Triangular Sets Research Articles

An integrated decision approach with triangular fuzzy neutrosophic sets for higher vocational education quality evaluation in the new era

China’s education has entered the era of quality, and in order for higher vocational colleges to develop better, effective measures must be taken to improve the quality of education. Facing the continuous innovation of education in higher vocational colleges, it is necessary to strengthen institutional construction and constrain educational management with institutional constraints, which is the key to ensuring the education quality. The higher vocational education quality evaluation in the new era is regarded as multiple attribute decision-making (MADM). Recently, the EDAS and CRITIC model has been employed to solve MAGDM. The triangular fuzzy neutrosophic sets (TFNSs) are constructed as an efficient tool for portraying the uncertain information during the higher vocational education quality evaluation in the new era. In this paper, the triangular fuzzy neutrosophic number EDAS (TFNN-EDAS) model based on the Hamming distance and Euclid distance is constructed to solve the MADM under TFNSs. The CRITIC method is utilized to obtain the weight information based on the Hamming distance and Euclid distance under INNSs. Finally, a numerical example of higher vocational education quality evaluation in the new era is constructed and some efficient comparisons are founded to verify the TFNN-EDAS method.

MADM framework based on the triangular Pythagorean fuzzy sets and applications to college public English teaching quality evaluation

The so-called “college English” teaching quality evaluation is to provide a basic, comprehensive, and realistic evaluation of the relevant aspects and management of teaching implementation on the basis of following the general laws of higher education; It is a comprehensive inspection of “College English” teaching and an important means of quality monitoring and policy adjustment for “College English”. As mentioned earlier, teaching evaluation is a comprehensive evaluation of teaching. Therefore, our evaluation of the quality of university public education is actually an examination of our specific measures in evaluating teaching, teaching methods and methods, teaching literature, and other aspects. The college public English teaching quality evaluation is a classical multiple attribute decision making (MADM). In this paper, we define the triangular Pythagorean fuzzy sets (TPFSs) and investigate the MADM problems under TPFSs. Based on the traditional dual generalized weighted Bonferroni mean (DGWBM) operator and dual generalized weighted geometric Bonferroni mean (DGWGBM) operator, some triangular Pythagorean fuzzy operators are proposed: triangular Pythagorean fuzzy DGWBM (TPFDGWBM) operator and triangular Pythagorean fuzzy DGWGBM (TPFDGWGBM) operator. Accordingly, we have took advantage of these operators to develop some approaches to work out the triangular Pythagorean fuzzy MADM. Ultimately, a practical example for college public English teaching quality evaluation is took advantage of to validate the developed approach, and an influence analysis of the parameter on the final results is been presented to attest its availability and validity.

Tuning of fuzzy controller with arbitrary triangular input fuzzy sets based on proximal policy optimization for time-delays system

The analytical structure of fuzzy controller is important for better understanding and effective design of the fuzzy control system, especially for explaining why fuzzy control system work well. In this paper, we develop a three-dimensional fuzzy controller with arbitrary triangular input fuzzy sets. Then, the analytical structure of this fuzzy controller is derived rigorously, and some properties of this controller are given. To determine the parameters of the fuzzy controller, a novel proximal policy optimization algorithm with recall pool (RP-PPO) is proposed. A recall pool is designed in proximal policy optimization algorithm to store the best ten sets of parameters to enhance the optimization results. Finally, simulation studies are carried out on two time-delay systems and a nonlinear chemical process with time-varying time delay to validate the applicability and effectiveness of the proposed fuzzy controller.

Uncertain support vector machine based on uncertain set theory

Support vector machine (SVM) is a supervised binary classifier with good generalization ability and excellent computational properties. It has been widely used in many fields such as image recognition, bioinformatics and so on. However, the traditional SVM requires the input data to be clear and knowable, while in the actual application process, there will be many cases that the input data is uncertain. In order to solve this problem, a new SVM model is proposed in this paper by combining the uncertainty theory with the SVM theory. The uncertainty theory was proposed by Liu in 2007, and it is often used to describe the uncertainty of things. The uncertain set in uncertainty theory is often used to model unsharp concepts. Therefore, this paper regards each uncertain data as an uncertain set and establishes a SVM model with uncertain chance constraints. However, the uncertain chance constraints are non-convex. Therefore, this paper gives the equivalent transformation process of constraint conditions when the input data are triangular uncertain sets. Finally, the non-convex constraint conditions are transformed into the linear constraint conditions, so that the model is transformed into a nonlinear programming model. In the numerical experiment, the Particle Swarm Optimization (PSO) algorithm is used to solve the problem, which proves the feasibility of the model.

