Triangular Sets Research Articles

Assessment of Near-Earth Asteroid Deflection Techniques via Spherical Fuzzy Sets

Extensions of fuzzy sets to broader contexts constitute one of the leading areas of research in the context of problems in artificial intelligence. Their aim is to address decision-making problems in the real world whenever obtaining accurate and sufficient data is not a straightforward task. In this way, spherical fuzzy sets were recently introduced as a step beyond to modelize such problems more precisely on the basis of the human nature, thus expanding the space of membership levels, which are defined under imprecise circumstances. The main goal in this study is to apply the spherical fuzzy set version of Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), a well-established multicriteria decision-making approach, in the context of planetary defense. As of the extraction of knowledge from a group of experts in the field of near-Earth asteroids, they rated four deflection technologies of asteroids (kinetic impactor, ion beam deflection, enhanced gravity tractor, and laser ablation) that had been previously assessed by means of the classical theory of fuzzy series. This way, a comparative study was carried out whose most significant results are the kinetic impactor being the most suitable alternative and the spherical fuzzy set version of the TOPSIS approach behaves more sensitively than the TOPSIS procedure for triangular fuzzy sets with regard to the information provided by our group of experts.

Design and analysis of inverse fuzzysoft set for triangular and trapezoidal membership function in decision making

Decision making plays a vital role within uncertain, imprecise situations in recent years. A numerical actualize called soft set by Molodtsov was started to bargain uncertain condition. Herein article, inverse triangular and trapezoidal fuzzy soft set theory is analyzed in decision making. Inverse fuzzysoft set theory is an extension of fuzzy soft set theory which gives deeper insights in decision making and gives a technique to portray the character of each object in the universe. An application of inverse triangular and trapezoidal fuzzy soft set is established in the course of case study in multi-criteria decision making.

Extension of TOPSIS method for group decision-making under triangular linguistic neutrosophic cubic sets

Extension of TOPSIS method for group decision-making under triangular linguistic neutrosophic cubic sets

Gaussian-Based Soft Computing Approach to Alternative Banking System for Sustainable Financial Sector

This study aims to identify the necessary strategies for the development of a sustainable financial system. For this purpose, a novel approach could be provided for soft computing with Gaussian-based fuzzy DEMATEL approach to understand the significant levels and impact-relation degrees of these criteria. For robustness check, this evaluation has also been performed for triangular and trapezoidal fuzzy sets. There are many novelties of this study. Firstly, computer science has a significant role in the decision-making process. Another specificity is that the use of Gaussian numbers with fuzzy numbers is a new area. In addition, interval type-2 fuzzy (IT2F) sets also provide a significant advantage in decision-making under uncertainty. The results demonstrate that the size of the alternative financing, investment in efficient projects, and effective advertisement/promotions are the most important items to affect the performance of alternative banking system of emerging economies. Hence, it is recommended that the new banking applications should be applied by the state or the state can provide some incentives for foreign banks to invest in emerging economies, such as tax reduction. Another important point is that the financial intermediations should generate a component financial analysis team to examine the financial reports of the companies in detail so that the financial credibility of these companies can be evaluated more effectively.

Imprecisely defined fractional-order Fokker–Planck equation subjected to fuzzy uncertainty

Fokker–Planck equation with interval and fuzzy uncertainty has been considered in this paper. Also the derivatives involved with respect to time and space are assumed to be fractional in nature. This problem has been solved using variational iteration method (VIM) along with the double parametric form of fuzzy numbers. For the analysis, both triangular and Gaussian normalised fuzzy sets are taken into consideration. Numerical results for different cases have been obtained and those are depicted in terms of plots and are also compared in special cases for the validation. Moreover, using an important method known as successive approximation method, it has also been verified that the obtained solutions are the same as that of VIM as both methods are equivalent.

PENENTUAN KELAY􏰁AKAN CALON PENERIMA BANTUAN 􏰂RTLH􏰃 MENGGUNAKAN LOGIKA FUZZY MAMDANI DI DESA TANJUNG GLUGUR

One of the strategies to increase the quality of basic facilities of residential districts based on livable house indicators is to repair the targetted houses using the RTLH (Rumah Tidak Layak Huni) grant. The nominees of the grant have been decided by the village officials subjectively by observing from the data of previous grants. The appropriateness of the nominees, however, must be reconsidered firmly in order to have the program done as purposed. The concept of Mamdani fuzzy logic can be used to determine the appropriateness of the candidates. The method used in this research is quantitative research by using secondary data collected by village officials of Tanjung Glugur. This research has one output variable, the appropriateness of the candidates, and four input variables: residential buildings, water needs, sanitation and waste management, and income. The Fuzzy numbers that are used are triangular and shoulder fuzzy sets, while for the defuzzification four methods are used, which are the Smallest of Maximum (SOM), Mean of Maximum (MOM), Bisector, and Centroid. By analyzing the data result, similar results are obtained from all four methods. There are 57 out of 76 candidates who are stated as successful.

