scholarly journals Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1272
Author(s):  
Bogdana Stanojević ◽  
Milan Stanojević ◽  
Sorin Nădăban
Keyword(s):  
Mathematical Programming ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Extension Principle ◽  
Fuzzy Arithmetic ◽  
Advantages And Disadvantages ◽  
Fuzzy Environment ◽  
Basic Ideas ◽  
Mathematical Programming Problems

Optimization problems in the fuzzy environment are widely studied in the literature. We restrict our attention to mathematical programming problems with coefficients and/or decision variables expressed by fuzzy numbers. Since the review of the recent literature on mathematical programming in the fuzzy environment shows that the extension principle is widely present through the fuzzy arithmetic but much less involved in the foundations of the solution concepts, we believe that efforts to rehabilitate the idea of following the extension principle when deriving relevant fuzzy descriptions to optimal solutions are highly needed. This paper identifies the current position and role of the extension principle in solving mathematical programming problems that involve fuzzy numbers in their models, highlighting the indispensability of the extension principle in approaching this class of problems. After presenting the basic ideas in fuzzy optimization, underlying the advantages and disadvantages of different solution approaches, we review the main methodologies yielding solutions that elude the extension principle, and then compare them to those that follow it. We also suggest research directions focusing on using the extension principle in all stages of the optimization process.

Fuzzy Reliability of an Imprecise Failure to Start of an Automobile Using Pseudo-Parabolic Fuzzy Numbers

2018 ◽  
Vol 14 (03) ◽  
pp. 323-341 ◽  
Author(s):  
F. Abbasi
Keyword(s):  
Fuzzy Systems ◽  
Fuzzy Numbers ◽  
Extension Principle ◽  
Fuzzy Arithmetic ◽  
Fuzzy Reliability ◽  
Fuzzy Environment ◽  
Distributed Components ◽  
Proposed Model ◽  
New Type ◽  
The Membership Function

In this paper, we propose the notion of pseudo-parabolic fuzzy numbers and the component failure probabilities are considered as a new type of fuzzy number as pseudo-parabolic to incorporate the uncertainties in the parameter, due to a more realistic estimate of them. Then, we analyze the reliability of fuzzy system (particularly, series and parallel system) with independent and non-identically distributed components, and using the new operations of TA [F. Abbasi et al., Journal of Intelligent and Fuzzy Systems 29 (2015) 851–861], due to the smaller results support, easier calculations and special properties than fuzzy arithmetic 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 provide a more realistic fuzzy reliability analysis. 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 pseudo-parabolic fuzzy numbers.

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

2020 ◽  
pp. 1-22
Author(s):  
F. Abbasi ◽  
T. Allahviranloo
Keyword(s):  
Fuzzy Systems ◽  
Fuzzy Numbers ◽  
Fault Tree Analysis ◽  
Extension Principle ◽  
Fuzzy Arithmetic ◽  
General Shape ◽  
Fuzzy Environment ◽  
Distributed Components ◽  
Proposed Model ◽  
Arithmetic Operator

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.

scholarly journals Application of Fuzzy Optimization Problem in Fuzzy Environment

2015 ◽  
Vol 62 (2) ◽  
pp. 119-125
Author(s):  
Shapla Shirin ◽  
Kamrunnahar
Keyword(s):  
Linear Programming ◽  
Simplex Method ◽  
Optimization Problem ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Daily Life ◽  
Fuzzy Optimization ◽  
Triangular Fuzzy Numbers ◽  
Fuzzy Environment ◽  
Optimum Solution

In this paper application of optimization problem has been introduced which belong to fuzzy environment. An attempt has been taken to find out a suitable option in order to obtain the optimum solutions of optimization problems in fuzzy environment. Optimum solutions of the proposed optimization problem computed by using three methods, such as Bellman-Zadeh’s method, Zimmerman’s Method, and Fuzzy Version of Simplex method, are compared to each other. In support of that the three algorithms of the above three methods have been reviewed. However, the main objective of this paper is to focus on the appropriate method and how to achieve good enough or optimum solution of linear programming using triangular fuzzy numbers with equal widths because of complex and undefined situations in our daily life. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21976 Dhaka Univ. J. Sci. 62(2): 119-125, 2014 (July)

scholarly journals An Extensive Operational Law for Monotone Functions of LR Fuzzy Intervals with Applications to Fuzzy Optimization

