Topological Data Analysis of m-Polar Spherical Fuzzy Information with LAM and SIR Models

: The concept of m-polar spherical fuzzy sets (mPSFS) is a combination of m-polar fuzzy sets (mPFS) and spherical fuzzy sets (SFS). An mPSFS is an optimal strategy for addressing multipolarity and fuzziness in terms of ordered triples of positive membership grades (PMGs), negative membership grades (NMGs), and neutral grades (NGs). In this study, the innovative concept of m-polar spherical fuzzy topology (mPSF-topology) is proposed for data analysis and information aggregation. We look into the characteristics and results of mPSF-topology with the help of several examples. Topological structures on mPSFSs help with both the development of new artificial intelligence (AI) tools for different domain strategies and the study of different kinds of uncertainty in everyday life problems. These strategies make it possible to recognise and look into a situation early on, which helps professionals to reduce certain risks. In order to address various group decision-making issues in the m-polar spherical fuzzy domain, one suggestion has been to apply an extended linear assignment model (LAM) along with the SIR method known as superiority and inferiority ranking methodology in order to analyze road accident issues and dispute resolution. In addition, we examine the symmetry of optimal decision and perform a comparative study between the research carried out using the suggested methodology and several existing methods.


Introduction
The conditions of our everyday life provide us with obstacles that arise from imprecise information and an absence of appropriate modelling, both of which can lead to imprecise answers and questionable reasoning. While human intelligence (HI), machine learning intelligence (MLI), and online social networks have only been around for a short time, they have changed the economic and social paradigms of society. Because of the rapid expansion of HI, MLI, and big data over the past few years, incredible improvements in our day-to-day lives have been made [1][2][3][4][5][6][7]. The theory of topology pertains to the interaction between spatial components or modules. It can be utilised to design large datasets with the ability to explain certain spatial functions. The definition of topological spaces is listed in Table 1.
We have extended the study of topology to a new hybrid approach of SFS and mPFS, i.e., m polar spherical fuzzy sets (mPSFS), by following the idea of Chang [5]. We define the concept of mPSF-topological space using an mPSF-null set, mPSF-absolute set, and certain basic operations of mPSFSs, such as union and intersection. Here, we investigate several basic properties and analyse them with the help of theorems and examples.
Introduced separation axioms and several of their equivalences as well as their relations with each other in fuzzifying topology 1995,1997 Coker [15,16] Invented the idea of intuitionistic fuzzy topological space 2011 Shabir and Naz [17] Soft topological spaces 2011 Cagman et al. [18] Soft topology 2019 Riaz et al. [19] N-soft topology and decision analysis 2019 Olgun et al. [20] Topological properties of Pythagorean fuzzy sets 2021 Alshammari et al. [21] Pythagorean fuzzy soft topological spaces Fuzzy mathematics differs from traditional mathematics, primarily in the field of set theory. Fuzzy mathematics was first developed only a few years ago, and is rich in ideas. In automobiles and traffic control systems, where logic circuits control anti-lock brakes, electronic systems, and other functions, its application is widespread. This is an even more specific form of the phrase "crisp set." It has the ability to decide either yes or no, which corresponds to the numbers 1 and 0. Several of the ideas that are discussed in this article were anticipated by an American philosopher by the name of Black [22], who lived thirty years ago. The author of [22] developed a theory that has as its primary building blocks "fuzzy sets," which are simply sets with "imperfect" bounds. An important work produced by Zadeh [23] in 1965 makes a crucial argument concerning the evolution of contemporary ideas around ambiguity, and was an immediate extension of the crisp set concept. Atanassov [24][25][26] suggested the idea of intuitionistic fuzzy sets (IFS) and intuitionistic fuzzy numbers (IFN) in 1983. Yager proposed "Pythagorean fuzzy sets (PFS)" [27,28], and Yager further introduced "q-rung orthopair fuzzy sets (q-ROFS)" [29]. Soft sets (SS) and SS theory were originated by Molodtsov [30,31]. Smarandache [32] suggested a new model called neutrosophic sets by introducing another grade to FS. Picture fuzzy sets (PiFS) were proposed by Cuong [33]. Spherical fuzzy sets (SFSs), which were recently introduced by Kutlu Gündodu and Kahraman [34], are the most recent extension of fuzzy sets; in these, the squared sum of the hesitancy-neutral degree, membership degree, and non-membership degree does not exceed 1. Ashraf et al. [35] and Mahmood et al. [36] introduced spherical fuzzy sets with certain operational rules and aggregation operations based on Archimedean t-norm and t-conorms. SFS covers a wider area as compared to picture fuzzy set (PFS). Feng et al. [37] proposed new score functions of q-rung orthopair fuzzy sets for further utilization in decision analysis. Deveci et al. proposed the idea of personal mobility in metaverse with autonomous vehicles based on q-ROFs [38]. Riaz et al. [39] presented an application of bipolar fuzzy soft sets (BFSSs) to supply chain management.
