Triangular Sets Research Articles

New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using nonlinear programming approach

In this paper, we propose a new method to obtain the eigenvalues and fuzzy triangular eigenvectors of a fuzzy triangular matrix $$\left( {\tilde{A}} \right)$$ , where the elements of the fuzzy triangular matrix are given. For this purpose, we solve 1-cut of a fuzzy triangular matrix $$\left( {\tilde{A}} \right)$$ to obtain 1-cut of eigenvalues and eigenvectors. Considering the interval system $$\left[ {\tilde{A}} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } = \left[ {\tilde{\lambda }} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$ as α-cut of the fuzzy system $$\tilde{A}\tilde{X} = \tilde{\lambda }\tilde{X}$$ , to determine the left and right width of eigenvalues $$\left[ {\tilde{\lambda }} \right]_{\alpha }$$ and eigenvector elements $$\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$ , we make a system of linear and nonlinear equations and inequalities. And we propose nonlinear programming models to solve the system of linear and nonlinear equations and inequalities and to calculate $$\left[ {\tilde{\lambda }} \right]_{\alpha }$$ and $$\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$$ . Furthermore, we define three other new eigenvalues (namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue) for a fuzzy triangular matrix (Ã) that the fuzzy eigenvalue and fuzzy eigenvector cannot be obtained based on interval calculations. Therefore, the fuzzy escribed eigenvalue which is placed in a tolerable fuzzy triangular eigenvalue set, the fuzzy peripheral eigenvalue placed in a controllable fuzzy triangular eigenvalue set, and the fuzzy approximate eigenvalue placed in an approximate fuzzy triangular eigenvalue set is defined in this paper. Finally, numerical examples are presented to illustrate the proposed method.

An Integrated Multi-Criteria Decision-Making Approach for Identification and Ranking Solar Drying Barriers under Single-Valued Triangular Neutrosophic Sets (SVTNSs)

Solar dryers utilized in agriculture for the drying of food and crops are also utilized for drying operations in industrial settings. They have the potential to be shown as a very helpful tool in terms of the management of energy saving. It not only helps conserve energy, but it also helps save a lot of time, consumes less space, makes the procedure more effective, enhances the standard of the product, and safeguards the surroundings. Due to the many associated potential barriers, the acceptance of solar dryers has not yet reached a benchmark, although this was an expectation. In this body of work, a methodical framework that makes use of the MCDM tools has been proposed to identify and rank several obstacles in descending order of importance.The AHP can identify both quantitative and qualitative aspects by using comparison matrices to assign weights to them and rank them in order of importance. The AHP technique is used to calculate the weights and relationship of the solar drying barrier. To account for the lack of clarity and coherence in the data that is available in the actual world, we tested the suggested model in a neutrosophic set. We used the single-valued triangular neutrosophic sets (SVTNSs). SVTNSs are a type of neutrosophic set, integrated into triangular neutrosophic sets and SVNSs. The application of applying the SVNSs-AHP is performed.

Ranking Renewable Energy Alternatives by using Triangular Neutrosophic Sets Integrated with MCDM

In this age of ecological sustainability, energy planning has grown more complicated as a result of the inclusion of numerous standards, including technological, political, financial, and environmental considerations. As a result, this places significant limitations on the ability of policymakers to independently and covertly optimize energy sources, which is particularly problematic for rural populations. In contrast, the constraints imposed by the topography of the land on renewable energy (REEN) systems, which are for the most part dispersed across the natural environment, make energy planning more difficult. In these kinds of situations, decision analysis plays a crucial part in the process of creating these kinds of systems by taking into account a wide range of requirements and goals, even at fragmented levels of digitization. Many criterion decision making, often known as MCDM, is a subfield of operational research that focuses on finding optimum outcomes in complicated situations that include various measures, competing goals, and multiple criteria. Because it enables decision-makers to make choices while simultaneously taking into account all of the standards and goals, this tool is gaining traction in the area of energy planning, which is one of the reasons why it is becoming more famous. In this paper, the TOPSIS MCDM methodology is integrated with the triangular neutrosophic sets to rank and select best source of REEN in Egypt. The neutrosophic sets used due to incomplete and uncertainty in this ranking.

