An algorithmic multiple attribute decision-making method for heart problem analysis under neutrosophic hypersoft expert set with fuzzy parameterized degree-based setting

A fuzzy parameterized neutrosophic hypersoft expert set (FpNHse-set) is one of the family members of fuzzy parameterized structures and a valuable extension of the neutrosophic soft expert set as well as the neutrosophic hypersoft set. This structure involves a multi-argument approximate function that has the ability to change the sub-characteristic pairs in the form of power set of universe. The main function of this structure is the classification of each character into sub-characteristic valued sets. Due to this prominent property, this mathematical structure is useful for uncertainties and also helps make the decision-making process more adaptable and dependable. By using the algebraic and basic ideas of the FpNHse-sets, a useful strategy, especially for medical diagnosis, known as Sanchez’s method has been used in this study. To see a reformed process for the medical diagnosis of heart disease, a useful combination of FpNHse-set and a modified Sanchez’s method has been made in this context. By using the real data from the Cleveland data set for heart disease, the implementation of the reform process has been made to see its veracity. Finally, a clear comparison of the used study with its existing studies has been made for the purpose of benefits.


INTRODUCTION
In different scientific directions, there are many mathematical techniques like the fuzzy set (F-set) (Zadeh, 1965) and the intuitionistic fuzzy set (IF-set) (Atanassov & Atanassov, 1999) for the solution of difficult situations having various types of uncertainties.The fuzzy set-like technique is for the truth value of an item belonging to a certain description space, and the second is for both the truth and false values of an item with the property of dependency on each other.Both of these techniques fail when there is an involvement of neutral value in an item that also has previous two values.To overcome such laziness, a new neutrosophic hypersoft set (NHs-sets).Using the accuracy function, they used decisionmaking strategies like TOPSIS to address issues in the actual world.Saeed et al. (2021a) and Saeed et al. (2021b) first introduced the idea of mappings, which is now employed in managerial for medicating analysis.Many researched faced problems of knowing the multi-decisive opinions while using a single structure.Then Ihsan, Rahman & Saeed (2021b) made useful extension in the structure of hypersoft set by introducing the experts opinions and formed a new structure called hypersoft expert set (Hse-set).Ihsan, Rahman & Saeed (2021a) and Ihsan, Saeed & Rahman (2022) developed fuzzy hypersoft expert set (FHse-set) and neutrosophic hypersoft expert set (NHse-set).All the abbreviation used in the introduction section shown in Table 1.

