Flat Fuzzy Numbers Research Articles

Estimation of Failure Using Fault Tree Analysis Based on New Operations on LR-Type Flat Fuzzy Numbers

In this paper, we analyze the reliability of fuzzy system (particularly, series and parallel system) with independent and non-identically distributed components using the new operations of TA [F. Abbasi, T. Allahviranloo and S. Abbasbandy, A new attitude coupled with fuzzy thinking to fuzzy rings and fields, Journal of Intelligent and Fuzzy Systems 29 (2015) 851–861.] due to the smaller results support, easier calculations and special properties than operations based on the extension principle (in the domain of the membership function) and the interval arithmetic (in the domain of the [Formula: see text]-cuts). we propose the new fuzzy arithmetic operations based on transmission average(TA) on LR type flat fuzzy numbers. In the proposed formulae, LR type flat fuzzy numbers are not restricted to have the same [Formula: see text] and [Formula: see text] shape functions. This allows arithmetic operator for arithmetic involving LR type flat fuzzy numbers of different and general shape. Finally, an imprecise failure to start of an automobile is considered in fuzzy environment. The reliability of components of the proposed model is considered as LR type flat fuzzy numbers.

Solving Fully Fuzzy Multi-objective Linear Programming Problem Using Nearest Interval Approximation of Fuzzy Number and Interval Programming

This paper focuses on Fully Fuzzy Multi-Objective Linear Programming (FFMOLP) problem in which all the coefficients and decision variables are LR flat fuzzy numbers, the more generalized version of fuzzy numbers and all the constraints are fuzzy inequalities. A new algorithm is proposed for solving FFMOLP problem which first converts it into the Multi-Objective Interval Linear Programming (MOILP) problem. Further, taking the help of fuzzy slack variable, fuzzy surplus variables, nearest interval approximation of fuzzy numbers and scalarization technique, MOILP is then converted into the Crisp Linear Programming (CLP) problem. It is shown that the optimal solution of CLP problem is the fuzzy Pareto optimal solution of FFMOLP problem. The main advantage of the proposed algorithm is that it transforms FFMOLP problem into Crisp Linear Programming problem. Moreover, to apply algorithm, only the knowledge of arithmetic operations of LR flat fuzzy numbers, centre and width of the closed intervals are required. At the end, to illustrate the proposed method and its effectiveness over the existing method, numerical examples are solved and compared.

Cost parametric analysis of linear programming problems with fuzzy cost coefficients based on ranking functions

Parametric analysis (PA) as a basic tool for studying perturbations in optimisation problems is applied to study the effect of the continuous variations of several parameters in the objective function coefficients on the optimum solution. To the best of our knowledge, till now there is no study in the literature to deal with the PA of the fuzzy linear programming problem in which the cost coefficients are represented by LR flat fuzzy numbers. In this paper, the parametric analysis for the same problem is investigated to determine the optimal solution and the fuzzy optimal objective values as a function of parameters when the fuzzy cost coefficients are perturbed along a new fuzzy cost vector. Then, its main advantages over the existing fuzzy sensitivity analysis problems are discussed. Finally, to show the application of fuzzy PA problem, an existing real-life fuzzy transportation problem is solved and the obtained results are explored.

New method for solving Fuzzy transportation problems with LR flat fuzzy numbers

Transportation problems have various applications in logistics and supply chains for reducing costs. In real–life situations, the parameters of transportation problems may not be known precisely because of uncontrollable factors. Herein, we propose a new method for solving fuzzy transportation problems (FTPs) in which the transportation costs and supply and demand are represented by non–negative LR flat fuzzy numbers. The FTP is converted into four transportation problems, which are solved using standard transportation simplex algorithms. The advantages of the proposed method over existing methods are discussed in the context of two application examples. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature.

Quotient algebraic structures on the set of fuzzy numbers

A. M. Bica has constructed in [6] two isomorphic Abelian groups, defined on quotient sets of the set of those unimodal fuzzy numbers which have strictly monotone and continuous sides. In this paper, we extend the results of above mentioned paper, to a larger class of fuzzy numbers, by adding the flat fuzzy numbers. Furthermore, we add the topological structure and we characterize the constructed quotient groups, by using the set of the continuous functions with bounded variation, defined on $[0,1]$.

