An Intelligent and Robust Framework towards Anomaly Detection, Medical Diagnosis, and Shortest Path Problems Based on Interval-Valued T-Spherical Fuzzy Information

The recent emerging advancements in the domain of the fuzzy sets are the framework of the T-spherical fuzzy set (TSFS) and interval valued T-spherical fuzzy set (IVTSFS). Keeping in view the promising signiﬁcance of the latest research trend in the fuzzy sets and the enabling impact of IVTSFS, we proposed a novelframework for decision assemblyusing interval valued TSFSbased upon encompassing the four impressive dimensions of human judgement including favor, abstinence, disfavor, and refusal degree. Another remarkable contribution is the optimization of information modeling and prevention of information loss by redeﬁning the concept of each membership in interval. Moreover, the proposed research made a worthy contribution work by demonstrating the eﬀective utilization of the interval valued TSFS based framework in anomaly detection, medical diagnosis, and shortest path problem. The proposed work demonstrates the eﬀective remedial measure for the anomaly detection problem based on several parameters using the aggregation operators of IVTSFS. Moreover, the interval valued T-spherical fuzzy relations and their composition are illustrated to investigate the medical diagnosis problem. Furthermore, the notion of interval valued T-spherical fuzzy graph is also presented and fundamental notions of graph theory are also demonstrated with the help of real world instances. In the context of interval valued T-spherical fuzzy graphs(IVTSFGs), amodiﬁed DijkstraAlgorithm(DA) isdevelopedand appliedto theshortest pathproblem.Thein-depthquantitative assessment and comparative analysis revealed that the proposed notion outpaces contemporary progressive approaches.


