Operations Of Fuzzy Numbers Research Articles

An effective estimation of distribution algorithm for the flexible job-shop scheduling problem with fuzzy processing time

Considering the fuzzy nature of the data in real-world scheduling, an effective estimation of distribution algorithm (EDA) is proposed to solve the flexible job-shop scheduling problem with fuzzy processing time. A probability model is presented to describe the probability distribution of the solution space. A mechanism is provided to update the probability model with the elite individuals. By sampling the probability model, new individuals can be generated among the search region with promising solutions. Moreover, a left-shift scheme is employed for improving schedule solution when idle time exists on the machine. In addition, some fuzzy number operations are used to calculate scheduling objective value. The influence of parameter setting is investigated based on the Taguchi method of design of experiment, and a suitable parameter setting is suggested. Numerical testing results and comparisons with some existing algorithms are provided, which demonstrate the effectiveness of the proposed EDA.

Modelling the running states of high-speed trains using triangular fuzzy number workflow nets

In this paper, we propose the use of workflow nets to study the issues associated with deriving the running states of trains. Triangular fuzzy numbers are introduced to express the train’s activity times and to aid in the modelling using workflow nets of the running processes of high-speed trains. A workflow net links all the activities of a train’s running processes into a sequence called a train running workflow net for high-speed trains. The main job of a triangular fuzzy number workflow net is triangular fuzzy number operation. The principles of this operation are discussed in this paper. The connotations of the two modes of operation of train running workflow nets—sequential mode and alternative mode—are also analysed. In addition, methods for calculating the time duration during which a triangular fuzzy number workflow net operates, both in sequential mode and alternative mode, for high-speed trains are proposed and proved. Train activity times and running status can be forecast by executing the proposed triangular fuzzy number workflow nets.

Intuitionistic fuzzy multi-criteria decision-making method based on evidential reasoning

An approach based on interval belief degrees and fuzzy evidential reasoning analytical algorithm is developed for multi-criteria decision problems with uncertainties. The criteria weights are represented by interval numbers and criteria values by triangular intuitionistic fuzzy numbers. The proposed approach does not need to utilize the theories such as arithmetic operations for Triangular intuitionistic fuzzy numbers, for it can remove the influence of the limitations existed in the arithmetic operations. A numerical example is also provided to illustrate the rationality and utility of the proposed method.

Notes on triangular intuitionistic fuzzy numbers

The notion of triangular fuzzy number is put forward on the basis of intuitionistic fuzzy sense. The basic algebra and some non-linear operations of triangular intuitionistic fuzzy numbers (TIFNs) are devised. The average ranking index is introduced to find out order relations between two TIFNs. A method is described to approximate a TIFN to a nearly approximated interval number. Applying this result and using interval arithmetic a bound unconstrained optimisation problem is solved whose coefficients are fixed TIFNs. Numerical examples are also given for illustration.

Approximate Solution of LR Fuzzy Sylvester Matrix Equations

The fuzzy Sylvester matrix equationAX~+X~B=C~in whichA,Barem×mandn×ncrisp matrices, respectively, andC~is anm×nLR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.

Fuzzy Approximate Solution of Positive Fully Fuzzy Linear Matrix Equations

The fuzzy matrix equationsA~⊗X~⊗B~=C~in whichA~,B~, andC~arem×m,n×n, andm×nnonnegative LR fuzzy numbers matrices, respectively, are investigated. The fuzzy matrix systems is extended into three crisp systems of linear matrix equations according to arithmetic operations of LR fuzzy numbers. Based on pseudoinverse of matrix, the fuzzy approximate solution of original fuzzy systems is obtained by solving the crisp linear matrix systems. In addition, the existence condition of nonnegative fuzzy solution is discussed. Two examples are calculated to illustrate the proposed method.

Approximate Solution ofnth-Order Fuzzy Linear Differential Equations

The approximate solution ofnth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

Square Operation of Triangular Fuzzy Number

Square Operation of Triangular Fuzzy Number

Multi-period production planning under fuzzy conditions

In the real-world multi-period production/operations management (MP-POM) problems, the parameters must be estimated and they are frequently given by interval estimates. But for most POM models, these interval estimates must be translated into single numbers. This often results in errors and in the loss of a considerable amount of information. The purpose of this paper is to develop, apply, and illustrate a new fuzzy approach using fuzzy numbers to solve the interval MP-POM problem. It consists of employing appropriate fuzzy numbers to represent the interval estimates in the multi-stage decision problems; using the operations of fuzzy numbers combined with dynamic programming to solve the problem; and determining the required minimum/maximum fuzzy number through ranking techniques. To demonstrate the application of this approach, three MP-POM problems with fuzzy costs and/or fuzzy demands are solved. The main advantages of this approach are fuzzy representative solutions for the optimal production schedule and the minimum total cost in terms of interval units rather than single numbers. This enables the production engineers and operations managers to manage the production flexibly and control the costs effectively. A significant original contribution of this research is the development of an effective technique to solve the fuzzy MP-POM problems that have not been addressed thus far. Suggestions for future research include extending the proposed general fuzzy approach to solve large scale multistage fuzzy problems using computers; and to solve problems with fuzzy goals and constraints defined in different spaces.