Research gap, motivation and novelty
To assess the research need, novelty, and purpose of the proposed study, consider the following factors: 1. Kirisci, Demir & Simsek (2021) presented an algorithm that can select patients at risk of developing heart disease based on cardiovascular data using S-set.A medical case was examined as a real-life application to see if the proposed method is applicable.Das et al. (2021) identified the most important diseases that are deadly as compared to other diseases by using Rough Set taken from soft computing technique, and then applied the same approach to identify the most important factor that causes heart diseases, before using Time Series to project the ailments and demonstrating their exponential growth rate.Kirisci (2020a) presented a strategy to make health choices that was connected to Celik-Yamak's fuzzy soft set using Sanchez's method and made use of the Cleveland data set.Muthukumar & Krishnan (2016) talked about some of the fundamental characteristics of IFs-sets and presented a new similarity measure for them.The suggested similarity score has also been used to demonstrate medical diagnosing issues in a fictitious investigation.To identify and keep track of patients with heart failure, Abdel-Basset et al. (2020) brought up a novel system based on IoT and computer-supported diagnosis.The suggested health care system made it possible to make diagnoses with greater accuracy despite ambiguous data.They recommended the use of a neutrosophic multi-criteria decision-making technique to help patients and doctors determine whether a patient has heart failure.Kirisci (2019) used the Riesz summability technique for heart disease and medical decision making purposes.Al-Sharqi, Ahmad & Al-Quran (2023) suggested a new technique based fuzzy parameterized interval complex neutrosophic soft set and applied to medical diagnoses especially in heart diseases.Hassan et al. (2017) developed a new fuzzy soft expert system to check the coronary artery disease with different parameters.Long, Meesad & Unger (2015) suggested a new method for the heart disease based on interval type-2 fuzzy logic system.As a decision support system for the diagnosis of cardiac disease, they suggested a very useful method.Sanz et al. (2014) introduced a new method for heart disease analysis by combing fuzzy set classification with interval valued fuzzy set based.Their method remained good due to high classification rate.Kumar, Inbarani & Azar (2015) developed a brand-new automatic classification system for use in taking decisions and heart rhythm evaluation.A back propagation neural network and a bijective soft set have been combined to create the described sorting technique.Lashari, Ibrahim & Senan (2017) developed a novel algorithm based on fuzzy soft set for the heart disease analysis.2. The Fse-set, IFse-set, Nse-set, and other techniques have been widely used to handle decision-making issues in a variety of contexts.However, under a number of circumstances, these structures show shortcomings in classifying the elements similar to their parameterized levels.Especially, it can be inferred that each parameter set element's parameterized degree is taken as one in the existing literature.The introduction of the fuzzy parameterized aspect was one of the key turning points in the evolution of soft sets, soft expert sets, and their generalisations.In particular, when combined with the more precise generalisations of soft and soft expert sets like Fse-set, IFse-set, and other hybrid models mentioned above, this new aspect has further improved the theories of soft and Ss-sets and made them better suited to be used in handling managerial difficulties.First, Cagman & Enginoglu (2011) presented the idea of a fuzzy parameterized soft set (FP-SS) and assigned a level of priority to each parameter in the set, establishing the fuzzy parameterized aspect.Bashir & Salleh (2012) introduced the concept of fuzzy parameterized soft expert sets(FP-SES).This element was then attached to the soft sets, soft expert sets, and fuzzy sets generalisations that already existed.As a generalisation of the work by Hazaymeh et al. (2012) introduced the concept of fuzzy parameterized fuzzy soft expert sets (FP-FSES).Selvachandran & Salleh (2016) introduced the concept of fuzzy parameterized intuitionistic fuzzy soft expert set (FP-IFSES) in their article.These sets, however, are limited in their ability to deal with inconsistent and imprecise information, which is typically present in real-world circumstances.The fuzzy parameterized single valued neutrosophic soft expert set (FP-SVNSES), developed by Al-Quran & Hassan (2016) to address this weakness, outperforms these models with three independent membership functions.Rahman et al. (2022a) converted into neutrosophic hypersoft set with fuzzy parameterizations settings.The fuzzy parameterized aspect of the FP-SVNSES model gives it additional advantages than single valued neutrosophic soft expert set in that it delivers more information, improving the quality of the information that single valued neutrosophic soft expert set presents, which in turn improves the accuracy of the final decision.The goal of the fuzzy parameterization notion is to give each characteristic (or sub-attribute) in the context of an approximate function with just one argument (or multiple arguments) a fuzzy grade.Using models that resemble soft sets, various scholars like (Yılmaz & Eraslan, 2012;Kirisci, 2020b), and have thoroughly investigated this concept.These models treat fuzzy subsets as items in their codomain and fuzzy parameters as elements inside the domain of soft approximate mapping.Other scholars have more recently expanded the idea of fuzzy parameterization to matrices in a soft set context (Rahman et al., 2022b;Rahman et al., 2022c).They have introduced and characterized a variety of new matrix-based features and operations and used them in decision-making, geographical study, and arranging of data that is numerical.
3. In contrast, there are many instances in real-world observations where the parameters are insufficient to allow for the proper decision-making and necessitate classification into the appropriate sets based on parametric values.By using the structure of hypersoft set having the function of multi-argument to handle these arrangements, such issues have been addressed.4. The NHse-set is a type of structure which covers of the fuzzy, intuitionistic fuzzy, neutrosophic-sets, and their hypersoft sets with expert sets.The NHse-set is created to address the certain limitations of the above described structures.For example, the Fse and IFse-structures ignore the property of function having multi-argument and degree of indeterminacy, whereas the NHse-set ignores both these properties.5.The considered structure, FpNHse-set is a peculiar design that not only generalises the current structures but also enhances them by utilising the function multi-argument.
To express how uncertain the neutrosophic numbers of the FpNHse-set are, a parameterized degree is assigned to them.In this sense, handling unknown data with caution is a more flexible and all-encompassing strategy.
The main contributions of this study are summarised in the following sentences: 1.The FpNHse-fundamental set's concepts and algebraic operations are described by the help of illustrative numerical.2. Contrary to prior access, the important values of attributes (sub-attributes) taking from the Cleveland data set are investigated first in terms of their operative and lingual responsibilities, and then these important values are transformed into parameterized values with the help of applicable procedure.3. The FpNHse-set environment modifies Sanchez's method, a traditional access to medicinal analysis, in order to create a connection between the decision-makers (medical specialists), the patients who are being observed, and the prescribed qualities.4. The Cleveland data set's necessary real data, the modified Sanchez's technique, and the FpNHse-sets concept are combined to create a reform process that is suggested for the medical identification of cardiac issues.5. Implementing the reform process in a scenario based on real-world problems allows for the validity of the approach to be evaluated.6.In order to assess the reliability, flexibility, and advantageous aspects of the proposed approach, two types of comparison are needed: structural comparison and computation-based comparison.The first one is meant to assess the flexibility of the proposed structure, whereas the second one is meant to check the reliability of the presented approach.