Introduction
Zadeh [1] developed a novel way of describing an entity's membership to a set using the idea of a fuzzy set (FS) which was further enhanced by Atanassov [2] who proposed the perception of an intuitionistic fuzzy set (IFS). In an IFS, the relation of an object to a set is described by the membership as well as nonmembership grades (NMGs) denoted by ş and d on the unit interval in a way that 0 ≤ sum(ş, d) ≤ 1. e condition 0 ≤ sum(ş, d) ≤ 1 on IFS makes it less effective in some cases because it does not allow us to choose the values of ş and d independently. is issue was addressed by the framework of the Pythagorean fuzzy set (PyFS) developed with a relatively stronger condition, i.e., 0 ≤ sum(ş 2 , d 2 ) ≤ 1.
e concept of PyFS also becomes limited in some cases and a new generalized ortho pair FS known as q-rung ortho pair FS (Q-ROPFS) introduced by Yager [3] with a condition that 0 ≤ sum(ş q , d q ) ≤ 1 such that q ≥ 1. e comparative evaluation of the spaces of IFS, PyFS, and Q-ROPFS is described in Figure 1. In Figure 1, the turquoise color stripe models the aggregated perimeter of IFSs, the light blue semicircle represents PyFS, and the red curve illustrates Q-ROPFS for q � 7. IFS, PyFS, and Q-ROPFS are only effective in situations with discrete resolutions, i.e., yes or no.
Cuong [4,5] proposed a new theory of picture fuzzy set (PFS) which described four types of grades ş, ı, d, and R denoting membership, nonmembership, abstinence, and refusal degree such that 0 ≤ sum(ş, ı, d) ≤ 1 and R � 1 − sum(ş, ı, d). Cuong's idea of PFS generalizes the theory of IFS but constrained by the condition 0 ≤ sum(ş, ı, d) ≤ 1 due to which grades could not be assigned independently. erefore, a novel idea of TSFS and SFS generalizing IFS, PyFS, Q-ROPFS, and PFS with a condition on membership grade (MG) 0 ≤ sum (ş n , ı n , d n ) ≤ 1 and R � ������������ � sum(ş n , ı n , d n ) n such that n ∈ Z + and n ≥ 1 was proposed by Mahmood [6]. e comparison of spaces of PFSs and SFSs is described in Figure 2 where the pink line represents the boundary space of PFSs, and the dark grey curves represent the boundary space of SFSs. One may also refer to [6,7] for further knowledge of PFSs, SFSs, and TSFSs.
In real-life problems, the MG of an object could not be assigned due to some type of uncertainty, and hence, [8] proposed IIVFS with the definition of entity's membership with closed subinterval (ş L , ş U ) of the unit interval. An IVFS easily reduces to FS if we take ş L � ş U . Following this phenomenon, the same type of structures has been proposed for IFSs, PyFSs, and PFSs. e concept of interval valued IFS (IVIFS) is proposed in [9], interval valued PyFS (IVPyFS) is developed in [10], and interval valued PFS (IVPFS) is proposed in [5]. e concepts of IVIFS, IVPyFS, and IVPFS are reduced to IFS, PyFS, and PFS. Keeping the importance of these concepts, Ullah [11] proposed the framework of IVTSFS. Ullah [11] also described the importance of describing membership degrees. All these concepts have been applied to various practical problems, such as [12,13] which described MADM problems using some normal interactive aggregation operators and VIKOR method based on IFSs, [14] which is based on multiple parameters based two-person zero sum game with IF information, and [15][16][17][18][19][20][21][22] which are dealing with MADM using different approaches. e concept of fuzzy graph (FG) is developed by [23] as a generalized crisp graph and comprehensively examined by Rosenfeld [24]. e idea of FG is indeed very helpful in various problems including classification, clustering, anomaly/malware detection, shortest path problems, networking problems, slicing, and in the description of group structures, etc. Due to the uncertainties lying in these problems, FG was further generalized to the concept of an intuitionistic fuzzy graph (IFG) where nodes and edges are assumed to be intuitionistic fuzzy numbers (IFNs) in [25]. e graphical aspects of PyFS are discussed in [25], the notion of PFG is developed in [26], and its operations are studied. Reference [27] developed the theory of IVFG and [28] is based on IVIFG. e TSFG's notion is also proposed in [29] which provided a more flexible ground for dealing with imprecise information. Some quality work on these graphs can be found in [30][31][32][33][34].
Malware/Anomaly detection is a critical cyber world problem, and it is crucial with malware's prevalence because it functions as threat intelligence and detection system against anomaly based cyber-attacks. In anomaly detection, we establish a process using aggregation operators regarding the classification of a given sample of Windows based Portable Executable (PE) file. e aggregation operators are applied to the decision matrix containing the attributes of the given sample to calculate the score values. e score value of each sample file is used by the problem solver to determine the nature of the portable executable file, i.e., either anomaly (malicious) or benign (nonmalicious). Some novel research work in the subject domain has been so far addressed in [35][36][37][38][39][40]. Medical diagnosis is also another important practical problem that has been widely discussed in the context of FSs and its algebraic extensions. In medical diagnosis, we establish some relations between patients, symptoms and symptoms, diagnosis. en by the composition of the previous two relations, we determine the diagnosis for patients. is type of phenomenon has been so far discussed in [6,[41][42][43]. Furthermore, the SP problem is also one of the interesting problems that have much impotence in computer sciences and fuzzy algebra. In this phenomenon, we find the SP in a network of nodes and edges from the source node to the destination node. Several algorithms have been developed for finding SP problems. Some notable work in this regard could be found in [44][45][46][47][48]. e problem of bounding the membership's aggregate, abstinence, and NMGs between one and zero is also  experienced by Cuong's PFS. e problem also revokes the privilege from the problem solvers to opt three aforementioned characteristic functions. Considering the problem, Mahmood [49] introduced a novel framework of the SFS.
ere is a dire need to develop innovative models keeping in view several characteristic functions with no shortcomings [53,254]. e significance of representing the MGs in terms of intervals strongly motivated us for our proposed research work. e research work proposed the IVTSFS with WA and WG aggregation operators and demonstration of its effective application for MADM. e notable benefactions of this manuscript are as follows: (1) To propose the novel concept of the IVTSFS that encompasses an event covering MGs in terms of intervals without any constraint (2) To propose novel aggregation tools and its effective application in those specialized domains which remain a problem for the application of IVFS, IVIFS, and IVPFS (3) To explore the problem of MADM and utilize the aggregation operators of IVTSFSs for decision making in real world scenarios e extensive model evaluation followed by indepth comparative assessment has demonstrated that the proposed aggregation operators with appropriate constraints outperform the preexisting aggregation operators. In this manuscript, we developed a new concept of interval valued T-spherical fuzzy graphs (IVTSFGs). e concepts of IVTSFSs and IVTSFGs are applied to the problem of anomaly detection, medical diagnosis, and SP problem. Also, it is proved that none of the existing tools could handle the data provided in IVTSFS and on the other hand, it is also proved that the structure of IVTSFSs and IVTSFG process the problem handsomely in lieu of FS or FG, IFS or IFG,  IVIFS or IVIFG, IVPFS, SFS, and TSFS. is manuscript is arranged as follows: Section 1 described a brief history of existing work and motivation for new work, followed by a review of fundamental definitions in Section 2. Sections 3 and 4 covered the concept of IVTSFSs and its significance, respectively. Section 5 discussed the notion of IVTSFG and its some graphical notions. Section 6 discussed three applications of IVTSFSs and IVTSFGs including anomaly detection, medical diagnosis, and shortest path based on interval valued T-spherical fuzzy relations (IVTSFRs). Section 7 narrates the concluding remarks followed by the future direction.