Research on core competitiveness evaluation of small and medium sized enterprises with hesitant triangular fuzzy multiple attribute decision making

There is a huge number of small and medium-sized enterprises in China, playing an important role in the development of the market economy and being the main body of the real economy. The products and services they create account for more than half of the gross domestic product. At present, China’s economy is gradually transitioning from a stage of rapid growth to a stage of high-quality development, and it is necessary to further stimulate its innovation drive and release innovation energy. As a component of the national economy, achieving high-quality development of small and medium-sized enterprises has become one of China’s important strategic tasks. At present, the construction of the innovation service system for small and medium-sized enterprises in China is still in the initial stage, and there are still certain problems in the construction that need to be further improved. The core competitiveness evaluation of small and medium sized enterprises is a classical multiple attribute decision making (MADM) problems. Recently, the TODIM and TOPSIS method has been used to cope with MADM issues. The hesitant triangular fuzzy sets (HTFSs) are used as a tool for characterizing uncertain information during the core competitiveness evaluation of small and medium sized enterprises. In this manuscript, the hesitant triangular fuzzy TODIM-TOPSIS (HTF-TODIM-TOPSIS) method is built to solve the MADM under HTFSs. In the end, a numerical case study for core competitiveness evaluation of small and medium sized enterprises is given to validate the proposed method.

Triangular Pythagorean fuzzy Bonferroni mean operators in multiple attribute decision making and applications to college English smart classroom teaching quality evaluation

At the stage of college education, English is a very important subject, which is directly related to the enhancement of college students’ English literacy and the improvement of their English level. From the perspective of mobile internet, the construction and application of college English intelligent classroom teaching mode can promote the reform and improvement of traditional English teaching mode. By giving full play to students’ subjectivity, English teaching can be carried out in a modern teaching classroom environment, which is conducive to strengthening students’ comprehensive English ability, which is also very important and meaningful for building an efficient college English classroom. The college English smart classroom (CESC) teaching quality evaluation is a classical multiple attribute decision making (MADM). In this paper, we define the triangular Pythagorean fuzzy set (TPFS) and investigate the multiple attribute decision making problems under it. Accordingly, we take advantage of traditional Bonferroni mean (BM) operator to develop some triangular Pythagorean fuzzy information operators: triangular Pythagorean fuzzy Bonferroni mean (TPFBM) operator and triangular Pythagorean fuzzy weighted Bonferroni mean (TPFWBM) operator. The dominating natures of these operators are researched. Accordingly, the TPFWBM operator is built for triangular Pythagorean fuzzy MADM. Ultimately, a practical example for CESC teaching quality evaluation is took advantage of to validate the developed approach.

Extended group decision making method for quality evaluation of mental health education of college students with hesitant triangular fuzzy information

Mental health education (MHE) is an indispensable and fundamental content for schools to cultivate high-quality, high-standard, and high-quality talents, which has attracted sufficient attention and widespread attention in the educational community. Due to the impact of various factors such as society, family, and oneself, the overall psychological quality of Chinese college students is poor, their thinking is relatively childish, they cannot calmly face new problems and situations in life, and they feel vague and confused about life, ideals, and goals Confusion and other psychological barriers are becoming more and more common. If effective psychological crisis intervention is not available, it will seriously affect their learning and personality development, and is not conducive to talent growth. In recent years, Huaibei Vocational and Technical College has placed psychological quality education at the top of its list, providing good psychological quality training for college students, improving their psychological endurance, eliminating negative attitudes and psychological barriers, and promoting their mental health and sustainable development. The quality evaluation of MHE of college students is a classical multiple attribute decision making (MADM) problems. Recently, the TODIM and VIKOR method has been used to cope with MAGDM issues. The hesitant triangular fuzzy sets (HTFSs) are used as a tool for characterizing uncertain information during the quality evaluation of MHE of college students. In this manuscript, the hesitant triangular fuzzy TODIM-VIKOR (HTF-TODIM-VIKOR) method is built to solve the MADM under HTFSs. In the end, a numerical case study for quality evaluation of MHE of college students is given to validate the proposed method.

