mapping. Abdullah et al. [26] proposed the concept of "cubic topology" on cubic sets. Sreedevi and Shankar [27] introduced "hesitant fuzzy soft topological spaces" and numerous topological properties, such as HF soft normal space, HF soft Hausdorff space, HF soft regular space, HF soft basis and first and second countable HF soft spaces. Riaz and Tehrim [28] described the concept of "bipolar fuzzy soft topology". Some applications of HFS and hybrid extensions of HFS are given in Table 2. "Multi-criteria decision making" (MCDM) is a sub-discipline of operations research that looks at how to make optimal decisions based on multiple competing criteria. MCDM is a multistage decision-making approach that ranks alternatives as well as selects a best possible alternative from a given set of feasible alternatives under several criteria. "TOPSIS" is a highly effective method for MCDM that is based on the premise of attempting to traverse the potentially best decision from the available options that is at a scaled-down distance from the "positive ideal solution" (PIS) and far from the "negative ideal solution" (NIS). TOPSIS has been studied and adopted for solving several real-life problems, as mentioned in Table 3. Table 2. Some applications based on hesitant fuzzy sets.

HFS Models Applications
IVH preference relations (Chen et al. [29]) Supply chain management IHFSs (Beg and Rashid [30]) Funds allocation to schools based on their performance Riaz and Hashmi [64] proposed a new extension of fuzzy sets, named linear Diophantine fuzzy set, which relax the strict constraints of existing fuzzy sets with the help of reference parameters. Kamaci [65] initiated the notion of linear Diophantine fuzzy algebraic structures and their application in coding theory. Kamaci [66] introduced the concept of complex linear Diophantine fuzzy sets, operational laws on complex linear Diophantine fuzzy sets and their application based on cosine similarity measures. Sreedevi and Shankar [67] proposed topological structure on HFS and defined the notion of HFStopology. Akram et al. [68] suggested a new mathematical model for group decision making with hesitant N-soft sets with application to expert systems. Mahmood et al. [69] initiated a new MCDM based on CHFSs and Einstein operational laws. Garg and Kaur [70] proposed an extended TOPSIS approach and nonlinear-programming for MCDM using cubic IFS information. Kamaci et al. [71] proposed dynamic aggregation operators and Einstein operations by using interval-valued, picture-hesitant fuzzy information and their application in multi-period decision making.
Topological structures with fuzzy models (see [72][73][74]) develop more effective and flexible approaches to seek solutions to various problems of artificial intelligence, engineering and information analysis (see [75][76][77]). A cubic hesitant fuzzy set (CHFS) is a hybrid of a hesitant fuzzy set (HFS) and a cubic set (CS). A CHFS is a new fuzzy model for data analysis, computational intelligence, neural computing, soft computing and other processes. The idea of cubic hesitant fuzzy topology defined on CHFS can be utilized to seek solutions of various problems of information analysis, information fusion, big data and decision analysis.
The main objectives of this study are: (1) to introduce the idea of P-cubic hesitant fuzzy topology with simultaneous P-order and R-cubic hesitant fuzzy topology with R-order; (2) to define certain properties of CHF topology under P(R)-order and their related results; (3) to develop an algorithm for the extension of MCDM methods based on CHF topology; (4) to demonstrate an application of proposed methodology towards medical diagnosis; and (5) to discuss the advantages, flexibility and validity of the proposed methodology.
The remainder of the paper is designed as follows. In Section 2, we discuss some fundamental notions, including, HFS, CS, CHFS and operations of CHFSs under P(R)order. In Section 3, we study "P-cubic hesitant fuzzy topology" (P-CHF topology), P-CHF subspace, P-CHF closure, P-CHF interior, P-CHF exterior, P-CHF frontier and P-CHF dense set. In Section 4, we define "R-cubic hesitant fuzzy topology" (R-CHF topology), R-CHF subspace, R-CHF closure, R-CHF interior, R-CHF exterior, R-CHF frontier and R-CHF dense set. In Section 5, Algorithms 1 and 2 are proposed for an extended cubic hesitant fuzzy TOPSIS method and topological data analysis, respectively. An application of the proposed methods for medical diagnosis is also given in Section 5. The conclusion and future directions are summarized at the end of manuscript.
where I(ξ) is an "interval-valued fuzzy set (IVFS)" in V and σ(ξ) is a "fuzzy set (FS)" in V. A CS can be expressed as C = I, σ .