Matrix games with dense fuzzy payoffs

A novel notion of dense fuzzy lock set was introduced as an extension of fuzzy sets. Learning experiences have a vital role in this fuzzy lock set instigation. The novel fuzzy lock set reduces the fuzziness of the situation. Occasionally, the players are bound to make little changes in their game strategies to reach their goal for some matrix game problems. It may lead to some changes in payoffs. In this scenario, a matrix game's payoffs are chosen as dense fuzzy lock sets to make the problem more realistic. This paper's prime intent is to develop a mathematical model of a matrix game that represents payoffs by triangular dense fuzzy lock sets. Initially, a new defuzzification function MagD(.) is defined to find a ranking order relation of the dense fuzzy lock sets. Then a pair of auxiliary dense fuzzy programming problems is established for two players. These two problems are transformed into two equivalent crisp linear programming problems applying the proposed defuzzification function and its linearity property. The reduced problems are solved using LINGO 17.0 software to determine each player's optimal strategies and the game values. One surprising fact of this approach is that the value of the game increases with the increment of the player's learning experience. The validity, applicability, and superiority of this proposed methodology are illustrated by considering a real-life media share problem.

An integrated production policy with defective items and stock-out based substitution under triangular dense fuzzy lock set environment

Brand substitution is common observed phenomenon in daily life. It is the decision makers’ economic understanding and potential scheme for business-industries. Also, it provides the flexibility in management and increases the ability to control the production. This works proposes an integrated supplier–retailer inventory model for substitutable products. Two suppliers work not works with two different brand products with their corresponding demand are involved and one retailer sells each of the products. To nullify the complexities of the joint optimization problem, we first develop a deterministic model for three cases: no substitution, partial substitution and full substitution, then we go for its fuzzification. Keeping the financial constraint of each producer, we have studied over the elasticity of the cost parameters by means of triangular dense fuzzy lock set approach with its locking and unlocking property for final decision making. Finally, sensitivity analysis and graphical illustrations are made to justify the model.

Decision Making Model for Strategy Choice in Higher Education Under Neutrosophic Environment

The importance of studying the relationships between colleges, businesses, and governments has grown as a result of the recent explosion of technical advancements. The potential for national economic growth is enhanced by the dissemination of new information discovered via research. When it comes to the generation of new information that can be used by established economies, universities play a crucial role. So, in this study we proposed a framework for select best strategy in higher education. This process contains many conflicting criteria, so the concept of multi-criteria decision making (MCDM) is used. The MCDM is integrated with the triangular neutrosophic sets to overcome the vague information. The COCOSO technique is proposed to rank the alternatives. Higher education officials, including government bureaucrats and academic administrators, may put the recommended approach to use.

The Emergence of a Brand: A Case of Jaffa Oranges from Mandate Palestine

The Shamouti oranges grown along the Mediterranean coast of Palestine under the British Mandate (1922-1948) sparkled in European markets in the 1920s and 1930s. The oranges were known as Jaffa Oranges, or Jaffas, and gained economic importance during the Mandate. Today, the Jaffa Oranges brand is contested and has opposite national meanings for Israelis and Palestinians. This chapter explores the emergence of the brand, when Jews and Arabs worked alongside each other and on occasion together, albeit with growing national tensions. The brand emerged in a multi-layered industry, with different kinds of competition in each of its segments: growers, traders, and foreign markets, and in a triangular legal setting. The law did not protect geographical indications, leaving JAFFA ORANGES free to all to use. The law did offer protection for distinctive trademarks. During the British Mandate, 102 trademarks for oranges were registered, by both Jewish and Arab growers and traders. Two-thirds of which used the geographical brand. Finally, a regulatory scheme established to assure the quality of the citrus exports, acted as a de facto certification, and resulted in numerous unregistered marks, known as markas. This chapter offers a legal history analysis of the JAFFA ORANGES brand, by examining archival material and data from a reconstructed trademark registry of the period, along a legal inquiry, contextualized within the economic and national situation.

