This chapter begins with an introduction that briefly describes the scope of Transgenders and Triangular fuzzy Cognitive Maps.
Transgenders is an umbrella term for persons whose gender identity, gender expression, or behaviour does not conform to that typically associated with the sex to which they were assigned at birth. Gender identity refers to a person's internal sense of being male or female. Gender expression refers to how a person communicates gender identity to others through behaviour, clothing, hairstyles, voice or body characteristics, etc. Transgender is in between categories of gender. They possess attributes of male and female. They are called transgender, multi-transgender, third-gender, etc. However, this third gender is not accepted by society anywhere in the world. It is due to different activities such as behaviour, talking, hairstyle, dressing, etc. Parents, colleagues, friends, and society should have equal rights in all activities. Gender identity refers to an internal feeling of being male or female gender expression. Transgender is a term used to describe people whose gender identity (sense of them as male or female) or gender expression differs from that usually associated with their birth sex. Many transgender people live part-time or full-time as members of the other gender. Broadly speaking, anyone whose identity, appearance, 18 or behaviour falls outside of conventional gender norms can be described as transgender. However, not everyone whose appearance or behaviour is gender-atypical will identify as a transgender person.
The fuzzy set provides us with a meaningful and powerful representation of the measurement of uncertainty with a meaningful representation of vague concepts expressed in natural language. Because every crisp set is fuzzy but not conversely. The mathematical embedding of conventional set theory into fuzzy sets is as natural as embedding real numbers into the complex plane. Thus, the idea of fuzziness is one of enrichment, not of replacement.
A Fuzzy graph is a convenient way of representing information involving relationships between objects. The objects are represented by vertices and relations by edges. When there is vagueness in the description of the objects or their relationships or both, it is natural that we need to design a Fuzzy Graph Model. Applications of fuzzy relations are widespread and important, especially in clustering analysis, neural networks, computer networks, pattern recognition, decision making and expert systems. In each of these, the basic mathematical structure is that of a fuzzy graph.
The membership function of the Triangular Fuzzy number = (a,b,c) is given by, where a≤b≤c. FCMs can successfully represent knowledge and human experience, introducing concepts to represent the essential elements and the cause-and-effect relationships among the concepts to model the behaviour of any system. It is a very convenient, simple and powerful tool, which is used in numerous fields such as social, economic, medical, etc., illustrated by Vasantha Kandasamy
Usually, in FCM, we analyze the number of states ON-OFF position. For this dissertation, Triangular Fuzzy Cognitive Maps (TFCM) gives the weight of the attributes and the ranking of the attributes. The membership function of the Triangular Fuzzy number = (a,b,c) is given by, where a≤b≤c.
TrFCM are more applicable when the data in the first place is an unsupervised one. The TrFCM works based on the opinions of three experts. TrFCM models the world as a collection of classes and causal relations between classes. It is a different process from FCM. Usually, the FCM only gives the ON-OFF position. However, this triangular fuzzy cognitive map is more precise, and it ranks the causes of the problem by using the weightage of the attribute. It is the main advantage of the new TrFCM.
Modelling-knowledge representation, decision making, enterprise resource management, socio-economic systems, engineering & technology management, adaptation and learning, classification tasks, robots and control, political and social fields, military planning, production systems, prediction capabilities, ecology and environmental.
The FCM models play an important role in finding the ranking with triangular fuzzy numbers. Several authors, such as Vidhya et al.
Vidhya et al.
Anand and Devadoss
Definition 1. When the TrFCM nodes are fuzzy sets, they are called fuzzy triangular nodes.
Definition 2. TrFCMs with edge weights or causalities from the set {-1, 0, 1} are called simple TrFCMs. Definition 3. A TrFCM is a directed graph with concepts like policies, events, etc., as nodes and causalities as edges; it represents causal relationships between concepts.
Definition 4. Consider the nodes/concepts are TrC1,TrC2……..TrCn of the TrFCM. Suppose the directed graph is drawn using edge weight Treij ∈{-1, 0,1}. The triangular matrix M be defined by Tr(M) = Treij where Treij is the triangular weight of the directed edge TrciTrcj.Triangular Matrix (Tr(M) is called the adjacency matrix of TrFCM, also known as the connection matrix of the TrFCM.
