Representation Of Fuzzy Numbers Research Articles

WEAK AND STRONG DUALITY THEOREMS FOR FUZZY CONIC OPTIMIZATION PROBLEMS

The objective of this paper is to deal with the fuzzy conic program- ming problems. The aim here is to derive weak and strong duality theorems for a general fuzzy conic programming. Toward this end, The convexity-like concept of fuzzy mappings is introduced and then a specic ordering cone is established based on the parameterized representation of fuzzy numbers. Un- der this setting, duality theorems are extended from crisp conic optimization problems to fuzzy ones.

Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment

A common algorithm to solve the shortest path problem (SPP) is the Dijkstra algorithm. In this paper, a generalized Dijkstra algorithm is proposed to handle SPP in an uncertain environment. Two key issues need to be addressed in SPP with fuzzy parameters. One is how to determine the addition of two edges. The other is how to compare the distance between two different paths with their edge lengths represented by fuzzy numbers. To solve these problems, the graded mean integration representation of fuzzy numbers is adopted to improve the classical Dijkstra algorithm. A numerical example of a transportation network is used to illustrate the efficiency of the proposed method.

An Interactive Method for Multi Stage Fuzzy Lattice decision making problems using Triangular fuzzy numbers

method for finding interactive multi- stage lattice fuzzy decision making problems in which all the parameters are represented by fuzzy numbers. In this interactive method a new algorithm is proposed to fuzzy lattice decision makings together with new representations of triangular fuzzy numbers. This paper will show the advantages of using proposed representations over the existing representations of fuzzy numbers and will present with great clarity the proposed method and illustrate numerical example.

Application of practical fuzzy arithmetic to fuzzy internal model control

This paper applies a new fuzzy arithmetic of interval calculus and fuzzy quantities to automatic control. Practical results are obtained which overcome those based on the extension principle or α-cuts. The proposed approach is based on a different representation of fuzzy numbers, though most common arithmetic operators cannot be directly applied for designing a fuzzy controller due to the unjustified overestimation effect. To avoid this phenomenon, a procedure based on an “exact” resolution calculus is proposed, whose solutions allow creating a fuzzy internal model control scheme. The validity of the new method is illustrated by a real-time educational engineering application on classical control design: a coupled tanks system.

Genetic algorithm based fuzzy weighted average for multi-criteria decision making problems

Ranking of alternatives on the basis of multi-criteria is an important issue in decision making. These criterion may be quantitative as well as qualitative in nature. In this paper, a method is developed using genetic algorithm for the ranking of alternatives on the basis of multi-criteria. Fuzzy numbers are used to evaluate the qualitative criteria. In this method, first, fuzzy weighted average of each alternative is formulated as a pair of non linear programs (in the form of α-cut representation of fuzzy numbers), which represent the left and right bounds of alternative. After that, membership value of fuzzy weighted average of each alternative is constructed by enumerating different values of α. The pair of non linear optimization problems for different values of α are solved using genetic algorithm. Finally, ranking order of alternatives is calculated by Yager’s ranking indices. A numerical example is solved successfully to demonstrate the proposed methodology.

A new algorithm for computing the fuzzy weighted average

A new fuzzy weighted average computation algorithm (NFWA) based on the α-cuts representation of fuzzy numbers is presented in this paper. For each α-cuts, the endpoints of the fuzzy weighted average (FWA) can be calculated from two particular switch points. In the proposed algorithm, these two switch points are computed with an opposite direction searching process, although recursive, which is remarkably efficient. The calculation complexity of the new algorithm is O(n). Experimental result demonstrates that compared with some commonly used FWA algorithms, the new algorithm approach requires the least CPU time, and then may be the fastest available FWA algorithm to date.

Conducting fuzzy division by using linear programming

Some approximation methods have been proposed for fuzzy multiplication and division in the literature. Instead of doing arithmetic operations using fuzzy membership functions for fuzzy numbers, parameterized representation of fuzzy numbers have been used in arithmetic operations. The most applied parameterized fuzzy numbers used in many of the research papers are symmetric and asymmetric triangular and trapezoidal fuzzy numbers. In this study, we propose a new approximation method based on linear programming for fuzzy division. In order to show the applicability of the proposed method, some examples are solved using the proposed method and the results are compared with those generated by other methods in the literature. The proposed method has produced better results than those generated by the others.

A Probability-Based Approach to Comparison of Fuzzy Numbers and Applications to Target-Oriented Decision Making

In this paper, we introduce a new comparison relation on fuzzy numbers based on their alpha-cut representation and comparison probabilities of interval values. Basically, this comparison process combines a widely accepted interpretation of fuzzy sets together with the uncertain characteristics inherent in the representation of fuzzy numbers. The proposed comparison relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of target-based evaluations. Some numerical examples are used to illuminate the proposed ranking technique as well as to compare with previous methods. More interestingly, according to the interpretation of the new comparison relation on fuzzy numbers, we provide a fuzzy target-based decision model as a solution to the problem of decision making under uncertainty, with which an interesting link between the decision maker's different attitudes about target and different risk attitudes in terms of utility functions can be established. Moreover, an application of the proposed comparison relation to the fuzzy target-based decision model for the problem of fuzzy decision making with uncertainty is provided. Numerical examples are also given for illustration.

A note on convexity and semicontinuity of fuzzy mappings

By using parameterized representation of fuzzy numbers, criteria for a lower semicontinuous fuzzy mapping defined on a non-empty convex subset of R n to be a convex fuzzy mapping are obtained.