PRELIMINARIES
To make sure that readers can understand the planned study, this section of written work introduces some basic concepts and definitions by reading the relevant literature.In this part, set will be expressed by ĥ and Z as a universe of discourse and set of experts is by X and N will be a set of opinions, C 1 = ĥ × X × N.While P(Z) will be used as a power set.Definition 2.1 (Ihsan, Rahman & Saeed, 2021b) An HSe-set ϒ HSe is defined by as ϒ HSe : → P(Z) where ⊆ C = P × X × N and

FUZZY PARAMETERIZED NEUTROSOPHIC HYPERSOFT EXPERT SET AND SET-THEORETIC OPERATIONS
Here, the definition of FpNHse-set and some basic operations are provided together with numerical examples.The real-world scenario, which necessitates the establishment of the FpNHse-set, is covered first.It is a frequent observation that a panel is assembled in any recruitment procedure to interview the initial inspected applicants.This panel typically includes a chairperson and several members with knowledge of the subject.All panel members are instructed to evaluate each candidate's aptitude and appropriateness for the open positions by taking into account predetermined assessing parameters and their sub-parametric values expressed as sets.They are also told to use their professional judgement in three dimensions when evaluating candidates for multi-argument tuples, i.e., to suggest, reject, or remain neutral.The chairperson has the power to rank the specialist's view points of the decision-makers in terms of their approach of approval.Three situations in this scenario must be handled by one model, briefly: 1.The circumstance that indicates the fundamental grouping of the properties into their associated sub-characteristic grades as various sets.2. It is necessary that the function of multi argument be able to handle the domain having a multi-argument function in which tuples are sub-parametric values in order for it to function.3. The condition that needs decision-makers to present their judgement as neutrosophic values, which guarantee the opinions three components truth, neutral value, and real non-membership.4. The situation that makes it necessary for the parameterized degree to be reflected in order to gauge the level of judgement for the things beneath discussion.
The currently available study is insufficient to offer any numerical structure that would address all of the aforementioned circumstances collectively in one structure.Our deficiency serves as the inspiration for this study.All of the aforementioned cases can be managed as a single structure using the suggested structure, the FpNHse-set.The FpNHse-set is made up of three components: (i) the fuzzy parameterized degree-based context (ii), neutrosophic context, and (iii) hypersoft context.The FpNHse context is necessary in a wide range of different real-world situations, including choosing products, diagnosing illnesses, choosing projects, analysing risks, etc. Definition 3.1 The structure FpNHse-set ϒ F can be defined as Example 3.2 Consider a scenario in which the medical director of a public hospital appoints a group of cardiologists to evaluate heart conditions by keeping track of the proper characteristics and their pertinent sub-characteristics grades for the purpose of study.A chairperson leads the committee and is in charge of making the final choice.The chairperson has the authority to carefully examine the received viewpoints in accordance with their acceptability level.Other committee members will contribute their professional (expert) opinions as decision-makers.The set of discourse { P 1 , P 2 , P 3 , P 4 } includes four sorts of cardiac problems (alternatives) that are taken into consideration.The committee members reached agreement on the parameters c 1 = chest pain type, c 2 = resting blood pressure (mmHg), and c 3 = serum cholesterol (mg/dL), before setting them.After careful examination, the characteristics are further divided into their respective related parametricvalued sets, J 1 = {c 11 = typicalangina,c 11 = atypicalangina}, J 2 = {c 21 = 150,c 22 = 180}, and J 3 = {c 31 = 320}.The cartesian product of the parameters with experts and their opinions is get the parametric pairs of characteristics.The committee persons are instructed to submit their comments in neutrosophic parts for every issue, while bearing in mind the preference of the parametric tuples.As functions of multi-argument of the FpNHse-set, the members' expert opinions are gathered along with the parameterized degree the monitor assigned for the side of acceptance of the opinions obtained.These elements are listed below: 2 ),( P2 / ≺ 0.2,0.7,0.3 ), ( P3 / ≺ 0.6,0.3,0.5 ),( P4 / ≺ 0.4,0.6,0.5 ) , 3 ),( P2 / ≺ 0.9,0.2,0.6 ), ( P3 / ≺ 0.3,0.7,0.6 ),( P4 / ≺ 0.8,0.1,0.2 ) , (ζ 4 /0.5, ˆ 2 ,1) = ( P1 / ≺ 0.3,0.7,0.9 ),( P2 / ≺ 0.4,0.8,0.5 ), ( P3 / ≺ 0.7,0.4,0.5 ),( P4 / ≺ 0.6,0.7,0.3 ) .
In above set, we see that this element ( P 1 / ≺ 0.2,0.3,0.4 ) represents the collective information given by the decision makers having 0.2(20%) as a membership value, 0.3 (30%) as an ambiguous value and 0.4 (40%), a non-membership value to disease P 1 for the taking side of special's judgement in this FpNHse-set, keeping in mind all subsequent estimates and their values are calculated in a manner similar to this.