Remark 1. A CHFSĤ
Remark 2. Some important observations are listed as follows.
(i) P-union of any two ICHFSs is an ICHFS.
(ii) P-intersection of any two ICHFSs is an ICHFS.
(iii) P-union of any two ECHFSs need not be an ICHFS.
(iv) P-intersection of any two ECHFSs need not be an ICHFS.
(v) P-union of any two ECHFSs need not be an ECHFS.
(vi) P-intersection of any two ECHFSs need not be an ECHFS.
(vii) R-union of any two ICHFSs need not be an ICHFS.
(viii) R-intersection of any two ICHFSs need not be an ICHFS.
(ix) R-intersection of any two ECHFSs need not be an ICHFS.
Definition 19. Let (V, T 1 ) and (V, T 2 ) be two PCHF topological spaces over identical universal set V. If T 1 ⊆ P T 2 then T 1 is said to be a PCHF coarser or a PCHF weaker than T 2 and T 2 is said to be a PCHF finer or a PCHF stronger than T 1 .
If T 1 P T 2 or T 2 P T 1 then these PCHF topologies are not comparable.
Thus, T 2 is a PCHF coarser than T 4 and T 4 is a PCHF finer than T 2 .
Definition 20. Assume a universal set V, and the assemblage of all CHFSs T is defined in V. Then T is a PCHF discrete topology on V and (V, T) is known as a PCHF discrete topological space in V.

Definition 21.
Suppose that V be a universal set and Proof.
Let us assume that {Ĥ α : α ∈ Ω} be an arbitrary family of CHFSs in whereY may be an absolute CHFS or any CHFS defined on Y. We can easily check that (Y,T) is a PCHF topological space. It is called a PCHF subspace of (V, T), andT is known as the PCHF subspace or PCHF relative topology on Y.
V is a PCHF subspace or PCHF relative topology on Y, and (Y,T) is a PCHF topological space known as PCHF subspace of (V, T).

Remark 4.
(i) Any PCHF subspace of a PCHF discrete space is PCHF discrete.
(iii) A PCHF subspaceǍ of a PCHF subspaceY of a PCHF topological space V is a PCHF subspace of V.
The P-complement of CHFSsĤ 1 andĤ 2 are given below
Since (Ĥ 1 ) • is a PCHF open set and is also the biggest PCHF open subset of itself, so From part (4), Proof.

1.
Consider a collection {Ĥ α : α ∈ Ω} of all PCHF open subsets of (Ĥ) c . We know utilizing the previous argument. We can now attain the desired result by taking the PCHF complement.

R-Cubic Hesitant Fuzzy Topology
Definition 28. Consider a universe of discourse V and a family of CHFSs in V is T. Then T is said to be an RCHF topology on V if it satisfies the following properties: Then (V, V ) is called an RCHF topological space. be CHFSs on V. Tables 6 and 7 show the R-union and R-intersection of the CHFSsĤ 1 andĤ 2 , respectively. Table 6. R-union of CHFSs. Table 7. R-intersection of CHFSs. are CHFSs on V. R-union and R-intersection of these CHFSs are given in Tables 8 and 9. Table 8. R-union of CHFSs. Table 9. R-intersection of CHFSs.
Proof. Proof is obvious.
Definition 31. Let (V, T 1 ) and (V, T 2 ) be two RCHF topological spaces over identical universal set V. If T 1 ⊆ R T 2 then T 1 is said to be an RCHF coarser or an RCHF weaker than T 2 , or T 2 is said to be an RCHF finer or an RCHF stronger than T 1 . If T 1 R T 2 or T 2 R T 1 , then these RCHF topologies are not camparable.

Definition 32. Assume a universal set V and the assemblage of all CHFSs T is defined in V. Then
T is an RCHF discrete topology on V, and (V, T) is known as RCHF discrete topological space in V.

Definition 33.
Suppose that V is a universal set and T = {0 V ,1 V }. Then T is an RCHF indiscrete topology on V, and (V, T) is an RCHF indiscrete topological space in V.

Definition 34.
Assume that an RCHF topological space is defined as (V, T), and Y is an RCHF subset of V. LetT consist of those RCHF subsetsD of Y for which there is anĤ in T such that D =Ĥ ∩ RY whereY may be an absolute CHFS or any CHFS defined on Y. We can easily check that (Y,T) is an RCHF topological space. It is called an RCHF subspace of (V, T) andT is known as RCHF subspace or RCHF relative topology on Y.

1.
Any RCHF subspace of an RCHF discrete space is RCHF discrete.

Proof. Proof is straightforward.
Theorem 11. Let (V, T) be an RCHF topological space in V andĤ be a CHFS, then Proof. It is obvious. ( Proof. Proof is obvious. Theorem 13. Let (V, T) be an RCHF topological space in V andĤ be a CHFS, then 1.
Proof. Proof is obvious.