Due to the sheer uncertainty and ambiguity of real-world circumstances, the approaches frequently employed in classical mathematics are not always suitable for solving real-world problems. Using a variety of multi-criteria decision-making (MCDM) techniques that analyse a collection of choices against a variety of methodologies, the reliability of human judgments can be determined [40]. Information aggregation and synthesis are fundamental to technological innovations such as deep learning, decision-making processes, remote sensing data, principal component analysis, and mathematical abilities. The process of combining individual preferences on a specific list of options into a single unified collective preference is an example of a type of MADM problem known as group decision-making. This type of MADM problem is typically thought of as the procedure by which group decision-making is carried out. The process of making decisions as a group involves the participation of multiple individuals, each of whom brings a unique set of skills, years of experience, and relevant information to bear on the analysis of the various facets of the issue. Multiple-Attribute Group Decision-Making (MAGDM) is the term used to describe this category of problems [41]. Several MADM algorithms have been changed to work with spherical fuzzy sets, and the applications of these changes have been examined in the literature. The conventional approach to decision analysis known as MCDM, utilises the linear assignment model (LAM) and the superiority and inferiority ranking (SIR) method in mPSFS to determine an order of liability for the causes of road accidents. In 1977, Bernardo and Blin [42] came up with the idea for what would become known as the linear assignment technique (LAM), which took its influence from an assignment problem in linear programming for MADM [43]. The fundamental concept behind the LAM is that the rankings of the criteria provide an overall preference ranking that, when combined with the other component rankings, results in the best possible compromise [44,45]. Many acandemics have extended LAM under a variety of fuzzy extensions and making use of a variety of modelling and solution methodologies to account for the unavoidable uncertainty that is inherent in real-life decision-making situations; several of these are mentioned in Table 2. In a similar way, the SIR method has been used by a number of researchers in different fields of FSs. First, in 2001, Xu [55] gave a presentation on the approach known as the superiority and inferiority ranking (SIR). In the SIR method, the placement of the options is determined by two different segmentation overviews, which are as follows. In this method, the various options are ordered according to the predominance positioning arrangement as well as the inadequacy positioning arrangement. Notably, the major degree of latitude afforded by the use of the SIR method merges the characteristics of various other MCDM procedures. Numerous researchers have added their expertise using this method in various fields and domains, which are discussed in Table 3. The following are the main objectives of the suggested framework: • Construction of the topological structure on mPSFFSs. • Analyzing the fundamental topology concepts of mPSF-open set, mPSF-closed set, mPSF-interior, mPSF-closure, mPSF-base, and mPSF-subbase, as well as their instances. • Introducing theorems and the proofs that support them in order to illustrate how mPSF-topology works; • Description of a case study entitled "Day by Day Increasing Road Accidents". • As an expansion of MCDM, LAM is used to elaborate the application for finding the causes of road accidents. • The SIR framework is a well-known and frequently used MCDM technique that is useful for finding the rankings of similar applications in order to obtain precise results. • A comparison analysis is provided to conclude our work.
The rest of the present paper is categorised in the following fashion across the remaining sections. Section 2 explores core mPSF principles. The most important findings of mPSF-Topology are outlined in Section 3. In Section 4, we demonstrate a SIR and LAM framework that is embedded inside the mPSFSs and has an application to vehicular mishaps. In Section 5, the most important results of the investigation are provided.

Preliminaries
In this section, we concisely review a few primary concepts related to different kinds of sets, including fuzzy sets, soft sets, spherical fuzzy sets and spherical fuzzy soft sets, that are employed during the rest of this paper.