Trapezoidal type-2 Pythagorean fuzzy TODIM approach for sensible decision-making with unknown weights in the presence of hesitancy

<abstract> <p>Motivated by the concept of type-2 fuzzy sets, we introduce a novel framework known as trapezoidal type-2 Pythagorean fuzzy sets (TRT-2-PFSs), an extension of triangular fuzzy sets. Basic operations like addition and scalar multiplication of two TRT-2-Pythagorean fuzzy numbers (TRT-2-PFNs) are defined. We also explore comparative analysis and distance measurements between two TRT-2-PFNs. A methodology for evaluating unknown weight vectors and criteria weights is proposed. Building upon TRT-2-PFSs, an extension of the TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method is developed to address intricate decision-making challenges. Ultimately, the newly introduced TRT-2-PFS-based TODIM technique is employed to tackle multi-criteria decision-making (MCDM) problems.</p> </abstract>

Analyses the least cost using Travelling Salesman problem through Neutrosophic Fuzzy system

The following paper introduces a methodology to calculate the least cost for a directed network through Travelling salesman problem. Dynamic programming method is used to find the minimum Cost. The recursion formula is used. The edge weights of the networks are being taken in terms of Triangular, Trapezoidal and Pentagonal Neutrosophic set. Score function for the Triangular, Trapezoidal and Pentagonal Neutrosophic sets are being defined for deneutrosophication. The least cost is estimated using all the above said Neutrosophic sets and the result is compared

An Integrated Approach of the AHP and Spherical Fuzzy Sets for Analyzing a Park-and-Ride Facility Location Problem Example by Heterogeneous Experts

Consensus is difficult to acquire among a group of transportation specialists with diverse areas of expertise over the most relevant set of criteria considering a transport related problem. For example, when establishing a Park-and-Ride (P&R) system, if an expert favors the public transport criterion, and another focuses on the economic area, the problem is considerably difficult to solve. Therefore, this research provides a methodological solution including the Analytic Hierarchy Process (AHP) and Spherical Fuzzy Sets (SFSs) with the purpose of addressing both sorts of difficulties concurrently, i.e., taking hesitant scores into account and synthesizing stakeholders’ viewpoints through a mathematical procedure. SFSs are preferable compared to other solutions due to their flexible specification of the belonging function. In current study, the spherical AHP method is applied to a P&R system location problem to evaluate the results of the participating transportation experts from diverse backgrounds. Additionally, similarities and differences between the obtained results and the fuzzy AHP calculation are emphasized. Based on the results public transport accessibility is the most important criterion when establishing a P&R system. However, when compared to the AHP Triangular Fuzzy Sets multi-criteria technique, the sub-criteria vary significantly. The results give urban transport planners a clear guideline that the implementation of the P&R system should run in parallel with the optimization of public transport.

Analyses the causes of alcoholism using triangular Neutrosophic soft set in a Neutrosophic environment.

This study suggested a novel method for identifying the patient most impacted by alcohol by employing the triangular neutrosophic soft set-in variable sense. Environmental, biological and psychological factors, among others, have an effect on alcoholics. Alcoholics are categorized according to the severity of the affliction, and the reasons for alcoholism are examined using a mixed set of data from various observers. The Neutrosophic soft set is a crucial resource for identifying the decision-making problems in the Neutrosophic domain. This categorization inquiry has taken the form of a comparison table. It is advantageous to classify objects and individuals in a neutrosophic environment according to their calibre, aptitude, performance, etc.

Unravelling the assignment problem under intuitionistic triangular fuzzy environment by the novel heuristic Dhouib-Matrix-AP1

The Assignment Problem (AP) can be stated as n activities to be assigned to n resources in such a way that the overall cost of assignment is minimized and each activity is assigned to one and only one resource. In real-life, the parameters of the AP are presented as uncertain numbers due to the lack of knowledge, experiences or any other (internal or external) factor. In this paper, the AP is considered under intuitionistic triangular fuzzy number and solved by the novel constructive heuristic Dhouib-Matrix-AP1 (DM-AP1) with a time complexity of O(n). Actually, this paper presents the first enhancement of the novel heuristic DM-AP1 to solve the AP under intuitionistic triangular fuzzy environment. DM-AP1 is composed of three simple steps: computing the total cost, selecting the highest value and finding the minimal element. These steps are repeated in n iterations with the use of a standard deviation statistical metric. Two case studies of AP under intuitionistic triangular fuzzy set are taken from the literature and a step-by-step application of the novel DM-AP1 heuristic is presented for more clarification.