Linear programming approach for solving fuzzy critical path problems with fuzzy parameters

To the best of our knowledge, there is no method in the literature to find the fuzzy optimal solution of fully fuzzy critical path (FFCP) problems i.e., critical path problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method is proposed for the same. Also, it is shown that it is better to use JMD representation of LR flat fuzzy numbers in the proposed method as compared to the other representation of LR flat fuzzy numbers.

General Form of Linear Programming Problems with Fuzzy Parameters

In this paper, it is pointed out that the existing general form of such fully fuzzy linear programming problems in whichall the parameters are represented by such flat fuzzy numbers for which is valid only if there is not a negative sign.However, if there is a negative sign, then the existing general form of fully fuzzy linear programming problems is notvalid. Thus, a new general form is proposed.

Mehar’s method for solving fully fuzzy linear programming problems with L-R fuzzy parameters

To the best of our knowledge, there is no method in literature for solving such fully fuzzy linear programming (FLP) problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers. Also, to propose such a method, there is need to find the product of unrestricted L-R flat fuzzy numbers. However, there is no method in the literature to find the product of unrestricted L-R flat fuzzy numbers.In this paper, firstly the product of unrestricted L-R flat fuzzy numbers is proposed and then with the help of proposed product, a new method (named as Mehar’s method) is proposed for solving fully FLP problems. It is also shown that the fully FLP problems which can be solved by the existing methods can also be solved by the Mehar’s method. However, such fully FLP problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers can be solved by Mehar’s method but can not be solved by any of the existing methods.

A NEW RANKING APPROACH AND ITS APPLICATION TO SOLVING FUZZY CRITICAL PATH PROBLEMS

There are several methods for solving fuzzy critical path problems in which ranking approaches are used to compare fuzzy numbers. In this paper, some fuzzy critical path problems are chosen to show that the results obtained with fuzzy critical path methods that use existing ranking approaches are not appropriate according to real life situations. To obtain appropriate results, a new ranking approach the Mehar ranking approach, is proposed for comparing LR flat fuzzy numbers. To show the advantages of the Mehar ranking approach over existing approaches, selected fuzzy critical path problems are solved by using the existing methods together with the Mehar ranking approach. It is shown that the obtained results are appropriate.

Optimization for different types of transportation problems with fuzzy coefficients in the objective function

To the best of our knowledge, there is no method in the literature for solving such fuzzy transportation problems in which the different cost parameters are represented by different type of LR flat fuzzy numbers. In this paper, a new method is proposed for solving similar type of fuzzy transportation problems. To illustrate the proposed method and also to show the advantages of the proposed method some fuzzy transportation problems which can not be solved by using any of the existing method are solved. Also, it is shown that the fuzzy transportation problems which can be solved by the existing methods can also be solved by using the proposed method and the same results are obtained.

Mehar’s method for solving fuzzy sensitivity analysis problems with LR flat fuzzy numbers

In published works on fuzzy linear programming there are only few papers dealing with stability or sensitivity analysis in fuzzy mathematical programming. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such fuzzy linear programming problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method, named as Mehar’s method, is proposed for the same. To show the advantages of proposed method over existing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solved by using the proposed method.

Application of Classical Transportation Methods to Find the Fuzzy Optimal Solution of Fuzzy Transportation Problems

To the best of our knowledge till now there is no method in the literature to find the exact fuzzy optimal solution of unbalanced fully fuzzy transportation problems. In this paper, the shortcomings and limitations of some of the existing methods for solving the problems are pointed out and to overcome these shortcomings and limitations, two new methods are proposed to find the exact fuzzy optimal solution of unbalanced fuzzy transportation problems by representing all the parameters as LR flat fuzzy numbers. To show the advantages of the proposed methods over existing methods, a fully fuzzy transportation problem which may not be solved by using any of the existing methods, is solved by using the proposed methods and by comparing the results, obtained by using the existing methods and proposed methods. It is shown that it is better to use proposed methods as compared to existing methods.

Comparison of methodologies for fuzzy and stochastic multi-objective programming

Methods for solving a multicriteria linear program with coefficients of the objective functions and the constraints being flat fuzzy numbers and those dealing with a stochastic multicriteria linear program where imprecision of some data is modelled by probability distributions are surveyed. The methodologies are compared and evaluated. Some ideas for future research are emphasized.