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

Jin, Huanhuan; Jah Rizvi, Syed Khurram; Mahmood, Tahir; Jan, Naeem; Ullah, Kifayat; Saleem, Shahzad

doi:https://doi.org/10.1155/2020/9656909

Mathematical Problems in Engineering

On this page

Abstract Introduction Applications Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Applied Mathematics for Engineering Problems in Biomechanics and Robotics 2020

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 9656909 | https://doi.org/10.1155/2020/9656909

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

Huanhuan Jin,¹Syed Khurram Jah Rizvi,²Tahir Mahmood,³Naeem Jan,³Kifayat Ullah,⁴and Shahzad Saleem²

Academic Editor: Vassilios Constantoudis

Received29 Mar 2020

Revised09 Sept 2020

Accepted23 Oct 2020

Published24 Nov 2020

Abstract

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 novel framework for decision assembly using interval valued TSFS based 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), a modified Dijkstra Algorithm (DA) is developed and applied to the shortest path problem. The in-depth quantitative assessment and comparative analysis revealed that the proposed notion outpaces contemporary progressive approaches.

1. 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 and on the unit interval in a way that . The condition on IFS makes it less effective in some cases because it does not allow us to choose the values of and independently. This issue was addressed by the framework of the Pythagorean fuzzy set (PyFS) developed with a relatively stronger condition, i.e., . The 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 such that . The 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 . 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 , and denoting membership, nonmembership, abstinence, and refusal degree such that and . Cuong’s idea of PFS generalizes the theory of IFS but constrained by the condition due to which grades could not be assigned independently. Therefore, a novel idea of TSFS and SFS generalizing IFS, PyFS, Q-ROPFS, and PFS with a condition on membership grade (MG) and such that and was proposed by Mahmood [6]. The 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 of the unit interval. An IVFS easily reduces to FS if we take . Following this phenomenon, the same type of structures has been proposed for IFSs, PyFSs, and PFSs. The 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].

The 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 [15–22] which are dealing with MADM using different approaches.

The concept of fuzzy graph (FG) is developed by [23] as a generalized crisp graph and comprehensively examined by Rosenfeld [24]. The 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]. The 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. The 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 [30–34].

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. The aggregation operators are applied to the decision matrix containing the attributes of the given sample to calculate the score values. The 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 [35–40]. 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. Then by the composition of the previous two relations, we determine the diagnosis for patients. This type of phenomenon has been so far discussed in [6, 41–43]. 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 [44–48].

The problem of bounding the membership’s aggregate, abstinence, and NMGs between one and zero is also experienced by Cuong’s PFS. The 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. There is a dire need to develop innovative models keeping in view several characteristic functions with no shortcomings [53, 254]. The significance of representing the MGs in terms of intervals strongly motivated us for our proposed research work. The research work proposed the IVTSFS with WA and WG aggregation operators and demonstration of its effective application for MADM. The 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

The 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). The 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.

This 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.

2. Basic Concepts

In our study, we denote a universal set by . Further, by , and we denote membership, abstinence, nonmembership, and refusal functions maps on interval.

Definition 1. (see [2]). An IFS is expressed as such that and are known as hesitancy index.

Definition 2. (see [4]). A PFS is expressed as such that and are termed as refusal degree.

Definition 3. (see [6]). A SFS is expressed as such that and are termed as refusal degree.

Definition 4. (see [6]). A TSFS is expressed as such that for and are termed as refusal degree.
The triplet is a picture fuzzy and spherical fuzzy and T-spherical fuzzy numbers, respectively, for Definitions 2–4.

Definition 5. (see [9]). An IVIFS is expressed as such that and provided that and are known as hesitancy index. Further, the duplet is an IVIFN.

Definition 6. (see [5]). An IVPFS is expressed as such that and provided that and are known as refusal degree. Further, the triplet is an IVPFN.

Example 1. Figure 3 corresponds to FG, while Figure 4 to its complement.

