Membership Degree Research Articles

A Study of n-ary Groupoids Based on Hesitant Fuzzy Sets

Fuzzy set theory plays a vital role in solving many complicated problems dealing with uncertainty. An [Formula: see text]-ary algebraic system as a generalization of algebraic structures that allow for operations involving more than two elements. They provide a natural framework for representing and manipulating complex relationships and interactions among multiple elements, which is essential for solving many real-world problems. Many authors have studied fuzzy set theory over an [Formula: see text]-ary algebraic systems and have provided many fruitful results. Hesitant fuzzy set theory is a further extension of fuzzy set theory, which allows for the representation of uncertainty in decision making where decision makers may have multiple possible choices or may be uncertain about the degree of membership of an element in a given set. No authors have studied, by combining hesitant fuzzy set theory with an [Formula: see text]-ary algebraic systems. This motivates us to study hesitant fuzzy set over an [Formula: see text]-ary algebraic system. In this paper, we have applied hesitant fuzzy set theory in an [Formula: see text]-ary algebraic systems and introduced the notions of hesitant fuzzy subgroupoid. We provide the characterization of hesitant fuzzy [Formula: see text]-ary subgroupoid over an [Formula: see text]-ary groupoids and have studied their related properties. As an application of hesitant fuzzy set over an [Formula: see text]-ary groupoids, we have developed the concept of normal hesitant fuzzy subgroupoids over an [Formula: see text]-ary groupoids and have studied their various properties.

Quantification of Expert Knowledge in Describing COLREGs Linguistic Variables

The International Regulations for Preventing Collisions at Sea 1972 (COLREGs) have been the cornerstone of maritime navigation since their introduction. Knowledge and implementation of these rules are paramount in collision avoidance at sea. However, terms found in these rules are sometimes imprecise or fuzzy, as they are written by humans for humans, giving them some freedom in interpretation. The term Very Large Ship used in Rule 7 of the COLREGs is, by its nature, fuzzy. While human navigators understand this term’s meaning, it could be challenging for machines or autonomous ships to understand such an unprecise expression. Fuzzy sets could easily describe unprecise terms used in maritime navigation. A fuzzy set consists of elements with degrees of membership in a set, making them perfect for interpreting some terms where boundaries are unclear. This research was conducted among 220 navigational experts to describe linguistic variables used in maritime regulations. This research consists of an internationally distributed questionnaire. Membership data were collected with the adapted horizontal method, and the results were statistically analyzed, followed by regression analyses to describe the range and shape of membership functions. A conceptual model of the implementation of linguistic variables is presented. The novelty of this study derives from the data collecting, modeling, and quantification of the important but neglected linguistic term Very Large Ship based on a large number of navigational experts. The same quantification method could be easily used for other COLREGs linguistic variables, which could easily lift barriers to advances in intelligent solutions based on fuzzy sets. The obtained quantified fuzzy sets can be used in decision support or control systems used by conventional or autonomous ships in the future.

Adaptive local neighborhood information based efficient fuzzy clustering approach

The purpose of clustering is to partition data similar with each other into a same group and partition data dissimilar with each other into different groups. However, in most existing fuzzy clustering approaches, the membership degrees of an individual belonging to different clusters are relied on the different distances between the individual and different cluster centroids, the similarity between the individual and data in different clusters are ignored, besides, the outliers cannot be distinguished effectively. For improving the clustering performance and addressing the problems in most existing fuzzy clustering approaches, based on the concept that data close to each other should be grouped together and be assigned to a same cluster, in this paper, we present a novel efficient fuzzy clustering approach, in which the adaptive local neighborhood information of each data referring to different clusters is taken into consideration; an entirely new idea that the membership degree values of an individual referring to different clusters should not only depend on the distances between the individual and different cluster centers, but also rely on the distances between the individual and several nearest neighbor data in different clusters is put forward; a new scheme to search for the outliers is presented, a method for identifying the different importance of different features is introduced. Experiments on synthetic and publicly real-world datasets were conducted, the clustering results show that the approach put forward can yield consistently satisfying clustering performance and has significant advantages.