2021 ◽  
Author(s):  
Mingxuan Zhao ◽  
Yulin Han ◽  
Jian Zhou
Keyword(s):  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Fuzzy Programming ◽  
Planning Problem ◽  
Monotone Functions ◽  
Solution Strategy ◽  
Deterministic Equivalent ◽  
The One ◽  
Short Time

Abstract The operational law put forward by Zhou et al. on strictly monotone functions with regard to regular LR fuzzy numbers makes a valuable push to the development of fuzzy set theory. However, its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which restricts its application in practice to a certain degree. In this paper, we propose an extensive operational law that generalizes the one proposed by Zhou et al. to apply to monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs), of which regular fuzzy numbers can be regarded as particular cases. By means of the extensive operational law, the inverse credibility distributions (ICDs) of monotone functions regarding regular LR-FIs can be calculated efficiently and effectively. Moreover, the extensive operational law has a wider range of applications, which can deal with the situations hard to be handled by the original operational law. Subsequently, based on the extensive operational law, the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs are presented. Furthermore, the proposed operational law is also applied to dispose fuzzy optimization problems with regular LR-FIs, for which a solution strategy is provided, where the fuzzy programming is converted to a deterministic equivalent first and then a newly-devised solution algorithm is utilized. Finally, the proposed solution strategy is applied to a purchasing planning problem, whose performances are evaluated by comparing with the traditional fuzzy simulation-based genetic algorithm. Experimental results indicate that our method is much more efficient, yielding high-quality solutions within a short time.

scholarly journals Optimisation of repair and maintenance costs for electrical equipment in agricultural enterprises

2021 ◽  
Vol 37 ◽  
pp. 00103
Author(s):  
Valery Zhdanov ◽  
Elena Logacheva ◽  
Viktor Yarosh ◽  
Alexander Ivashina
Keyword(s):  
Mathematical Programming ◽  
Spatial Data ◽  
Optimization Problems ◽  
Cost Optimization ◽  
Electrical Equipment ◽  
Mathematical Methods ◽  
Advantages And Disadvantages ◽  
Industrial Enterprises ◽  
Repair Operation ◽  
Method Of Solution

Application of mathematical methods of cost optimization for repair and maintenance of electrical equipment of agro-industrial enterprises is one of the important and promising directions for increasing the efficiency of electrical equipment operation management in agriculture. Mathematical programming systems use graphical and related attributive information in solving optimization problems. As graphical information in these systems we used maps, plans, diagrams, schedules of preventive measures from which the list of equipment for certain types of repair and maintenance, their labor intensity for individual objects, types of equipment and in total for the enterprise are established. Databases of electrical equipment are used as attributive information to describe electrical equipment of agro-industrial enterprises. Due to joint processing of graphical and attributive information in optimization systems, all stages of work with spatial data are more operative. Beginning from spatial data search, selection and analysis we can make a specific decision during the operation control of electrical equipment. This article considers maintenance and repair operation (MR) as a task of mathematical programming with cost optimization and deals with three approaches to the organization of this task. The expediency of using each method of solution is analyzed. The structural schemes, equations describing mathematical models, advantages and disadvantages of the presented models are given. We marked prospect of using linear programming programs for the decision of the given optimization problem by means of the inverse matrix method, i.e. the modified simplex method and computing algorithm with a standard sequence of operations.