Definition 7. (see [24]). An IFG is defined by and are the collections nodes and edges, respectively.(1)For all is categorized by and denoting the MG and NMG of s.t . The term denotes the hesitancy level of such that .(2)Every is attributed and denoting the MG and the NMG defined as follows:s.t. . denotes the hesitancy degree of such that .

Example 2. Figure 5 represents IFG.

3. Interval-Valued T-Spherical Fuzzy Sets

The recently introduced concept of IVTSFSs with properties is discussed and the notion of IVTSF relation is also presented with results and discussion.

Definition 8. (see [11]). An IVTSFS A where and are from to subinterval of , i.e., and with a condition and . Moreover, the interval represents the level of refusal , i.e., and .
The following theorem proves the generalization of IVTSFSs over TSFSs, IVPFSs, PFSs, IVIFSs. IFSs, IVFSs, and FSs.

Theorem 1. In Definition 8 of IVTSFSs, if we(1)Taking and , then it reduces to TSFSs.(2)Taking , then it reduces to the definition of IVSFS.(3)Taking , , and , then it reduces to the definition of SFS.(4)Taking , then it reduces to the definition of IVPFS.(5)Taking , and , then it reduces to the definition of PFS.(6)Taking and , then it reduces to the definition of IVPyFS.(7)Taking , and , then it reduces to the definition of PyFS.(8)Taking and , then it reduces to the definition of IVIFS.(9)Taking , and , then it reduces to the definition of IFS.(10)Taking and , then it reduces to the definition of IVFS.(11)Taking and , then it reduces to the definition of FS.

Definition 9. For IVITSFSs and :(1) if and , where(2) iff and .(3), where(4) where(5).(6).The following theorem shows that IVTSFSs preserve the basic set theoretic properties.

Theorem 2. For three IVTSFSs and :(1)The transitive law w.r.t. inclusion holds true(2)The union and intersection are commutative(3)The 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

Proof. Let , and be three IVTSFSs. The proof of the results is as follows:(1)Transitivity: let us assume and . To prove . As so and . Also, as so and . Now, and . Also, and . Similarly, and . Hence, .(2)Commutativity of union: for IVTSFSs and , Commutativity of intersection is similar.(3)Distributive law of union over intersection: to prove . Distributive law of intersection over union is similar to prove.(4)De Morgan’s Laws: to prove . Similarly, can be proved.

Definition 10. For IVITSFSs and , we defineIn the following, the concept of Cartesian product and relations for IVTSFSs are proposed, and their properties are studied. These operations are the generalization of operations of IVIFSs and IVPFSs.

Definition 11. The Cartesian product of two IVTSFSs and over two universes and is of the following form:

Definition 12. An IVTSFR is an IVTSF fuzzy subset of and is of the following form:Provided that and .

Definition 13. For an IVTSFR , is defined as follows:where .

Proposition 1. The following results for three IVTSFRs , and are Boolean positive.(1)(2)If , then (3) and (4) and (5)If and , then (6)If and , then

Definition 14. For two IVTSFRs on and on , their composition is denoted by and defined as follows:where ,

Definition 15. The score function for an IVTSFN is defined as follows:

Remark 1. Replacing and reduces the defined function for score calculation.

4. Importance of Structures: Interval-Valued Fuzzy

The 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]. The 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.

The main decision in the following instance is regarding the selection of the best candidate among five candidates (), with six attributes () and weight vector . In this case, Table 1 illustrates the evaluation data.

The proposed aggregation tool by Xu and Cai [29] for the data provided in Table 1 is as follows:

The aggregated results are as follows:

The score of IVFN is defined as follows [31]:

Clearly,

Hence, according to the MADM method, is considered to be the most appropriate selection candidate. The 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]. The decision matrix is illustrated in Table 2.

The weighted averaging aggregation operator of IFSs, proposed by Xu [22], is given by the following:

For ranking IFN , the score function developed in [22] is given by

Using this score functions, we have the following:

Clearly,

Thus, according to the MADM approach, is the appropriate candidate with IVIFWA operators. The analysis of the results revealed the significance of the aggregation of IVIFS in contrast to the one obtained using IFS. The 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. The 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.

