Duality in fuzzy linear programming with symmetric trapezoidal numbers (original) (raw)
Related papers
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.
Fuzzy linear programs with trapezoidal fuzzy numbers
2006
The objective of this paper is to deal with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting results are obtained which in turn lead to a solution of fuzzy linear programming problems without converting them to crisp linear programming problems.
A Generalized Model for Fuzzy Linear Programs with Trapezoidal Fuzzy Numbers
2017
In this paper, we generalize a linear programming problem with symmetric trapezoidal fuzzy number which is introduced by Ganesan and et al. in [3] to a general kind of trapezoidal fuzzy number. In this way, we first establish a new arithmetic operation for multiplication of two trapezoidal fuzzy numbers. Then in order to preparing a method for solving the fuzzy linear programming as well as the primal simplex algorithm, we use a general linear ranking function as a convenient approach in the literature. In fact, our main contribution in this work is based on 3 items: 1) Extending the current fuzzy linear program to a general kind which is not essentially including the symmetric trapezoidal fuzzy numbers , 2) Defining a new multiplication role of two trapezoidal fuzzy numbers, 3) Establishing a fuzzy primal simplex algorithm for solving the generalized model. We in particular emphasize that this study can be used for establishing fuzzy dual simplex algorithm, fuzzy prima...
Bounded Linear Programs with Trapezoidal Fuzzy Numbers
International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 2010
Recently Ganesan and Veeramani introduced a new approach for solving a kind of linear programming problems involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems. But their approach is not efficient for situations in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds. In this paper, by a natural extension of their approach we obtain some new results leading to a new method to overcome this shortcoming.
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
Applied Mathematical Modelling, 2011
In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305-315] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems.
Applied Mathematics, 2011
Two existing methods for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems are the fuzzy primal simplex method proposed by Ganesan and Veeramani [1] and the fuzzy dual simplex method proposed by Ebrahimnejad and Nasseri . The former method is not applicable when a primal basic feasible solution is not easily at hand and the later method needs to an initial dual basic feasible solution. In this paper, we develop a novel approach namely the primal-dual simplex algorithm to overcome mentioned shortcomings. A numerical example is given to illustrate the proposed approach.
A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers
Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.
Journal of Mathematics, 2021
We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.
Duality in fuzzy variable linear programming
2011
Abstract—Recently, fuzzy variable linear programming problems have attracted some interests. We introduce the dual of fuzzy variable linear programming problem, and then deduce some important duality results. Keywords—Duality, fuzzy number, fuzzy variable linear programming, ranking functions. I.