Definition 1 ([23]
). Let⨿ be a classical set and µ A :⨿ → [0, 1] be the membership function. Then, a fuzzy set A can be written in the form The value of the mapping µ A at ρ ∈⨿ i.e., µ A (ρ) denotes the degree of membership of ρ to fuzzy set A. The aggregate of all fuzzy sets in⨿ is designated as F (⨿). Definition 2 ( [34][35][36]). Let U be the universe of discourse; then, a spherical fuzzy set (SFS)¯U is an object with the form¯= where µ(⨿) ∈ [0, 1] is called the "degree of membership (MG) of⨿ in U", γ(⨿) ∈ [0, 1] is called the "degree of hesitancy (HG) of⨿ in U", and η(⨿) ∈ [0, 1] is called the "degree of non-membership (NMG) of⨿ in U"; furthermore, µ(⨿), γ(x) and η(⨿) satisfy the following condition: is called "refusal degree" of⨿ ∈ U. A spherical fuzzy number (SFN) can be simply represented as Definition 3. Let U be a universal set of discourse. An m-polar fuzzy set on U is defined on interval [0, 1] m and can be expressed as

m-Polar Spherical Fuzzy Set
Now we define m-polar spherical fuzzy sets (mPSFSs) as a hybrid structure comprised of both spherical fuzzy sets (SFS) and m-polar fuzzy sets (mPSFS). For m = 1, an mPSFS turns into an SFS.
And matrix representation of m-polar spherical fuzzy set(mPSFS) is Then, 5PSFS in M is expressed in Table 5.
} and letS Γ A andS Γ B be two 5PSFSs defined as shown in the Table 6 and Table 7, respectively.
can be defined as shown in the Table 8.
can be defined as shown in the Table 9. And the complement ofS Γ c A can be calculated as shown in the Table 10.
If e i and e j are two spherical fuzzy numbers, then 1. Given s(e i ) < s(e j ) , e i precedes e j i.e., e i ≺ e j , 2.
Given s(e i ) > s(e j ) , e i succeeds e j i.e., e i ≻ e j , 3.
Given s(e i ) = s(e j ) , e i ∼e j .
Definition 7. The accuracy function for any SFN e = ( µ e ), j = 1, 2, 3, ..., m is defined as where 0 ≤ s(e) ≤ 1. If e i and e j are two spherical fuzzy numbers, then 1. If s(e i ) and s(e j ) coincide and a(e i ) exceeds a(e j ), then e i ≻ e j , 2.
If both s(e i ), s(e j ) and a(e i ), a(e j ) coincide, then e i ∼e j Definition 8. An m-PSFS is said to be an empty or null m- It is denoted as∅ and scripted as It is denoted asX and scripted as

m-Polar Spherical Fuzzy Topology
This section describes the idea of mPSF-topology in relation to a mPSF set. The concepts of mPSF-union and mPSF-intersection are used as building blocks while designing mPSF topology. Several elements of the mPSF-topology are defined, and the accompanying illustrations are presented in an optimized way. Definition 10. Assume X Γ is a set which is non-empty and mPSFS(X Γ ) is the collection of all the mPSF-subsets. A sub-collection T Ω of mPSFS(X Γ ) is called m-polar Spherical fuzzy topology (mPSFT) on X Γ if the following properties hold true: The couplet (X, T Ω ), or simply T Ω , is known as m-polar spherical fuzzy topological space, shortened as mPSFTS.
Example 3. LetX = {⨿ 1 ,⨿ 2 ,⨿ 3 } be a universal set and S 1 , S 2 , and S 3 be as shown in Tables 11-13 below  Definition 11. Let T Ω 1 and T Ω 2 be two mPSF-topologies over X. If every member of T Ω 1 is also a member of T Ω 2 , i.e., T Ω 1 ⊆ T Ω 2 , then T Ω 1 and T Ω 2 are comparable. In such situations, T Ω 1 is purported to be coarser or weaker than T Ω 2 and T Ω 2 is purported to be finer or stronger than T Ω 1 . In Example 3, T Ω 3 is coarser than T Ω 2 , and hence these are comparable.

Remark 1.
The mPSFTS intersection is always an mPSFTS, however, its union does not have to be an mPSFTS.