It is important to note that all matrices associated with a TrFCM are always square matrices with diagonal entries as zero.
Definition 5. Let TrC1, TrC2,…,TrCn be the nodes of an TrFCM. A = (a1, a2,…,an) where Treij∈{-1, 0,1}. A is called the instantaneous state vector and denotes the node's on-off position at an instant.
ai
Tr = 1Maximum(weight), Instantaneous vector = {
ai
Tr = 0 Otherwise.
Definition 6. Let TrC1,TrC2,…….…,TrCn be the triangular nodes of and TrFCM.
Let⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗⃗ , ⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗, ⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗, ⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗ be the edges of the TrFCM (i≠j). ATrFCM is said to be cyclic if
Tr 1 Tr 2 Tr 2 Tr 3 Tr 3 Tr 4 Tr i Tr j
it possesses a directed cycle. A TrFCM is said to be acyclic if it does not possess any directed cycle.
Definition 7. A TrFCM is said to be cyclic and is said to have feedback.
Definition 8. When there is feedback in a TrFCM, i.e., when the causal relations flow through a cycle in a revolutionary way, the TrFCM is called a dynamical system.
Definition 9. Let ⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗⃗ ⃗,⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗⃗ ⃗,⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗⃗ , … ⃗,⃗⃗⃗⃗⃗⃗C ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗C ⃗⃗⃗ be a cycle. When TrCi is switched ON and if
Tr 1 Tr 2 Tr 2 Tr 3 Tr 3 Tr 4 Tr n−1 Tr n
the causality flows through the triangular edges of a cycle and if it again causes ci, we say that the dynamical system goes round and round. This is true for any triangular node Trci for i =1,2,…,n. The equilibrium state for this dynamical system is called the hidden pattern.
Definition 10. If the equilibrium state of a dynamical system is a unique state vector, then it is called a fixed point. Consider a TrFCM with TrC1, TrC2,…, and TrCn as nodes.
For example, let us start the dynamic system by switching on TrC1.
Let us assume that the TrFCM settles down with TrC1 and TrCn is ON, i.e., in the state, the vector remains as (1, 0, 0,……0) is called fixed point.
Definition 11. If the TrFCM settles down with a state vector repeating in the form A1→A2→…→Ai→A1, then this equilibrium is called a limit cycle.
Definition 12. A triangular fuzzy number A ̃ = (a , a , a ) is said to be non-negative iff a ≥ 0.
1 2 3 1
Definition 13. A fuzzy set ̃ defined on X is called a normal fuzzy set if there exists at least one x∈ X such that ̃(x) = 1.
Definition 14. Given a fuzzy set ̃ defined on ‘X’ and any α ∈[0,1] the α–cut is denoted by A ̃ (α) and is defined as, ̃(α) = {x, µ(x) ≥ α, α ∈ [0,1]}.
Definition 15. Arithmetic in triangle fuzzy numbers.
Let's say ã is represented by 〈x ̅ , l , r 〉 and ̃ is represented by 〈x ̅ , l , r 〉 both being triangular fuzzy numbers.
a a a b b b
Additionally, x belongs to the set of real numbers. The addition, subtraction, and multiplication operations of fuzzy numbers are defined as follows in the real number set.
a ̃ + b ̃ = 〈x ̅ + x ̅ , l + l , r + r 〉,
a b a b a b
a ̃ − b ̃ = 〈x ̅ − x ̅ , l − l , r − r 〉,
a b a b a b
〈xx ̅a, xla, xra〉, x ≥ 0, xa ̃ = { 〈xx ̅a, −xla, −xra〉, x < 0.
Linguistic values of Triangular Fuzzy Numbers
The linguistic values.
This table is shown in the PDF version of the article.
The New TrFCM models play an important role in finding the ranking with triangular fuzzy numbers. Several authors, such as Anand and Devadoss
FCM works on unsupervised data. It is a simple and effective tool to analyze social, economic, and political problems. The idea of a fuzzy set was introduced by Zadeh
Vasantha Kandasamy
The TrFCM, introduced by Anand and Devadoss