An Intuitionistic Fuzzy Graph Model: Matrix Representations and Applications

The concept, tools and techniques, of an intuitionistic fuzzy graph have found many applications in different areas such as topology, all kinds of systems and networks, computer science, etc.An intuitionistic fuzzy graph is a generalized structure of a fuzzy graph that provides more flexibility, adaptability and compatibility to real human-centric systems than a simpler fuzzy graph.So, in this paper, we study some results in the intuitionistic fuzzy graph (IFG) and present some basic definitions.We investigate several kinds of arcs, for example, -strong, -strong, -arc in an intuitionistic fuzzy graph and analyse some properties.Also, we give intuitionistic fuzzy bridge, intuitionistic fuzzy cut nodes and some interesting properties of an intuitionistic fuzzy bridge.

Rough sets and intuitionistic fuzzy (IF) sets are two separate mathematical frameworks designed to model and manage incomplete or uncertain knowledge. By integrating these models, an IF rough framework is constructed, offering enhanced expressiveness and flexibility for representing and processing incomplete data within information systems. In this paper, we introduce a new hybrid model utilizing minimal IF neighborhoods. This model, based on any two IF binary relations defined on a non-empty universe, leads to the development of two novel IF graph approximation spaces aimed at reducing the boundary region of fuzzy uncertainty and increasing the precision degree of the fuzzy approximations. Furthermore, key results pertaining to both types of IF graph approximations are established. The relationships between the existing IF approximation methods are derived, and comparisons are made to demonstrate that the proposed approaches are more general than previous models. Finally, we explore an application of these IF graph approximation spaces in decision-making contexts and propose an algorithm to facilitate solving such problems.

Intuitionistic Fuzzy sets combine the ideas of uniformity, membership, and non-membership grades of the elements. Similarly, Intuitionistic Fuzzy Graphs are the generalization of simple fuzzy graphs. Depending on the uniformity of the fuzzy graphs (USIF) they can be categorized in different ways via membership values. From the idea of uniform fuzzy topological indices, we have developed the concepts of uniform intuitionistic fuzzy topological indices for uniform intuitionistic fuzzy graphs. This idea provides a more adaptable and nuanced representation of structural properties in graphs or networks. According to the theory of fuzzy topological indices, the importance of topological indices changes depending on the circumstance and the specific problem at hand. When the interactions between nodes are uncertain but not always hesitant, fuzzy graph theory and its adjusted topological indices are sufficient to capture and assess the underlying structure. In such cases where uncertainty is more complicated and hesitation is a major problem then there are better ways to address by intuitionistic fuzzy graph theory and the topological indices that go along with it. This article, developed the concept of uniform intuitionistic fuzzy graphs afresh and proposed Intuitionistic Fuzzy Topological Indices. We determine these indices using the topological indices and labeling of crisp graphs, rather than relying on the degrees of intuitionistic fuzzy graphs and edge portions. This approach is then applied to find intuitionistic fuzzy topological indices. Also, we have provided the MATLAB algorithm to illustrate the concept of IF labeling of cellular neural networks of any order. An example is given to explain the idea and approach towards one kind of uniform intuitionistic fuzzy graph represented by Cellular Neural Networks and graphical plots of the indices involved are also made.

Resilience is a topic receiving much attention in relation to energy systems, with particular attention being paid to the supply of electricity. As a result of the growing interest in energy sector resilience, research communities have proposed a plethora of candidate resilience indicators and metrics, most of which remain immature at different scales and segments within the energy system. A necessary focus of the research community lies in implementing, testing, and validating resilience metrics and analysis approaches in energy sector models, which will be invaluable for informing resilience planning and investment decisions. Recognizing these challenges that need to be addressed, we explore how to effectively integrate resilience considerations into energy sector models and tools. The overarching goal of the effort was to evaluate the data needs, methodologies, and outcomes - including consequences and/or changes in investment or operational decisions due to avoided consequences - based on resilience analysis in a range of existing tools. In particular, we selected five models originally built at NREL to explore non-resilience energy research questions to implement and exercise resilience metrics and analysis approaches. To demonstrate the importance of perspective, we selected models that represent different segments of the energy sector, geographic scales, and modeling approaches. A second important aspect of our effort was the development of generalized power interruption scenarios. These scenarios were intended to help establish a framework for simulating the effects of real-world threats in terms of their impacts on system components and, in turn, power interruption.