Type-II fuzzy collaborative intelligence for assessing cloud manufacturing technology applications

• Nine cloud manufacturing applications are compared. • Efficient approximating alpha-cut operations method is proposed to enhance the derivation of fuzzy priorities. • Type-II fuzzy VIKOR is applied to assess a cloud manufacturing application. Cloud Manufacturing (CMfg) is a new manufacturing paradigm that can reduce costs, improve data analysis, increase efficiency and flexibility, and provide manufacturers with closer partnerships. However, most previous studies on CMfg focused on the planning and scheduling of a hypothetical CMfg system or its infrastructure. In addition, the cost effectiveness of a CMfg application has rarely been evaluated. Consequently, manufacturers are not certain about adopting specific CMfg applications. To address this issue, this research proposed a type-II fuzzy collaborative intelligence approach to assist in the selection of suitable CMfg applications in a factory. In the proposed methodology, first, an efficient approximating alpha-cut operations (exACO) method was proposed to enhance the efficiency of the derivation of the fuzzy priorities of factors, which is an innovative treatment. Subsequently, the membership function of a type-II fuzzy priority was derived from the overall and partial consensuses among decision makers using layered fuzzy intersection (LFI), which is also a novel approach. As the type-II fuzzy priority cannot be described using a type-II triangular or trapezoidal fuzzy set, a type-II fuzzy VIšekriterijumskoKOmpromisnoRangiranje (VIKOR) method based on the α -cuts of fuzzy priorities was employed to assess and compare nine CMfg applications. The experimental results revealed that “migrating legacy system to CMfg services” and “cloud-based customization environment” were the most and least preferable CMfg applications, respectively.

Q-rung orthopair triangular fuzzy linguistic Frank aggregation operators and their application

The aim of this paper is to introduce a Frank operator in the q-rung orthopair triangular fuzzy linguistic environment on the basis of the notion of the Frank operator and the q-rung orthopair fuzzy set. Firstly, the concept of a q-rung orthopair triangular fuzzy linguistic set (q-ROTrFLS) is proposed, then several basic operations, score, and accuracy functions to compare the q-ROTrFLS values are defined. Secondly, a series of q-rung orthopair triangular fuzzy linguistic Frank aggregation operators are developed, such as q-rung orthopair triangular fuzzy linguistic Frank weighted average (q-ROTrFLWA)operator,q-rung orthopair triangular fuzzy linguistic Frank weighted geometric (q-ROTrFLWG) operator, and we introduce several relevant properties of these operators and prove their validity, and show the relevant relationship between some operators. Thirdly, two different decision-making approaches are constructed in the q-rung orthopair triangular fuzzy linguistic environment. Furthermore, a practical example is given to explain the developed methods. Finally, a comparative study is conducted, and the relevant sensitivity analysis is also discussed, and the outcome shows the prominence and the effectiveness of the developed methods compared to previous studies.

A Newly Proposed and Comparative Study on Multi - Criteria Decision Making in Interval Valued Bipolar Triangular Neutrosophic Set by PROMETHEE II

Objectives: To propose the operational laws and operators in interval valued bipolar triangular neutrosophic sets and also to resolve multi criteria decision making problem using PROMETHEE II. Methods: Based on the combination of bipolar neutrosophic sets and interval valued triangular neutrosophic sets, an interval valued bipolar triangular neutrosophic set is proposed followed by its operational laws, score function, accuracy and certainty function. Findings: The arithmetic operators, operational laws and also the score function, accuracy and certainty functions have been proposed for interval valued bipolar triangular neutrosophic sets. The PROMETHEE II technique discovered and employed an aggregated weighted arithmetic operator to address the multi-criterion decision-making problem. Novelty: By the concept of interval valued bipolar triangular neutrosophic set, the results can be viewed with more accuracy for the linguistic variables rather than the crisp values. An illustrative example has been explained for the interval valued bipolar triangular neutrosophic numbers by which the effectiveness is tested with the comparative approach by using PROMETHEE II method. Keywords : Interval valued neutrosophic numbers, Bipolar neutrosophic numbers, Interval valued bipolar triangular neutrosophic numbers, Score function, PROMETHEE II method.

Two-Stage Fuzzy Interactive Multi-Objective Approach under Interval Type-2 Fuzzy Environment with Application to the Remanufacture of Old Clothes

In this study, a two-stage approach is introduced to obtain a more interactive and flexible solution to deal with the multi-objective programming under interval type-2 fuzzy environment. In the first stage, the fuzzy multi-objective chance-constrained programming with regular symmetric triangular interval type-2 fuzzy set parameters is proposed and transferred into its crisp equivalent form. In the second stage, we use the fuzzy interactive approach to address the crisp multi-objective programming obtained in the first stage by introducing the trade-off rate, which helps the decision maker react via updating their reference member values to obtain a satisfactory solution. Finally, taking a remanufacture of old clothes problem as an example, the comparison of experimental results obtained using a non-interactive method and interactive method shows that the proposed approach is conducive to obtaining satisfactory solutions effectively and efficiently, which broadens the application scope of the multi-objective programming with regular symmetric triangular interval type-2 fuzzy set parameters for sustainable manufacturing.

