triangular fuzzy numbers
در نشریات گروه ریاضی-
In this work, we defne a new sequence denominated by fuzzy Leonardo numbers. Some algebraic properties of this new sequence are studied and several identities are established. Moreover, the relations between the fuzzy Fibonacci and fuzzy Lucas numbers are explored, and several results are given. In addition, some sums involving fuzzy Leonardo numbers are provided.
Keywords: Triangular Fuzzy Numbers, Fuzzy Fibonacci Numbers, Fuzzy Lucas Numbers, Leonardonumbers, Algebraic Properties, Identities, Sum Identities -
In the real world, all available data are not definitive and are considered based on quality. Estimating the values of the inputs when we change the values of the outputs as desired is one of the important applications of inverse data envelopment analysis. If we want to estimate the level of inputs (outputs) among a group of decision-making units (DMUs), when some or all of its outputs (inputs) are changed so that cost efficiency is maintained or improved, inverse data envelopment analysis is used. In this article, cost efficiency is investigated by increasing desired outputs along with triangular fuzzy data. The problem of inverse data envelopment analysis with fuzzy data is presented for the cost efficiency of the DMU under evaluation. Also, in this connection, the results of the proposed model will be examined in a numerical example.
Keywords: Inverse Data Envelopment Analysis, Improving Cost Efficiency, Increasing Outputs, Membership Function, Triangular Fuzzy Numbers -
Journal of Computational Algorithms and Numerical Dimensions, Volume:3 Issue: 4, Autumn 2024, PP 252 -276COVID-19 is undoubtedly the major crisis that the world has witnessed in the last many decades. The pandemic spread its effect on the socio-economic and cultural environment across the nations. The present study aims to introspect the performance of leading global indices vis-à-vis the recent pandemic. In this regard, the ongoing research demonstrates a Fuzzy Multi-Criteria Decision-Making (F-MCDM) framework to assess the early effects of the COVID-19 pandemic on nine leading global stock indices. The underlying intention is to examine whether the concern for lives and livelihoods affected the relative ranking of leading global indices. To this end, in this paper, the researchers put forth a fuzzy entropy (for determining criteria weights) and Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) methodology utilizing time series data for the performance analysis of global indices during July 01, 2019, to May 31, 2020, distributed in two phases such as phase 1: July 01, 2019, to December 31, 2019 (pre-crisis) and phase 2: January 01, 2020, to May 31, 2020 (crisis). During the study period, daily data has been collected to indicate variables (criteria) such as momentum, volume, spread, and historical and implied volatility. It is observed that global indices could maintain their positions without much variation. Singapore and Hong Kong stock exchanges show consistent top performance, while Brazil and the Indian market record poor performance. Surprisingly, despite being the country of origin of COVID-19, the Chinese market showed better resilience and improved its position. The findings also reveal that no significant herding exists. The reliability and stability of the fuzzy-entropy-MARCOS framework are demonstrated through comparison with other MCDM models and sensitivity analysis.Keywords: COVID-19, Stock Market Performance, Triangular Fuzzy Numbers, Entropy, Marcos
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This paper presents a proposed method, Triangular Fuzzy MEREC (TFMEREC), which combines Triangular Fuzzy Numbers (TFNs) and Method Based on The Removal Effects of Criteria (MEREC). This integration aims to create an efficient implementation process that provides an efficient solution for complex MCDM issues, focusing on assessing halal suppliers. To demonstrate the feasibility and effectiveness of TFMEREC, this study will use an illustrative example to employ three different normalization methods with three different distance methods. Key findings from this research show that TFMEREC improves the accuracy and reliability of criteria weight determination. TFMEREC gives decision-makers more accurate weights, allowing for more informed decision-making processes. Furthermore, the sensitivity analysis provides insights into the impact of various normalization and distance methods on overall results, which improves the method's applicability and reliability. The TFMEREC method is a promising approach for dealing with imprecise and uncertain information in decision-making contexts, with potential applications in various domains. Overall, the findings underline the importance of methodological breakthroughs in enhancing decision-making processes, and the study is relevant to both experts and researchers.Keywords: Multi-Criteria Decision-Making, Objective Weighing Model, MEREC, Uncertainty, Vagueness, Triangular Fuzzy Numbers, Halal Supplier Selection