Example 4.
Let X={⨿ 1 ,⨿ 2 } be the universal set andS Γ 1 andS Γ 2 be 3PSFSs as provided below in Tables 14 and 15, Then, fails to be a 4PSFT over X.
is open. This shows that ∩ q∈QS Γ q is a closed mPSFS. (iii) BecauseS Γ q is closed for q=1,2,3,...,n, is an open mPSFS. Thus, ∪ n q=1S Γ q is a closed mPSFS. T Y also known as the relative mPSF topology or induced mPSF topology on Y .
Now, let the absolute 4PSF on Y = {⨿ 1 ,⨿ 3 } ⊆ X be is shown in the Table 18. Working is shown in Table 19 and Table 20 .
Definition 13. Let (X, T Ω ) be an mPSFTS andS Γ ⊆ mPSFS(X); then, the interior ofS Γ is indicated asS Γ • and described as the union of all open mPSF subsets included inS Γ . Alternatively, it can be defined as the largest open mPSFS that includesS Γ . Example 6. Let X={⨿ 1 ,⨿ 2 ,⨿ 3 }. Let us consider 4PSFSsS Γ 1 andS Γ 2 in X such that T Ω = {∅, S 1 , S 2 ,X} is the topology on X.S Γ 1 is given in Table 21,S Γ 2 is given in Table 22 andS Γ is given in Table 23. Then,  Theorem 3. Let (X, T Ω ) be an mPSFTS and letS Γ ,S Γ 1 , andS Γ 2 ∈ mPSFS(X). Then, the following results hold: Let (X, T Ω ) be an mPSFTS and letS Γ ∈ mPSFS(X); then, closure ofS Γ is indicated asS Γ cl and is described as the intersection of all closed mPSF supersets ofS Γ ; it can also be defined as the smallest closed mPSFS ofS Γ .
Suppose that the collection of all open sets in (X, T Ω ) which are contained in the setS Γ is provided byȮ :⨿ ∈ X , where i ∈ I.

Now, the interior ofS Γ is provided bȳ
Therefore, For this purpose, assume that ma⨿ i∈I µ Hence, (S Γ • ) c and (S Γ c ) cl both lie in the mPSF domain. We can prove (ii) in a similar way.

Proposition 1.
Let (X, T Ω ) be an mPSFTS and letS Γ ∈ mPSFS(X). Then, Definition 15. Let (X, T Ω ) be an mPSFTS and letS Γ ∈ mPSFS(X). Then, the frontier ofS Γ is indicated as Fr(S Γ ), and is described as Definition 16. Let (X, T Ω ) be an mPSFTS and letS Γ ∈ mPSFS(X). Then, the exterior ofS Γ is indicated as Ext(S Γ ) and is described as Example 8. In Example 7, the frontier ofS Γ is provided as follows:

Multi-Criteria Group Decision Making by m-PSF Topology
Choice-making occurs in trading, business, and sciences, and often faces huge challenges. It differentiates between a minimal regime's constant operational analyses and leaders' extensive formulation and management. While collected data can be analyzed to reach inferences that can have major or unfavorable implications, ultimately officials must show a logical operational strategy for winning. Before reaching a choice, decision-makers should assess a wide range of factors. Consequently, while coming to a conclusion, it is imperative that all of these variables are factored into the equation. In legal terms, a systematic approach to decision-making is essential to guarantee that all of the key data and information are properly considered. Among several other methods, mathematics can assists in reaching conclusions based on the evidence provided by science. This excerpt provides an illustration of a strategy for handling the problem of multiple criterion group decision-making by employing the superiority inferiority ranking (SIR) and linear assignment model (LAM) within an mPSF framework.
MCDM with SIR method.
The essential steps for the extended SIR approach for mPSFS are outlined below.
1 Determine the individual measure degree ℶ k (k = 1, 2, ..., l), then obtain the relative closeness coefficient using the formula provided below: where W − D R and W + D R denote the relative minimum and relative maxima, respectively, and can be calculated as follows: Normalize ℶ k (k = 1, 2, ..., l) using the formulā and denote them as an individual measure degree bȳ W = (w 1 , w 2 , . . . , w n )

3:
Acquire the amalgamated mPSF decision matrix and weight vector utilizing the mPSF operator to class individual viewpoints together into an aggregated frame of reference.
• the mPSFWAM aggregation operator for integrating individual decision matrices is • the mPSFWAM aggregation operator for integrating th weights of individual attributes is Build a performance relationship based on relative performance using the formula and the inferiority matrix Activate where F P K = 0.01 if 0⟨P κ ⟨1 and F(P κ ) = 0 if 0 ≥ P κ or P κ ≥ 1