PROPOSED TECHNIQUE AND ALGORITHMIC USAGE
The following study uses real data from the Cleveland data set (O'Brien et al., 2008) in the FpNHse-set environment to diagnose heart problems using the medical diagnosis approach of Sanchez (1979) with some minor adjustments.Figure 1 shows a visual depiction of the entire technique that was used to create the publication.

Modified Sanchez's method
A traditional way for connecting a model's parameters, universal sets, and decision-makers's perspectives is Sanchez's method (Sanchez, 1979).To establish this relationship, it makes use of the idea of matrix theory.Although it is typically used for medicinal analysis, it may be used in any other situation with minor adjustments.It needs primary matrices, which are transitively built from characteristics, universal sets, and decision-makers's viewpoints.We now provide this method with adjustments for our suggested model.

Cleveland data set
Heart issue diagnostic research is designed to use the Cleveland data set (UCI Machine Learning Repository, 2010), 303 diseased persons were investigated for the analysis of heart issues utilising 76 characteristics and five outcomes in this data collection (14 of which, however, can be used in experiments and analysis).In the Cleveland data set there are 14 significant parameters related to heart disease.Nine of the most pertinent attributes are taken into consideration while choosing six people for heart disease diagnosis, bearing for the further classification of properties into their corresponding disjoint set of attributevalued.Only the 9 parameters from the Cleveland data set are selected from the total of 14 parameters.This is so that parameters can be further categorised into their relevant sub-parametric values in the form of disjoint sets, as required by the suggested structure FpNHse-set.(attributes or characteristics).The necessary condition has already been discussed by many researchers based on this large dataset.These researches are not based on the concept of classification, that's why the whole population has been included (all patients and attributes), i.e., reduction in the size of the dataset is not the demand of such research.2. However, the sufficient condition (conversely) is interpreted in such a way that the universal set of 303 patients has been categorized into six disjoint groups based on the evaluating indicators.Consequently, six patients (Patient 1, Patient 2, Patient 24, Patient 25, Patient 75, and Patient 303) are selected from these groups by taking one from each group.The same mechanism has been adopted while screening out the number of attributes and sub-attributes, i.e., the selection of 9 attributes out of 76 attributes.3.As the proposed approach is in fact a part of medical classification which itself demands statistical data based on sampling.That's why the authors have reduced the size of the dataset from (a = 303; b = 76) to (a = 6; b = 9) to fulfill the demand of the medical classification scenario.In the mentioned pair, ''a'' stands for the number of patients considered for diagnosis, and ''b'' stands for the number of attributes considered for such evaluation.

