Novel decision-making method based on bipolar neutrosophic information

In 1998, Smarandache originally considered the concept of neutrosophic set from philosophical point of view. The notion of a single-valued neutrosophic set is a subclass of the neutrosophic set from a scientific and engineering point of view and an extension of intuitionistic fuzzy sets. A bipolar single-valued neutrosophic set is an extension of a bipolar fuzzy set, which provides us an additional possibility to represent uncertainty, imprecise, incomplete and inconsistent information existing in real situations. In this research article, we apply the concept of bipolar single-valued neutrosophic sets to graph structures and present a novel framework for handling bipolar neutrosophic information by combining bipolar neutrosophic sets with graph structures. Several basic notions concerning bipolar single-valued neutrosophic graph structures are introduced, and some related properties are investigated. We also consider the applications of bipolar single-valued neutrosophic graph structures in decision making. In particular, efficient algorithms are developed to solve decision-making problems regarding recognition of each country’s participation in its conspicuous relationships, detection of psychological improvement of patients in a mental hospital and uncovering the undercover reasons for global terrorism.


Introduction
Fuzzy set theory Zadeh (1965) is meticulous research field having abundant applications in diverse disciplines.Fuzzy set theory is a strong tool for dealing approximate reasoning problems.On those complex phenomenons while classical set theory does not represent reasonably, it is necessary to consider implicit counter property.Bipolar fuzzy sets Zhang (1994) are an extension of fuzzy sets whose membership degree range is [− 1,1].Bosc and Pivert (2013) discussed the is a branch of philosophy which studies the origin, nature and scope of neutralities as well as their interactions with different ideational spectra.For convenient and advantageous usage of neutrosophic sets in science and engineering, Myithili et al. (2016) proposed the single-valued neutrosophic (SVNS) sets, whose three independent components have values in standard unit interval.Deli et al. (2015) narrated the notion of bipolar neutrosophic sets as an extension of the fuzzy sets, bipolar fuzzy sets and neutrosophic sets.
Fuzzy graph theory is finding an increasing number of applications in modeling real-time systems where the level of information inherent in the system varies with different levels of precision.Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems.Kaufmann's initial definition of a fuzzy graph Kauffman (1973) was based on Zadeh's fuzzy relations Zadeh (1971).Rosenfeld Rosenfeld (1975) described the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts.Bhattacharya (1987) gave some remarks on fuzzy graphs.Mordeson and Peng (1994) discussed some operations on fuzzy graphs.The complement of a fuzzy graph was defined by Mordeson and Peng (1994) and further studied by Sunitha and Vijayakumar (2002).Akram (2011Akram ( , 2013) ) worked on certain notions of bipolar fuzzy graphs.Further theory of bipolar fuzzy graphs was developed in Akram and Anam (2017), Akram and Sitara (2017), Akram and Sarwar (2017), Sarwar and Akram (2017), Singh and Kumar (2014a), Singh and Kumar (2014b) and Yang et al. (2013).On the other hand, Dinesh and Ramakrishnan (2011) introduced the concept of the fuzzy graph structure.Akram and Akmal (2016) considered the notion of bipolar fuzzy graph structures.Akram and Shahzadi (2017) first dealt with neutrosophic soft graphs.Single-valued neutrosophic minimum spanning tree and its clustering method were studied by Ye (2014).The same author gave many clustering algorithms based on different methods, including similarity measures, netting method and distance-based similarity measures (Ye 2013(Ye , 2014a, b), b).In this research article, we apply the concept of bipolar singlevalued neutrosophic sets to graph structures and present a novel framework for handling bipolar neutrosophic information