LEXICOGRAPHIC METHOD FOR MATRIX GAMES WITH PAYOFFS OF TRIANGULAR FUZZY NUMBERS

2008 ◽  
Vol 16 (03) ◽  
pp. 371-389 ◽  
Author(s):  
DENG-FENG LI
Keyword(s):  
Linear Programming ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Order Relation ◽  
Programming Models ◽  
Matrix Game ◽  
Triangular Fuzzy Numbers ◽  
Matrix Games ◽  
Lexicographic Method

The purpose of the paper is to study how to solve a type of matrix games with payoffs of triangular fuzzy numbers. In this paper, the value of a matrix game with payoffs of triangular fuzzy numbers has been considered as a variable of the triangular fuzzy number. First, based on two auxiliary linear programming models of a classical matrix game and the operations of triangular fuzzy numbers, fuzzy optimization problems are established for two players. Then, based on the order relation of triangular fuzzy numbers the fuzzy optimization problems for players are decomposed into three-objective linear programming models. Finally, using the lexicographic method maximin and minimax strategies for players and the fuzzy value of the matrix game with payoffs of triangular fuzzy numbers can be obtained through solving two corresponding auxiliary linear programming problems, which are easily computed using the existing Simplex method for the linear programming problem. It has been shown that the models proposed in this paper extend the classical matrix game models. A numerical example is provided to illustrate the methodology.

Interval Methods and Fuzzy Optimization

1997 ◽  
Vol 05 (03) ◽  
pp. 239-249 ◽  
Author(s):  
Weldon A. Lodwick ◽  
K. David Jamison
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Interval Analysis ◽  
Fuzzy Set Theory ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Interval Methods

In this paper, we describe interval-based methods for solving constrained fuzzy optimization problems. The class of fuzzy functions we consider for the optimization problems is the set of real-valued functions where one or more parameters/coefficients are fuzzy numbers. The focus of this research is to explore some relationships between fuzzy set theory and interval analysis as it relates to optimization problems.

scholarly journals Fuzzy Number Addition with the Application of Horizontal Membership Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Andrzej Piegat ◽  
Marcin Pluciński
Keyword(s):  
Interval Arithmetic ◽  
Fuzzy Numbers ◽  
Order Relation ◽  
Extension Principle ◽  
Fuzzy Arithmetic ◽  
Multidimensional Approach ◽  
Membership Functions ◽  
Multidimensional Information ◽  
Rdm Arithmetic ◽  
Horizontal Membership Functions

The paper presents addition of fuzzy numbers realised with the application of the multidimensional RDM arithmetic and horizontal membership functions (MFs). Fuzzy arithmetic (FA) is a very difficult task because operations should be performed here on multidimensional information granules. Instead, a lot of FA methods useα-cuts in connection with 1-dimensional classical interval arithmetic that operates not on multidimensional granules but on 1-dimensional intervals. Such approach causes difficulties in calculations and is a reason for arithmetical paradoxes. The multidimensional approach allows for removing drawbacks and weaknesses of FA. It is possible thanks to the application of horizontal membership functions which considerably facilitate calculations because now uncertain values can be inserted directly into equations without using the extension principle. The paper shows how the addition operation can be realised on independent fuzzy numbers and on partly or fully dependent fuzzy numbers with taking into account the order relation and how to solve equations, which can be a difficult task for 1-dimensional FAs.

scholarly journals Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions

2020 ◽  
Vol 18 (1) ◽  
pp. 1451-1477
Author(s):  
Ting Xie ◽  
Zengtai Gong ◽  
Dapeng Li
Keyword(s):  
Fuzzy Number ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Directional Derivative ◽  
Fuzzy Optimization ◽  
Optimality Criteria ◽  
Generalized Derivatives ◽  
Duality Theorems ◽  
Generalized Derivative ◽  
Partial Orderings

Abstract In this paper, we present the concepts of generalized derivative, directional generalized derivative, subdifferential and conjugate for n-dimensional fuzzy-number-valued functions and discuss the characterizations of generalized derivative and directional generalized derivative by, respectively, using the derivative and directional derivative of crisp functions that are determined by the fuzzy mapping. Furthermore, the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for n-dimensional fuzzy-number-valued functions are investigated. Finally, under two kinds of partial orderings defined on the set of all n-dimensional fuzzy numbers, the duality theorems and saddle point optimality criteria in fuzzy optimization problems with constraints are discussed.

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