Decision Making on Teachers’ adaptation to Cybergogy in Saturated Interval-valued Refined Neutrosophic overset /underset /offset Environment

,

Smarandache [27] introduced neutrosophic oversets, offsets and undersets which are the special kinds of neutrosophic sets with values beyond [0,1]. Overset is characterized with membership values greater than 1, underset is characterized with membership values less than 0 and the combination of both these sets is offset. Smarandache justified the practical implications of these special kinds of sets with real life illustrations. These kinds of neutrosophic sets highly influenced and motivated us to propose a fuzzy relational decision making model with saturated refined interval-valued neutrosophic oversets, undersets and offsets based on application of refined neutrosophic sets in medical diagnosis by Deli et al [28]. Smarandache conceptualized n-valued refined neutrosophic sets and these sets are used in decision making model of medical diagnosis. Broumi [29] extended the model of Deli et al by applying correlation measure. Various distance measures are used to make optimal decisions without changing the neutrosophic representations. In their model relation between symptoms and diseases was represented by neutrosophic sets; relation between patients and symptoms was represented by refined neutrosophic sets over certain interval period of time. In this decision making model the representation of the symptoms of the patients varies from time to time. But on profound analysis, the effects of treatment on the status and the degree of symptoms lack representation. This deficit in the decision maing model paved the way for developing a novel decision making model with new kind of representations.The same model is extended to fuzzy relation decision making model on teacher's adaptation to cybergogy in this research work. A relation between digital teaching skills and training methods is represented by neutrosophic sets and the relation between different kinds of teachers and the acquisition of digital skills after continuous stages of training is represented by refined neutrosophic oversets, undersets and offsets. Such kinds of representations are made to reflect the impact of training on skill acquisition rate by the teachers. The degree of digital skill acquisition by the teacher greatly depends on the personal interest, trainer's approach and training environment. The self-interest of the teachers may induce them to spend additional time other than the specified training time; also the disinterest of the teachers or dislike of trainer's approach may make them to refrain from the training and their participation rate is disturbed. At such circumstances refined neutrosophic oversets, underset and offset are used to represent such impacts. Also a new concept of saturated refined sets is introduced in this paper. The refined neutrosophic overset, underset and offset values remain to settle to a particular value over a consecutive period of time then it is called as saturated. The existences of situations where the degree of digital skill acquisition is confined and attained the maximum value and also there is no chance of further change over a period of training can be represented by saturated refined neutrosophic sets. The apt method of training to different kinds of teachers is determined by using hamming distance, normalized hamming distance, Euclidean distance and normalized Euclidean distance measures. The practical implications of neutrosophic overset, underset and offset are not explored to the best of the knowledge and so this research work will certainly fill the gap and it is intended to do so.
The paper is organized as follows: section 2 presents the basic definitions; section 3 describes about saturated refined neutrosophic sets; section 4 consists of the application of the proposed model; section 5 discusses the results and the last section concludes the work.

Definition 2.2 [27]
Let X be the universe of discourse with a generic element in X is denoted by x. An interval valued neutrosophic set (IVNS) A in X is defined by A={ ,

Definition2.3 [27]
Let U be a universe of discourse. A neutrosophic refined set (NRS) A on U can be defined as follows

Definition 2.4 [27]
Let U be the universe of discourse. A neutrosophic set A1 in U which consist the membership function T(x),I(x),F(x) that define true, Indeterminacy and falsity respectively, of a generic element x∈ , is called over limit.
A single valued neutrosophic over set A1 is defined as A1={x,< ( ); ( ); ( )>x∈ } such that in the neutrosophic components contains there exist atleast one element in A1 is >1 and no element is < 0.

Definition 2.5[27]
Let U be the universe of discourse. A neutrosophic set A2 in U which consist the membership function T(x), I(x), F(x) that define true, Indeterminacy and falsity respectively, of a generic element x∈ , A single valued neutrosophic underset A2 is defined as An interval valued neutrosophic offset A3 is defined as A3 = {x, < ( ); ( ); ( )> x∈ } such that in the neutrosophic components contains atleast one is partially or totally above 1 and atleast another is partially or totally below 0.

3.Euclidean distance between A and B is denoted as dE(A,B) and is defined by
1. If any of the membership values is saturated it is partial in nature and it is also a saturated refined set.
2.The saturated refined neutrosophic sets can be extended to overset, underset and offset.
3.The interval -valued refined neutrosophic sets are also extended to saturated interval-valued refined neutrosophic sets and the saturated values varies from interval sets to single valued sets over a period of time.

Application of the proposed decision making model
A decision making model together with fuzzy relational matrix and saturated refined neutrosophic overset, underset and offset is validated with the following illustration.

Decision Making Environment
Presently COVID -19 has brought a paradigm shift in teaching and learning process, the teaching fraternity is expected to possess digital teaching skills to face the post quarantine period. The developing nations have begun to encourage online educational system with the motive of unlocking learning during lock down. In this juncture the teachers are categorized based on their attributes and exposed to different kinds of training method to foster the acquisition of digital skills. The ultimate aim of this decision making model is to determine the suitable training method to the different kinds of teachers. This training programme is conducted to train the teachers to acquire online teaching skills. The expected outcome is enhancement of teacher's online teaching skills. The effectiveness of the programme is evaluated based on certain attributes and these attributes duly play crucial role in the enhancement of teacher's online teaching skills.
The attributes are A1 Trainer's efficiency-Refers to mastery A2 Teacher's interest A3 Teacher's duration of participation -present throughout the sessions A4 Teacher's grasping ability -how quick they understand A5 Trainer's Approach -Refers to inter personal relationship/ social skills The teachers are made to undergo four phases of training namely I, II, III, IV and they are grouped into four categories and their characteristic features are presented in Table 4.1 Table 4.

Types of Teachers & Attributes
The training to teachers are given using the following modes such as Self-paced learning, Blended learning, Adaptive learning, Virtual classes. The digital skills that are focussed in this training programme are Online skills, Digital literacy skills, Administrative skills of Learning Management System (LMS), Technology skills, Organization skills.
The relation between digital skills and training methods are presented in Table 4.2 Adaptative mode to type III teachers and blended mode to type IV teachers. This optimal relation between teachers and methods are highly pragmatic as it has incorporated the influence of external and internal factors of the training programme. The various methods of distance measures are used to determine the feasible method of teaching and on comparative analysis, the results obtained by using the different methods, are same. The proposed decision making model with saturated refined neutrosophic sets of different kinds can be extended further with other representations of neutrosophic sets, also other kinds of distance measures can be applied to find the optimal method of teaching. This model also has certain limitations as neutrosophic oversets, undersets and offsets of representations are used only specifically and these special kinds of representations cannot be applied at all circumstances. This decision -making model caters to particular needs.

Conclusion
In this research work the concept of saturated refined neutrosophic sets, interval -valued saturated refined neutrosophic sets and its extension to neutrosophic overset, underset and offset are proposed. A decision making model with fuzzy relational matrix and saturated refined neutrosophic overset, underset and offset is proposed in this paper. The model is validated with a real life application. This research work will certainly enlighten the researchers to explore in deep about the concepts of neutrosophic overset, underset and offset. The profound extension of these concepts will disclose new portals of neutrosophic research.