by combining bipolar neutrosophic sets with graph structures.We introduce several basic notions concerning bipolar single-valued neutrosophic graph structures and investigate some related properties.We also consider the applications of bipolar single-valued neutrosophic graph structures in decision making.In particular, efficient algorithms are developed to solve decision-making problems regarding recognition of each country's participation in its conspicuous relationships, detection of psychological improvement of patients in a mental hospital and uncovering the undercover reasons for global terrorism.Bipolar single-valued neutrosophic graphs, extension of bipolar fuzzy graph, handle the phenomenons concerning bipolar information in which indeterminacy is explicitly quantified.We deal with multiple relations in graph structure model; a bipolar neutrosophic graph structure is more advantageous and utilitarian.Obviously, a bipolar neutrosophic graph structure can deal with more than one factors involved for particular type of relationship between any two vertices, which is properly explained in this research paper with some interesting real-life applications.For other notations, terminologies and applications are not mentioned in this research article; the readers are referred to Alcantud (2016), Ali et al. (2016), Dinesh (2011), Greco and Kadzinski (2018), Luo et al. (2018), Majumdar and Samanta (2014), Mordeson and Nair (2001), Myithili et al. (2016), Pramanik et al. (2016), Sayed et al. (2018), Smarandache (1999), Turksen (1986), Wu et al. (2017) and Zhan et al. (2018).
2 Bipolar single-valued neutrosophic graph structures Sampathkumar (2006) introduced the graph structure which is a generalization of undirected graph and is quite useful in studying some structures like graphs, signed graphs, labeled graphs and edge-colored graphs.
Definition 2.1 Sampathkumar (2006) A graph structure Ǧs = (V , R 1 , . . ., R m ) consists of a non-empty set V together with relations R 1 , R 2 , . . ., R m on V which are mutually disjoint such that each R k , 1 ≤ k ≤ m, is symmetric and irreflexive.
One can represent a graph structure G = (V , R 1 , ..., R m ) in the plane just like a graph where each edge is labeled as R k , 1 ≤ k ≤ m.Smarandache (1998) introduced neutrosophic sets as a generalization of fuzzy sets and intuitionistic fuzzy sets.A neutrosophic set is identified by three functions called truth membership (T ), indeterminacy membership (I ) and falsity membership (F) whose values are real standard or nonstandard subset of unit interval ] − 0, 1 + [, where − 0 = 0− , 1 + = 1 + , is an infinitesimal number.To apply neutrosophic set in real-life problems more conveniently, Smarandache (1998) and Myithili et al. (2016) defined single-valued neutrosophic sets which take the value from the subset of [0,1].Thus, a single-valued neutrosophic set is an instance of neutrosophic set and can be used expediently to deal with real-world problems, especially in decision support.Definition 2.2 Smarandache (1998) A neutrosophic set N on a non-empty set V is an object of the form where T N , I N , F N : V →]0 − , 1 + [ and there is no restriction on the sum of T N (v), I N (v) and F N (v) for all v ∈ V .Definition 2.3 Myithili et al. (2016) A single-valued neutrosophic set N on a non-empty set V is an object of the form where T N , I N , F N : V → [0, 1] and sum of T N (v), I N (v) and F N (v) is confined between 0 and 3 for all v ∈ V .Definition 2.4 Deli et al. (2015) A bipolar single-valued neutrosophic set B on a non-empty set V is an object of the form where denote the truth, indeterminacy and falsity membership values of an element v ∈ V , whereas negative values indicate the implicit counterproperty of truth, indeterminacy and falsity membership values of an element v ∈ V .Definition 2.5 A bipolar single-valued neutrosophic graph on a non-empty set V is a pair G = (B, R), where B is a bipolar single-valued neutrosophic set on V and R is a bipolar single-valued neutrosophic relation in V such that for all b, d ∈ V .
We now define bipolar single-valued neutrosophic graph structure.