A new theory of triangular intuitionistic fuzzy sets to solve the two-sided matching problem

In view of the limitations of triangular intuitionistic fuzzy numbers (TIFNs) in value selection and application, the purpose of this paper is to redefine and demonstrate the algorithm, expectation function and distance measure of TIFNs, and to put forward a new theory of triangular intuitionistic fuzzy sets (TIFSs). The new theory of TIFSs is then applied to the field of two-sided matching decision-making. To solve the two-sided matching problem based on TIFSs, the agent behavior factors of both sides are calculated by this theory. On this basis, a multi-objective matching model under TIFS preference is constructed. Furthermore, the optimal matching scheme is obtained by transforming and solving the model according to the expected score, the linear weighting and the algorithm of “being as good as possible”. Finally, the practicability of the proposed two-sided matching method is verified by a transaction matching example. The novelty and key idea of the proposed method are as follows: (1) it combines the idea of closeness in TOPSIS method and the expected score in the new theory of TIFSs to measure the behavior factors of two-sided agents; (2) it builds a two-sided matching model considering the behavior factors under the preference of TIFSs; (3) according to the idea of “being as good as possible”, a novel algorithm for solving the two-sided matching model under TIFS preference is proposed. The major conclusion and advantage of this paper is that the obtained BM scheme using the proposed method can reflect agent behavior factors and maximize agents’ satisfaction as much as possible.

Marine ship’s course stabilization based on an autopilot with a simple fuzzy controller

A simple method for generating a fuzzy course controller for a marine ship is presented. The controller is built without using training data. The new fuzzy controller uses conventional triangular sets, no complex overlaps, no complex expert judgments inherent in other types of fuzzy controllers. This approach makes it possible to synthesize a controller based on unified rules. The system is not hybrid and does not use other methods such as neural networks and reference models. The applicability of the proposed approach is demonstrated by an application for controlling the course of a marine ship in various modes. It is shown by means of simulation that the exchange rate stabilization system synthesized with the new fuzzy controller has robust properties.

Optimal solution of neutrosophic linear fractional programming problems with mixed constraints

In this paper, linear fractional programming problems have been extended to neutrosophic sets, and the operations and functionality of these laws are studied. Moreover, the new algorithm is based on aggregation ranking function and arithmetic operations of triangular neutrosophic sets. Furthermore, in this paper, we take up a problem where the constraints are both equality and inequality neutrosophic triangular fuzzy number. Lead from genuine issue, a few numerical models are considered to survey the legitimacy, profitability and materialness of our technique. At last, some numerical trials alongside one contextual analysis are given to show the novel techniques are better than the current strategies.

A reliable GIS-based FAHP-FTOPSIS model to prioritize urban water supply management scenarios: A case study in semi-arid climate

• Developed a GIS-based-MCDM method to select optimum urban water management scenario. • The paper considers triangular fuzzy sets to reduce uncertainty in decision-making. • The developed model is robust and flexible to manage required urban water demands. • A hydrologic model utilizing WEAP is designed to assess the developed model scenarios. The growing problem of urban water shortage and its sustainable management methods is one of today's critical research needs globally. This paper applies an integration of the Geographic Information System (GIS) and Multi-Criteria Decision-Making (MCDM) with the help of triangular fuzzy sets to manage urban water supply priorities in a semi-arid region. Hence, a group decision-making approach was proposed by combining the Fuzzy AHP (FAHP) and the Fuzzy TOPSIS (FTOPSIS) models based on quantitative and qualitative criteria. Therefore, a hierarchical model-based GIS and AHP was developed to classify the effective criteria and determine the Importance Weights (IWs) of criteria according to the experts’ opinions and using data layers based on spatial criteria. The priority ranking of scenarios was obtained using the FTOPSIS method. Different criteria such as environmental, geographical, geological, economic, climatic, and social conditions were considered in this study. Furthermore, to assess the water supply scenarios under the essential water demands of the basin cities and suburbs, a hydrologic model utilizing WEAP software was designed. Among the considered criteria, total expense, climatic conditions, and geographic conditions with the IWs values of 0.9, 0.85, and 0.83 were determined as the most effective criteria for evaluating the scenarios' ranking, respectively.