5. Interval-Valued T-Spherical Fuzzy Graphs

The IVTSFS is a generalization of FS, IFSs, PFSs, IVIFSs, and IVPFSs, and therefore, it is of great importance. The 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 , is node set and edges’ collection and(1)For all is attributed with three functions and degree of membership, abstinence, and nonmembership of . Basically, and are subintervals of the unit interval [0, 1] with a condition that for . The R denotes the refusal degree of and and .(2)For all is attributed with three functions and degree of membership, abstinence, and nonmembership of . Basically and are as follows: and s.t . and s.t . and s.t . With a condition for , the term denotes the refusal degree of and and .

Theorem 3. IVTSFG is a generalization of IVPFG, IVIFG, and IVFG.(1)If we take , the IVTSFG diminishes to IVPFG(2)If we take and , the IVTSFG diminishes to IVIFG(3)If we take , and , the IVTSFG diminishes to IVFGThe outcome reveals the significance of the novel concept due to its generalization.

Example 3. The following Figure 6 is an IVTSFG with a set of nodes and a set of edges ).

Definition 17. An IVTSFG is IVTSFSG of graph iff and

Example 4. The IVTSFSG of IVTSFG provided in Figure 6 is depicted in Figure 7

Definition 18. The complement, i.e., IVTSFG is represented by where and the MGs of are defined by the following:

Theorem 4. For IVTSFG , .

Proof. Let be an IVTSFG. The proof is as follows:. Similarly, we can prove that

Example 5. Figure 8 represents the IVTSFGs graph, whereas Figure 9 represents its complement.

Definition 19. An IVTSFG is known as follows:(1)Semi S strong: if and (2)Semi I strong: if and (3)Semi D strong: if and (4)Strong: if (1), (2), and (3) hold true.

Example 6. Figure 10 represents IVTSFG.

Definition 20. In an IVTSFG, a set of nonidentical nodes is regarded as a path provided an edge exists between every two vertices and for . The nonidentical nodes are equivalent to a path provided at least one of the following conditions holds true.(1) is a nonzero subinterval of (2) is a nonzero subinterval of (3) is a nonzero subinterval of Consequences of Definition 20:(1)The length of a path is m−1 if it has 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,(1) is a path(2) is a cycle

6. Applications

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

6.1. Anomaly Detection

The 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. Therefore, 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. The algorithm for malware detection is shown in Algorithm 1, and the algorithm for medical diagnosis is shown in Algorithm 2.

(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.
	,
	where I1, I2, I3 …Im are interval-valued T-spherical fuzzy numbers.
(3)	Calculate the rank/score of IVTSFNs acquired in the previous step.
(4)	Analyze the score of the given PE sample, i.e., SC (S) takes the classification decision based upon the following threshold based criteria:

(1)	Formation of relation
(2)	Formation of relation
(3)	Computation of as a composition of and
(4)	Investigation of diagnosis for patients using the score function

Consider a sample software file being investigated by anomaly analyst under attributes as . The problem solvers investigate the sample file under attributes and represent the findings in the form of IVTSFNs. Further, corresponds to the weight vector of attributes. The classification process is given as follows:

The 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. There 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. S₁: Windows-based PE Sample # 1, S₂: Windows-based PE Sample # 2, S₃: Windows-based PE Sample # 3, and S₄: 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: A₁: entropy of the PE sample, A₂: resource size of the PE sample, A₃: virtual Size of the PE sample, and A₄: section mean entropy of the PE sample. All the four attributes are most significant ones for anomaly detection. Let be the weight vector and represent their opinions in IVTSFNs format. The 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 (A₁, A₂, A₃, and A₄) from given PE samples (S₁, S₂, S_3, and S₄), a decision matrix D_4 × 4 is constituted. The representation of attributes (A₁, A₂, A₃, and A₄) of PE samples (S₁, S₂, S_3, and S₄) in matrix D_4 × 4 is given as in Table 3:The assessment analysis of individual score against the threshold shows that is greater than given criteria, i.e., 0.5; therefore, only among all samples is declared as malicious, i.e., anomaly. Moreover, as the 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.