Operational role of selected attributes
To support their choice for the diagnosis of cardiac disorders, the operational role of the chosen traits is addressed in this section of the paragraph: 1. Age: Heart illnesses are independently at risk due to ageing.Although elderly adults (60 years or more) have a higher chance of developing heart disease, younger people may also be at risk if certain additional variables are present.The pressure that the blood applies to the arterial walls is known as blood pressure.Systolic and diastolic pressure are additional categories for such pressure.The first occurs when the heart releases blood into the blood vessels, while the second occurs when the heart is resting and occurs inside the arteries.When the blood pressure is abnormally high during cardiac compression or when the arteries are relaxing, this condition is known as hypertension.There could be more resistance to blood flow in the arteries.In mm Hg, both pressures are expressed.Systolic blood pressure should be less than 120 and diastolic should be less than 80 (120/80).If your blood pressure is higher than 120 to 129, and diastolic below 80. 4. Serum cholesterol: A type of fat is cholesterol.A lipid is another name for it.It circulates in our bloodstream as tiny particles encased in proteins.Lipoproteins are the name given to these bundles.One of the main types of lipoproteins in human blood is LDL.High-density lipoproteins (HDLs) are the other main type.Triglycerides, a third class of lipid, also circulate in our blood.Our complete blood cholesterol, also known as serum cholesterol, is calculated by estimating our HDL (the ''good'' cholesterol), LDL (the ''bad'' cholesterol), and triglycerides.Although our bodies require cholesterol to create healthy cells, having too much of it can increase our chance of developing heart disease.Our blood arteries may become fatty with excessive cholesterol levels.Over time, these deposits thicken and restrict the amount of blood that can pass through our arteries.HDL and LDL cholesterol levels are added, along with 20 percent of triglycerides, to determine serum cholesterol levels.It fluctuates between 126 and 564 mg/dL. 5. Fasting blood sugar Due to the ''stress reaction'', a significant portion of persons with heart disease have elevated glucose levels.This implies that those who do not have diabetes may nevertheless have high blood sugar.A healthy person's range is between 120 mg/dL to 140 mg/dL.6. Maximum heart rate achieved: In individuals with ischemic heart disease, heart rate plays a significant role in determining oxygen consumption.Its highest value, which can be reached, falls between 71 and 195 b/m. 7. Old peak and slope: Exercise-induced Shock, Toxicity depression compared to rest is regarded as a trustworthy electrocardiogram (ECG) finding for the identification of obstructive coronary diseases.It ranges from 0.0 to 0.5 mm and is measured in mm.The exercise's peak's incline The three types of ST-segments are up sloping, flat (horizontal), and down sloping.8. Thal: This is brought on by the blood disorder thalassemia.Four categories can be made from it: null, fixed defect, no blood flow in some heart locations, normal blood flow, and reversible defect, blood flow is present but is aberrant.It's common practise to ignore the first group when diagnosing heart issues.

Parameterized grades corresponding to selected attributes
The mandated values (values assigned in the Cleveland data set) of the selected characteristics are converted into the associated parameterized grades in this stage using an appropriate transformational criterion.The potential grade for each attribute is determined by dividing its prescribed value by its highest prescribed value.The maximum prescribed value for chest pain is 4 and the other attributes in the Cleveland data set have highest values.After this the fuzzy parameterized valued is obtained by subtracting the highest value from the lowest value and divide by 2. This is shown in Table 3.It can viewed by the following formula.Let ¨ i be the multi-argument multi-decisive tuples corresponding non-overlapping parametric sub classes and parameters then their corresponding approximated fuzzy grades are denoted by µ( ¨ i ).Let µ max ( ¨ ) and µ min ( ¨ ) be the maximum and minimum approximated fuzzy grades then fuzzy parameterized grade corresponding to ¨ can be computed with the help of the following formulation (1)

Scenario and statement of the problem
The clinical diagnosis of particular diseases through mathematical modelling is a topic that scholars are getting more and more interested in.This modelling may make use of real or made-up facts or information.Since the creation of N-set, researchers have become interested in neutrosophic modelling for clinical analysis with ambiguous circumstances.Numerous modifications and generalisations have been made to N-sets.The FpNHse-set is one of them.It not only generalises the existing structures but also addresses their flaws in terms of how they handle the use of parameterized grades collectively and the new classification of characteristics into disjoint sets of attributed values.There are a few research that have been published in the study that parameterize numerical structures for clinical analysis using real data and fuzzy set extensions.This study used fresh conditions of the FpNHse-set for the medical analysis of certain people with cardiac problems using actual data from the Cleveland data set.