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Iranian Journal of Numerical Analysis and Optimization, Volume:14 Issue: 4, Autumn 2024, PP 1106 -1139Numerous optimization problems comprise uncertain data in practical circumstances and such uncertainty can be suitably addressed using the concept of fuzzy logic. This paper proposes a computationally efficient solution methodology to generate a set of fuzzy non-dominated solutions of a fully fuzzy multi-objective linear programming problem, which incorporates all its parameters and decision variables expressed in form of triangular fuzzy numbers. The fuzzy parameters associated with the objective functions are transformed into interval forms by utilizing the fuzzy-cuts, which subsequently generates the equivalent interval valued objective functions. The concept of centroid of triangular fuzzy numbers derives the deterministic form of the constraints. Furthermore, the scalarization process of weighting sum approach and certain concepts of interval analysis are used to generate the fuzzy non-dominated solutions from which the compromise solution can be determined based on the corresponding real valued expressions of fuzzy optimal objective values resulted due to the ranking function. Three numerical problems and one practical problem are solved for illustration and validation of the proposed approach. The computational results are also discussed as compared to some existing methods.Keywords: Multi-Objective Optimization, Fully Fuzzy Programming, Triangular Fuzzy Numbers, Interval Valued Functions, Weighting Sum Approach
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Recently, a unique extension of fuzzy sets known as hesitant fuzzy sets has been established to address hesitant cases that previous methods were unable to manage adequately. In this paper, the triangular hesitant fuzzy sets approach has been employed to explore the inherent uncertainty within the parameters of the life distribution. Two essential reliability measures, triangular hesitant fuzzy reliability and the hazard rate function designed for the Pareto Type I life distribution, have been established. Moreover, the triangular hesitant fuzzy reliability measure is utilized to assess the reliability of series and parallel systems. Furthermore, the weighted averaging operator has been used on both the series and parallel systems, making them more reliable and giving much better results than hesitant fuzzy sets. Finally, a numerical example demonstrating the use of these techniques is provided, and the results are presented in tabular and graphical formats.Keywords: Hesitant Fuzzy Sets, Triangular Fuzzy Numbers, Hesitant Fuzzy Reliability, Pareto Type I Distribution, Weighted Averaging Operator
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Facility location problems are one of the most important issues for healthcare organizations and centers to achieve social welfare and respond to customer needs. Proper distribution of health and treatment facilities in cities is vital to minimize costs and improve the efficiency of health centers. The main contribution of the current article is dealing with the uncertainty issue in the p-median location-efficient problem. In this article, the p-median location problem along with network data envelopment analysis (Network DEA) is used in parallel mode to calculate the efficiency of health and treatment centers. In this issue, health centers are considered as parallel networks with two departments that operate independently. Due to the precision of the input and output values, triangular fuzzy numbers and the α-level fuzzy method have been used. The primary results that consider the uncertainty provide efficient solution and suggestions for the potential location of health centers in our case study.