Modified GRA methodology for MADM under triangular fuzzy neutrosophic sets and applications to blended teaching effect evaluation of college English courses

Modified GRA methodology for MADM under triangular fuzzy neutrosophic sets and applications to blended teaching effect evaluation of college English courses

A novel approach based on neutrosophic Bonferroni mean operator of trapezoidal and triangular neutrosophic interval environments in multi-attribute group decision making

Neutrosophic multicriteria is a method of decision-making that uses indeterminacy to combine several criteria or elements, frequently with incomplete or ambiguous information, to find a solution. The neutrosophic multicriteria analysis enables the assessment of qualitative and subjective aspects and can assist in resolving conflicting goals and preferences. In the Neutrosophic Multi-Attribute Group Decision Making (NMAGDM) problems, all the information provided by the decision makers (DMs) is expressed as single value neutrosophic triangular and trapezoidal numbers examined in this study which can provide more flexibility and accuracy in capturing uncertainty and aggregating preferences. We offer a novel approach for determining the neutrosophic possibility degree of two and three trapezoidal and triangular neutrosophic sets and the concepts of neutrosophic possibility mean value. The trapezoidal and triangular neutrosophic Bonferroni mean (TITRNBM) operator and the trapezoidal and triangular neutrosophic weighted Bonferroni mean (TITRNWBM) operator are two aggregation methods we then create. Further, we examine the TITRNBM and TITRNWBM attributes and their uniqueness. The NMAGDM approach with trapezoidal and triangular information is suggested based on the TITRNWBM operator and possibility degree. Finally, a concrete example of manufacturing companies searching for the best supplier for assembling the critical parts is provided to validate the established strategies and show their practical applicability and efficacy.

Failure mode and Effect Analysis of personal fall arrest system under the intuitionistic fuzzy environment

Background and aimsIntuitionistic fuzzy sets (IFS) theory is more powerful than classic fuzzy sets theory in handling uncertainty. A new approach for Failure Mode and Effect Analysis (FMEA) was developed based on IFS and group decision-making (known as IF-FMEA) for investigating Personal Fall Arrest System (PFAS). MethodFMEA parameters, including occurrence, consequence, and detection, were re-defined based on a seven-point linguistic scale. Each linguistic term was associated with an intuitionistic triangular fuzzy set. Opinions on the parameters were gathered from a panel of experts, integrated using the similarity aggregation method, and defuzzified utilizing the center of gravity approach. ResultsNine failure modes were identified and analyzed using both FMEA and IF-FMEA. The risk priority numbers (RPNs) and prioritization obtained from the two approaches were different, highlighting the importance of using IFS. The highest RPN was associated with the lanyard web failure, while the failure of the anchor D-ring had the least RPN. Detection score was higher for metal parts of the PFAS, suggesting that failures in these parts are harder to detect. ConclusionIn addition to being economical in terms of calculations, the proposed method was efficient in handling uncertainty. Different parts of a PFAS create different levels of risk.

Operational efficiency evaluation of Urban and rural residents’ basic pension insurance system based on the triangular fuzzy neutrosophic GRA method

At present, the basic pension insurance system for urban and rural residents in China has played a positive role in guaranteeing the basic life of the elderly in urban and rural areas. At present, the basic pension insurance system for urban and rural residents is not perfect, and there is still a great lag in the formulation of cross-system and cross-regional policies. There are differences in treatment between groups, between regions and between urban and rural areas. The coverage is not comprehensive enough and there are still some people who are not included in the basic protection system, etc. People urgently need a social pension insurance system that can provide reliable and sustainable protection in their old age. The operational efficiency evaluation of urban and rural residents’ basic pension insurance systems is viewed as the multi-attribute decision-making (MADM). In this paper, the triangular fuzzy neutrosophic numbers grey relational analysis (TFNN-GRA) method is built based on the traditional grey relational analysis (GRA) and triangular fuzzy neutrosophic sets (TFNSs). Finally, a numerical example for operational efficiency evaluation of urban and rural residents’ basic pension insurance systems has been given and some comparisons are used to illustrate advantages of 2TLNN-GRA method.

A Neutrosophic Framework for Assessment of Distributed Circular Water to Give Neighborhoods Analysis to Prepare for Unexpected Stressor Events

The global water issue is caused by a number of factors, including extreme weather, population increase, and industrial activity. One of the reasons we're running out of water as a planet is because of the outdated take-make-use-throw-away linear paradigm of handling water. It has been suggested that the circular economy may help alleviate water shortages by inspiring a fundamental change in municipal water infrastructure. Reduced consumption, recovered natural resources, and minimal waste are the three pillars of a circular water supply. A dispersed water supply is more adaptable and robust than a centralized one because it gives communities more time to be ready for emergencies. Nonetheless, there have been no extensive studies of the most important elements that influence the choice to build distributed water supplies. In order to better inform the planning process, this research seeks to identify critical selection factors that influence the evaluation of viable choices. This study proposed the triangular neutrosophic set with the Decision-Making Trial and Evaluation Laboratory (DEMATEL) method to compute the weights of criteria. The neutrosophic set is used to deal with uncertain data.

