Duality in fuzzy variable linear programming (original) (raw)

Method for Optimizing the Dual of Linear Fuzzy Programming Problems

JOURNAL OF ADVANCES IN MATHEMATICS, 2016

This article’s goal is to support the existence of the dual in a Linear Fuzzy Real environment and focus on its application to Linear fuzzy program problems. This concept will apply to linear fuzzy programming problems that contain fuzzy constraints with a crisp objective function, crisp constraints with a fuzzy objective function, or fuzzy constraints with a fuzzy objective function. It is also proposed here that optimizing fuzzy constraints and objectives of the dual linear program that consist of a triplet and are much like triangular fuzzy numbers, but differ in that they are a hybrid fuzzy number that contains characteristics that are both fuzzy and crisp.

On duality in linear programming under fuzzy environment

Fuzzy Sets and Systems, 2002

A pair of linear primal-dual problems is introduced under fuzzy environment and appropriate results are proved to establish the duality relationship between them. Possible extensions are also suggested.

Some duality results on linear programming problems with symmetric fuzzy numbers

Fuzzy Information and Engineering, 2009

Recently, linear programming problems with symmetric fuzzy numbers (LPSFN) have considered by some authors and have proposed a new method for solving these problems without converting to the classical linear programming problem, where the cost coefficients are symmetric fuzzy numbers (see in [4]). Here we extend their results and first prove the optimality theorem and then define the dual problem

Fuzzy linear programming duality

2012

The word ”duality” has been used in various areas of science for long time. Nevertheless, in general, there is a lack of consensus about the exact meaning of this important notion. However, in the field of optimization, and particularly in linear programming, the notion of duality is well understood and remarkably useful. Various attempts to develop analogous useful duality schemes for linear programming involving fuzzy data have been appearing since the early days of fuzzy sets. After recalling basic results on linear programming duality, we give examples of early attempts in extending duality to problems involving fuzzy data, and then we discuss recent results on duality in fuzzy linear programming and their possible application.

A primal-dual method for solving linear programming problems with fuzzy cost coefficients based on linear ranking functions and its applications

International Journal of Industrial and Systems Engineering, 2012

There are two important approaches based on linear ranking functions for solving linear programming problems with cost coefficients as an auxiliary problem to obtain a fuzzy solution of fuzzy variable linear programming problem. The first approach uses the primal simplex method that assumes an initial primal feasible basic solution is at hand. The second approach is based on dual simplex method that begins with a basic dual feasible basic solution and proceeds by pivoting through a series of dual basic solutions until the associated complementary primal basic fuzzy solution is feasible. In this paper, we propose a new method called the primal-dual algorithm, which is similar to the dual simplex method and begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

New Approach to Solve Fuzzy Linear Programming Problems by the Ranking Function

Bonfring

In this paper, a new method is proposed to find the fuzzy optimal solution of fully fuzzy linear programming problems with triangular fuzzy numbers. A computational method for solving fully fuzzy linear programming problems (FFLPP) is proposed, based upon the new Ranking function. The proposed method is very easy to understand and to apply for fully fuzzy linear programming problems occurring in real life situations as compared to the existing methods. To illustrate the proposed method numerical examples are solved

Fuzzy linear programming problems: models and solutions

Soft Computing, 2019

We investigate various types of fuzzy linear programming problems based on models and solution methods. First, we review fuzzy linear programming problems with fuzzy decision variables and fuzzy linear programming problems with fuzzy parameters (fuzzy numbers in the definition of the objective function or constraints) along with the associated duality results. Then, we review the fully fuzzy linear programming problems with all variables and parameters being allowed to be fuzzy. Most methods used for solving such problems are based on ranking functions, α-cuts, using duality results or penalty functions. In these methods, authors deal with crisp formulations of the fuzzy problems. Recently, some heuristic algorithms have also been proposed. In these methods, some authors solve the fuzzy problem directly, while others solve the crisp problems approximately.

New ranking function for fuzzy linear programming problem and system of linear equations

Journal of Information and Optimization Sciences, 2018

Linear Programming problems and System of Linear equations have many applications in various science and engineering problems like network analysis, operations research etc. In general Linear Programming Problem (LPP) and the system of linear equations contain crisp parameters that is real numbers or complex numbers as their coefficients and constants, but in real life applications, LPP and system of equations may contain the constrains or the parameters as uncertain. These uncertain values are not the exact real numbers but vary within some range of values, the values may vary within an interval or can be considered as fuzzy number. In this paper, we have developed a new Ranking function (which converts the fuzzy number into crisp) to solve a fully fuzzy LPP and System of equations. Unlike the previous ranking functions, the proposed ranking function uses fuzzy number itself improving the accuracy of the solution. The ranking function is derived by replacing the non-parallel sides of the trapezoidal fuzzy number with non-linear functions. Various numerical examples are included and compared with the pre-existing methods.

Fuzzy Dual Programming: An Introduction

To tackle optimization problems with uncertain parameters and variables, this communication introduces the concepts of fuzzy dual numbers, vectors and matrices and considers basic operations which compose fuzzy dual calculus applied to fuzzy quantities. The formulation of optimization problems using this new formalism is discussed. It is shown that each fuzzy dual programming problem generates a finite set of classical optimization problems, even in the case in which the feasible set is defined using fuzzy dual LMI constraints.