Keywords: Facility Location Problem, Network DEA, Healthcare, Health Centers, Triangular Fuzzy Numbers -
A new similarity/distance measure based on the centre of nine-point circle of the isosceles triangular fuzzy numbers is recommend in this paper. Extend the similarity/distance measure based on centroid, orthocenter, circumcenter, incenter and nine-point circle center of the isosceles triangles. It is proved that this general similarity/distance measure conforms to the properties of distance. Subsequently, some examples are presented to justify the superiority and validity of the proposed similarity/distance measure between IFSs based on the center of nine-point circle, which demonstrate that this measure overcomes the disadvantage of the existing similarity measures. The application of the proposed similarity measure to deal with pattern recognition problems is described, and the results are correlated with those reported in some prevailing studies. In addition, a clustering technique to classify objects based on the proposed similarity measure is discussed. Through a detailed comparative analysis of some existing measures, it is concluded that some of the existing measures fail to discriminate the results obtained under different circumstances, such as zero division or counter intuitive cases; in contrast, the proposed similarity measure successfully overcomes this weakness.Keywords: Intuitionistic fuzzy set, Similarity, Distance Measures, Triangular fuzzy numbers, decision-making, Nine-Point Circle
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Journal of Computational Algorithms and Numerical Dimensions, Volume:1 Issue: 4, Autumn 2022, PP 147 -154Zero-Base Budgeting (ZBB) is a very well-known method for the selection and management of budgets and is widely used by companies and government agencies. In this paper, a new method for modelling ZBB in fuzzy environment is described. Triangular fuzzy numbers are used for describing the imprecise budget data. In addition, an alternative approach is proposed for people who need to be more precise in their requirements. The efficiency of the proposed method is illustrated by numerical example using triangular fuzzy numbers and possibility theory.Keywords: Zero-base budgeting, triangular fuzzy numbers, Interval confidence, Fuzzy threshold
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عموما سازمانها و موسسات دارای بخشهای متعدد داخلی بوده که عملکرد کلی سازمان، نتیجهای از عملکرد هریک از این بخشها یا مراحل است. هر بخش، دارای عوامل ورودی و خروجی خاص خود هم چنین عوامل ارتباط دهنده بین مراحل است. شاخصهای بین مراحل را شاخصهای میانی مینامند. در یک ساختار دومرحله ای، شاخصهای میانی خروجیهای مرحله اول بوده که به عنوان ورودی مرحله دوم به کار میروند. ارزیابی عملکرد یک سازمان باید با درنظر گرفتن عملکرد هریک از بخشهای آن تعیین گردد. تحلیل پوششی داده ها یکی از روش های مناسب برای ارزیابی عملکرد براساس چند شاخص است. در عمل تعیین این شاخصها با مقادیر قطعی امکانپذیر نمیباشد. این مقاله به ارایه مدلی جهت تعیین عملکرد موسسات دوبخشی در محیط فازی همچنین تخصیص منابع به آن میپردازد. با استفاده از تحلیل پوششی داده های معکوس، یک مدل برنامه ریزی چندهدفه پیشنهاد شده است که با افزایش خروجیهای واحد تحت ارزیابی، میزان افزایش ورودیهای مرحله اول و میانی را بنحوی تعیین کند که کارایی آن حفظ شود. سپس مدل پیشنهادی برای تخصیص منابع به شعب بانک استفاده شده است.
کلید واژگان: عبارات و کلمات کلیدی: تحلیل پوششیداده ها، تحلیل پوششیداده های معکوس، تخصیص منابع، مدل دومرحله ای، عدد فازی مثلثیOrganizations and institutions have internal departments that the overall performance of the organization is the result of the performance of each of these departments or stages. Each stage has its own input and output factors as well as connecting factors between stages. The factors between stages are called intermediate indicators. In a two-stage structure, the intermediate indicators are the outputs of the first stage, which are used as the input of the second stage. Evaluation performance of an organization should be determined by considering the performance of each of its stages. Data envelopment analysis is one of the appropriate methods to evaluate performance based on several indicators. In practice, it is not possible to determine these indicators with exact values. In this paper, we proposed a model for determining the performance of two-stager systems in a fuzzy environment, also as well as the allocation of resources to it. We apply inverse data envelopment analysis and propose a multi-objective programming model that, by increasing the output of the unit under evaluation, determines the rate of increase of the first stage inputs and intermediate indicators in a way that maintains its efficiency. Then we illustrate the proposed model to allocate resources to bank branches.
Keywords: Data Envelopment Analysis, Inverse data envelopment analysis, Allocation resource, two stage model, triangular fuzzy numbers -
Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of fully fuzzy transportation problem involving trapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a two-step method for solving fuzzy transportation problem where all of the parameters are represented by triangular fuzzy numbers i.e. two interval transportation problems. Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers. To illustrate the proposed approach four application examples are solved. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature.