Parametric representation of fuzzy numbers and application to fuzzy calculus

We present several models to obtain simple parametric representations of the fuzzy numbers or intervals, based on the use of piecewise monotonic functions of different forms. The representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers (we use the standard α - cuts setting, but also the membership functions are obtained immediately) and can be used directly to obtain error-controlled-approximations of the fuzzy calculus in terms of a finite set of parameters. The general setting is the Hermite-type interpolation, where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces (and of the parameters) by which the results are obtained. The representations are designed to model the lower and the upper extremal values of each α - cut (compact) intervals of the fuzzy numbers and are able to produce almost any possible configuration (differentiable, continuous or with a finite number of discontinuity points) by using parametric monotonic functions of different types. We show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics. The proposed model is called the Lower–Upper representation and we denote the associated fuzzy numbers or intervals as LU-fuzzy.

Convex fuzzy mapping and operations of convex fuzzy mappings

Convex and concave fuzzy mappings are considered in this paper. Based on the seminal work of Nanda and Kar on the definition of convex fuzzy mapping, the operations of convex fuzzy mapping are established. Some of the important concepts such as positively homogeneous, infimal convolution, right scalar multiplication, and convex hull are introduced and the corresponding theorems are demonstrated by using the parametric representation of fuzzy numbers.

Simulation of fuzzy dynamical systems using the LU-representation of fuzzy numbers☆

We suggest the use of the parametric LU-representation of the fuzzy numbers, introduced in Guerra and Stefanini [Guerra ML, Stefanini L. Approximate fuzzy arithmetic operations using monotonic interpolations. Fuzzy Sets Syst 2005;150:5–33] and improved in Stefanini et al. [Stefanini L, Sorini L, Guerra ML. Parametric representations of fuzzy numbers and applications. Working Paper Series EMS, 95, University of Urbino, 2004], in the simulation of fuzzy dynamical systems or fuzzy iterated maps. We show the computational advantages of the LU-representation in extending some well known standard maps to the fuzzy context, allowing the simulation by the Zadeh’s extension principle in the general case of fuzzy parameters.

Solving a multiobjective possibilistic problem through compromise programming

Real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory. Here we propose a method for solving these kinds of models through a fuzzy compromise programming approach. To formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced. This concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their α-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis.

Fuzzy compromise programming for portfolio selection

The aim of this paper is to solve a portfolio selection problem using Sharpe's single index model in a soft framework. Estimations of subjective or imprecise future beta for every asset can be represented through fuzzy numbers constructed on the basis of statistical data and the relevant knowledge of the financial analyst; the model, therefore, works with data that contain more information than any classical model and dealing with it does not involve a great extra computational effort. In order to solve the portfolio selection problem we have formulated a Fuzzy Compromise Programming problem. For this task we have introduced the fuzzy ideal solution concept based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their @a-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and using discrepancy between fuzzy numbers in our analysis. A major feature of this model is its sensitivity to the analyst's opinion as well as to the decision-maker's preferences. This allows interaction with both when it comes to design the best portfolio.

Fuzzy arithmetic and solving of the static governing equations of fuzzy finite element method

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables, was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.

A fuzziness measure for fuzzy numbers: Applications

Extending the ideas of Delgado et al. (1995) to include the notion of fuzziness, we obtain canonical representations of fuzzy numbers. In particular, we show that any fuzzy number can be represented by either a trapezoidal or a “quasitrapezoidal” number having the same basic attributes as the original fuzzy number.

Representing and comparing two kinds of fuzzy numbers

As a mathematical representation of the linguistic variable, fuzzy number theory with its applications was extensively developed. We investigate two popular representations of fuzzy numbers, symmetric-triangular and bell-shaped, and compare their arithmetic, dimensions and metrics. It is shown that, as far as linear structures are concerned, both representations are equivalent.

On a canonical representation of fuzzy numbers

Fuzzy numbers, and more generally linguistic values, are approximate assessments, given by experts and accepted by decision-makers when obtaining more accurate values is impossible or unnecessary. To simplify the task of representing and handling fuzzy numbers, several authors have introduced real indices in order to capture the information contained in a fuzzy number. In this paper we propose two parameters, value and ambiguity, for this purpose. We use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision-making problems. Several examples illustrate these ideas.

Representation and application of fuzzy numbers

In this paper we present the theoretical background of fuzzy numbers connected with the possibility and Dempster-Shafer theories. We describe some types of representation of fuzzy numbers and we study the notions of the distance and orders between fuzzy numbers based on these representations. The second part is devoted to the application of fuzzy numbers in data analysis, artificial intelligence and decision making.

A parametric representation of fuzzy numbers and their arithmetic operators

Direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex. Implementation of the extension principle is equivalent to solving a nonlinear programming problem. To overcome this difficulty many applications limit the membership functions to certain shapes, usually either triangular fuzzy numbers (TFN) or trapezoidal fuzzy numbers (TrFN). Then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers, thus making the calculations trivial. Unfortunately the TFN shape is not closed under multiplication and division. The result of these operators is a polynomial membership function and the triangular shape only approximates the actual result. The linear approximation can be quite poor and may lead to incorrect results when used in engineering applications. We analyze this problem and propose six parameters which define parameterized fuzzy numbers (PFN), of which TFNs are a special case. We provide the methods for performing fuzzy arithmetic and show that the PFN representation is closed under the arithmetic operations. The new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as TFNs. The new method has better accuracy and similar computational speed to using TFNs and appears to have benefits when used in engineering applications.