The 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: Step 3: score values are computed as follows:

6.1.1. Comparative Analysis

The 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. The 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 and . The IVTSFWA operator for and becomes as follows:

The aggregation results are as follows:

Using the score function, we get the following:

The analysis of each given sample is summarized as follows:

The assessment analysis of individual scores shows that is greater than given criteria, i.e., 0.5; therefore only samples are declared as Malicious file, i.e., anomaly. Moreover, as the 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 . The classification of anomaly sample by using the IVq-OFNs method is Malicious which was previously classified as benign. The sample is actually benign. This occurrence of wrong judgement about sample is also known as False Positive; i.e., a benign PE file is classified as malicious. Therefore, the proposed claim using IVTSFWA operators is a worthy concept.

6.2. Medical Diagnosis

In this subsection, the process of medical diagnosis is carried out in the context of IVTSFSs. The 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. Then, 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. The proposed medical diagnosis problem is demonstrated through the following algorithm where and represents a set of patients, symptoms, and diagnosis, respectively.

The proposed algorithm is depicted in Figure 13.

Example 9. The example that we are discussing here is adapted from [6], and instead of using TSFNs, we used IVTSFNs. All the data is here being in the form of IVTSFNs, where , , and denote the sets of patients, symptoms, and diagnosis, respectively. The detailed stepwise demonstration is as follows:(1)Formation of relation (see Table 5)(2)Formation of relation (see Table 6)(3)Computation of as a composition of and (4)Investigation of diagnosis for patients using the score functionThe scores of Table 8 represent that and , i.e., Amjad and Naeem have allergy problems. Zeeshan having symptoms of malaria and typhoid, while Ali is a patient of malaria too, as indicated by his symptoms.

6.2.1. 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 interval instead of closed subinterval of , 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]. This type of data can be shifted into equivalent IVTSFRs of closed subintervals without disturbing the grades.

The information in Tables 9 and 10 is similar to that in Tables 11 and 12. The 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.

6.3. 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. The 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. The detailed algorithm for finding SP using DA is further described.

6.3.1. 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 [44–48]. Here, we discussed the DA for the information provided as IVTSFNs. The IVTSF modified DA [46] (IVTSFDA) is described as follows:(1)Identify the source as permanent node (P) and assign it the label as by default this node is included in low cost path and distance covered at this stage is .(2)For every node whose path is from node , compute the label if is not a permanent node. Further, if is identified as via some other node, then replace by only if is less than .(3)The algorithm terminates if all the nodes are permanently labeled. Else, select having low cost distance and reiterate Step 2 by making .(4)Find the low cost path from SN to DN.

The flowchart is portrayed in Figure 14.

Remarks 2. The label means that we reached from node and covered a distance . Furthermore, it is important to note that we cannot proceed to a permeant node but can go reverse. For two directly connected adjacent nodes and respectively, node is considered as the predecessor of node if the path connecting them is directed from to .

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 to DN .
The edges involved in this network are listed in Table 13.
The step-by-step computations of modified DA is as follows:

Step 1. Identify node 1 as permanent due to its lowest cost.

Step 2. The 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.
Now we compute the scores of and .As the score of is less than , we mark node 3 as and labeled it permanent.

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.
Now, we compute the scores of and as follows:As the score of is less than , therefore, we mark the node 5 as
and labeled it permanent.

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. Therefore, we mark node 6 as and labeled it permanent.

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.
The 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 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 in place of abstinence and nonmembership grade. Therefore, using the proposed approach of modified DA, we can solve a low cost path problem.

7. 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. The 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. The 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).

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the manuscript.

Acknowledgments

This paper was supported by the Natural Science Foundation of Zhejiang Province (no. LQ20G010001), China Postdoctoral Science Foundation (no.2020M673195), and Ningbo Province Natural Science Foundation (no. 2019A610037).