Fuzzy Economic Order Quantity for Items with Imperfect Quality and Inspection Errors in an Uncertain Environment on Fuzzy Parameters

This article investigates the inventory problem for item received with imperfect quality and inspection errors in an uncertain environment. Two inventory models are discussed with fuzzy parameters for crisp order quantity, or for fuzzy order quantity. Function principle is proposed as an arithmetic operation of fuzzy trapezoidal number to obtain fuzzy economic order quantity and fuzzy annual profit. Graded mean integration method is used for defuzzification of the annual profit. Extension of Lagrangian method is used to find optimal order quantity. Numerical examples are provided to illustrate the results of proposed models.

Extension of the DEMATEL method in an uncertain linguistic environment

The decision making trial and evaluation laboratory (DEMATEL) method is a useful tool for analyzing correlations among factors using crisp values. However, the crisp values are inadequate to model real-life situations due to the fuzziness and uncertainty that are frequently involved in judgments of experts. The aim of this paper is to extend the DEMATEL method to an uncertain linguistic environment. In this paper, the correlation information among factors provided by experts is in the form of uncertain linguistic terms. A formula is first presented to transform correlation information from uncertain linguistic terms to trapezoidal fuzzy numbers. Then, we aggregate the transformed correlation information of each expert into group information using the operations of trapezoidal fuzzy numbers. The importance and classification of factors are determined via fuzzy matrix operations. Furthermore, a causal diagram is constructed to vividly show the different roles of factors. Finally, an example is used to illustrate the procedure of the proposed method.

The Temperature and Humidity Based on the Embedded Implementation of Fuzzy Control System

For the characteristics and needs of temperature and humidity control in current industrial environment, the paper proposes a fuzzy-based embedded temperature and humidity control system，which enables consumers monitor and control temperature and humidity parameters through the PDA when moving. The article first introduces the hardware structure of the temperature and humidity fuzzy control system, then based on the theory of fuzzy set and the fuzzy number operations of l - cut set, focusing on the transition rules of fuzzy controlling fuzzy conditions of the temperature and humidity control systems, and finally introduce the key technologies of embedded software system of the PDA,. The experimental results show that the system is very practical and flexible.

A Novel Algorithm For Finding Equilibrium Strategy In Two Person Zero Sum Game With Fuzzy Strategy Sets And Payoffs

This paper investigates a two person zero sum matrix game in which the payoffs and strategy are characterized as random fuzzy variables. Using the operations of triangular fuzzy numbers, the fuzzy payoffs for all synthetic outcomes are calculated. Based on random fuzzy expected value operator, a random fuzzy expected minimax equilibrium strategy to the game is defined. After that, based on the constraints, the feasible strategy string sets of the players for multi conflict situations are constructed. Then an iterative algorithm based on random fuzzy simulation is designed to seek the minimax equilibrium strategy. Using a linear ranking function, the aggregation model can be solved by transforming it into a crisp bimatrix game. Then, the fuzzy synthetic aggregation model is established and solved by transforming it into a crisp bimatrix game. Finally, a military example is provided to illustrate the practicality and effectively of the model.

A fuzzy AHP approach based on the concept of possibility extent

The main purpose of this paper is to develop a fuzzy AHP method for tackling the uncertainty and imprecision existing in multi-criteria decision process. The proposed method uses fuzzy pair-wise comparison judgments in place of exact numerical values of the comparison ratios. The geometric mean technique is used to integrate all decision-makers’ opinions and construct the fuzzy positive reciprocal matrices. The algebraic operations of triangular fuzzy numbers are utilized to calculate the fuzzy suitability indices of all alternatives. The extent analysis method is used to compute the degree of possibility of priority among fuzzy suitability indices. Besides, two principles are presented to solve the multi-criteria decision problem in a fuzzy decision environment. Principle I provides a partial preorder, and Principle II gives a total preorder on the set of the possible alternatives. Finally, a numerical example of selecting the company with optimal performance in performing customer relationship management is used to demonstrate the decision process of proposed method.

