Applications of maximum matching by using bipolar fuzzy incidence graphs.

The extension of bipolar fuzzy graph is bipolar fuzzy incidence graph (BFIG) which gives the information regarding the effect of vertices on the edges. In this paper, the concept of matching in bipartite BFIG and also for BFIG is introduced. Some results and theorems of fuzzy graphs are also extended in BFIGs. The number of operations in BFIGs such as augmenting paths, matching principal numbers, relation between these principal numbers and maximum matching principal numbers are being investigated which are helpful in the selection of maximum most allied applicants for the job and also to get the maximum outcome with minimum loss (due to any controversial issues among the employees of a company). Some characteristics of maximum matching principal numbers in BFIG are explained which are helpful for solving the vertex and incidence pair fuzzy maximization problems. Lastly, obtained maximum matching principal numbers by using the matching concept to prove its applicability and effectiveness for the applications in bipartite BFIG and also for the BFIG.

Correction to: A study of plithogenic graphs: applications in spreading coronavirus disease (COVID-19) globally.

[This corrects the article DOI: 10.1007/s12652-022-03772-6.].

Connectivity Concepts in Intuitionistic Fuzzy Incidence Graphs with Application.

In this research article, we presented the idea of intuitionistic fuzzy incidence graphs (IFIGs) along with connectivity concepts. IFIGs are the generalization of fuzzy incidence graphs (FIGs). Specific ideas analogous to intuitionistic fuzzy cut-vertices and intuitionistic fuzzy bridges in intuitionistic fuzzy graphs, intuitionistic incidence cut-vertices, and intuitionistic incidence bridges are explored. The notion of intuitionistic incidence gain and intuitionistic incidence loss for intuitionistic incidence paths and pairs of vertices is also initiated. In the case of FIGs, we have only membership value, and we do not have non-membership value (NMSV). Therefore, we use IFIGs because they are more reliable, valuable, and helpful than FIGs. Also, we can not apply graphs, fuzzy graphs, and FIGs to the application provided in Sect. 3 due to the non-availability of NMSV. An application in selecting the best paint company for investment among different companies by using IFIG is also obtained.

A study of plithogenic graphs: applications in spreading coronavirus disease (COVID-19) globally.

During the last two decades, the world has experienced three major outbreaks of Coronaviruses, namely severe acute respiratory syndrome (SARS- CoV), middle east respiratory syndrome (MERS-CoV), and the current ongoing pandemic of severe acute respiratory syndrome 2 (SARS-CoV-2). The SARS-CoV-2 caused the disease known as Coronavirus Disease 2019 (COVID-19). Since its discovery for the first time in Wuhan, China, in December 2019, the disease has spread very fast, and cases have been reported in more than 200 countries/territories. In this study, the idea of Smarandache's pathogenic set is used to discuss the novel COVID-19 spread. We first introduced plithogenic graphs and their subclass, like plithogenic fuzzy graphs. We also established certain binary operations like union, join, Cartesian product, and composition of pathogenic fuzzy graphs, which are helpful when we discuss combining two different graphs. In the end, we investigate the spreading trend of COVID-19 by applying the pathogenic fuzzy graphs. We observe that COVID-19 is much dangerous than (MERS-CoV) and (SARS-CoV). Moreover, as the SARS-CoV and MERS-CoV outbreaks were controlled, there are greater chances to overcome the current pandemic of COVID-19 too. Our model suggests that all the countries should stop all types of traveling/movement across the borders and internally too to control the spread of COVID-19. The proposed model also predicts that in case precautionary measures have not been taken then there is a chance of severe outbreak in future.

Cyclic connectivity index of fuzzy incidence graphs with applications in the highway system of different cities to minimize road accidents and in a network of different computers.

A parameter is a numerical factor whose values help us to identify a system. Connectivity parameters are essential in the analysis of connectivity of various kinds of networks. In graphs, the strength of a cycle is always one. But, in a fuzzy incidence graph (FIG), the strengths of cycles may vary even for a given pair of vertices. Cyclic reachability is an attribute that decides the overall connectedness of any network. In graph the cycle connectivity (CC) from vertex a to vertex b and from vertex b to vertex a is always one. In fuzzy graph (FG) the CC from vertex a to vertex b and from vertex b to vertex a is always same. But if someone is interested in finding CC from vertex a to an edge ab, then graphs and FGs cannot answer this question. Therefore, in this research article, we proposed the idea of CC for FIG. Because in FIG, we can find CC from vertex a to vertex b and also from vertex a to an edge ab. Also, we proposed the idea of CC of fuzzy incidence cycles (FICs) and complete fuzzy incidence graphs (CFIGs). The fuzzy incidence cyclic cut-vertex, fuzzy incidence cyclic bridge, and fuzzy incidence cyclic cut pair are established. A condition for CFIG to have fuzzy incidence cyclic cut-vertex is examined. Cyclic connectivity index and average cyclic connectivity index of FIG are also investigated. Three different types of vertices, such as cyclic connectivity increasing vertex, cyclically neutral vertex and, cyclic connectivity decreasing vertex, are also defined. The real-life applications of CC of FIG in a highway system of different cities to minimize road accidents and a computer network to find the best computers among all other computers are also provided.