Extended Cubic Hesitant Fuzzy TOPSIS Method
In this section, we discuss various kinds of heart disease by providing a brief but exhaustive overview of this critical disease, including the symptoms for each category, and we use the proposed TOPSIS technique to rank patients having heart disease.

Case study
The heart is at the core of your vascular system, which is a series of vessels that conduct blood to all parts of your body. Blood delivers oxygen and other vital nutrients that all vital tissues requires to keep fit and adequately healthy. Heart failure, also known as congestive heart failure, transpires when the heart muscle fails to pump blood as well as it should. When this transpires, blood can pool in the lungs, and fluid can accumulate, causing respiratory distress. Cardiac disorders, such as narrowed carotid arteries (coronary artery disease) or high cholesterol, cause the heart to stiffen over time or to weaken, rendering it unable to fill and pump blood effectively. Heart failure is a potentially fatal condition. People that suffer from heart failure can suffer drastic indications, and some can undergo heart surgery or receive a ventricular assist device (VAD). Heart failure can be persistent (chronic) or develop abruptly (acute). Heart failure symptoms may include the following (https://www.mayoclinic.org/diseases-conditions/heart-failure/symptomscauses/syc-20373142 accessed on 4 April 2022): Heart failure frequently arises as a result of the heart being injured or weakened by another disorder. Heart failure can also occur if the heart becomes overly rigid. In heart failure, the crucial pumping chambers of the heart (the ventricles) may stiffen and fail to fill properly between beats. Ejection fraction is used to assist categorization and heart failure diagnosis. In a healthy heart, the ejection fraction is 50 percent or greater-this means that with each beat, more than half of the blood in the ventricle is pushed out. Heart failure can develop even with a normal ejection fraction. This occurs when the heart muscle stiffens due to factors such as high blood pressure. Other causes of abrupt (acute) cardiac failure include (https://www.healthline.com/health/heart-failure accessed on accessed on 4 April 2022):

•
Allergic reactions • Disease affecting the entire body • Blood clots in the lungs Taking some medications • Germs that wreak havoc on the cardiac muscle Types of heart failure (https://www.webmd.com/heart-disease/heart-failure/heartfailure-overview accessed on accessed on 4 April 2022) are given in Table 10. Table 10. Types of heart failure.

Types of Heart Failure Description
Systolic heart failure Left ventricle is unable to eject blood forcefully, indicating a pumping issue Right-sided heart failure Swelling can result from fluid backing up into the abdomen, legs and feet Left-sided heart failure Shortness of breath may occur due to fluid buildup in the lungs

Preserved ejection fraction heart failure
Left ventricle is unable to relax or fill completely, indicating a filling issue A lone risk factor may be sufficient to induce heart failure, but a fusion of risk factors raises your chances. Heart failure risk factors include: irregular heartbeats, alcohol use, viruses, coronary artery disease, heart attack, congenital heart disease, sleep apnea, obesity, heart valve disease, diabetes, smoking or using tobacco, etc.
Heart failure is classified into four stages (https://https://www.topdoctors.co.uk/ medical-articles/understanding-4-stages-heart-failure accessed on 4 April 2022): A, B, C and D, which vary from higher risk of getting heart failure to progressive disease.
Stage A Stage A is classified as pre-heart failure. It implies you are at risk of developing heart failure because you have had heart failure before or have one or more of the following medical conditions: hyperpiesis and coronary artery disease, rheumatic fever history, diabetes, metabolic disorder, a history of alcoholism and/or family history of cardiomyopathy.

Stage B
Asymptomatic or silent heart failure is considered Stage B. It indicates that you have systolic left ventricular dysfunction but have never experienced heart failure symptoms. The majority of people with Stage B heart failure have an ejection fraction (EF) of 40 percent or less on an echocardiogram (echo). People in this cluster have heart failure and low EF (HF rEF) for any reason.

Stage C
People in stage C heart failure have been diagnosed with heart failure and are experiencing (or have previously experienced) signs and symptoms. The most common heart failure symptoms are: wheezing, fatigue, reduced exercising capability and/or swelling of the feet, ankles, lower legs and abdomen.

Stage D
Patients in Stage D have severe symptoms that do not improve with intervention. This is the most severe stage of heart failure. They exhibit symptoms under modest or minor activity or even at rest.
For making a unanimous judgment, we employ the well-known TOPSIS technique for selecting an optimal alternative from a set of feasible alternatives, with the goal that the chosen solution is next to the ideal solution and farthest away from the worst answer. The TOPSIS approach is suitable to address hesitancy and vagueness in MCDM problems. The proposed extended cubic hesitant fuzzy TOPSIS is an efficient mathematical model for medical diagnosis and other MCDM problems.
We begin by developing the suggested technique step-wise, as shown below Table 11: Table 11. Dialectal/linguistic variables and fuzzy weights of the stages.