5:
Compute the superiority and inferiority indexes as follows: Determine score functions of ′ (P κ α ) and γ (P κ α ). 7: By following the rules below, determine the flow of superiority and inferiority. Superiority flow rules (SFRs): ) and S( γ (P κ α )) > S( γ (P κ t )) Inferiority flow rules (IFRs): Obtain the best possible outcome by coupling SFRs and IFRs next to each other MCDM with LAM. On the basis of attributes and structures, the classical linear assignment method (LAM) can be extended to mPSFs. The following algorithm explains the numerous processes that form the proposed mPSFLAM.

1:
Take into account the scores and alternative weights of decision-makers.

2:
Aggregate the score matrices and weights of decision-makers.

3:
Compute the score values of the aggregated decision matrix.

4:
Determine the rank frequency non-negative matrix Υ i k 5: Establish a weighted rank frequency ℶ, where the ℶ i k receives the alternative contribution to the total ranking. The value is . The LAM can be expressed in a linear programming format, as mentioned below: Find the best possible answer by working through the linear assignment model and analyzing it. 8: Rank the optimal alternatives.

Case Study
Numerous kinds of vehicles, including automobiles, buses, tractors, motorcycles, mopeds, pedestrians, animals, taxis, and others, use roads throughout the world. The invention of the vehicle is directly responsible for the growth of many nations' economies and social structures. Nonetheless, every year, millions of people are killed and wounded in vehicle-related incidents. Each year, 1.35 million people are killed on the roads and highways of the world. Every day, more than 3,700 people are killed in accidents involving cars, buses, motorcycles, bicycles, lorries, and pedestrians around the globe. More than half of those killed are cyclists, pedestrians, or motorcyclists. Gopalakrishnan has presented these statistics in [71]. According to estimates, crash-related injuries are the eighth-leading cause of death around the globe for people of all ages. Today, automotive accidents kill more people than HIV/AIDS. The global death toll is depicted in Figure 1. Transportation system quality is one of the key indicators of the standard of living. The transportation system is essential to human survival. Automobile crashes are a major worry, as they risk people's lives, health, and property. In addition, the management of the transportation system may be compromised by road mishaps. Due to the dangerous nature of this system, it might not be able to work properly. Therefore, traffic accident prediction models may be useful for comprehending the elements that lead to accidents as well as their frequency under different conditions. Accidents are the largest cause of injury and death worldwide, and as a result, they contribute to a vast array of social and economic problems. Here, we use Albania as a case study for the creation of a fuzzy model for accident prediction. The conditions of the road and the flow of traffic are two major contributors to road accidents. It is feasible to reduce the number of accidents by investigating these factors. Research on road safety has revealed the accuracy of collision prediction models.
Accidents on the road are the leading cause of hospitalisation for teenagers; drivers between the ages of 18 and 24 account for 23% of all fatalities resulting from automotive accidents. Similar to the incidence rate, the recurrence rate is rather high. Following an initial accident, one in every four teenagers goes on to experience a recurrence during the subsequent year. Cognitive deficiencies are known to be present in teenagers, and can be a contributing factor in the development of risky behaviours. Risky behaviours are defined as repeated engagement in potentially dangerous situations, such as getting into vehicle accidents. There are two distinct groups of factors that appear to be linked to road accidents: (1) characteristics that are unique to the environment in which the accident occurred and (2) "human" factors, which appear to be the most significant of the two. Both of these groups of factors are discussed further below. Additionally, the development of a tighter connection to high speed driving increases the risk of becoming involved in a car accident, which is a risk that can be amplified by the pursuit of extreme sensations. Additional variables, such as the use of drugs (including alcohol, opioids, or "binge drinking"), have been identified as risk factors. In addition, it would appear that both using a cell phone while driving and having attention deficit disorder, whether or not it is accompanied by hyperactivity, are major factors in the chance of being involved in a car collision.
Below, we describe the factors that can potentially cause vehicle accidents.