Techniques for Finding Analytical Solution of Generalized Fuzzy Differential Equations with Applications

Engineering and applied mathematics disciplines that involve differential equations include classical mechanics, thermodynamics, electrodynamics, and general relativity. Modelling a wide range of real-world situations sometimes comprises ambiguous, imprecise, or insufficient situational information, as well as multiindex, uncertainty, or restriction dynamics. As a result, intuitionistic fuzzy set models are significantly more useful and versatile in dealing with this type of data than fuzzy set models, triangular, or trapezoidal fuzzy set models. In this research, we looked at differential equations in a generalized intuitionistic fuzzy environment. We used the modified Adomian decomposition technique to solve generalized intuitionistic fuzzy initial value problems. The generalized modified Adomian decomposition technique is used to solve various higher-order generalized trapezoidal intuitionistic fuzzy initial value problems, circuit analysis problems, mass-spring systems, steam supply control sliding value problems, and some other problems in physical science. The outcomes of numerical test applications were compared to exact technique solutions, and it was shown that our generalized modified Adomian decomposition method is efficient, robotic, and reliable, as well as simple to implement.

A Learning-Based Optimal Decision Scenario for an Inventory Problem under a Price Discount Policy

This paper aims to design an inventory model for a retail enterprise with a profit maximization objective using the opportunity for a price discount facility given by a supplier. In the profit maximization objective, the demand should be increased. The demand can be boosted by lowering the selling price. However, lowering the selling price may not always give the best profit. Impreciseness plays a vital role during such decision-making. The decision-making and managerial activities may be imprecise due to some decision variables. For instance, the selling price may not be deterministic. A vague selling price will make the retail decision imprecise. To achieve this goal, the retailer must minimize impreciseness as much as possible. Learning through repetition may be a practical approach in this regard. This paper investigates the impact of fuzzy impreciseness and triangular dense fuzzy setting, which dilutes the impreciseness involved with managerial decisions. Based on the mentioned objectives, this article considers an inventory model with price-dependent demand and time and a purchasing cost-dependent holding cost in an uncertain phenomenon. This paper incorporates the all-units discount policy into the unit purchase cost according to the order quantity. In this paper, the sense of learning is accounted for using a dense fuzzy set by considering the unit selling price as a triangular dense fuzzy number to lessen the impreciseness in the model. Four fuzzy optimization methods are used to obtain the usual extreme profit when searching for the optimal purchasing cost and sale price. It is perceived from the numerical outcomes that a dense fuzzy environment contributes the best results compared to a crisp and general fuzzy environment. Managerial insights from this paper are that learning from repeated dealing activities contributes to enhancing profitability by diluting impreciseness about the selling price and demand rate and taking the best opportunity from the discount facility while purchasing.

A two-sided matching decision method based on interval triangular fuzzy sets

The research work on two-sided matching decision problem considering interval triangular fuzzy information is very scarce, and it needs to be further studied. Based on this, this paper proposes a two-sided matching model based on interval triangular fuzzy sets, with the background of the two-sided matching problem in the interval triangular fuzzy set environment. Firstly, the theory of two-sided matching and interval triangular fuzzy sets is given; Secondly, the comprehensive mean value formula is defined, and the interval triangular fuzzy evaluation matrix is transformed into the comprehensive mean value matrix by using the comprehensive mean value formula; Thirdly, a two-sided matching model is built with the goal of maximizing the satisfaction of each subject; Finally, the feasibility and effectiveness of the proposed method are verified by examples of investment fund institutions and financing enterprises.

Pythagorean probabilistic hesitant triangular fuzzy aggregation operators with applications in multiple attribute decision making

As a generalization of fuzzy set, hesitant probabilistic fuzzy set and pythagorean triangular fuzzy set have their own unique advantages in describing decision information. As modern socioeconomic decision-making problems are becoming more and more complex, it also becomes more and more difficult to appropriately depict decision makers' cognitive information in decision-making process. In order to describe the decision information more comprehensively, we define a pythagorean probabilistic hesitant triangular fuzzy set (PPHTFS) by combining the pythagorean triangular fuzzy set and the probabilistic hesitant fuzzy set. Firstly, the basic operation and scoring function of the pythagorean probabilistic hesitant triangular fuzzy element (PPHTFE) are proposed, and the comparison rule of two PPHTFEs is given. Then, some pythagorean probabilistic hesitant triangular fuzzy aggregation operators are developed, and their properties are also studied. Finally, a multi-attribute decision-making (MADM) model is constructed based on the proposed operators under the pythagorean probabilistic hesitant triangular fuzzy information, and an illustration example is given to demonstrate the practicability and validity of the proposed decision-making method.