Definition 2.6 Ǧbn
> are bipolar single-valued neutrosophic (BSVN) sets on V and R k , respectively, such that for all b, d ∈ V .We note that 0 ≤ T P k (bd) We illustrate the concept of BSVNGS with an example.
Example 2.7 Consider a graph structure ) is a bipolar single-valued neutrosophic graph structure, as shown in Fig. 1.
Remark 2.19 Every self-complementary BSVNGS is always a totally self-complementary BSVNGS.
Remark 2.21 Converse of theorem 2.20 may not be true; for example, a BSVNSGS depicted in Fig. 7 is totally strong self-complementary BSVNGS; it is also strong BSVNGS with a totally strong self-complementary underlying GS but

Recognition of each country's participation in its conspicuous relationships
Naturally, any living organism cannot be 100% self-reliant, especially man who is the noblest of all creatures but he relies In a relationship, both countries play their role so that these relations can be represented using BSVNGS, which will show that between two particular countries which type of relationship is the strongest and what are the degrees of its strength from both countries.There are some relationships which are not good for the universe in some aspects.BSVNGS will highlight that which one country of those two countries is strongest participant.To make that relationship weak, we will hit strong participant for that relationship.We can see that which relationship is prevailing and which are strong participants of it.We can also observe that which country is the most favorite for a particular relationship.Let V be the set of eight countries in the universe: V = {China, Turkey, Bangladesh, America, India, Russia, Pakistan, Afghanistan}.
Consider a BSVN set B on V as shown in Table 3.In Table 3, T P , F P of a country indicate its positive and negative approach for the universe and I P shows indeterminacy/ambiguity of its approach, whereas T N and F N denote its positive and negative reputation in the universe and I N stands to represent the percentage of its uncertain reputation.We will use the following abbreviations for country names: Ch = China, TU = Turkey, Ba = Bangladesh, Am = America, In = India, Ru = Russia, Pak = Pakistan, Afg = Afghanistan.
Each pair of countries of set V has T P , I P , F P , T N , I N and F N values of different relationships in Tables 4,5,6,7,8,9,10,11 and 12.For each pair of countries, T P , T N of any particular relationship indicate strength of that relationship from first country and second country, respectively.Similarly, F P , F N represent percentage of weakness and I P , I N demonstrate percentage of ambiguity of that relationship from first country and second country, respectively.
Since many relations exist on set V , we define six relations on set V as: We will define the relations with those elements such that ) is a GS.Every element of a relation denotes a highly worthwhile relationship in that pair of countries.Since ) is a GS, every relation has unique elements; that is, they are present in that relation only.Hence, any pair of countries will be an element of that relation for which its value of T P , I N and F N is high, and T N , I P and F P values are comparatively low, according to the data of Tables 4, 5, 6, 7, 8, 9, 10, 11 and 12. Write down the T P , I P , F P , T N , I N and F N values of all elements in the relations, using the data of Tables 24,25,26,27,28,29,30,31 and 32,such    In BSVNGS, represented in Fig. 8, an edge depicts most dominating and attention-getting relationship of contigu-ous countries.For instance, between Turkey and Pakistan, most dominating and attention-getting relationship is project investment, participation of Turkey in this relationship's strength, weakness and indeterminacy is 70%, 10% and 10%, respectively, and participation of Pakistan in this relationship's strength, weakness and indeterminacy is 50%, 10% and 20%, respectively.This BSVNGS represents complete status of a relationship from its both participants.It can be very utilitarian for some particular relationships, like war-   It also highlights those countries whose attention-getting relationship is nuclear weapons and spotlights those countries which are its strong participants than their contiguous countries.For example, America and India have nuclear weapons as their dominating relationship and America is strong participant, as its participation for its strength is high.So, this BSVNGS indicates that in order to reduce nuclear proliferation, United Nations should pressurize America and Russia.We now present the general procedure of our method which is used in our application in the following algorithm.do j from 1 → m 6.

Detection of Psychological Improvement of Patients in a Mental Hospital
A society is an enduring and cooperating social group whose members have developed organized patterns of relationships through interaction with one another.In a society, interaction of its members is very important; this compactness causes all types of relationships among society members.As Homo sapiens have complex nature, for grouping of human beings we have to focus on many psychological aspects and behaviors.We use BSVNGS to judge behavior and other social aspects of any group of society.Using BSVNGS, we may get aware of conspicuous behavioral property of any two group members with each other.Most frequent occurrence of a particular behavioral property reveals that this group is good or bad in that perspective.This BSVNGS is very helpful for subgrouping and assigning of tasks.It may be very utilitarian in mental hospitals, where in a group of patients we estimate what is recent stage of their psychological behavior and psychiatrist can decide which patient is having high violence-based relationship with others.He may decrease his interaction with others and can decide either mental stage of his patients is getting better or worse.It helps a psychiatrist in taking precautionary measures and providing suitable environment for his patients.He can decide looking at the BSVNGS that which patient can properly survive in the society with better behavior and which one is still not capable of showing moderate behavior.A psychiatrist may use many types of psychological behaviors to estimate improvement in mental state of his patients including violence, friendliness, jealousy, helping, forgiving, communication and problem sharing, etc.Consider set V of eight patients in a mental hospital: V = {Albert, Charles, Burton, Calvert, Christopher, David, Chapman, Joseph}.
Consider a BSVN set B on V as shown in Table 13.
In BSVNGS, depicted in Fig. 9, every edge depicts most conspicuous behavioral property of adjacent vertices (patients).For instance, most conspicuous behavioral property between Charles and Burton is helping, Charles' participation in its strength, weakness and indeterminacy is 50%, 40% and 50%, respectively, and Burton's participation in its strength, weakness and indeterminacy is 40%, 40% and 60%, respectively.This BSVNGS also accentuates that vertex Albert has highest degree for violence.This points out that Albert is most violent mental patient among these eight patients.So, his psychiatrist must focus on Albert's violent behavior, as Albert's most violent behavior is with Chapman and his positive behavior is with just one person, that is, Charles.So psychiatrist should try to minimize his interaction with Chapman to avoid his most violent behavior and should maximize his interaction with Charles to improve his good psychological behavior.Similarly, a psychiatrist can jude the behavior of all of his patients and can take better decisions for their well-being.He can make two BSVNGSs with a gap of some time period and can estimate their behavioral improvement by comparing those two BSVNGSs; it also helps him in continuation or cancelation of his decisions for his patients.General procedure regarding this application is deciphered through following algorithm.
then label u k u l as R i .6. T P R i , F P R i , I P R i values of an edge between two different vertices (patients) u k and u l show participation of vertex(patient) u k in strength, weakness and indeterminacy of most conspicuous psychological behavior R i between them, whereas T N R i , F N R i , I N R i are corresponding values of vertex (patient) u l .7. End