Characteristic decomposition of polynomial sets

An ordered pair $(\pset{G},\,~\pset{C})$ is called the characteristic pair of a polynomial ideal if $\pset{G}$ is the reduced lexicographic grobner basis of the ideal and $\pset{C}$ is the minimal triangular set contained in $\pset{G}$. A characteristic pair $(\pset{G},\,~\pset{C})$ is said to be regular or normal if $\pset{C}$ is regular or normal; it is said to be strong if the ideal generated by $\pset{G}$ and the saturated ideal of $\pset{C}$ are the same. A strong regular or normal characteristic decomposition of a polynomial set means a decomposition of the polynomial set into finitely many (strong) regular or normal characteristic pairs with an associated zero or ideal decomposition. This paper presents two algorithms for characteristic decomposition of any given polynomial set, one producing strong normal characteristic pairs with ideal splitting according to pseudo-divisibility relations between polynomials in reduced lexicographic grobner bases and the other producing strong regular characteristic pairs by constructing nontrivial ideals that can be divided out of given ideals by using ideal quotient. Novel theorems are established for regularizing characteristic pairs and strengthening regular and normal characteristic pairs by using ideal saturation. Strong regular or normal characteristic pairs produced by the algorithms possess nice properties and furnish two representations, one in terms of lexicographic grobner bases and the other in terms of triangular sets, for the zeros of the input set of multivariate polynomials. The paper includes a brief review on triangular decompositions of various kinds and discussions about inherent connections among triangular sets, Ritt characteristic sets, and reduced lexicographic grobner bases. Examples and experimental results are also provided to illustrate the method of characteristic decomposition and its applicability and performance.

Conflict analysis based on three-way decision for triangular fuzzy information systems

Triangular fuzzy numbers (TFNs) can not only provide the range of fuzzy points, but contain the three most representative fuzzy points, which play an essential role in describing fuzzy information. Combining with conflict situations, the agents' caution in the decision-making process can be well depicted by TFNs. However, little effort has been paid to conflict analysis on triangular fuzzy information systems (TFISs). In this paper, we study three-way conflict analysis based on TFISs. First, we propose the concept of TFISs and then establish a TFIS for conflict analysis. Further, relative area ΔS is defined to describe concrete attitudes of agents about issues. Second, since the importance of the same issue may vary for agents, we attach fuzzy weights to issues according to triangular fuzzy symmetric judgment matrices (TFSJMs) of agents. Third, we determine the total attitude of each agent to all issues. Then we illustrate how to compute thresholds α and β using the theory of triangular fuzzy decision-theoretic rough sets. Finally, a tri-partition of agents is obtained based on the thresholds.

An extended TODIM approach for group emergency decision making based on bidirectional projection with hesitant triangular fuzzy sets

With the recent Covid-19 outbreak, group emergency decision-making (GEDM), as a new management model to pursue both social stability and decrease the negative impact of emergencies, has become highly popular. Evaluating and choosing the best emergency response is the core of the GEDM and selecting the choices can be regarded as a multi-attribute group decision-making (MAGDM) problem. Due to the increasing complexity and fuzziness of emergency decision-making environment, decision-makers (DMs) often cannot express completely rational preference information in many real EDM situations. At the same time, the existing methods seldom consider the DM’s psychological mindset at the point of decision making. In this paper, an extended TODIM (an acronym for interactive and multi-criteria decision-making in Portugese) method based on bidirectional projection is proposed to solve the GEDM problem in the context of hesitant triangular fuzzy sets (HTFSs) and the novel method is applied to a case study and compared with other existing methods. The validity and applicability of the proposed method are discussed.

Regional Water Resources Security Evaluation Based on a Hybrid Fuzzy BWM-TOPSIS Method.

Nowadays, water resource security is becoming increasingly prominent, and this problem is a primary bottleneck restricting China’s future sustainable development. It is difficult to come to a unified conclusion on water resources security, and applications of highly feasible evaluation methods are lacking in practice. In this paper, a novel evaluation methodology is proposed for regional water resources security evaluation. First, water security is divided into two aspects: water quantity security and water quality security. The disposal rate of harmless household garbage, the excellent water resources proportion, and the functional water body loss proportion are creatively considered as indicators of water quality security in the evaluation system. In addition, a Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is used to evaluate the water security levels in different regions. For distinguishing the importance of different indicators, a Best–Worst Method (BWM) is employed to calculate the indicator weights, as triangular fuzzy linguistic sets can more flexibly describe the preferences of decision makers (DMs) regarding the indicators; therefore, it is embedded in BWM to determine indicator weights. Moreover, the fuzzy BWM-TOPSIS method is applied to evaluate the water security levels of six regions in North China, a comparison analysis with the equal weight TOPSIS method as well as the fuzzy BWM-AHP method, and a sensitivity analysis for indicator weights are presented to illustrate the effectiveness of this proposed method. Finally, some suggestions based on the evaluation results are given for effective and rational utilization of water resources in North China.

Computing strong regular characteristic pairs with Gröbner bases

The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Gröbner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Gröbner basis, C is the W-characteristic set of the ideal 〈G〉, the saturated ideal sat(C) of C is equal to 〈G〉, and C is regular. In this paper, we show that for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (G,C) by computing Gröbner bases such that I⊆sat(C)=〈G〉 and sat(C) divides I, so the ideal I can be split into the saturated ideal sat(C) and the quotient ideal I:sat(C). Based on this strategy of splitting by means of quotient and with Gröbner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Gröbner bases and the other in terms of regular triangular sets. We present some properties about strong regular characteristic pairs and characteristic decomposition and illustrate the proposed algorithm and its performance by examples and experimental results.