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
View at: Publisher Site | Google Scholar
R. R. Yager, “Generalized orthopair fuzzy sets,” IEEE Transactions on Fuzzy Systems, vol. 25, no. 5, pp. 1222–1230, 2017.
View at: Publisher Site | Google Scholar
B. C. Cuong and V. Kreinovich, “Picture Fuzzy Sets-a new concept for computational intelligence problems,” in Proceedings of the 2013 Third World Congress on Information and Communication Technologies (WICT 2013), pp. 1–6, IEEE, Hanoi, Vietnam, 2013, December.
View at: Publisher Site | Google Scholar
B. C. Cuong and V. Kreinovich, “Picture fuzzy sets,” Journal of Computer Science and Cybernetics, vol. 30, no. 4, pp. 409–420, 2014.
View at: Publisher Site | Google Scholar
T. Mahmood, K. Ullah, Q. Khan, and N. Jan, “An approach towards decision making and medical diagnosis problems using the concept of spherical fuzzy sets,” Neural Computing and Applications, vol. 31, pp. 7041–7053, 2019.
View at: Publisher Site | Google Scholar
K. Ullah, T. Mahmood, and N. Jan, “Similarity measures for T-spherical fuzzy sets with applications in pattern recognition,” Symmetry, vol. 10, no. 6, pp. 1–14, 2018.
View at: Publisher Site | Google Scholar
M. B. Gorzałczany, “A method of inference in approximate reasoning based on interval-valued fuzzy sets,” Fuzzy Sets and Systems, vol. 21, no. 1, pp. 1–17, 1987.
View at: Publisher Site | Google Scholar
K. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343–349, 1989.
View at: Publisher Site | Google Scholar
X. Peng and Y. Yang, “Fundamental properties of interval-valued pythagorean fuzzy aggregation operators,” International Journal of Intelligent Systems, vol. 31, no. 5, pp. 444–487, 2016.
View at: Publisher Site | Google Scholar
K. Ullah, N. Hassan, T. Mahmood, N. Jan, and M. Hassan, “Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators,” Symmetry, vol. 11, no. 3, p. 357, 2019.
View at: Publisher Site | Google Scholar
P. Liu, “Multiple attribute decision-making methods based on normal intuitionistic fuzzy interaction aggregation operators,” Symmetry, vol. 9, no. 11, p. 261, 2017.
View at: Publisher Site | Google Scholar
J. Zhao, X.-Y. You, H.-C. Liu, and S.-M. Wu, “An extended VIKOR method using intuitionistic fuzzy sets and combination weights for supplier selection,” Symmetry, vol. 9, no. 9, p. 169, 2017.
View at: Publisher Site | Google Scholar
Z. Bashir, J. Wątróbski, T. Rashid, W. Sałabun, and J. Ali, “Intuitionistic-fuzzy goals in zero-sum multi criteria matrix games,” Symmetry, vol. 9, no. 8, p. 158, 2017.
View at: Publisher Site | Google Scholar
G. Kaur and H. Garg, “Multi-attribute decision-making based on bonferroni mean operators under cubic intuitionistic fuzzy set environment,” Entropy, vol. 20, no. 1, p. 65, 2018.
View at: Publisher Site | Google Scholar
K. Rahman, S. Abdullah, M. Shakeel, M. S. Ali Khan, and M. Ullah, “Interval-valued Pythagorean fuzzy geometric aggregation operators and their application to group decision making problem,” Cogent Mathematics, vol. 4, no. 1, p. 1338638, 2017.
View at: Publisher Site | Google Scholar
D. Joshi and S. Kumar, “Improved accuracy function for interval-valued intuitionistic fuzzy sets and its application to multi-attributes group decision making,” Cybernetics and Systems, vol. 49, no. 1, pp. 64–76, 2018.
View at: Publisher Site | Google Scholar
H. Garg, “A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making,” International Journal of Intelligent Systems, vol. 31, no. 9, pp. 886–920, 2016.
View at: Publisher Site | Google Scholar