Uncovering the undercover reasons for global terrorism
Terrorism is the process of using violence or threat of violence to purport a religious, ideological or political change.It is committed by some non-state actors and may undercover personnel servings on the behalf of respective governments of non-state actors.Terrorism achieves more than immediate target victims and also directed at targets covering a large spectrum of society.It is considered as fourth-generation warfare and violent crime.Nowadays, terrorism is major threat to the society and is illegal under anti-terrorism laws in all jurisdictions.It is considered as a war crime under laws of war, when it is used to target the noncombatants, including neutral military personnel, civilians or enemy prisoners of the war.Many political organizations have practiced the terrorism to achieve their objectives.It is being practiced by both leftwing and right-wing political organizations, religious groups, nationalist groups, revolutionaries and ruling governments.Terrorism can be used to exploit human fear to achieve these goals.According to Global Terrorism Database, non-state terrorism incidents are more than 61,000, claiming more than 140,000 lives from 2000 to 2014.Nowadays, many countries are blamed to be responsible for global terrorism.That's why many countries are doing war against terror.When a country blames other country to be responsible for terrorism in its territory, many factors are responsible for it.But there are some reasons or factors which are strongly responsible for that particular incident and cannot be ignored.We can use bipolar neutrosophic graph structure to highlight those factors which are highly responsible for terrorism between any two countries or global terrorism due to those countries.There are many undercover reasons highly responsible for global terrorism, including religious factors, military interference/drone attacks, political benefits, to pressurize other government, to occupy other country, benefits of powerful countries, to revenge past political occupation.Let V be the set of eight countries/estates on the globe which are either victims of terrorism or said to be responsible for global terrorism: V = {Pakistan, India, Afghanistan, America, Iraq, Israel, Palestine, Kashmir}.
Consider a BSVN set B on V as shown in Table 23.In Table 23, T P , T N of a country depict the percentage of its actual involvement in terrorism and the percentage of its involvement on the basis of other countries' point of view, respectively, whereas F P , F N of a country represent the percentage of its actual noninvolvement and of reputation of its noninvolvement, respectively.Similarly, I P , I N indicate ambiguity in its involvement and reputation with respect to terrorism.
Every pair of countries in set V has T P , I P , F P , T N , I N and F N values of different undercover reasons in Tables 24,25,26,27,28,29,30,31 and 32.For each pair of countries, T P , T N of any terrorism reason show the percentage of responsibility from first country and second country, respectively.Similarly, F P , F N indicate percentage of being not responsible and I P , I N denote the percentage of ambiguity of being responsible for that  There are many relations on the set V ; here, we define six relations on set V as: R 1 = Religious factors, R 2 = Political benefits, R 3 = To pressurize other government, R 4 = To occupy other country, R 5 = Benefits of powerful countries, R 6 = Military interference/drone attacks.We will define the relations with those elements such that ( ) is a graph structure, elements in every relation are unique; that is, they belong to just that relation.Hence, we will consider any pair of countries as an element of that particular relation, for which its value of T P , I N and F N is high, and T N , I P and F P values are low as compared to other relations, according to the data of Tables 24,25,26,27,28,29,30,31 and 32.Write down the  Obviously, (B, B 1 , B 2 , B 3 , B 4 , B 5 , B 6 ) is a BSVNGS as shown in Fig. 10.In BSVNGS, shown in Fig. 10, every edge demonstrates most dominating and attention-getting undercover reason for global terrorism due to those contiguous countries.For instance, attention-getting undercover reason for global terrorism due to Pakistan and America relationship is the drone attacks of America on Pakistan, participation of America in this reason's strength, weakness and indeterminacy is 40%, 20% and 20%, respectively, and participation of Pakistan in its strength, weakness and indeterminacy is 20%, 10% and 60%, respectively.This Pakistan-America edge does not represent direct terrorism from contiguous countries but it points out dominating reason for global terrorism due to Pakistan-America relationships.This BSVNGS also highlights that sovereignty issues of some countries or estates are also increasing global terrorism and tells its intensity, weakness and uncertainty due to both occupying and occupied countries.For instance, sovereignty issues of Kashmir and Palestine are also causing global terrorism.Furthermore, it also helps any country's government that what is main reason for terrorism due to its relationships with other countries and what is percentage of its strength, weakness and ambiguity from its side and other country's side.
Moreover, most frequent relation in this BSVNGS is the benefits of powerful countries.It shows that powerful countries are destroying global peace to achieve their political benefits.A BSVNGS of all countries can be very helpful for anti-terrorism organizations that which perspective is to be considered to control global terrorism.According to this BSVNGS, they should pressurize powerful countries and solve sovereignty issues to maintain global peace.General procedure of this application is explained by the following algorithm.