Proposed algorithm and implementation
The idea of aggregating the FpNHse-set is used in this part to suggest an algorithm for the diagnosis of heart disorders in patients.Different types of software and programming languages are used in multidisciplinary research, for example, mathematicians prefer LATEX, MATLAB, MSWORD, etc., computer science researchers prefer Python, C++, etc, for computer researchers, while scientific workspace for physicists and chemists.Online, there are numerous converters available for converting these programmes and codes amongst one other.It is important to use an approach that is simple to grasp but thorough to describe the suggested algorithm's procedural flow so that researchers can  adapt it to their area of interest and improve the comprehension of the suggested work.
Due to the multidisciplinary nature of the proposed study, a straightforward explanation of the proposed algorithm's procedural flow was established for the benefit of the researchers (readers).The aforementioned technique is demonstrated using the example below.
Figure 2 shows a flowchart that summarises the processes of Algorithm 1 and Fig. 3 presents the hierarchical structure of the proposed decision design.
Example 4.1 By using the data from Cleveland, select six patients for heart issue analysis; these will make up the set represents the group of heart specialist (experts) who will evaluate the diagnosing course and their decisions = {0 = disagree,1 = agree} and the set of parameters is L = {L 1 ,L 2 ,L 3 ,L 4 ,L 5 ,L 6 ,L 7 ,L 8 ,L 9 } where each parameters has its own identity such as
Step 9: The matrix W 8 of rank 6 × 4 is created by multiplying W 6 and W 7 using the traditional concept of matrix multiplication.This matrix is shown in Table 11.
Final stage step 10: Final scores are obtained by adding the each row.Figure 4 and Table 12 demonstrate that patient ˆ 303 is more likely than the rest to have cardiac disease.