1.
Speeding (‫ג‬Y 1 ) Speeding continues to be the major cause of automobile accidents. According to the Royal Society for the Prevention of Accidents (RoSPA), improper speed leads to around 11% of reported injury incidents and 24% of fatal collisions. All collisions caused by speeding are fully preventable; drivers and riders must respect traffic regulations to ensure the safety of others. Keep in mind that speed restrictions govern the maximum speed, not the minimum speed; therefore, modifying speed to the road or weather conditions is perfectly reasonable. In addition to being aware of their speed, one of the most crucial pieces of advice for drivers and riders is to maintain a safe distance from the vehicle in front, sometimes known as the "two-second rule." Keeping a safe distance from other vehicles dramatically lessens the probability of being involved in an accident. The following may occur in an accident due to speeding.
A. Lessened ability to drive and stop the car There is a correlation between high rates of speed and the likelihood of becoming involved in an accident. The likelihood of a collision increases proportionally with the absolute speed of travel. Consider the fact that the driver needs a regular amount of time to respond effectively to unplanned events. As the speed of the vehicle rises, so does the distance it travels before reacting. When travelling at high speeds, there is less time to adapt to changes in the surroundings and less room to manoeuvre. In addition, as illustrated in Figure 2, the stopping distance is larger.

B.
Increased the likelihood of roll-over When driving at high speeds, there is a significantly increased risk of becoming engaged in an accident involving a roll-over. It is highly uncommon for a car to flip over when it is engaged in an accident that occurs at a low speed, such as one that occurs in a parking lot. If even one of the vehicles is moving at a fast rate of speed, the probability that one of them may roll over is greatly increased, regardless of whether it is a car travelling at high speed or a lower-speed car struck by the high-speed vehicle.
Non-adherence to lane driving. According to official statistics, this is one of the top five causes of accidents on Indian roads. There are 1214 car accidents every single day in India, and one person loses their life as a result of a road accident every four minutes. Figure 3 shows the number of accidents annually due to changing and/or not following the lane. The number of deaths, which has been rising over the last few years, did not decrease despite the fact that there were fewer road traffic incidents overall. Instead, the number of fatalities has been steadily rising.

B.
Overtaking in an incorrect manner. If you have been driving for a long time and often go on lengthy journeys, you have likely witnessed accidents caused by drivers who failed to properly pass another car or who came perilously close to being struck by an oncoming vehicle when passing another vehicle. When attempting to pass another car on the road, it is common for drivers to underestimate the necessary space and time. This happens more often in rural areas. One of the most common errors that can lead to accidents when overtaking another vehicle is failing to give the proper signal to the cars in front of and behind them. Overtaking is one of the leading causes of accidents, as illustrated in Figure 4.

3.
Pedestrian negligence contributes to road accidents and endangers their own lives (‫ג‬Y 3 ) A.
Running or Darting into the Road. It is not unusual for children and adolescents to run into the street without first checking to see if there are vehicles approaching from the opposite direction. Because of this, there is a chance of an accident occurring in nearly any part of the city. It is likely that a motorist will not have the time to react in the appropriate manner when a pedestrian unexpectedly walks into the street. Because of this, they stand a good risk of colliding with the pedestrian, which may lead to serious injuries. B.
Standing, Lying, Playing, or Working in a Roadway. When a person is standing or loitering in the middle of the roadway, it can be difficult for a car to avoid colliding with the pedestrian. Even if a driver swerves to avoid striking a pedestrian, the driver behind that vehicle may not see the pedestrian in time. Pedestrians who are standing, lying down, or playing in the roadway at the time of an accident may be held liable. The negligence of drivers may endanger the lives of roadside workers. In order to comply with safety standards, employees are frequently required to show road signs and wear apparel that is highly visible to drivers. Similarly, motorists are expected to exhibit heightened caution when driving close to a construction site.

Decision-Making
Consider the CCTV footage of an accident shown in Figure 5.