Basic Concepts
In our study, we denote a universal set by X. Further, by ş, ı, d, and R we denote membership, abstinence, nonmembership, and refusal functions maps on [0, 1] interval.
Definition 5 (see [9]). An IVIFS is expressed as Definition 6 (see [5]). An IVPFS is expressed as are known as refusal degree. Further, the triplet Figure 3 corresponds to FG, while Figure 4 to its complement.
Definition 7 (see [24]). An IFG is defined by G � (V, E) V and E are the collections nodes and edges, respectively.
(1) For all v ∈ V is categorized by ş and d denoting the Every e ∈ E is attributed s and Ð denoting the MG and the NMG defined as follows: Example 2. Figure 5 represents IFG.

Interval-Valued T-Spherical Fuzzy Sets
e recently introduced concept of IVTSFSs with properties is discussed and the notion of IVTSF relation is also presented with results and discussion.

e union and intersection are commutative (3) e union and intersection are associative (4) Distributive laws of the union over intersection and intersection over union hold true (5) DE Morgan's laws hold true
IVTSFSs. e proof of the results is as follows: (1) Transitivity: let us assume A ≤ B and B ≤ C. To prove (2) Commutativity of union: for IVTSFSs A and B,

Commutativity of intersection is similar. (3) Distributive law of union over intersection: to prove
Distributive law of intersection over union is similar to prove.
Mathematical Problems in Engineering In the following, the concept of Cartesian product and relations for IVTSFSs are proposed, and their properties are studied. ese operations are the generalization of operations of IVIFSs and IVPFSs.

Definition 11.
e Cartesian product of two IVTSFSs A and B over two universes X 1 and X 2 is of the following form: Definition 12 An IVTSFR R is an IVTSF fuzzy subset of X × Y and is of the following form: Provided that 0 ≤ ş n (8, y) + ı n (8, y) + d n (8, y) ≤ 1 and n ∈ Z + . Definition 13. For an IVTSFR R, R − 1 is defined as follows:

Proposition 1.
e following results for three IVTSFRs R, R 1 , and R 2 are Boolean positive.
Definition 14. For two IVTSFRs R 1 on X × Y and R 2 on Y × Z, their composition is denoted by R 2 ∘ R 1 and defined as follows:

Definition 15. e score function for an IVTSFN
is defined as follows: Remark 1. Replacing ı L � ı U � 0 and n � 1 reduces the defined function for score calculation.