Operations and Measurement of Interval-valued Fuzzy Numbers Based on the Method of Structured Element

In this paper we introduced the definition of interval-valued fuzzy numbers based on the definition of interval-valued fuzzy set and the operations of real fuzzy numbers by extension principle, defined the operations of interval-valued fuzzy numbers and their distance, and gave their structured element representations and operations of them by using the fuzzy structured element method. The work of this paper is an expansion of the research of real fuzzy numbers.

Solution of the Fuzzy Equation <i>A + X = B</i> Using the Method of Superimposition

Fuzzy equations were solved by using different standard methods. One of the well-known methods is the method of -cut. The method of superimposition of sets has been used to define arithmetic operations of fuzzy numbers. In this article, it has been shown that the fuzzy equation AX B   , where A, X, B are fuzzy numbers can be solved by using the method of superimposition of sets. It has also been shown that the method gives same result as the method of -cut.

A method based on distance measure for interval-valued intuitionistic fuzzy group decision making

In this paper we introduce some relations and operations of interval-valued intuitionistic fuzzy numbers and define some types of matrices, including interval-valued intuitionistic fuzzy matrix, interval-valued intuitionistic fuzzy similarity matrix and interval-valued intuitionistic fuzzy equivalence matrix. We study their properties, develop a method based on distance measure for group decision making with interval-valued intuitionistic fuzzy matrices and, finally, provide an illustrative example.

Quasi-linear Fuzzy Number and Its Application in Fuzzy Programming

Abstract System of fuzzy equations is a widespread problem in many applied fields, such as production planning, resource management, optimization decision, and so on, and the variation description of fuzzy variables is the bottle neck problem of realizing solution operation. Starting from the structure-feature of fuzzy information and the essential characteristic of fuzzy decision, this paper proposes the concept of quasi-linear fuzzy number, and establishes a solution model based on metric and uncertainty restriction for system of fuzzy equations (denoted by FESM-M+U, for short); further, on the basis of quasi-linear fuzzy number and principle operation strategy, then gives a genetic algorithm named by FGA-QL+PO; finally, considers its convergence using Markov chain theory and analyzes its performance through an example. All these indicate that, FGA-QL+PO can not only effectively reflect decision consciousness, but has good global convergence, and be suitable for all systems of fuzzy equations, so it can be widely used in many problems.

The extension of fuzzy QFD: From product planning to part deployment

By focusing on listening to the customers, quality function deployment (QFD) has been a successful analysis tool in product design and development. To solve the uncertainty or imprecision in QFD, numerous researchers have attempted to apply the fuzzy set theory to QFD and have developed various fuzzy QFD approaches. Their models usually concentrate on product planning, the first phase of QFD. The subsequent phases (part deployment, process planning, and production planning) of QFD are seldom addressed. Moreover, their models often use algebraic operations of fuzzy numbers to calculate the fuzzy sets in QFD. Biased results are easily produced after several multiplicative or divisional operations. Aiming to solve these two issues, the objective of this study is to develop an extended fuzzy quality function deployment approach (E-QFD) which expands the research scope, from product planning to part deployment. In product planning, a more advanced method for collecting customer requirements is developed while the competitive analysis is also considered. In part deployment, the original part deployment table is enhanced by including the importance of part characteristics (PCs) and the bottleneck level of PCs. A modified fuzzy k-means clustering method is proposed to classify various bottleneck (or importance) groups of PCs. The failure mode and effects analysis (FMEA) is conducted for the high bottleneck (or high importance) group of PCs through the fuzzy inference approach. Moreover, E-QFD employs a more precise method, α-cut operations, to calculate the fuzzy sets in QFD instead of algebraic operations of fuzzy numbers. Finally, a case study is given to explain the analysis process of the proposed method.

Multiplication Operation on Fuzzy Numbers

A fuzzy number is simply an ordinary number whose precise value is somewhat uncertain. Fuzzy numbers are used in statistics, computer programming, engineering, and experimental science. The arithmetic operators on fuzzy numbers are basic content in fuzzy mathematics. Operation of fuzzy number can be generalized from that of crisp interval. The operations of interval are discussed. Multiplication operation on fuzzy numbers is defined by the extension principle. Based on extension principle, nonlinear programming method, analytical method, computer drawing method and computer simulation method are used for solving multiplication operation of two fuzzy numbers. The nonlinear programming method is a precise method also, but it only gets a membership value as given number and it is a difficult problem for solving nonlinear programming. The analytical method is most precise, but it is hard to α-cuts interval when the membership function is complicated. The computer drawing method is simple, but it need calculate the α -cuts interval. The computer simulation method is the most simple, and it has wide applicability, but the membership function is rough. Each method is illuminated by examples.