An extended TDM method under probabilistic interval-valued hesitant fuzzy environment for stock selection.

Generally, in real decision-making, all the pieces of information are used to find the optimal alternatives. However, in many cases, the decision-makers (DMs) only want "how good/bad a thing can become." One possibility is to classify the alternatives based on minimum (tail) information instead of using all the data to select the optimal options. By considering the opportunity, we first introduce the value at risk (VaR), which is used in the financial field, and the probabilistic interval-valued hesitant fuzzy set (PIVHFS), which is the generalization of the probabilistic hesitant fuzzy set (PHFS). Second, deemed value at risk (DVaR) and reckoned value at risk (RVaR) are proposed to measure the tail information under the probabilistic interval-valued hesitant fuzzy (PIVHF) environment. We proved that RVaR is more suitable than DVaR to differentiate the PIVHFEs with example. After that, a novel complete group decision-making model with PIVHFS is put forward. This study aims to determine the most appropriate alternative using only tail information under the PIVHF environment. Finally, the proposed methods' practicality and effectiveness are tested using a stock selection example by selecting the ideal stock for four recently enrolled stocks in China. By using the novel group decision-making model under the environment of PIVHFS, we see that the best stock is E4 when the distributors focus on the criteria against 10% certainty degree and E1 is the best against the degree of 20%, 30%, 40% and 50% using the DVaR method. On the other hand when RVaR method is used then the best alternative is E4 and the worst is E2 against the different certainty degrees. Furthermore, a comparative analysis with the existing process is presented under the PHF environment to illustrate the effectiveness of the presented approaches.

Hybrid BW-EDAS MCDM methodology for optimal industrial robot selection.

Industrial robots have different capabilities and specifications according to the required applications. It is becoming difficult to select a suitable robot for specific applications and requirements due to the availability of several types with different specifications of robots in the market. Best-worst method is a useful, highly consistent and reliable method to derive weights of criteria and it is worthy to integrate it with the evaluation based on distance from average solution (EDAS) method that is more applicable and needs fewer number of calculations as compared to other methods. An example is presented to show the validity and usability of the proposed methodology. Comparison of ranking results matches with the well-known distance-based approach, technique for order preference by similarity to ideal solution and VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) methods showing the robustness of the best-worst EDAS hybrid method. Sensitivity analysis performed using eighty to one ratio shows that the proposed hybrid MCDM methodology is more stable and reliable.

Modeling of linear programming and extended TOPSIS in decision making problem under the framework of picture fuzzy sets.

Picture fuzzy sets (PFSs) are comparatively a new extension of fuzzy sets which describe the human opinions that has more answers like acceptance, rejection, neutral and desist, which cannot be correctly presented in fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). The PFSs are categorized by three objects, the degree of belonging, the degree of neutral belonging and the degree of non- belonging such that the total of these three degrees must not be more than one. So far, there is no such work presented in the literature which deals with unknown weights of criteria based on PFSs. In the present work, we have developed a linear programming (LP) model to find the exact weights from the given constraints of weights for the criteria and construct a modified distance based on similarity measure between picture fuzzy sets. Then we have utilized this similarity measure to achieve the best option in the multiple criteria decision making (MCDM) problem. Lastly, two practical examples for the selection of alternatives are presented to compare the obtained results with the existing similarity measures.

Multicriteria group decision making by using trapezoidal valued hesitant fuzzy sets.

The concept of trapezoidal valued hesitant fuzzy set is introduced. Notion for distance between any two trapezoidal valued hesitant fuzzy elements is given. Using this proposed distance measure, we extend the technique for order preference by similarity to ideal solution for trapezoidal valued hesitant fuzzy sets. An example is constructed to show usefulness of this extension for multicriteria group decision making, where the opinions about the criteria values are expressed as trapezoidal valued hesitant fuzzy set.