Dialectal/Linguistic Variables Fuzzy Weights
Stage A: Healthy heart (SA) 0.100 Stage B: Silent heart failure (SB) 0.300 Stage C: Moderate heart failure (SC) 0.600 Stage D: Severe heart failure (SD) 0.900 A summary of the stages of heart failure are shown in Figure 1. Step 4: Suppose that the four decision makers give the following P-CHF matrices, where the patients are given by the rows and the criteria are expressed in the columns; the (i, j) th position are given in Tables 12-15.  is given in Table 16. whereκ θ jk = k ×κ θ jk is given in Table 17. Step 7, 8: The distance of each patient from the P-CHF PIS and the P-CHF NIS and the corresponding relative closeness are given in Table 18. Step 9: So, the patients' preferred order is This shows that patient κ θ 4 is in a critical situation.
Step 2: Now we establish P-CHF topology given by where D 3 and D 4 are the null P-CHFS and absolute P-CHFS, respectively.   Tables 19 and 20. Step 5: The aggregated decision table can be assessed by adding decision tables D 1 and D 2 . The resulting decision table is given in Table 21. 0.704 Step 6: The final ranking is given by This implies that patient κ θ 4 is in critical condition.

Comparison Analysis
We notice that the optimal alternative remains identical by use of Algorithms 1 and 2. However, the ranking of alternatives is not exactly the same. This shows that the optimal alternative is selected unanimously. The numerical values of alternatives are very close by using Algorithm 1. However, the numerical values of alternatives have clear differences by using Algorithm 2. The comparative analysis of ranking of alternatives by using Algorithms 1 and 2 is shown in Table 22 and Figure 2.

Algorithm 1 Extended cubic hesitant fuzzy TOPSIS
Step 1: Assume that 'n' is the number of doctors/decision-makers D 1 , D 2 , . . . , D n , provided with 'l' number of patients κ θ 1 , κ θ 2 , . . . , κ θ l and 'm' number of symptoms/criteria e 1 , e 2 , . . . , e m . Step 2: The DMs have to grant preference weights to the symptoms/criteria. Let w ij be the weight given by ith DM to jth attribute. The dialectal/linguistic variables are given in Table 11. For our convenience, we establish weighted parameter matrix P = [w ij ] n×p .
Step 3: The normality of the weights matrix must be ensured. If the weights w ij are not normal, we will normalize by the formulaw ij = w ij The normalized values are shown as N = [ñ ij ] n×p . Afterwards, the weight vector can be obtained as K = ( j : j = 1, 2, ..., p), Step 4: Each DM gives a CHF matrix D i = (κ θ i jk ) n×p , i = 1, 2, . . . , n, where κ θ i jk is the value which DM 'i' assigns to attribute 'k' corresponding to alternative 'j'. Then the mean proportional matrix A = [κ θ jk ] n×p is obtained by averaging the CHFEs.
Step 6: In this step, the positive ideal solution (PIS) and negative ideal solution (NIS) of P-order or R-order (whichever is suitable) are obtained in CHF domain by using 1 Figure 2. Comparative bar chart of ranking of patients.

Conclusions
A cubic hesitant fuzzy set (CHFS) is a hybrid of a hesitant fuzzy set (HFS) and a cubic set (CS). A CHFS is a new fuzzy model for data analysis, computational intelligence, soft computing and other processes. Cubic hesitant fuzzy topology defined on a CHFS can be utilized to seek solutions of various problems of information analysis, information fusion, big data and decision analysis. We proposed the notions of P-CHF topology with P-order and R-CHF topology with R-order. Certain properties of P-CHF topology and R-CHF topology are defined, such as CHF open set, CHF closed set, CHF closure, CHF interior, CHF exterior, CHF frontier, CHF dense set, CHF neighborhood and CHF basis. Algorithms 1 and 2 were proposed for extended cubic hesitant fuzzy TOPSIS and CHF topology method, respectively. The symmetry of the optimal decision was analyzed by computations with Algorithms 1 and 2. The numerical values of alternatives were very close using Algorithm 1. However, the numerical values of alternatives had clear differences by using Algorithm 2. We applied the proposed methodology for medical diagnosis. A comparative analysis was given to discuss the advantages and validity of the proposed methodology.
For forthcoming analysis, due to flexibility of CHF topology towards data analysis and information analysis, one can extend this work to develop new MCDM techniques with CHF VIKOR, CHF AHP, CHF ELECTRE, CHF aggregation operators, etc.