A new soft clustering method for traffic prediction in bike-sharing systems

For the efficient management of bike-sharing systems (BSSs), accurate demand predictions are crucial to address the uneven distribution of bikes at various stations. Recent studies have explored a hierarchical prediction framework using cluster-level models to more accurately estimate demand at the station level. However, in frameworks based on hard clustering, where each station is exclusively assigned to one of several clusters, prediction accuracy tends to be lower for stations at the cluster boundaries. To improve accuracy for such stations, this study proposes a novel soft clustering algorithm for BSSs. The key idea is to allow stations to belong to multiple clusters, calculating the membership degree for each station based on transitions between stations and clusters obtained through hard clustering. This study also investigated the impact of restricting clusters to which individual stations belong based on distance or usage history. Two approaches, distance- and usage-based, were employed to determine the clusters to which each station belongs. Experimental results using Seoul Bike data demonstrate the effectiveness of the proposed method in enhancing traffic prediction accuracy within the hierarchical prediction framework. Notably, excluding clusters with minimal usage for each station using the usage-based approach yielded the best performance.

Trust Improvement Consensus Model Considering Unreliability Degree of Opinions with Hesitant Fuzzy Sets

Trust is a crucial element in the consensus decision-making process, as it significantly impacts a group’s capacity to achieve consensus. Nevertheless, low trust by experts may result in detrimental decision-making behaviors that decrease group consensus degree. Meanwhile, it is important to note that experts’ opinions are not always reliable, and ignoring the unreliability degree of opinions may have an impact on the result of consensus decision-making. To deal with these issues, this paper proposes a trust improvement consensus model considering unreliability degree of opinions. First, considering that the discreteness of membership degree has a much smaller impact on the hesitancy degree in hesitant fuzzy elements than the number of membership degrees in extant research, an improved approach is suggested to quantify the unreliability degree of opinions. Second, a trust improvement method is proposed based on the score function for trust propagation path and the trust propagation operator considering unreliability degree of opinions. Third, a method for adjusting the opinions of experts is suggested, considering unreliability degree of opinions. Finally, the feasibility, effectiveness, and advantages of the proposed trust improvement consensus model are verified through a case study on the selection of cascade utilization alternatives for power lithium-ion batteries, as well as simulation and comparative analysis. The results show that: (1) The improved method exhibits a significant impact of the discreteness of membership degree on the hesitancy degree; (2) Trust improvement can facilitate group consensus, and the score function of trust propagation path can enhance trust degree among experts more effectively; (3) The unreliability degree of opinions not only affects the degree of trust improvement but also affects the ranking of alternatives. By taking into account the unreliability degree of opinions throughout the process of adjusting opinions, it is possible to promote group consensus and decrease the unreliability degree of opinions.

Application of Research on Risk Assessment of Roadway Roof Falls Based on Combined Weight Matter Element Extension Model

Roof falls in coal mine roadways are the main causes of many casualties, shutdowns and production plan delays. To understand the relationship between the influencing factors of roadway roof fall accidents and the importance ranking of the accidents, we will reduce safety accidents in coal mines. To enable the timely prediction and control of roadway roof fall risks, based on the investigation of many roadway roof fall risk factors, 12 evaluation indexes such as the roadway roof rock thickness, geological conditions and roadway section shape were selected. An evaluation index system of roadway roof fall risks is constructed. A risk degree standard of roadway roof falls is proposed. The risk evaluation model of roadway roof falls was established by using the combination weight of the analytic hierarchy process (AHP), entropy weight method (EW) and matter element extension theory. According to the principle of the maximum membership degree, the risk degree of roadway roof falls is determined. Based on Java Web, a risk assessment system for roadway roof falls was developed. We name the system Multiple Weight-Material Element Web (MW-MEW). The MW-MEW system was used to evaluate the risk degree of roof falls in the C9 return airway of the Xingu Coal Mine. Compared with the evaluation results of the AHP matter element extension model, it is found that the evaluation results of the MW-MEW system are more in line with the actual engineering conditions. The successful application of the MW-MEW system will provide new avenues for the quantitative evaluation of roof fall risks in coal mine roadways.