H. Garg, “A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem,” Journal of Intelligent & Fuzzy Systems, vol. 31, no. 1, pp. 529–540, 2016.
View at: Publisher Site | Google Scholar
H. Garg, “Some picture fuzzy aggregation operators and their applications to multicriteria decision-making,” Arabian Journal for Science and Engineering, vol. 42, no. 12, pp. 5275–5290, 2017.
View at: Publisher Site | Google Scholar
G. Wei, “Picture fuzzy aggregation operators and their application to multiple attribute decision making,” Journal of Intelligent & Fuzzy Systems, vol. 33, no. 2, pp. 713–724, 2017.
View at: Publisher Site | Google Scholar
T. Mahmood, K. Ullah, M. Ullah, N. Jan, I. Deli, and Q. Khan, “Some aggregation operators for bipolar-valued hesitant fuzzy information based on einstein operational laws,” Journal of Engineering and Applied Sciences, vol. 36, no. 2, pp. 62–73, 2017.
View at: Google Scholar
A. Kaufmann, “Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs: Éléments théoriques de base,” International Journal of General Systems, vol. 2, 120 pages, 1975.
View at: Publisher Site | Google Scholar
A. Rosenfeld, Fuzzy Graphs, in Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Elsevier, Amsterdam, Netherlands, 1975.
R. Verma, J. M. Merigó, and M. Sahni, “Pythagorean fuzzy graphs: some results,” 2018, https://arxiv.org/abs/1806.06721.
View at: Google Scholar
T. Al Hawary, T. Mahmood, N. Jan, K. Ullah, and N. Hussain, “On intuitionistic fuzzy graphs and some operations on picture fuzzy graphs,” Italian Journal of Pure and Applied Mathematics, 2018.
View at: Google Scholar
M. Akram and W. A. Dudek, “Interval-valued fuzzy graphs,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 289–299, 2011.
View at: Publisher Site | Google Scholar
S. Mishra and A. Pal, “Product of interval valued intuitionistic fuzzy graph,” Annals of Pure and Applied Mathematics, vol. 5, no. 1, pp. 37–46, 2013.
View at: Google Scholar
A. Guleria and R. K. Bajaj, “T-spherical fuzzy graphs: operations and applications in various selection processes,” Arabian Journal for Science and Engineering, vol. 45, no. 3, pp. 2177–2193, 2019.
View at: Publisher Site | Google Scholar
S. Shahzadi and M. Akram, “Graphs in an intuitionistic fuzzy soft environment,” Axioms, vol. 7, no. 2, p. 20, 2018.
View at: Publisher Site | Google Scholar
J. Mordeson and S. Mathew, “t-Norm fuzzy incidence graphs,” Mathematics, vol. 6, no. 4, p. 62, 2018.
View at: Publisher Site | Google Scholar
J. N. Mordeson and P. S. Nair, “Applications of fuzzy graphs,” in Fuzzy Graphs and Fuzzy Hypergraphs, pp. 83–133, Springer, Berlin, Germany, 2000.
View at: Publisher Site | Google Scholar
M. Akram and W. A. Dudek, “Intuitionistic fuzzy hypergraphs with applications,” Information Sciences, vol. 218, pp. 182–193, 2013.
View at: Publisher Site | Google Scholar
M. Akram and W. A. Dudek, “Regular bipolar fuzzy graphs,” Neural Computing and Applications, vol. 21, no. 1, pp. 197–205, 2012.
View at: Publisher Site | Google Scholar
Y. Li, S. C. Sundaramurthy, A. G. Bardas et al., “Experimental study of fuzzy hashing in malware clustering analysis,” in Proceedings of the 8th Workshop on Cyber Security Experimentation and Test (CSET 15), USENIX Association, Washington, DC, USA, August 2015.
View at: Google Scholar
M. L. Bernardi, M. Cimitile, F. Martinelli, and F. Mercaldo, “A fuzzy-based process mining approach for dynamic malware detection,” in Proceedings of the 2017 IEEE International Conference on Fuzzy Systems (FUZZ- IEEE), pp. 1–8, Naples, Italy, July 2017.
View at: Publisher Site | Google Scholar