Keywords: fully fuzzy linear programming, Transportation problem, Trapezoidal fuzzy numbers, triangular fuzzy numbers -
Transportation Problem (TP) is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total transportation cost of a commodity in order to satisfy demands at destinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to the transportation problem. The aim of this paper is to introduce a formulation of TP involving Triangular fuzzy numbers for the transportation costs and values of supplies and demands. We propose a two-step method for solving fuzzy transportation problem where all of the parameters are represented by non-negative triangular fuzzy numbers i.e., an Interval Transportation Problems (TPIn) and a Classical Transport Problem (TP). Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers. To illustrate the proposed approach two application examples are solved. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature.
Keywords: Interval Numbers, Triangular Fuzzy Numbers, Fully Fuzzy Transportation Problem -
Recently, Srinivasan [On solving fuzzy linear fractional programming inmaterial aspects, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.04.209] proposed a method to solve fractional linear programmingproblem under fuzzy environment based on ranking and decompositionmethods. Srinivasan also claimed that the proposed method solved fractionallinear programming problem with inequality and equality constraints. In thisnote, we point out that the paper entitled above suffers from certainmathematical mistakes for solving these problems. Hence, the mentionedmethod and example are not valid. Further the exact method is stated and solvedthe problem.
Keywords: Fractional programming, Fuzzy linear programming, Triangular Fuzzy Numbers -
The aim of this paper is to present algebraic method which is called Wu's method to solving fuzzy complex systems of linear equations. Wu's method is used as a solution procedure for solving the crisp polynomial equations system. This algorithm leads to solving characteristic sets that are amenable to easy solution. To illustrate the easy application of the proposed method, numerical examples are given and the obtained results are discussed.Keywords: Triangular fuzzy numbers, Fuzzy complex numbers, Fuzzy complex systems of linear equations, Characteristic sets, Wu's algorithm
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This paper addresses a type of fully fuzzy bilevel linear programming (FFBLP) wherein all the coefficients and decision variables in both the objective function and constraints are triangular fuzzy numbers. This paper proposes a new simple-structured, efficient method for FFBLP problems based on crisp bilevel programming that yields fuzzy optimal solutions with unconstraint variables and parameters. some examples have been provided to illustrate these methods.
Keywords: Bilevel linear programming, Triangular fuzzy numbers, Ranking function, Optimal solution, Unconstrain variables -
Ranking fuzzy numbers is a very important decision-making procedure in decision analysis and applications. The last few decades have seen a large number of methods investigated for ranking fuzzy numbers, yet some of these approaches are non-intuitive and inconsistent. The most commonly used approaches for ranking fuzzy numbers are ranking indices based on centroid of fuzzy numbers. Despite their merits, there are some weakness associated with these indices. This paper review several recent fuzzy numbers ranking methods based on centroid points, then proposes a new centroid index ranking method that is capable of effectively ranking various types of fuzzy numbers. The presented method is compared with the given attitude by the way of centroid point. The contents herein present several comparative examples demonstrating the usage and advantages of the proposed centroid index ranking method for fuzzy numbers. Meanwhile, it can overcome the drawback of other methods.
Keywords: Centroid points, center of gravity, comparing, defuzzification, trapezoidal fuzzy numbers, triangular fuzzy numbers, ordering, ranking -
در این مقاله ما یک مدل شبکه عصبی برای تشخیص واحدهای تصمیم گیرنده کارا در تحلیل پوششی داده ها معرفی می کنیم.