DISCUSSION, AND SENSITIVITY ANALYSIS AND COMPARISON ANALYSIS
Numerous researchers have already studied the medical diagnosis of various diseases using models that resemble neutrosophic sets in the literature.These methods employ   hypothetical information with broad conclusions.Instead, the suggested study used the Cleveland data set, a genuine data collection, to identify and assess the risk of cardiac illnesses.In fact, present techniques overlook the scenario of medical diagnosis, which necessitates more classification of certain characters into their relevant values of sub-parameters and these values are found in sets with no common elements.Such a classification provides accurate outcomes and judgements.Meanwhile, Sanchez's approach, which is simple but effective, was used for this diagnosis to reduce the computational complexity.As previously mentioned, the Cleveland data set has 74 attributes, and it would be difficult to take them all into account for the computations.As a result, only the nine  attributes that played the most important roles were selected, after consulting with medical experts.According to the chosen data set, the values of sub-parameters and nature of these qualities are real.Following this debate, the primary benefits of the suggested study are listed as follows: 1.The adopted method used the FpNHse-set and parameterization to address the issues with modern decision-making.The alleged parameterized degree mimics the ambiguous nature of the level of acknowledgment; as a result, this has extraordinary potential for legitimacy inside the computational domain.2. The Cleveland data set's attributes and their sub-attributive values with real values were converted to fuzzy memberships using the proper transformation criteria.3. The emphasis on thorough observation of the parameters and their corresponding sub-parametric values in the proposed model offers better, more dependable, and adaptable findings from medical doctors as decision-makers.Because no researcher has previously used the suggested model, the FpNHse-set, to identify cardiac issues, the statistical outcomes of the recent work are not comparable with any previous structure.Though, some characteristics, including the indeterminacy score (IS), membership score (MS), non-membership score (NMS), score of parameterized (SOP), single argument-function (SIAF), multi-argument-function (MIAF), and expert's opinions (EOP) are thought to be sufficient for comparing the proposed model with the most pertinent keep going structures.It is common to see in multi-attribute decisionmaking that some experts choose to avoid making any expert judgements on the characteristics for the things beneath study.This impartiality of the decision-makers is ensured by the incorporation of the IS.Similar to this, the SOP assures that the level of acceptability of the received expert judgements.The absence of these characteristics could result in a biased judgement.Table 13 is chosen for the comparison of this structure, which demonstrates clearly that the suggested model meets all of the requirements individually  Smarandache & Öztürk (2021) IFHs-set in one model whereas the existing models are missing one or more features.Therefore, it is safe to say that the proposed model is more generalised, adaptable, and trustworthy than the ones that now exist.The symbols ⇑ and ⇓ in Table 13 stand for Yes and No, respectively.
In the Table 13, the arrows are used to indicate whether certain attributes or factors are present or not in the respective structures being compared.An upward arrow signifies the presence or existence of a particular attribute, while a downward arrow indicates the absence or non-existence of that attribute.In relation to the Table 13, the following is an explanation of the arrow symbols: 1. ⇑ (upward arrow): Represents the existence or favourable existence of a property.An upward arrow, for instance, indicates that a property has that property or demonstrates that property in the relevant structure or procedure.2. ⇓ (downward arrow): Indicates a quality's lack or unfavourable presence.If a property is shown by a downward arrow, it signifies that the relevant structure or method lacks the property or does not display the property.
The arrows represent the presence or absence of the matching attribute in the framework suggested by each author.The upward arrows indicate the presence of the attribute, while the downward arrows indicate its lack.The structure ''FHs-set'' in the row for ' 'Debnath (2021)'', for instance, has an upward arrow next to ''structure'', suggesting that it has the attribute of structure.In a similar manner, it has upward arrows for IS, MS, SIAF, and MIAF, signifying the existence of these traits, and downward arrows for NMS and EOP, signifying the absence of them.Anyone can determine if each attribute is present or absent in the comparing structures or techniques by interpreting the arrows in this way.The suggested structure is compared with the others structures based on ranking and is shown in Table 14.Some parameters like number of attributes (NA), number of patients (NP), Sanchez's method, ranking, and use of sub-attributes (UOSA) are used for this comparison.Kirisci, Demir & Simsek (2021) and Kirisci ( 2019) have not used sub-attributes criteria and used 11 attributes with no use of Sanchez's method.Rahman et al. (2022b) used sub-attributes criteria with 9 attributes.He used Sanchez's method in one but not in other structure.Ranking of all the structures have been show.
In order to evaluate the fluctuation of these score values and rank them, statistical techniques like the Pythagorean mean (arithmetic, geometric, and harmonic means)

Benefits of the proposed structure
The suggested algorithmic MADM method outperforms existing methods in the following specific areas and aspects for heart problem analysis using a neutrosophic hypersoft expert set with fuzzy parameterized degree-based settings: 1. Comprehensive analysis: To give a thorough examination of heart issues, the method blends a number of techniques, including fuzzy sets, neutrosophic hypersoft sets, and parameterized degree-based settings.This all-encompassing strategy enables the evaluation of numerous factors, such as speculative facts, professional judgements, and personal preferences, leading to a more thorough understanding of heart diseases.2. Handling uncertainty and vagueness: Working with confusing and speculative data is necessary for cardiac condition analysis.Neutrosophic hypersoft sets and fuzzy sets can be used to express and manage data ambiguity and uncertainty.This decreases the impact of erroneous or incomplete information and enables the detection of heart problems with greater accuracy.3. Expert knowledge integration: The methodology includes expert knowledge through the use of expert sets.Expert ideas and perspectives are crucial to an objective analysis of cardiac disorders.The approach takes use of domain experts' insights to improve decision-making and increase analytical accuracy.4. Parameterized degree-based setting: A versatile framework for accounting for the fuzzy value assigned to various aspects (parameters) or criteria in heart problem analysis is provided by the parameterized degree-based setup.This makes it possible for decision-makers to assign diverse factors relative weights or priority based on their significance, leading to more focused and relevant analysis results. 5. Algorithmic approach: Algorithms are used in the methodology, which provides a scientific and structured framework for examining cardiac issues.As a result, it is simpler to deliver reliable results that are repeatable and consistent, ensuring that the analytical process is transparent and open to independent verification.6. Applicability to heart problem analysis: Because the method was established expressly for cardiac condition analysis, it has been modified to address the unique problems and considerations in this area.Since the method incorporates neutrosophic hypersoft sets, fuzzy sets, and parameterized degree-based setting to take into consideration the complexities of cardiac illnesses, it is better suited for accurate and effective analysis.7. Classification of parameters: Through the use of disjoint sets, the suggested strategy successfully divides parameters into their corresponding sub-parametric values.As a result, the results are more accurate and pertinent because each parameter is taken into account in its appropriate context.8. Validation through real-world applications: The Cleveland data set is used to validate the algorithm in real-world settings, specifically in the diagnosis of heart disease.This validation shows how well the model functions in practical decision-making situations.Overall, the superior performance of the proposed method can be attributed to its ability to manage uncertainty, account for various expert opinions, classify parameters, handle multi-argument approximations, and show successful applications in real-world decision-making scenarios related to the diagnosis of heart disease.