Importance of Structures: Interval-Valued Fuzzy
e worthy concept of modeling a MG of a FS using an interval was the inception of Gorzalczany [2], whereas the concepts of IFS, PyFS, Q-ROPFS, and PFS are established in [5,6,15,50]. e IVFS's concept has shown great significance, especially in those situations where an occurrence is difficult to be attributed by a crisp number and therefore, the significance of interval valued frameworks is elaborated in more detail. e main decision in the following instance is regarding the selection of the best candidate among five candidates (g , i , i � 1, 2, 3, 4), with six attributes (g , j , j � 1, 2, 3, 4) and weight vector w � (0.22, 0.34, 0.27, 0.17) T . In this case, Table 1 illustrates the evaluation data. e proposed aggregation tool by Xu and Cai [29] for the data provided in Table 1 is as follows: e aggregated results are as follows: is defined as follows [31]: Hence, according to the MADM method, ģ 2 is considered to be the most appropriate selection candidate. e assignment of a crisp value to membership and NMG causes the information loss in Table 1 and can be aggregated using the aggregation operators Xu [22]. e decision matrix is illustrated in Table 2.
e weighted averaging aggregation operator of IFSs, proposed by Xu [22], is given by the following: For ranking IFN I, the score function developed in [22] is given by SC0(I) � Z − η.
Using this score functions, we have the following: Clearly, SC0(g , us, according to the MADM approach, C 4 is the appropriate candidate with IVIFWA operators. e analysis of the results revealed the significance of the aggregation of IVIFS in contrast to the one obtained using IFS. e concept of TSFS [49] is a generalization of Q-ROPFS, SFS, PyFS, PFS, IFS, as well as FS, and therefore, the interval-valued structure for TSFSs with its diverse structure would have an outstanding impact. e concept of IVTSFS [6,7,49] gives further convincing strength to the notion of TSFSs due to its usefulness and flexibility towards the assignment of MGs of TSFS in terms of intervals.

Interval-Valued T-Spherical Fuzzy Graphs
e IVTSFS is a generalization of FS, IFSs, PFSs, IVIFSs, and IVPFSs, and therefore, it is of great importance. e notion of graph for IVTSFS, i.e., IVTSFG is proposed as a generalization of FG, IFG, PFG, IVIFG, and IVPFG along with fundamental terms and definitions with applications in Section 6.

Definition 16. An IVTSFG is � (V, E), V is node set and E edges' collection and
(1) For all v ∈ V is attributed with three functions ş, ı and d degree of membership, abstinence, and (2) For all e ∈ E is attributed with three functions S,Î and Ð degree of membership, abstinence, and .
denotes the refusal degree of e ∈ E anď Theorem 3. IVTSFG is a generalization of IVPFG, IVIFG, and IVFG.
(1) If we take n � 1, the IVTSFG diminishes to IVPFG (2) If we take n � 1 and e outcome reveals the significance of the novel concept due to its generalization. Example 3. e following Figure 6 is an IVTSFG with a set of nodes v i (i � 1, 2, 3, . . . , 6) and a set of edges e i i (i � 1, 2, 3, 4 and 5)).
e IVTSFSG of IVTSFG provided in Figure 6 is depicted in Figure 7 Table 1: Decision matrix: evaluation data.
Mathematical Problems in Engineering

Theorem 4. For IVTSFG
e proof is as follows: □ Example 5. Figure 8 represents the IVTSFGs graph, whereas Figure 9 represents its complement.
Definition 19. An IVTSFG is known as follows:  1, 2, , 3, . . . , m) is regarded as a path provided an edge exists between every two vertices v i and v j for i, j � 1, 2, , 3, . . . , m. e nonidentical nodes v i (i � 1, 2, , 3, . . . , m) are equivalent to a path provided at least one of the following conditions holds true.
Consequences of Definition 20: (1) e length of a path is m− 1 if it has m vertices (2) If the first and last vertices of a path coincide, then it will be a cycle (3) If two vertices are joined by a path, then they will be regarded as connected Example 7. In Figure 11,

Applications
Several applications of IVTSFSs and IVTSFGs in anomaly detection, medical diagnosis, and SP problem are discussed in this section. e comparative assessment of the results is also presented.