مدل شبکه عصبی پیشنهادی از یک مسئله بهینه سازی نامقید حاصل می شود. از دیدگاه تئوری نشان داده می شود شبکه عصبی پیشنهادی دارای پایداری لیاپانف و همگرای سراسری می باشد. مدل پیشنهادی تک لایه می باشد. شبیه سازی نشان می دهد مدل پیشنهادی قادر به تشخیص واحدهای کارا در تحلیل پوششی داده ها می باشد.کلید واژگان: تابع تبدیل، برنامه ریزی خطی تماما فازی، برنامه ریزی خطی چند هدفه، اعداد فازی مثلثی و ذوزنقه ایRecently, fuzzy linear programming problems have been considered by many. In the literature of fuzzy linear programming several models are offered and therefore some various methods have been suggested to solve these problems. One of the most important of these problems that recently has been considered; are Fully Fuzzy Linear Programming (FFLP), which all coefficients and variables of the problem are the same kind of fuzzy numbers. One of most common of them is the model in which all fuzzy parameters are discussed by triangle numbers. In this paper, we first define a fully fuzzy linear programming with trapezoidal numbers and then suggest a new method based on reducing the original problem to the problem with triangle number. Specially, a conversion function for converting two trapezoidal and triangular numbers to each other is offered. Finally, the mentioned method is illustrated by a numerical example.Keywords: Conversion function, Fully fuzzy linear programming, Multi, objective linear programming, Trapezoidal, triangular fuzzy numbers -
مسئله حمل و نقل، یک مساله برنامه ریزی خطی است که حداقل هزینه را برای حمل یک محصول ازتعدادی مبدا به تعدادی مقصد از قبیل کارخانه به انبار یا انبار به سوپر مارکت لحاظ می کند. برای حل این مساله الگوریتم سیمپلکس درنظر گرفته شده است. در پروژه های عملی هزینه و مقدار عرضه و تقاضا اعداد فازی هستند و انتظار می رود جوابهای بهینه که مقدار کالایی که از مبدا به مقصد را تعیین می کند، به صورت فازی تعیین شود بنابراین در ابتدا ایده در شرایط کاملا فازی و سپس الگوریتمی که اولویت بندی برای حل چنین مساله ای است ارائه می شود. در این مقاله الگوریتمی جدید برای حل مسائل حمل و نقل در حالت کاملا فازی پیشنهاد می شود. این الگوریتم، مساله ی حمل و نقل در حالت کاملا فازی را به یک مساله سه هدفه تبدیل و سپس از روش وزن دار شده در حل مسائل چندهدفه استفاده می کند و سپس مساله ی جدید را با روش سیمپلکس حل می کند. در خاتمه روش پیشنهادی برای داده های واقعی استفاده می شود.کلید واژگان: مساله حمل و نقل کاملا فازی، اعداد فازی مثلثی، مساله برنامه ریزی خطی چند هدفهTransportation problem is a linear programming which considers minimum cost for shipping a product from some origins to other destinations such as from factories to warehouse, or from a warehouse to supermarkets. To solve this problem simplex algorithmis utilized. In real projects costs and the value of supply and demands are fuzzy numbers and it is expected that optimal solutions for determining the value of commodities transported from a source to a destination be obtained as a fuzzy. So the first idea is to present the in the full fuzzy condition and then an algorithmwhich is of importance for solving such a problem. In this article, a new algorithm is suggested for solving fully fuzzy transportation problem. This algorithm transforms the fully fuzzy transportation problem into a triple-objective problem and then it utilizes a weighted method for solving multi-objective problems and solves the new problem using simplex transportation method. At the end, the suggested method is utilized for the real data.Keywords: fully fuzzy transportation problem, triangular fuzzy numbers, linear multi objective programming
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In the literature hardly any attention is paid to solving a fuzzy fixed charge transportation problem. In this paper, we consider the fully fixed-charge transportation problem and try to find both the lower and upper bounds on the fuzzy optimal value of such a problem in which all of the parameters are triangular fuzzy numbers. To illustrate the proposed method, a numerical example is presented.Keywords: Fixed charge transportation, Triangular fuzzy numbers, Fuzzy transportation problem, Ranking ýfunction
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The SST decomposition method for solving system of linear equations make it possible to obtain the values of roots of the system with the specified accuracy as the limit of the sequence of some vectors. In this topic we are going to consider vectors as fuzzy vectors. We have considered a numerical example and tried to find out solution vector x in fuzzified form using method of SST decomposition.Keywords: Triangular fuzzy numbers, System of Linear Equations, SST Decomposition Method
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