scholarly journals Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 72 ◽  
Author(s):  
Naeem Jan ◽  
Kifayat Ullah ◽  
Tahir Mahmood ◽  
Harish Garg ◽  
Bijan Davvaz ◽  
...  
Keyword(s):  
Decision Making ◽  
Essential Role ◽  
Real Life ◽  
Current Definition ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Decision Making Problem ◽  
Neutrosophic Graph ◽  
Definition Of ◽  
Interval Valued

Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined; i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work.

scholarly journals A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 551 ◽  
Author(s):  
Liangsong Huang ◽  
Yu Hu ◽  
Yuxia Li ◽  
P. K. Kishore Kumar ◽  
Dipak Koley ◽  
...  
Keyword(s):  
Graph Theory ◽  
Optimization Problems ◽  
Real Life ◽  
Road Transport ◽  
Fuzzy Graph ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Real World Problem ◽  
Neutrosophic Graph ◽  
Definition Of

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here.

scholarly journals Graphical Analysis of Covering and Paired Domination in the Environment of Neutrosophic Information

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Sami Ullah Khan ◽  
Abdul Nasir ◽  
Naeem Jan ◽  
Zhen-Hua Ma
Keyword(s):  
Graph Theory ◽  
Optimization Problems ◽  
Domination Number ◽  
Real Life ◽  
Graphical Analysis ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Neutrosophic Graph ◽  
Paired Domination ◽  
Intuitionistic Fuzzy Graphs

Neutrosophic graph (NG) is a powerful tool in graph theory, which is capable of modeling many real-life problems with uncertainty due to unclear, varying, and indeterminate information. Meanwhile, the fuzzy graphs (FGs) and intuitionistic fuzzy graphs (IFGs) may not handle these problems as efficiently as NGs. It is difficult to model uncertainty due to imprecise information and vagueness in real-world scenarios. Many real-life optimization problems are modeled and solved using the well-known fuzzy graph theory. The concepts of covering, matching, and paired domination play a major role in theoretical and applied neutrosophic environments of graph theory. Henceforth, the current study covers this void by introducing the notions of covering, matching, and paired domination in single-valued neutrosophic graph (SVNG) using the strong edges. Also, many attention-grabbing properties of these concepts are studied. Moreover, the strong covering number, strong matching number, and the strong paired domination number of complete SVNG, complete single-valued neutrosophic cycle (SVNC), and complete bipartite SVNG are worked out along with their fascinating properties.

scholarly journals Multiple Criteria Decision Making Based on Probabilistic Interval-Valued Hesitant Fuzzy Sets by Using LP Methodology

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
M. Sarwar Sindhu ◽  
Tabasam Rashid ◽  
Agha Kashif ◽  
Juan Luis García Guirao
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Real Life ◽  
Multiple Criteria Decision Making ◽  
Multiple Criteria ◽  
Decision Makers ◽  
Occurrence Probability ◽  
Hesitant Fuzzy Sets ◽  
Life Problems ◽  
Interval Valued

Probabilistic interval-valued hesitant fuzzy sets (PIVHFSs) are an extension of interval-valued hesitant fuzzy sets (IVHFSs) in which each hesitant interval value is considered along with its occurrence probability. These assigned probabilities give more details about the level of agreeness or disagreeness. PIVHFSs describe the belonging degrees in the form of interval along with probabilities and thereby provide more information and can help the decision makers (DMs) to obtain precise, rational, and consistent decision consequences than IVHFSs, as the correspondence of unpredictability and inaccuracy broadly presents in real life problems due to which experts are confused to assign the weights to the criteria. In order to cope with this problem, we construct the linear programming (LP) methodology to find the exact values of the weights for the criteria. Furthermore these weights are employed in the aggregation operators of PIVHFSs recently developed. Finally, the LP methodology and the actions are then applied on a certain multiple criteria decision making (MCDM) problem and a comparative analysis is given at the end.

scholarly journals Certain Properties of Vague Graphs with a Novel Application

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1647
Author(s):  
Yongsheng Rao ◽  
Saeed Kosari ◽  
Zehui Shao
Keyword(s):  
Social Systems ◽  
Sufficient Conditions ◽  
Real Life ◽  
Laplacian Matrix ◽  
Fuzzy Graph ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Real World Problem ◽  
Laplacian Energy ◽  
Definition Of

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital.

scholarly journals A note on different types of product of neutrosophic graphs

2021 ◽  
Author(s):  
Kartick Mohanta ◽  
Arindam Dey ◽  
Anita Pal
Keyword(s):  
Decision Making ◽  
Fuzzy Set ◽  
Real World ◽  
Real Life ◽  
Neutrosophic Set ◽  
Total Degree ◽  
Decision Making Problem ◽  
Different Types ◽  
Neutrosophic Graph ◽  
Real World Problems

AbstractFuzzy set and neutrosophic set are two efficient tools to handle the uncertainties and vagueness of any real-world problems. Neutrosophic set is more capable than fuzzy set to deal the uncertainties of a real-life problem. This research paper introduces some new concept of single-valued neutrosophic graph (SVNG). We have also presented some different operations on SVNG such as rejection, symmetric difference, maximal product, and residue product with appropriate examples, and some of their important theorems are also described. Then, we have described the concept of total degree of a neutrosophic graph with some interesting examples. We have also presented an efficient approach to solve a decision-making problem using SVNG.

scholarly journals New Ranking Functions for Interval-Valued Intuitionistic Fuzzy Sets and Their Application to Multi-Criteria Decision-Making Problem