Anomaly Detection.
e Anomaly/Malware based threat has become an irresistible concern for businesses and organizations. Malware is an abbreviated form of "malicious software." It is a file or a set of instructions intended to bring controlled or catastrophic damage to organizations, facilities, processes, cloud infrastructure, industrial processes, and digital systems. In 2018, the number of users attacked with banking Trojans was 889,452, an increase of 15.9% in comparison with 767,072 in 2017, whereas in 2019, more than 100 million different hosts were attacked in H1 2019 as reported by Kaspersky Labs. Moreover, as per the statistics collected during research conducted by Juniper revealed that the cost of data breaches would rise from $3 trillion each year to over $5 trillion in 2024. erefore, anomaly detection is one of the most critical elements for cyber security. As IVTSFS is a generalization of TSFSs and IVIFSs, therefore we used the approach of IVTSFSs using aggregation operators to handle anomaly detection problem that is proved to be of great significance. An anomaly is considered to be a file with malicious content or it can be Portable Executables (PE) file. In order to make a decision regarding anomaly detection problems, i.e., information is obtained about a given PE file and associated attributes/features are represented in an appropriate format for subsequent analysis. After the representation, aggregation operators are applied by the problem solvers. Subsequently, the score value is computed using the score function for IVTSFSs to make a decision regarding the classification of the anomaly, i.e., PE sample either as malicious or benign. e algorithm for malware detection is shown in Algorithm 1, and the algorithm for medical diagnosis is shown in Algorithm 2.
Consider a sample software file S being investigated by anomaly analyst under k attributes as A k . e problem solvers investigate the a sample file under k attributes and represent the findings in the form of IVTSFNs. Further, w � (w 1 , w 2 , w 3 , . . . , w k ) T corresponds to the weight vector of attributes. e classification process is given as follows: e proposed algorithm is illustrated in Figure 12.
Example 8. Consider a multinational cyber security consultancy firm who has to make a decision about the classification of a given zero-day anomaly. A zero-day malware/ anomaly is an anomaly/malware one that is either unknown or not yet addressed by anyone. ere is a team of domain experts in the research lab of that consultancy firm to investigate the problem and make a decision regarding the classification, i.e., malicious or benign (not malicious) of a given sample. Initially, assume there are four samples to be examined. S 1 : Windows-based PE Sample # 1, S 2 : Windowsbased PE Sample # 2, S 3 : Windows-based PE Sample # 3, and S 4 : Windows-based PE Sample # 4. In order to correctly analyze the samples, their attributes/features have to be extracted. So for this purpose, there are four features to be extracted and analyzed: A 1 : entropy of the PE sample, A 2 : resource size of the PE sample, A 3 : virtual Size of the PE sample, and A 4 : section mean entropy of the PE sample. All the four attributes are most significant ones for anomaly detection. Let w � (0.23, 0.35, 0.26, 0.16) T be the weight vector and represent their opinions in IVTSFNs format. e stepwise demonstration of the process is as follows: Step 1: in the first step, the analyst examines the given samples S i and extracts the four attributes/features including entropy, resource size, virtual size, and section mean entropy. After the extraction of the aforementioned four attributes (A 1 , A 2 , A 3 , and A 4 ) from given PE samples (S 1 , S 2 , S 3, and S 4 ), a decision matrix D 4 × 4 is constituted. e representation of attributes (A 1 , A 2 , A 3 , and A 4 ) of PE samples (S 1 , S 2 , S 3, and S 4 ) in matrix D 4 × 4 is given as in Table 3: e assessment analysis of individual score against the threshold shows that Score ofS 3 is greater than given criteria, i.e., 0.5; therefore, only S 3 among all samples is declared as malicious, i.e., anomaly. Moreover, as the score of S 1, S 2 , and S 4 is below the defined threshold criteria, i.e., 0.5; therefore, they will be classified as benign in nature. Hence, it is shown that how can a critical and complex problem; i.e., Anomaly Detection can be solved using the proposed intelligent and robust decision making approach. e classification decision of the given sample (s) either as malicious or benign is carried out by using the IVTSFWA operators. Only for q � 5 values are IVTSFNs.
Step 2: the IVTSFWA operators to the matrix gives the following output:      (1) Analyst will perform the extraction of features/attributes (A) from the given PE sample (S) and constitute a decision Matrix.
(2) Put on the following aggregation tools to the attribute matrix obtained in Step 1.  e resolution of comparative assessment reiterates that none of the existing structures including FSs, IVFSs, IFSs, IVIFSs, PyFSs, IVPyFSs, q-OFS, IVq-OFS, PFSs, and IVPFSs processed the data except the proposed approach and its effectiveness is demonstrated by solving the similar problem using the existing IVq-OFNs method in Table 4.
In this section, previously discussed four similar Windows based PE samples are used. e decision is required regarding their anomaly classification, i.e., either as malicious or benign effectively addressed using the aggregation operators of IVq-OFNs by taking q � 2 and i l � i u � i � 0. e IVTSFWA operator for q � 2 and i l � i u � i � 0 becomes as follows: e aggregation results are as follows:

(26)
Using the score function, we get the following: e analysis of each given sample is summarized as follows: e assessment analysis of individual scores shows that Score ofS 1 and S 3 is greater than given criteria, i.e., 0.5; therefore only S 1 and S 3 samples are declared as Malicious file, i.e., anomaly. Moreover, as the Score ofS 2 and S 4 is below the defined threshold, i.e., 0.5; therefore, they will be classified as benign.
Comparing the respective score of all samples in both sections reveals an instance of wrong judgement about sample S 1 . e classification of anomaly sample S 1 by using the IVq-OFNs method is Malicious which was previously classified as benign. e sample S 1 is actually benign. is occurrence of wrong judgement about sample S 1 is also known as False Positive; i.e., a benign PE file is classified as malicious. erefore, the proposed claim using IVTSFWA operators is a worthy concept.

Medical Diagnosis.
In this subsection, the process of medical diagnosis is carried out in the context of IVTSFSs. e medical diagnosis process based on TSFSs is established in [6] and proved to be of great importance as IFSs and PFSs failed to process such type of information. As IVTSFS is a generalization of TSFSs and IVIFSs so we used the approach of IVTSFRs to handle medical diagnosis problem.
In a medical diagnosis problem, some information is obtained about patients and symptoms and symptoms and diagnosis in terms of IVTSFRs showing the relation between them. en, with the help of the composition of IVTSFRs, a relation between patient and diagnosis is obtained which is further analyzed using the score function for IVTSFSs. e proposed medical diagnosis problem is demonstrated through the following algorithm where α, β and c represents a set of patients, symptoms, and diagnosis, respectively. e proposed algorithm is depicted in Figure 13.

Comparative Analysis.
In this subsection, our aim is to show that if we consider the structures of FSs, IVFSs, IFSs, IVIFSs, PyFSs, IVPyFSs, PFSs, and IVPFS, they all will turn out be a fail case. Conversely, if we have the data in the existing structures, we are able to deal with it using the operations of IVTSFSs. For example, if we convert the data available in Tables 5 and 6 in the form of TSFNs by choosing MGs as a single number from [0, 1] interval instead of closed subinterval of [0, 1], then we will obtain the following relations shown in Table 9: Now, the data provided in Tables 9 and 10 are purely in the form of TSFNs as described in [6]. is type of data can be shifted into equivalent IVTSFRs of closed subintervals without disturbing the grades. e information in Tables 9 and 10 is similar to that in Tables 11 and 12. e process of medical diagnosis proposed in an intuitionistic and picture fuzzy environment [43,51] can also be carried out using the proposed approach. Hence, we can use the tools of IVTSFSs to solve problems of TSFSs and consequently IVPFSs, PFSs, IVIFSs, IFSs, IVFSs, and FSs.

Shortest Path (SP) Problem.
It is one of the important applications of graph theory and got great attention in past decades. A SP from a source to a destination node is required in most of the fields of engineering and other sciences. As briefly explained in the first section of this manuscript, the problem is highly valued in different fuzzy algebraic structures and several new approaches are developed to resolve it. We are going to follow the famous DA for finding an appropriate path problem in the context of IVTSFG. A directed network of finite nodes and edges is considered to find the SP form a source node (SN) to destination node (DN) based on alternative nodes and edges, respectively. e edges denoted the path from one vertex to the adjacent vertex. In real-life, the shortest paths mean a path having low cost to travel on or a path that required less time to reach a destination or a path having the least distance between SN and DN and the information about these parameters is provided in the context of IVTSFNs. e detailed algorithm for finding SP using DA is further described.