Novel strict intuitionistic fuzzy similarity measures-based on fuzzy negation and their applications

The concept of similarity measure is fundamental for evaluating the consistency between intuitionistic fuzzy sets (IFSs). Several researchers have developed various intuitionistic fuzzy similarity measures (IFSimMs) utilizing intuitionistic fuzzy distance measures (IFDisMs) or combinations of membership and non-membership degrees of IFSs. Nevertheless, there is still ample room for the improvement, as some of these measures do not satisfy the axiomatic properties of IFSimMs and produce unreasonable results. In this paper, we first establish a dual relationship between IFDisMs and IFSimMs based on fuzzy negations, enabling the construction of an infinite number of IFSimMs from a given IFDisM. We then propose two novel IFSimMs by directly manipulating the membership and non-membership degrees of IFSs and prove that they satisfy the axiomatic properties of strict IFSimMs (SIFSimMs). Furthermore, a comparative analysis reveals that the proposed SIFSimMs do not yield any unreasonable results in various scenarios. Finally, we apply these SIFSimMs to pattern recognition, medical diagnosis, and clustering analysis problems, demonstrating their superior performance compared to some existing measures.

Electrical Sensor Calibration by Fuzzy Clustering with Mandatory Constraint.

Electrical tomography sensors have been widely used for pipeline parameter detection and estimation. Before they can be used in formal applications, the sensors must be calibrated using enough labeled data. However, due to the high complexity of actual measuring environments, the calibrated sensors are inaccurate since the labeling data may be uncertain, inconsistent, incomplete, or even invalid. Alternatively, it is always possible to obtain partial data with accurate labels, which can form mandatory constraints to correct errors in other labeling data. In this paper, a semi-supervised fuzzy clustering algorithm is proposed, and the fuzzy membership degree in the algorithm leads to a set of mandatory constraints to correct these inaccurate labels. Experiments in a dredger validate the proposed algorithm in terms of its accuracy and stability. This new fuzzy clustering algorithm can generally decrease the error of labeling data in any sensor calibration process.

Well Selection for CO2 Huff-n-Puff in Unconventional Oil Reservoirs Based on Improved Fuzzy Method

The implementation of CO2 huff-n-puff in unconventional oil reservoirs represents a green development technology that integrates oil recovery and carbon storage, emphasizing both efficiency and environmental protection. A rational well selection method is crucial for the success of CO2 huff-n-puff development. This paper initially identifies eight parameters that influence the effectiveness of CO2 huff-n-puff development and conducts a systematic analysis of the impact of each factor on development effectiveness. A set of factors for well selection decisions is established with seven successful CO2 huff-n-puff cases. Subsequently, the influencing factors are classified into positive, inverse, and moderate indicators. By using an exponential formulation, a method for calculating membership degrees is calculated to accurately represent the nonlinearity of each parameter’s influence on development, resulting in a dimensionless fuzzy matrix. Furthermore, with the oil exchange ratio serving as a pivotal parameter reflecting development effectiveness, recalibration of weighting factors is performed in conjunction with the dimensionless fuzzy matrix. The hierarchical order of weighting factors, from primary to secondary, is as follows: porosity, reservoir temperature, water saturation, formation pressure, reservoir thickness, crude oil density, crude oil viscosity, and permeability. The comprehensive decision factor and oil exchange ratio exhibit a positive correlation, affirming the reliability of the weighting factors. Finally, utilizing parameters of the Ordos Basin as a case study, the comprehensive decision factor is calculated, with a value of 0.617, and the oil exchange ratio is predicted as 0.354 t/t, which falls between the Chattanooga and Eagle Ford reservoirs. This approach, which incorporates exponential membership degrees and recalibrated weighting factors derived from actual cases, breaks the limitations of linear membership calculation methods and human factors in expert scoring methods utilized in existing decision-making methodologies. It furnishes oilfield decision-makers with a swifter and more precise well selection method.

A Statistical Evaluation Method Based on Fuzzy Failure Data for Multi-State Equipment Reliability

For complex equipment, it is easy to over-evaluate the impact of failure on production by estimating the reliability level only through failure probability. To remedy this problem, this paper proposes a statistical evaluation method based on fuzzy failure data considering the multi-state characteristics of equipment failures. In this method, the new reliability-evaluation scheme is firstly presented based on the traditional statistical analysis method using the Weibull distribution function. For this scheme, the failure-grade index is defined, and a fuzzy-evaluation method is also proposed by comprehensively considering failure severity, failure maintenance, time, and cost; this is then combined with the time between failures to characterize the failure state. Based on the fuzzy failure data, an improved adaptive-failure small-sample-expansion method is proposed based on the classical bootstrap method and the deviation judgment between distributions of the original and newborn samples. Finally, a novel reliability-evaluation model, related to the failure grade and its membership degree, is established to quantify the reliability level of equipment more realistically. Example cases for three methods of the scheme (the failure-grade fuzzy-evaluation method, the sample-expansion method, and the reliability-evaluation modeling method) are presented, respectively, to validate the effectiveness and significance of the proposed reliability-evaluation technology.