Triangular Fuzzy-Rough Set Based Fuzzification of Fuzzy Rule-Based Systems

Abstract In real-world approximation problems, precise input data are economically expensive. Therefore, fuzzy methods devoted to uncertain data are in the focus of current research. Consequently, a method based on fuzzy-rough sets for fuzzification of inputs in a rule-based fuzzy system is discussed in this paper. A triangular membership function is applied to describe the nature of imprecision in data. Firstly, triangular fuzzy partitions are introduced to approximate common antecedent fuzzy rule sets. As a consequence of the proposed method, we obtain a structure of a general (non-interval) type-2 fuzzy logic system in which secondary membership functions are cropped triangular. Then, the possibility of applying so-called regular triangular norms is discussed. Finally, an experimental system constructed on precise data, which is then transformed and verified for uncertain data, is provided to demonstrate its basic properties.

Triangular interval type-2 fuzzy soft set and its application

Decision-making is an essential task in Science and Engineering. Since most of the real-world problems have uncertainty in nature, making the decision is challengeable one for the decision makers. Soft set has the advantage of free from the deficiency of the parameterization tools of existing theories, namely probability, fuzzy theory and the theory of rough sets. Linguistic terms mean different things to different people, so variability in expert's acceptance degree is possible. Here usage of type-1 fuzzy leads to noisy and uncertain, and the parameters also may be noisy and hence type-2 fuzzy sets may be used to address the mentioned issues. Therefore, a triangular interval type-2 fuzzy soft set has been considered in the present work by combining triangular interval type-2 fuzzy set and soft set. In this paper, a triangular interval type-2 fuzzy soft weighted arithmetic operator (TIT2FSWA) has been proposed with its desired mathematical properties; also applied the proposed methodology in a decision-making problem for profit analysis. Further comparative analysis has been made with the existing methods to show the effectiveness of the proposed method.

Dimension of the repeller for a piecewise expanding affine map

In this paper, we study the dimension theory of a class of piecewise affine systems in euclidean spaces suggested by Michael Barnsley, with some applications to the fractal image compression. It is a more general version of the class considered in the work of Keane, Simon and Solomyak [The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition. {\it Fund. Math.}, {\bf 180}(3):279-292, 2003] and can be considered as the continuation of the works [On the dimension of self-affine sets and measures with overlaps. {\it Proc. Amer. Math. Soc.}, {\bf 144}(10):4427-4440, 2016], [On the dimension of triangular self-affine sets. {\it Erg. Th. \& Dynam. Sys.}, to appear.] by the authors. We also present some applications of our results for the generalized Takagi functions and fractal interpolation functions.

Sugeno Intuitionistic Fuzzy Generator Based Computational Technique for Crude Oil Price Forecasting

Crude oil being a significant source of energy, change of crude oil price can affect the global economy. In this paper, a new approach based on the intuitionistic fuzzy set theory has been implemented to predict the crude oil price. This paper presents the intuitionistic fuzzy time series forecasting algorithm to enhance the efficacy of time series forecasting which includes fuzzy c-means clustering to obtain the optimal cluster centers. Further, a computational technique is proposed for the construction of triangular fuzzy sets and these fuzzy sets are converted to intuitionistic fuzzy sets with the help of Sugeno type intuitionistic fuzzy generator. The popular benchmark dataset of West Texas Intermediate crude oil spot price is used for the validation process. The numerical results when compared with existing methods notify that the proposed method enhances the accuracy of the crude oil price forecasts.

A modified weighted method of time series forecasting in intuitionistic fuzzy environment

In this paper, we present a modified weighted method of time series forecasting using intuitionistic fuzzy sets. The proposed weighted method provides a better approach to extent of the accuracy in forecasted outputs. As it is established that the length of interval plays a crucial role in forecasting the historical time series data, so a new technique is proposed to define the length of interval and the partition of the universe of discourse into unequal length of intervals. Further, triangular fuzzy sets are defined and obtain membership grades of each datum in historical time series data to their respective triangular fuzzy sets. Based on the score and accuracy function of intuitionistic fuzzy number, the historical time series data is intuitionistic fuzzified and assigned the weight for intuitionistic fuzzy logical relationship groups. Defuzzification technique is based on the defined intuitionistic fuzzy logical relationship groups and provides better forecasting accuracy rate. The proposed method is implemented to forecast the enrollment data at the University of Alabama and market share price of SBI at BSE India. The results obtained have been compared with other existing methods in terms of root mean square error and average forecasting error to show the suitability of the proposed method.