Computing zero-dimensional tropical varieties via projections

We present an algorithm for computing zero-dimensional tropical varieties using projections. Our main tools are fast monomial transforms of triangular sets. Given a Gröbner basis, we prove that our algorithm requires only a polynomial number of arithmetic operations, and, for ideals in shape position, we show that its timings compare well against univariate factorization and backsubstitution. We conclude that the complexity of computing positive-dimensional tropical varieties via a traversal of the Gröbner complex is dominated by the complexity of the Gröbner walk.

Initial experience using a novel algorithm to calculate omnipolar electrograms using a high-density grid-style catheter

Abstract Funding Acknowledgements Type of funding sources: None. Background/Introduction A novel mapping algorithm recently became available in Europe and select countries in the Asia Pacific region. EnSite Omnipolar Technology (OT) utilizes the Advisor HD Grid Mapping Catheter, Sensor Enabled (HD Grid) to calculate electrograms in 360 degrees from two bipolar and three unipolar electrograms acquired from a triangular set of three electrodes. This software also enables calculation of wavefront characteristics including maximum voltage, activation direction and wave speed. Procedural characteristics and clinical utilization of this novel software have not yet been reported. Purpose To examine the clinical utility and procedural characteristics associated with the use of this novel mapping algorithm among participating centers. Methods Acute procedural data were prospectively collected in 386 cases at 52 centers utilizing the newly approved mapping software in the initial phases of commercialization in Europe and the Asia Pacific region. Procedural characteristics recorded included indication for mapping/ablation, navigation mode, mapping reference, and maps and tools used to diagnose/locate ablation targets. Results A total of 14 indications for mapping and ablation were represented including AF, atypical flutter, and VT (Table 1). VoXel (magnetic primary) navigation mode was used in most cases (n=325, 84%). The CS catheter was used as the map reference in 280 cases (73%). A variety of workflows were used with OT to diagnose and locate ablation targets including voltage maps (n=275, 71%), LAT maps (n=226, 59%), OT activation vectors (n=124, 32%), propagation maps (n=94, 24%), and sparkle maps (n=69, 18%); note: total exceeds 100%, multiple workflows per case observed. EnSite LiveView Dynamic Display (LiveView) was used in 129 cases (33%) to diagnose and locate ablation targets, including 28 (22%) where OT activation vectors were used with LiveView. Several benefits of OT were reported including rapid identification of ablation target(s) (n=133, 35%), higher point density (n=82, 21%), better signals (n=69, 18%), and improved procedure efficiency (n=48, 12%). Conclusion(s) Initial experience with this novel mapping software in Europe and the Asia-Pacific region illustrated utility in a variety of arrhythmias to diagnose and locate ablation targets. The use of OT provided several benefits including rapid identification of ablation targets and higher point density, however further examination of its impact on procedural efficiencies and acute outcomes is likely warranted.

Empirical process of concomitants for partly categorial data and applications in statistics

On the basis of independent and identically distributed bivariate random vectors, where the components are categorial and continuous variables, respectively, the related concomitants, also called induced order statistic, are considered. The main theoretical result is a functional central limit theorem for the empirical process of the concomitants in a triangular array setting. A natural application is hypothesis testing. An independence test and a two-sample test are investigated in detail. The fairly general setting enables limit results under local alternatives and bootstrap samples. For the comparison with existing tests from the literature simulation studies are conducted. The empirical results obtained confirm the theoretical findings.

A Double Triangular Feature-Based Sensor Sequence Coding Approach for Identifying Chinese Liquors Using an E-Nose System

Herein, a novel model-free double triangular feature-based sensor sequence coding approach is developed to identify Chinese liquors using a self-designed electronic nose (e-nose) system. In the feature extraction stage, the response curve of each gas sensor of the e-nose is averagely divided into ten segments, followed by a summation of the response voltages in each segment to form the voltage features. Two triangular feature sets, of which the number of the feature layers is equal to the number of sensors used, are subsequently constructed for each segment of each liquor sample based on voltage features. The feature’s sensor sequence (SS) codes are generated by arranging features corresponding to different sensors or different positions of the same layer in each triangular feature set in ascending order. The sample SS codes are obtained when all the feature SS codes within a sample are connected. The SS codes of each liquor type are then constructed by integrating the coincident SS codes with the inconsistent SS codes of all the training samples within the liquor type, where the latter are marked as zero. In the identification stage, the liquor type whose SS codes are the most similar to those of the testing samples is selected as the predicted result of the sample. The proposed method has an average accuracy of 96.3%, which is considerably higher than traditional classification algorithms, such as random forest, support vector machine, naive Bayesian, k nearest neighbor, voting-extreme learning machine, long short-term memory, and gated recurrent unit.