Imminent climate change impacts call for stronger energy system modeling approaches in order to design resilient communities. This study presents a flexible framework to integrate resilience analysis within the scope of long-term energy system optimization models (ESOMs). It employs a multi-objective resilience metric approach for energy system design, which allows for the independent representation and treatment of resilience and sustainability metrics. Several energy system-characterizing resilience and sustainability metrics are identified and integrated into a composite resilience metric, which is maximized as the objective function in an open-source ESOM. The cost performance of the resulting energy system design is tested across a range of short-term resilience scenarios, capturing different shocks. The method is demonstrated on two municipal case studies (located in China and Ghana). Three energy systems are designed and compared based on cost, emission, and resilience optimization objectives. Results illustrate a wide range of cost impacts depending on the system and resilience scenario. Systems designed based on a resilience objective offer more flexibility to adapt to and absorb shocks, thus reducing damage costs. Case study findings illustrate the value of incorporating resilience analysis into conventional ESOM and energy planning approaches in order to build more resilient communities.

Objectives: To remove the hesitation that appears in choosing membership degree of an element from some possible values. Methods: Through Intuitionistic Fuzzy Graph and Intuitionistic Double Layered Fuzzy Graph, we developed a new graph to represent the hesitancy degree. Findings: Further we have proved some theoretical concepts and principle properties which have the roots from Double Layered Fuzzy Graph and Intuitionistic Double layered Fuzzy Graph. Conclusion: In this paper,a new fuzzy graph called Hesitancy Fuzzy Graphs (HFGs) is introduced.

Graph Isomorphism is the illustration of same graph in more than one pattern. In order to check that the graphs are isomorphic it should follow some properties. Graphs have many real life applications. Here we are considering Intuitionistic Fuzzy Graphs. As it is extended from crisp graph, Intuitionistic fuzzy graphs have all the properties of crisp graphs. Intuitionistic Fuzzy graph theory is tracking down an expanding number of uses in demonstrating genuine frameworks and circumstances where the level of data is uncertain. In the field of networks, one has to take into account whether two networks are similar or not. In this context studying isomorphism of Intuitionistic Fuzzy graphs become important. In this paper we propose algorithms to check weak isomorphism and co-weak isomorphism between Intuitionistic Fuzzy Graphs. We have applied this algorithms to detect Intuitionistic Fuzzy Graph isomorphism with the suitable examples.

Grid resilience is a concept related to a power system's ability to continue operating and delivering power even in the event that low probability, high-consequence disruptions such as hurricanes, earthquakes, and cyber-attacks occur. Grid resilience objectives focus on managing and, ideally, minimizing potential consequences that occur as a result of these disruptions. Currently, no formal grid resilience definitions, metrics, or analysis methods have been universally accepted. This document describes an effort to develop and describe grid resilience metrics and analysis methods. The metrics and methods described herein extend upon the Resilience Analysis Process (RAP) developed by Watson et al. for the 2015 Quadrennial Energy Review. The extension allows for both outputs from system models and for historical data to serve as the basis for creating grid resilience metrics and informing grid resilience planning and response decision-making. This document describes the grid resilience metrics and analysis methods. Demonstration of the metrics and methods is shown through a set of illustrative use cases.

Resilience is a topic receiving much attention in relation to energy systems, with particular attention being paid to the supply of electricity. As a result, research communities have proposed a plethora of candidate indicators and metrics for resilience, most of which remain immature at different scales and segments within the energy system. Given the complexity of resilience analyses and mitigation strategies, there is limited value in attempting to identify a single resilience metric, as no one metric can quantify resilience or its associated value for all stakeholders. Instead, a necessary focus of the research community should lie in implementing, testing, and validating resilience metrics and analysis approaches in energy sector models, which will be invaluable for informing resilience planning and investment decisions. Recognizing that implementing, testing, and validating resilience metrics are challenges that need to be addressed, the National Renewable Energy Laboratory (NREL) dedicated staff and time to researching how to effectively integrate resilience considerations into energy sector models and tools, as part of the Laboratory Directed Research and Development (LDRD) program. The overarching goal of the effort was to evaluate the data needs, methodologies, and outcomes - including consequences and/or changes in investment or operational decisions due to avoided consequences - based on resilience analysis in a range of existing tools. In particular, we selected five models originally built at NREL to explore non-resilience energy research questions to implement and exercise resilience metrics and analysis approaches.

Group decision-making (GDM) is crucial in various components of graph theory, management science, and operations research. In particular, in an intuitionistic fuzzy group decision-making problem, the experts communicate their preferences using intuitionistic fuzzy preference relations (IFPRs). This approach is a way that decision-makers rank or select the most desirable alternatives by gathering criteria-based information to estimate the best alternatives using a wider range of knowledge and experience. This article proposes a new statistical measure in a fuzzy environment when the data is ambiguous or unreliable to solve a decision-making problem. This study uses the variation coefficient measure combined with intuitionistic fuzzy graphs (IFG) and Laplacian energy (LE) to solve a GDM problem that utilizes intuitionistic fuzzy preference relations (IFPRs) to select a reliable alliance partner. Initially, the Laplacian energy determines the weight of individual standards, and the obtained weight average further estimates the overall criterion weight vector. We establish the authority criteria weights using the variation coefficient measure and then ultimately rank the alternatives for each criterion using the same measure. We examine four distinct companies Alpha, Beta, Delta, and Zeta to conduct a realistic GDM to choose which alliance partner would be ideal. We successfully implemented the suggested technique, determining that Alpha satisfies company standards and is ranked first among other companies. Moreover, this technique is useful for all kinds of Intuitionistic fuzzy group decision-making problems to select optimal ones.