Algorithm 3
1. Begin 2. Input membership values B(u i ) of n number of countries u 1 , u 2 , . . ., u n which are either victims of terrorist or said to be responsible for global terrorism.3. Input the adjacency matrix of countries with respect to R 1 , R 2 , . . ., R m terrorism reasons due to relationships among all pairs of vertices(i.e., pairs of countries).4. Follow steps 3 and 4 of Algorithm 2. 5. T P R i , F P R i , I P R i values of an edge between two different vertices (countries) u k and u l show participation of vertex (country) u k in strength, weakness and indeterminacy of most dominating and attention-getting terrorist reason R i due to relationships between them, whereas are corresponding values of vertex (country) u l .6. End.

Conclusions
Graph theory is widely used for dealing with structural information in different domains, including computer science, electrical engineering, operations research, network routing, and transportation, pattern recognition and image interpretation.However, in many cases, some aspects of a graph-theoretic problem may be uncertain.In such cases, it is natural to deal with the uncertainty using the methods of fuzzy sets and bipolar single-valued neutrosophic sets.Bipolar single-valued neutrosophic models are more flexible and practical than fuzzy models.A bipolar neutrosophic graph constitutes a generalization of the notion bipolar fuzzy graph.In this research paper, we have introduced the idea of bipolar single-valued neutrosophic graph structure and discussed many relevant notions.We also have investigated worthwhile applications of bipolar single-valued neutrosophic graph structure in decision making.In the future, we aim to generalize our notions to (1) rough neutrosophic structures, (2) soft rough neutrosophic graph structures, (3) bipolar neutrosophic soft graph structures and (4) rough neutrosophic hypergraph structures.

Fig. 10 A
Fig. 10 A BSVNGS highlighting undercover reasons for global terrorism due to any pair of countries

Table 1
BSVN set B on vertex set V

Table 3
BSVN set B of eight countries in the universe

Table 9
BSVN set indicating relationships of Russia with some other countries in V

Table 12
BSVN set indicating relationships of some countries in V

Table 13
BSVN set B of eight patients in a mental hospital

Table 14
BSVN set indicating psychological behaviors of Albert with some other patients in V

Table 23
Bipolar single-valued neutrosophic set B of set of countries V

Table 29
BSVN set indicating reasons for terrorism due to relationships of Israel and other countries Every element in a relation indicates most prominent undercover reason for global terrorism due to that pair of countries.Since (

Table 32
BSVN set indicating reasons for terrorism due to relationships of some countries