Similarity measure on intuitionistic fuzzy sets based on Benchmark Line and it’s diverse applications

An intuitionistic fuzzy set (IFS) is an extension of fuzzy sets that can deal with inconsistent and uncertain information more accurately. In this paper, a novel similarity measure is presented that might be applied to several application issues in decision-making, pattern recognition, and medical diagnostics. IFSs are sources of information that contain both the membership and non-membership degrees of the elements in the set. As such, similarity measures based on geometric concepts might occasionally be misleading. Hence, five parameters are specified to develop the similarity measure. These parameters are membership information similarity, non-membership information similarity, maximum cross-information similarity, minimum cross-information similarity, and product cross-information similarity. Most of the existing similarity measures cannot reasonably determine the appropriate similarity degree when the IFS points lie on the benchmark line. On the other hand, product cross-information similarity is a new parameter introduced in this study that has a significant role in handling benchmark problems more effectively and reasonably. The proposed similarity measure compensates for several counterintuitive problems related to decision-making and pattern recognition. Complexity analysis illustrates the complexity level of the proposed measure, which is reliable. Finally, various challenges with applications in decision-making, pattern recognition, and medical diagnosis difficulties are used to validate the suggested similarity measure.

Energy efficiency as a driver of the circular economy and carbon neutrality in selected countries of Southern Europe: a soft computing approach

BackgroundThe main goal of the paper is to define the level of energy efficiency in the economies of selected countries in the Balkan region that have opted for the EU Green Deal, a circular economy, and a transition to carbon neutrality. Energy efficiency, as a determinant of carbon neutrality, was selected as an indicator for analysis because it records particularly unfavorable indicators in the region under observation. The research was carried out on a sample of seven Balkan countries and their surrounding areas. An initial qualitative analysis was followed by a quantitative analysis based on a combination of statistical methods and soft computing. Six indicators were selected for the analysis covering a period of 30 years (1990–2020).ResultsA significant obstacle to the green transition and the region’s transition to a circular economy and carbon neutrality is energy efficiency and energy related pollution—the reliance of most countries on coal-fired thermal power plants for electricity generation. The research results showed the following: (a) the degree of economic development and membership in the European Union are not significantly related to the level of energy efficiency; (b) most of the sampled countries are in the initial stages of introducing activities to achieve carbon neutrality; and (c) only Slovenia has documented consistent indicators and evident advancements in its efforts to achieve carbon neutrality. Based on the research findings, proposals for improvements were made in the direction of policymaking and in a methodological sense.ConclusionsThe implementation of circularity and carbon neutrality as a long-term goal of the European Union is not necessarily related to the level of economic development, nor can its trajectory be exclusively ascertained by means of data processing and monitoring. A more precise understanding of a carbon-neutral future can be achieved through the incorporation of qualitative data to a greater extent, a realistic evaluation of historical facts and their repercussions, as well as projections of the effects that reality and global developments after 2022 will have on each country.

New information measures for linear Diophantine fuzzy sets and their applications with LDF-ARAS on data storage system selection problem

In addition to membership and non-membership degrees, linear Diophantine fuzzy set (LDFS) incorporates two reference parameters in fuzzy set definition to represent human judgment more comprehensively. It also provides flexibility in updating the decision setting by changing the meaning of the reference parameters when required by the decision-maker. Information measures in fuzzy sets have many application areas in the literature. Especially, distance and entropy measures are utilized in multiple attribute decision-making (MADM) field. Distance measures are needed by distance-based methods like TOPSIS and EDAS because their alternative ranking procedure is based on measuring the distances between alternatives and some artificial ideal or average solutions. Entropy measure is a very important tool being used in objective attribute weighting procedures of MADM applications. Thus, new distance and entropy measure propositions for LDFS are the first contribution of the study. Also, Additive Ratio Assessment (ARAS) method is extended into LDFS (LDF-ARAS) for the first time in the literature as a second contribution. In this extension, we selected ARAS as one of the most cited tools because it is found flexible and easily adaptable by researchers who do not have any background in MADM. The proposed entropy measure is integrated with the objective attribute weighting procedure of the novel LDF-ARAS approach. The third contribution is the newly developed decision model proposed for a real-life data storage system (DSS) selection problem. Three DSS options were evaluated with respect to thirteen attributes by five skilled and well-educated experts and LDF-ARAS was performed to rank alternatives. The entropy-based LDF-ARAS approach was validated by comparing the results of six different applications. In the first three, different entropy formulas are used and in the last three, these entropy measures are integrated with LDF-TOPSIS. All six comparisons gave similar alternative rankings.

