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.
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.
Duality in fuzzy linear programming with symmetric trapezoidal numbers
2010
Linear programming problems with trapezoidal fuzzy numbers have recently attracted much interest. Various methods have been developed for solving these types of problems. Here, following the work of Ganesan and Veeramani and using the recent approach of Mahdavi-Amiri and Nasseri, we introduce the dual of the linear programming problem with symmetric trapezoidal fuzzy numbers and establish some duality results. The results will be useful for post optimality analysis.
Fuzzy variable linear programming with fuzzy technical coefficients
Pakistan Journal of Statistics and Operation Research, 2012
Fuzzy linear programming is an application of fuzzy set theory in linear decision making problems and most of these problems are related to linear programming (LP) with fuzzy variables. In this paper, an approximate but convenient method for solving these problems with fuzzy non-negative technical coefficient and without using the ranking functions, is proposed. With the help of numerical examples, the method is illustrated.
Fuzzy Sets and Systems, 2007
Linear programming problems with trapezoidal fuzzy variables (FVLP) have recently attracted some interest. Some methods have been developed for solving these problems by introducing and solving certain auxiliary problems. Here, we apply a linear ranking function to order trapezoidal fuzzy numbers. Then, we establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results. In particular, we prove that the auxiliary problem is indeed the dual of the FVLP problem. Having established the dual problem, the results will then follow as natural extensions of duality results for linear programming problems with crisp data. Finally, using the results, we develop a new dual algorithm for solving the FVLP problem directly, making use of the primal simplex tableau. This algorithm will be useful for sensitivity (or post optimality) analysis when using primal simplex tableaus.
The Comparative Relation and Its Application in solving Fuzzy Linear Programming Problem
This paper considers linear programming problem whose objective function is fuzzy numbers vector. In order to solve this problem, we first present a new definition of comparative relation on the set of fuzzy numbers. Based on this definition, we state a method to compare fuzzy numbers directly and then, by the related theorems and lemmas, we build an algorithm to solve fuzzy presented problem.
Solving a Fully Fuzzy Linear Programming Problem by Ranking
International Journal of Mathematics Trends and Technology, 2014
In this paper, we propose a new method for solving Fully Fuzzy linear programming Problem (FFLP) using ranking method .In this proposed ranking method, the given FFLPP is converted into a crisp linear programming (CLP) Problem with bound variable constraints and solved by using Robust’s ranking technique and the optimal solution to the given FFLP problem is obtained and then compared between our proposed method and the existing method. Numerical examples are used to demonstrate the effectiveness and accuracy of this method.
Fuzzy Integer Linear Programming with Fuzzy Decision Variables
In this paper a new method for dealing with Fuzzy Integer Linear Programming Problems (FILPP) has been proposed. FILPP with fuzzy variables model was taken for solution. This solution method is based on the fuzzy ranking method. The proposed method can serve deci-sion makers by providing the reasonable range of values for the fuzzy variable, which is comparatively better than the currently available solu-tions. Numerical examples demonstrate the effectiveness and accuracy of the proposed method.
Three models of fuzzy integer linear programming
1995
In this paper we study some models for dealing with Fuzzy Integer Linear Programming problems which have a certain lack of precision of a vague nature in their formulation. We present methods to solve them with either fuzzy constraints, or fuzzy numbers in the objective function or fuzzy numbers defining the set of constraints. These methods are based on the representation theorem and on fuzzy number ranking methods.
A Constructive Proof of Fundamental Theory for Fuzzy Variable Linear Programming Problems
2012
Two existing methods for solving fuzzy variable linear programming problems based on ranking functions are the fuzzy primal simplex method proposed by Mahdavi-Amiri et al. (2009) and the fuzzy dual simplex method proposed by Mahdavi-Amiri and Nasseri (2007). In this paper, we prove that in the absence of degeneracy these fuzzy methods stop in a finite number of iterations. Moreover, we generalize the fundamental theorem of linear programming in a crisp environment to a fuzzy one. Finally, we illustrate our proof using a numerical example.
International Journal of Applied Science and Engineering, 2010
Ranking of fuzzy numbers play an important role in decision making problems. Fuzzy numbers must be ranked before an action is taken by a decision maker. Chen (Operations on fuzzy numbers with function principal, Tamkang Journal of Management Science 6 (1985) 13-25) pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. In this paper two phase method is proposed for solving a special type of fuzzy linear programming (FLP) problems using generalized fuzzy numbers. To illustrate the proposed method a numerical example is solved and the advantages of the proposed method are discussed. Since the proposed method is a direct ex- tension of classical method so it is very easy to understand and apply the proposed method to find the fuzzy optimal solution of FLP problems occurring in the real life situations.
Introduction to Fuzzy Dual Mathematical Programming
2016
In this communication the formulation of optimization problems using fuzzy dual parameters and variables is introduced to cope with parametric or implementation uncertainties. It is shown that fuzzy dual programming problems generate finite sets of deterministic optimization problems, allowing to assess the range of the solutions and of the resulting performance at an acceptable computational effort.