P. Ganeshkumar and N. Pandeeswari, “Adaptive neuro-fuzzy-based anomaly detection system in cloud,” International Journal of Fuzzy Systems, vol. 18, no. 3, pp. 367–378, 2016.
View at: Publisher Site | Google Scholar
H. Ge, D. Yue, X. Xie, and S. Hu, “Security control of networked T-S fuzzy system under intermittent DoS jamming attack with event-based predictor,” International Journal of Fuzzy Systems, vol. 21, no. 3, pp. 700–714, 2019.
View at: Publisher Site | Google Scholar
F. Li, D. Zheng, T. Zhao, and W. Pedrycz, “A novel approach for anomaly detection in data streams: fuzzy-statistical detection mode,” Journal of Intelligent & Fuzzy Systems, vol. 30, no. 5, pp. 2611–2622, 2016.
View at: Publisher Site | Google Scholar
C. Zhu, L. Jin, R. Mesiar, R. R. Yager, D. Paternain, and H. Bustince, “Nested formulation paradigms for induced ordered weighted averaging aggregation for decision-making and evaluation,” International Journal of Intelligent Systems, vol. 34, no. 11, pp. 3046–3057, 2019.
View at: Publisher Site | Google Scholar
J. F.-F. Yao and J.-S. Yao, “Fuzzy decision making for medical diagnosis based on fuzzy number and compositional rule of inference,” Fuzzy Sets and Systems, vol. 120, no. 2, pp. 351–366, 2001.
View at: Publisher Site | Google Scholar
K.-C. Hung and H.-W. Tuan, “Medical diagnosis based on intuitionistic fuzzy sets revisited,” Journal of Interdisciplinary Mathematics, vol. 16, no. 6, pp. 385–395, 2013.
View at: Publisher Site | Google Scholar
S. K. De, R. Biswas, and A. R. Roy, “An application of intuitionistic fuzzy sets in medical diagnosis,” Fuzzy Sets and Systems, vol. 117, no. 2, pp. 209–213, 2001.
View at: Publisher Site | Google Scholar
S. Okada and T. Soper, “A shortest path problem on a network with fuzzy arc lengths,” Fuzzy Sets and Systems, vol. 109, no. 1, pp. 129–140, 2000.
View at: Publisher Site | Google Scholar
Y. Deng, Y. Chen, Y. Zhang, and S. Mahadevan, “Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment,” Applied Soft Computing, vol. 12, no. 3, pp. 1231–1237, 2012.
View at: Publisher Site | Google Scholar
N. Bhatia and A. G. Kumar, “Algorithms for some fuzzy network problems using ranking function,” TIET–Thapar Institute of Engineering and Technology, Patiala, India, 2009, Ph. D thesis.
View at: Google Scholar
M. ., G. Karunambigai, P. Rangasamy, K. Atanassov, and N. Palaniappan, “An intuitionistic fuzzy graph method for finding the shortest paths in networks,” in Theoretical Advances and Applications of Fuzzy Logic and Soft Computing, pp. 3–10, Springer, Berlin, Germany, 2007.
View at: Publisher Site | Google Scholar
S. Mukherjee, “Dijkstra’s algorithm for solving the shortest path problem on networks under intuitionistic fuzzy environment,” Journal of Mathematical Modelling and Algorithms, vol. 11, no. 4, pp. 345–359, 2012.
View at: Publisher Site | Google Scholar
Z. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 6, pp. 1179–1187, 2007.
View at: Publisher Site | Google Scholar
R. R. Yager and A. M. Abbasov, “Pythagorean membership grades, complex numbers, and decision making,” International Journal of Intelligent Systems, vol. 28, no. 5, pp. 436–452, 2013.
View at: Publisher Site | Google Scholar
P. H. Phong, D. T. Hieu, R. T. Ngan, and P. T. Them, “Some compositions of picture fuzzy relations,” in Proceedings of the 7th National Conference on Fundamental and Applied Information Technology Research (FAIR’7), Thai Nguyen, Vietnam, June 2014.
View at: Google Scholar

Copyright

Copyright © 2020 Huanhuan Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

402

Downloads

638

Citations