[formula omitted] Neutrosophic aggregation operators and their applications in the software site selection

In this paper, we expended the concept of neutrosophic sets (NS) by introducing the idea α,β,γ− neutrosophic set (α,β,γ− NS). The existing models under conventional NSs, fail to adequately address the management of membership degree influence during the aggregation process. While the proposed framework manages the influence of membership degree (MD), indeterminacy membership degree (IMD), and non-membership degree (NMD) by incorporating parameters α, β, and γ. Furthermore, we defined some fundamental operational laws for α,β,γ− NSs and introduced a series of aggregation operators (AOs) to effectively combine α,β,γ− neutrosophic information. Based on these AOs, a new Multiple Criteria Decision Making (MCDM) model is proposed for solving real-life decision-making (DM) challenges.An illustrative case study is presented to showcase the effectiveness of the proposed model in selecting an optimal location for a software office. The article concludes by validating the proposed model's authenticity and effectiveness through a comparative analysis with existing approaches.

The application of cosine similarity measures with Laplacian energy to q-rung orthopair fuzzy graphs in decision-making problems

Q-rung orthopair fuzzy sets (q-ROFS), which are better than the intuitionistic and Pythagorean fuzzy sets, are a significant tool for expressing ambiguous information. Their key feature is that their ability to represent a larger space of uncertain information is based on the fact that the product of the qth power of the membership degree and the qth power of the degrees of non-membership is equal to or less than 1. Under these circumstances, we train group decision-making problems in the study using q-rung orthopair fuzzy inclination associations. Through the calculation of the standard deviation of one separable q-rung orthopair inclination association to the others and the unclear evidence of q-rung orthopair fuzzy inclination connections, we propose a novel approach to estimate the qualified reputation weights of authority. The “internal” and “impartial” evidence of authority is taken into consideration by this new mindset. Subsequently, we included the weights of authorities into the q-rung orthopair fuzzy inclination relations and used a relative similarity approach to determine the relevance of replacements and the best substitutes. The planned techniques' usefulness and realism are demonstrated by the contrast analysis with additional methods through mathematical demonstrations, both of which show the fuzzy set’s membership degree and non-membership degree, respectively.

Minimum nonprobabilistic entropy deconvolution for fault diagnosis of rolling element bearings

The blind deconvolution methods (BDMs) is one of the most common methods for fault diagnosis of rolling bearings, and it is essential to maintain the safe and reliable operation of mechanical equipment. However, noise interference and the need for prior periods limit the scope of application of the BDMs. In this paper, a new minimum nonprobabilistic entropy deconvolution (MNPED) method is proposed. According to the correlation between fault impact and non-Gaussianity, the Gaussian membership function in fuzzy set theory is used to map the sample points to the membership degree of Gaussian distribution, and then the nonprobabilistic entropy (NPE) is formed to measure the impact characteristics of the signal. Then NPE is incorporated into the iterative process of solving the filter coefficient. Finally, the target signal and the optimal filter coefficient are selected based on the criterion of minimum NPE. MNPED is capable of adaptively extracting the periodic pulse of a signal without requiring prior knowledge of the period, even in the presence of strong noise interference. The effectiveness and robustness of the proposed approach are validated through simulation and experimental data.