It is known that Intuitionistic fuzzy models give more precision, flexibility and compatibility to the system as compared to the classic and fuzzy models. Intuitionistic fuzzy tree has an important role in neural networks, computer networks, and clustering. In the design of a network, it is important to analyze connections between the levels. In addition, the intuitionistic fuzzy tree is becoming increasingly significant as it is applied to different areas in real life. The study proposes the novel concepts of intuitionistic fuzzy graph (IFG) and some basic definitions. We investigate the types of arcs, for example, alpha _{mu }-strong, beta _{mu }-strong, and delta _{mu }-arc in an intuitionistic fuzzy graph, and introduce some of their properties. In particular, the present work develops the concepts of intuitionistic fuzzy bridge (IFB), intuitionistic fuzzy cut nodes (IFCN) and some important properties of an intuitionistic fuzzy bridge. Next, we define an intuitionistic fuzzy cycle (IFC) and an intuitionistic fuzzy tree (IFT). Likewise, we discuss some properties of the IFT and the relationship between an intuitionistic fuzzy tree and an intuitionistic fuzzy cycle. Finally, an application of intuitionistic fuzzy tree is illustrated in other sciences.

In this paper, we provide three operations on Intuitionistic fuzzy graphs, namely direct product, semi strong product and strong product on Intuitionistic fuzzy graphs. We give sufficient condition for each one of them to be complete and we show that if any of these product is complete, then at least one factor is a complete intuitionistic fuzzy graph. Moreover, we introduced and study the notion and sufficient conditions for the preceding products of two Intuitionistic fuzzy balanced graphs to be balanced and we prove that any isomorphic Intuitionistic fuzzy graph to a balanced Intuitionistic fuzzy graph must be balanced.

Wastewater treatment facilities’ main goal is to protect the public and environment from the hazardous and poisonous materials found in wastewater. Water treatment facilities were developed to speed up the natural process of cleansing water. A novel cosine similarity measure across intuitionistic fuzzy graphs has been proven to be more effective than certain present ones in group decision-making issues using example verification. This paper provides a unique approach for calculating expert-certified, well-known scores by finding the ambiguous information of intuitionistic fuzzy preference relations as well as the regular cosine similarity grades from one separable intuitionistic fuzzy preference relation to another. The new technique considers both "objective" and "subjective" information provided by experts. Using intuitionistic fuzzy preference relations, we provide workable techniques for judging experts’ eligible reputational ratings. This can be used to raise or decrease the relevance of the stated criteria in an evaluation that takes into account several competing elements. We give a solution to a decisional problem by using two effective methods: the newly constructed cosine similarity measure and the Seidel Laplacian energy (SLe+) of an intuitionistic fuzzy graph. Finally, two working procedures and circumstances are offered to show the effectiveness and superiority of the proposed techniques.

In this paper, the algorithm for finding a Hamiltonian cycle in an intuitionistic fuzzy graph (IFG) is proposed, based on the theories of intuitionistic fuzzy sets (IFSs) and of index matrices (IMs). The aim of the paper is to extend the algorithm to find a fuzzy Hamiltonian cycle (FHC) in an IFG to the intuitionistic fuzzy (IFHC) using the IFSs and IMs concepts. An intuitionistic fuzzy graph example about network of Wizz air airlines is modeled by the extended IM to illustrate the proposed algorithm. In the paper also are introduced for the first time three index-type operations over IMs.

The authors have designed and developed algorithms for pattern recognition and clustering techniques using intuitionistic fuzzy (IF) sets, IF operators, IF logic (IFL) – shortest path in networks using IF graphs and IF hypergraphs – video processing using temporal IF sets, RGB image representation through IF index matrices, and molecular structure representation through IF directed hypergraphs. The three major steps involved in the above-said modeling processes via IFSs are (i) intuitionistic fuzzification, (ii) modification of membership and non- membership values (using IF logic/operators/rules/relations) and (iii) intuitionistic defuzzification. While developing these algorithms, parameter tuning was one of the major limitations, and hence specific values were assigned to complete the running process. To overcome this, it is necessary to introduce a toolbox in MATLAB so that the users can select the appropriate tools and parameterize them. Hence, in the long process of contributing a full-pledged intuitionistic fuzzy logic toolbox, namely IFL Toolbox in MATLAB, the membership and non-membership functions gallery has been developed initially, as one of the modules which is the foundation for any IFL control system. This module contains functions, codes, examples and figures/graphs, which will be available on the MATLAB creation page. The proposed module is compared with the existing fuzzy logic toolbox in MATLAB and verified.