Dijkstra Algorithm (DA) for Finding SP in Interval-
Valued T-Spherical Fuzzy Environment. DA is a baseline algorithm for the subject problem and for some quality work on fuzzy SP using DA, one may refer to [44][45][46][47][48]. Here, we  discussed the DA for the information provided as IVTSFNs.
(2) For every node j whose path is from node i, compute the label e flowchart is portrayed in Figure 14.

Remarks 2.
e label [v i ⊕d ij , i] means that we reached from node i and covered a distance v i ⊕d ij . Furthermore, it is  important to note that we cannot proceed to a permeant node but can go reverse. For two directly connected adjacent nodes i and j respectively, node i is considered as the predecessor of node j if the path connecting them is directed from i to j.
Example 10. Consider Figure 15 where a network of 6 nodes with 8 edges is portrayed. Our aim is to apply modified DA to this network and find the low cost path from SN (N 1 ) to DN (N 2 ). e edges involved in this network are listed in Table 13.
e step-by-step computations of modified DA is as follows: Step 1. Identify node 1 as permanent due to its lowest cost.
e ways directed from node 1 are also possible; i.e., we may either move to node 2 or to node 3. List of nodes is in Table 14.   Table 9: T-spherical fuzzy relation showing the relation between patients and symptoms.    Tables 5 and 6 showing the diagnosis for a patient.
As the score of ( Step 3. Also, dual routes directed from node 3, i.e., we may either move to node 4 or to node 5. List of nodes is in Table 15 Step 4. One way from node 5 exists only, i.e., we can only move to node 6 depicted in Table 16.
As there is only one way from nodes 5 to 6. erefore, we mark node 6 as   Step 5. Nodes 2 and 4 are the remaining temporary nodes, so their status is changed to permanent and depicted in Table 17.
Step 6. From Table 18, the lowest cost path pattern is from SN to DN, i.e., from nodes 1 to 6.
e SP computed as per modified DA is as follows: 6.3.2. Comparative Study. Consider a network based IVIF environment portrayed in Figure 16. All the path values are         in the form of IVIFNs that can be converted into IVTSFNs if we insert the abstinence grade [0.0, 0.0] in it and therefore by using the proposed approach of modified DA, we can find the SP. Now, consider a network depicted in Figure 17, where all the path values are in the form of IVFNs. As an IVFN can be regarded as IVTSFN, if we insert [0.0, 0.0] in place of abstinence and nonmembership grade. erefore, using the proposed approach of modified DA, we can solve a low cost path problem.

Conclusion
In this manuscript, a new concept of IVTSFSs which generalizes all the existing concepts like FSs, IVFSs, IFSs, IVIFSs, PyFSs, IVPyFS, PFSs, IVPFSs, SFSs, and TSFSs, is proposed. Such a concept is necessary when we have four types of opinion and information is not exact; i.e., information is provided in the form of intervals. Similarly, we developed the concept of IVTSFG which generalizes all the existing definitions of graph theory so far existed. e basic operations and related terms of IVTSFSs and IVTSFGs are defined with their properties. We utilized the aggregation operators of IVTSFSs in anomaly detection problems which is a challenging problem from computer science. We defined relations for IVTSFSs along with their compositions and applied them to solve anomaly detection and medical diagnosis problem. We also proposed a modified DA to solve a low cost path problem in a network based on IVTSFGs. e comparative evaluation demonstrated that the proposed structure is novel and addressed the shortcomings of the existing ideas. In the future, we shall try to develop some aggregation operators for IVTSFSs to deal with a scenario where the other tools fail to be applied.

Data Availability
All data, models, or codes generated or used during the study are available in a repository or online in accordance with funder data retention policies (full citations that include URLs or DOIs are provided).