Simulation-Based Resilience Evaluation for Urban Rail Transit Transfer Stations

Disturbances often occur in transfer stations; however, little is known about the weaknesses of transfer stations and their ability to cope with passenger flows. Therefore, this paper introduces resilience into the study of transfer stations to enhance their emergency response processes and improve the sustainability of URT networks. It establishes a two-level fuzzy evaluation model, using the G1 weighting method, to assess resilience across various scenarios (daily operation, heavy passenger flow, and emergencies) and identify weaknesses; then, corresponding enhancement strategies are proposed. First, factor sets are established according to resilience stages, including rapidity before disturbance, robustness, redundancy, resourcefulness, and rapidity after disturbance. Using the G1 method, the weight matrix for each factor is calibrated, and a membership degree matrix is determined based on their affiliation with the review set. Multiplying the weight matrix and membership degree matrix yields the resilience value. We apply these steps to a representative station with the assistance of Anylogic simulation in calculating the hard-to-obtain data, yielding a peak-hour resilience value of 0.3425, which indicates a “poor” rating in the review set. By combining the peak-hour resilience with resilience curves under different multiples of peak-hour flows, an enhancement prioritization strategy is proposed for the station, which can act as a reference for the management of URT transfer stations.

Statistical Reliability Assessment with Generalized Intuitionistic Fuzzy Burr XII Distribution

Intuitionistic fuzzy sets provide a viable framework for modelling lifetime distribution characteristics, particularly in scenarios with measurement imprecision. This is accomplished by utilizing membership and non-membership degrees to accurately express the complexities of data uncertainty. Nonetheless, the complexities of some cases necessitate a more advanced approach of imprecise data, motivating the use of generalized intuitionistic fuzzy sets (GenIFSs). The use of GenIFSs represents a flexible modeling strategy that is characterized by the careful incorporation of an extra level of hesitancy, which effectively clarifies the underlying ambiguity and uncertainty present in reliability evaluations. The study employs a methodology based on generalized intuitionistic fuzzy distributions to thoroughly examine the uncertainty related to the parameters and reliability characteristics present in the Burr XII distribution. The goal is to provide a more accurate evaluation of reliability measurements by addressing the inherent ambiguity in the distribution’s shape parameter. Various reliability measurements, such as reliability, hazard rate, and conditional reliability functions, are derived for the Burr XII distribution. This extensive analysis is carried out within the context of the generalized intuitionistic fuzzy sets paradigm, improving the understanding of the Burr XII distribution’s reliability measurements and providing important insights into its performance for the study of various types of systems. To facilitate understanding and point to practical application, the findings are shown graphically and contrasted across various cut-set values using a valuable numerical example.

Research on a Classification Method of Goaf Stability Based on CMS Measurement and the Cloud Matter–Element Model

The evaluation and classification of goaf stability are fuzzy and random. To address this problem, a new classification method is proposed. A cavity monitoring system is used to detect the goaf, 3DMine and FLAC3D software are used to conduct the 3D visual modeling of the scanning results, and numerical simulation analysis is performed on the goaf. According to the analysis results, the stability classification standard of the goaf is constructed, and the characteristics of each classification are described. The evaluation indicator system of goaf stability is constructed in accordance with similar engineering experience, and the evaluation indicator is weighted by using the analytic hierarchy process. The cloud–element coupling evaluation model is built, the field measured values of indicators are collected, the cloud correlation degree of goafs belonging to each stability level is calculated, the stability level is evaluated according to the principle of maximum membership degree, and the results are compared with the numerical simulation to analyze the reasons for the differences in the stability evaluation results obtained by the two methods and to improve the accuracy of the evaluation of goaf stability. The pillar stress and surrounding rock deformation are monitored in Room 1# of the inclined mining area of Shirengou Iron Mine. The monitoring results are consistent with the evaluation results, which proves the accuracy of the proposed goaf stability classification method.

Some operations on intuitionistic fuzzy graphs via novel versions of the Sombor index for internet routing

Topological indices (TIs) are numerical structures that are associated with a graph to identify its topology. TIs are highly popular in the literature with a wide range of applications from chemistry to economics. However, TIs have limitations in representating complex relations within the graphs creating some uncertainities. Fuzzy graph (FG) and intuitionistic fuzzy graph (IFG) are introduced to overcome these uncertainities. While a FG a describes degree of membership of an object in a graph, IFG delineate information on membership or nonmembership under uncertainity. This study aims to introduce novel TIs such as the general second Zagreb index, the Sombor index of the third version, and the Sombor index of the fourth version in the IFG framework in order to improve practicality of FG and IFG applications. Some properties of the proposed indices and their upper bounds are provided as well. Proposed TIs are applied to an internet routing network as a case study. Results of the study show that adding more internet routers in the network can increase internet speed and the strength of the entire system. Finally, comparative studies for the Sombor index of the third version and the Sombor index of the fourth version are also revealed.