Consider a set of experts
with the credibility weights = , criterion ‫ג‬Y = {‫ג‬Y 1 , ‫ג‬Y 2 , ‫ג‬Y 3 }, and alternatives U ¶ = U ¶ 1 , U ¶ 2 , U ¶ 3 , which are highlighted in the Figure 5, namely, a red vehicle (U ¶ 1 ), two pedestrians (U ¶ 2 ), and a white vehicle (U ¶ 3 ). Let P k αβ be the mPSFN assigned to αth potential cause of accident with respect to βth by the k th expert. Construct the mPSF decision matrix P(k). Assume that w k β represents the mPSF weights of symptom S P β provided by the criteria decision matrix . The most effect patient is filtered using the proposed techniques.
The weights associated with experts' assessments are provided in Table 27. The minimum reference weight and maximum reference weight are given in the Table 28:

1.
Calculate the relative proximity coefficient using the following formula; for this purpose, we first need to determine the distances, as calculated in the Table 29.
The relative proximity coefficients are computed in are calculated in the Table 30. Furthermore, the aggregated mPSF-decision matrix is calculated and presented in the Table 35. The aggregated weights are given in the Table 36.

4.
Calculate the relative performance using the values of + and − given in the Table 37 are shown in the Table 38.
S 11 = 0.01 First calculations are performed in order to get superiority and inferiority presented in the Table 39. Further calculation are performed and the superiority index and inferiority indexes using the values listed in Table 39 are given in the Table 40.  6.
The superiority rankings are provided in the Table 41. Table 41. Superiority rankings.

Score Value Rank
The inferiority rankings are provided in the Table 42. Table 42. Inferiority rankings.

Score Value Rank
The optimal Solution is P k 2 . On the basis of this optimal value U ¶ 2 is the main cause of the accident.
Numerical computation using LAM: Step 1, Step 2 and Step 3 are the same as in the calculations for SIR. Calculation of score function is given in the Table 43 :

4.
The ranking of each alternative based on each criterion is provided in the Table 44.
The optimal rank order is P k 2 > P k 1 > P k 3 . On the basis of this optimal rank the alternatives are ranked as U ¶ 2 > U ¶ 1 > U ¶ 3 . Hence U ¶ 2 is the main cause of accident whose CCTV footage is shown in the Figure 5.

Discussion
The first benefit of the proposed method (LAM) is that it lowers the subjectivity of decision-makers by explicitly calculating the proximity of each alternative to the best choice. This is the initial benefit. Using a collection of criterion-specific rankings as inputs, the linear assignment method can yield a general preference ranking of the available options. The SIR technique generalises the concepts of superiority and inferiority by taking into account the distinctions between criterion values and other forms of generalised criteria. As a consequence, it presents a model that is dependable, efficient, and objective for a full investigation of the relative importance of alternatives. The capacity of the decisionmaker (DM) to alter the aggregation procedures depending on his or her perspective on compensation, trade-offs, and the types of information gathered is possibly the most significant aspect of the SIR method. As a result, the SIR methodology makes use of the qualities shared by the majority of MCDM techniques when working with unquantifiable, cardinal, and ordinal data. In order to address the MCDM issues, the beginning SIR technique has been used; real figures are used to evaluate the criteria.

Conclusions
Topics associated with mPSF topology have been examined in this work. Combining the notions of the mPSF-union and mPSF-intersection as well as the mPSF-absolute set and the mPSF-null set produces the mPSF-topology. Meanwhile, several components of mPSF-topology are specified, including mPSF-open sets, mPSF-closed sets, the mPSFinterior, mPSF-closure, and the mPSF-exterior. Other features include mPSF-open and mPSF-closed sets. In addition, this study emphasises the mPSF-base and the mPSF subbase. mPSF-topology is a type of fuzzy topology that is an extension of m-polar fuzzy topology and spherical fuzzy topology. In this article, we have demonstrated real-world applications of MCGDM using mPSF-sets and mPSF-topologies. SIR and LAM are two well-known and commonly used approaches that we employ. One of the phases is to create the appropriate algorithms and flowcharts to make the process easier to visualise. We extend the SIR and LAM to mPSFSs and apply it to obtain a ranking to find the main causes of road accidents. In addition, we include a case study illustrating the use of both techniques for evaluating the main reasons for roadside accidents.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: mPSFS m-polar spherical fuzzy set mPSFN m-polar spherical fuzzy number mPSF m-polar spherical fuzzy mPSF-topology m-polar spherical fuzzy topology HI human intelligence MLI machine learning intelligence LAM linear assignment model SIR superiority and inferiority ranking methodology MCDM multicriteria decision making Notations (ρ, µ A (ρ)) membership of ρ in A.
γ(⨿) degree of hesitancy of⨿ U ¶ i i-th alternative E ℸ i i-th expert W E ℸ i credibility weight of i-th expert