CONCLUSIONS
By approaching the context of novel mathematical structure, i.e., FpNHse-set, the flaws of already developed researches are settled.These flaws may be of different nature like entitlement of multi-argument approximate domain, the gathering of expert's opinions in terms of neutrosophic values and the provision of arrangement that is capable of assessing the vague nature of attributes and sub attributes.The aggregation operations of FpNHse-set are utilized to establish a decision based system to judge the risk for heart diseases.The proposed framework employs Sanchez's medical technique to establish a relationship among the patients, medical experts (decision makers) and fuzzy parameterized sub parametric tuples by matrix representation.The attributes and their related sub attributive values are opted from the Cleveland data set so that the genuineness of the proposed framework may be over viewed.The fuzzy parameterized grades for sub parametric tuples are determined by using arithmetical formulation to ease the understanding.The suggested algorithm is consisting of streamline flow of understandable knowledge which leads to easy computational results and thus ranks the patients.Moreover, the flexibility is judged by comparing the suggested framework with already presented researches.In order to seek further extensions of this framework, the structural modifications may be proceeded in future.

Figure 2
Figure 2 Flowchart of algorithm.

Figure 3
Figure 3 Hierarchical structure of decision problem.

.
1 and W 2 are actually the values of FpNHse-set, these important values are made to change in fuzzy values with the help of method |t−f −i| 3 To ensure a discrete judgement, FpNHse values are transformed to fuzzy values.Fuzzy parameterized values for the sub-parametric tuples have been calculated by subtracting the highest fuzzy value from the lowest fuzzy value and divided by 2. Then these values have been used in Ihsan et al. (2023), PeerJ Comput.Sci., DOI 10.7717/peerj-cs.160715/26 Figure 4 Representation of alternatives.

Definition 2.3 (Al-Quran & Hassan, 2016) A
Six diseased persons have been selected because Sanchez's technique, and including extra diseased persons could have led to ant complicated problem.But, with the appropriate planning, such complexity might be overcome.These nine qualities are explained in Table 2 along with the suggested values for each (data set).Note: 1.The Cleveland data set is one of the prominent large datasets.This consists of observations obtained by considering 303 patients with 76 evaluating indicators Ihsan et al. (2023), PeerJ Comput.Sci., DOI 10.7717/peerj-cs.16079/26

1607 10/26 variety
Four categories of ageing have been established by medical professionals: up to 20, 40, 60 and above.2. Chest pain type: It may be the utmost common logic for people to go to the trauma of harmless typical factors.Atypical angina (ATA), non-anginal pain, typical angina (TA), and asymptomatic chest pain are different types of heart-related chest pain (AM).Angina is typically characterised by substernal chest pain or discomfort that is (1) brought on by physical activity or mental stress and (2) alleviated by rest or nitroglycerin (or both).When two of the three traditional angina criteria are met, atypical (probable) angina chest pain is present.Non-anginal pain is used to identify hospitalised individuals or showing signs of a myocardial ischemia.3. Resting blood pressure: department is chest pain.Depending on the person, it varies.It also varies in terms of quality, force, span, and area.It could be a mild pain or an acute, painful feeling.It can be a sign of an actual cardiac condition.It could have also been induced by a Ihsan et al. (2023), PeerJ Comput.Sci., DOI 10.7717/peerj-cs.