The cross-association relation based on intervals ratio in fuzzy time series

<span lang="EN-US">The fuzzy time series (FTS) is a forecasting model based on linguistic values. This forecasting method was developed in recent years after the existing ones were insufficiently accurate. Furthermore, this research modified the accuracy of existing methods for determining and the partitioning universe of discourse, fuzzy logic relationship (FLR), and variation historical data using intervals ratio, cross association relationship, and rubber production Indonesia data, respectively. The modified steps start with the intervals ratio to partition the determined universe discourse. Then the triangular fuzzy sets were built, allowing fuzzification. After this, the FLR are built based on the cross-association relationship, leading to defuzzification. The average forecasting error rate (AFER) was used to compare the modified results and the existing methods. Additionally, the simulations were conducted using rubber production Indonesia data from 2000-2020. With an AFER result of 4.77%<10%, the modification accuracy has a smaller error than previous methods, indicating very good forecasting criteria. In addition, the coefficient values of </span><em><span lang="EN-US">D<sub>1</sub></span></em><span lang="EN-US"> and </span><em><span lang="EN-US">D<sub>2</sub></span></em><span lang="EN-US"> were automatically obtained from the intervals ratio algorithm. The future works modified the partitioning of the universe of discourse using frequency density to eliminate unused partition intervals.</span>

Correction: Skin Cancer Segmentation Based on Triangular Intuitionistic Fuzzy Sets

Correction: Skin Cancer Segmentation Based on Triangular Intuitionistic Fuzzy Sets

Canonical triangular interval type-2 fuzzy set linguistic distribution assessment TODIM approach: A case study of FMEA for electric vehicles DC charging piles

Linguistic distribution assessment (LDA) plays a prominent role in describing the subjective uncertainties and objective distribution of decision-maker (DM) opinions. Although there have been several studies on LDA, few of these studies have paid attention to the different understandings of language semantics provided by DMs. To address the issue, type-2 (T2) fuzzy sets (T2FSs) are a well-known approach because T2FSs can accurately model the understanding of individuals with linguistic uncertainty. A trapezium (or trapezoidal) interval type-2 fuzzy set (TrIT2FS) is a common application of T2 fuzzy logic. Accordingly, this paper discusses a novel LDA called a canonical triangular interval type-2 fuzzy set (CTriIT2FS)-LDA in a T2 fuzzy environment. The CTriIT2FS-LDA combines a CTriIT2FS simplified from a TrIT2FS with LDA, which can simplify the computation while retaining their advantages. To facilitate the calculation of the CTriIT2FS-LDA, the basic operations of the CTriIT2FSs and the CTriIT2F-LDA, such as the defuzzied (crisp) function, distance measurement, and entropy measurement, are defined. Based on the TODIM approach, a CTriIT2F-LDA-TODIM is developed for FMEA for interval type-2 fuzzy environments. Furthermore, the proposed approach is applied in a practical case for FMEA for the electric vehicle DC charging piles of the State Grid Corporation of China.

Intelligent replacement analysis using picture fuzzy sets: Defender-challenger comparison application

Replacement analysis is based on the principle that it may be more economical to retirement of an equipment earlier than its physical life. This replacement time, called the economic life, is usually calculated by annual cash flow analysis. These calculations made with deterministic estimations in classical analyzes are not realistic because they do not take into account the uncertainties in their estimations. Picture fuzzy sets (PFSs) introduced to the literature in recent years can provide a more detailed estimation by using the yes, abstain, no, and refusal degrees of the estimations. In this paper, the economic lives of defender and challenger, which are two alternatives of replacement analysis, are calculated using PFSs and a replacement decision was made. Estimates are made by three experts in the PF environment, and these estimates are aggregated by using PF aggregation operators. The optimistic, normal, and pessimistic estimations made with PFSs are converted into ordinary triangular fuzzy sets in the further stages of the analysis, and the economic life is determined for both the defender and the challenger by defuzzification of the triangular fuzzy numbers obtained with the center of gravity method. It has been revealed that the proposed fuzzy replacement analysis can be used as a decision support model for managers under uncertainty. A multi-expert comprehensive example has shown that PFSs can be used successfully in a real replacement application. The robustness of the given decisions by the replacement analysis is confirmed by a sensitivity analysis.