2021 ◽  
Vol 21 (1) ◽  
pp. 3-18
Author(s):  
Melda Kokoç ◽  
Süleyman Ersöz
Keyword(s):  
Decision Making ◽  
High Performance ◽  
Intuitionistic Fuzzy Set ◽  
Real Life ◽  
Score Function ◽  
Multi Criteria Decision Making ◽  
Ranking Functions ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Interval Valued

Abstract Many authors agree that the Interval-Valued Intuitionistic Fuzzy Set (IVIFS) theory generates as realistic as possible evaluation of real-life problems. One of the real-life problems where IVIFSs are often preferred is the Multi-Criteria Decision-Making (MCDM) problem. For this problem, the ranking of values obtained by fuzzing the opinions corresponding to alternatives is an important step, as a failure in ranking may lead to the selection of the wrong alternative. Therefore, the method used for ranking must have high performance. In this article, a new score function SKE and a new accuracy function HKE are developed to overcome the disadvantages of existing ranking functions for IVIFSs. Then, two illustrative examples of MCDM problems are presented to show the application of the proposed functions and to evaluate their effectiveness. Results show that the functions proposed have high performance and they are the eligibility for the MCDM problem.

scholarly journals Graphical Structures of Cubic Intuitionistic Fuzzy Information

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Sami Ullah Khan ◽  
Naeem Jan ◽  
Kifayat Ullah ◽  
Lazim Abdullah
Keyword(s):  
Decision Making ◽  
Future Study ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Graph ◽  
Fuzzy Information ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Graphs ◽  
Decision Making Problem ◽  
Intuitionistic Fuzzy Graphs ◽  
Interval Valued

The theory developed in this article is based on graphs of cubic intuitionistic fuzzy sets (CIFS) called cubic intuitionistic fuzzy graphs (CIFGs). This graph generalizes the structures of fuzzy graph (FG), intuitionistic fuzzy graph (IFG), and interval-valued fuzzy graph (IVFG). Moreover, several associated concepts are established for CIFG, such as the idea subgraphs, degree of CIFG, order of CIFG, complement of CIFG, path in CIFG, strong CIFG, and the concept of bridges for CIFGs. Furthermore, the generalization of CIFG is proved with the help of some remarks. In addition, the comparison among the existing and the proposed ideas is carried out. Finally, an application of CIFG in decision-making problem is studied, and some future study is proposed.

Interval-Valued Intuitionistic Fuzzy Sets based Method for Multiple Criteria Decision-Making

2016 ◽  
Vol 5 (4) ◽  
pp. 192-210 ◽  
Author(s):  
Bhagawati Prasad Joshi
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Real Life ◽  
Multiple Criteria Decision Making ◽  
Intuitionistic Fuzzy Sets ◽  
Multiple Criteria ◽  
Decision Makers ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Interval Valued

Due to the huge applications of fuzzy set theory, many generalizations were available in literature. Atanassov (1983) and Atanassov and Gargov (1989) introduced the notions of intuitionistic fuzzy sets (IFSs) and interval-valued intuitionistic fuzzy sets (IVIFSs) respectively. It is observed that IFSs and IVIFSs are more suitable tools for dealing with imprecise information and very powerful in modeling real life problems. However, many researchers made efforts to rank IVIFSs due to its importance in fusion of information. In this paper, a new ranking method is introduced and studied for IVIFSs. The proposed method is compared and illustrated with other existing methods by numerical examples. Then, it is utilized to identify the best alternative in multiple criteria decision-making problems in which criterion values for alternatives are IVIFSs. On the basis of the developed approach, it would provide a powerful way to the decision-makers to make his or her decision under IVIFSs. The validity and applicability of the proposed method are illustrated with practical examples.

A New Ranking Approach for Interval Valued Intuitionistic Fuzzy Sets and its Application in Decision Making

2019 ◽  
Vol 8 (2) ◽  
pp. 110-125 ◽  
Author(s):  
Pranjal Talukdar ◽  
Palash Dutta
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Model Uncertainty ◽  
Real Life ◽  
Intuitionistic Fuzzy Sets ◽  
Multi Criteria Decision Making ◽  
Numerical Examples ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Interval Valued

Ranking of interval valued intuitionistic fuzzy sets (IVIFSs) plays an important role because of its attraction and applicability to model uncertainty in real life problems. In this article, an attempt has been made to devise a new method for ranking of IVIFSs based on exponential function. The significance of the method is illustrated with the help of some numerical examples and the results are compared with other existing methods. Furthermore, a multi criteria decision making method is presented here to evaluate the final ranking of the alternatives using the proposed ranking method and discussed the consistency of so obtained results.

scholarly journals ACO with Intuitionistic Fuzzy Pheromone Updating Applied on Multiple-Constraint Knapsack Problem

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1456
Author(s):  
Stefka Fidanova ◽  
Krassimir Todorov Atanassov
Keyword(s):  
Decision Making ◽  
Knapsack Problem ◽  
Propositional Logic ◽  
Optimization Problems ◽  
Real Life ◽  
Combinatorial Optimization Problems ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Multiple Constraint

Some of industrial and real life problems are difficult to be solved by traditional methods, because they need exponential number of calculations. As an example, we can mention decision-making problems. They can be defined as optimization problems. Ant Colony Optimization (ACO) is between the best methods, that solves combinatorial optimization problems. The method mimics behavior of the ants in the nature, when they look for a food. One of the algorithm parameters is called pheromone, and it is updated every iteration according quality of the achieved solutions. The intuitionistic fuzzy (propositional) logic was introduced as an extension of Zadeh’s fuzzy logic. In it, each proposition is estimated by two values: degree of validity and degree of non-validity. In this paper, we propose two variants of intuitionistic fuzzy pheromone updating. We apply our ideas on Multiple-Constraint Knapsack Problem (MKP) and compare achieved results with traditional ACO.

Sign in / Sign up

Export Citation Format

Share Document