Solving Fuzzy Linear Programming Problems Research Articles

Solving Fuzzy Linear Programming Problem Using Defuzzification Method

Linear Programming (LP) has been one of the efficient, reliable and time tested techniques in Optimization. Conventional LP is not suitable for many real time problems which involve data with inherent vagueness or impreciseness. Fuzzy set theory is proved to be quite good in addressing the inherent vagueness or impreciseness and thus Fuzzy Linear Programming (FLP) is brought to light and developed over the years. A quite good number of techniques have been proposed for solving FLP problems to obtain optimal solution for real world problems involving fuzzy (vague or imprecise) environment. In this paper, “Extended Geometric Mean Defuzzification” is defined and based on it, a method is proposed for solving FLP problems. To showcase the advantages of the proposed method, different problems of FLP, available in the literature, are discussed. Numerical comparisons are also provided to validate the authentication of the proposed method.

Two Phase Method for Solving Fuzzy Linear Programming Problems using Ranking of Generalized Fuzzy Numbers

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.

Applications of Fuzzy Linear Programming to Business Problems

In large and complex systems it is often very difficult to find exact solutions to the Linear Programming problems with the standard techniques, since the necessary data cannot be easily determined precisely and therefore estimates of them are used in practice. Consequently, fuzzy techniques have been developed for achieving better results. In this work we introduce a simple method for solving Fuzzy Linear Programming problems with the help of Triangular Fuzzy Numbers and we apply it for solving business problems.

On New Centroid Based Techniques for Solving Fuzzy Linear Programming Problems

On New Centroid Based Techniques for Solving Fuzzy Linear Programming Problems

Solving Fuzzy Linear Programming Problems with Fuzzy Decision Variables

The numerical method for solving the fuzzy linear programming problems with fuzzydecision variables is proposed in this paper. The difficulty for solving this kind of problem is thatthe decision variables are assumed to be nonnegative fuzzy numbers instead of nonnegative realnumbers. In other words, the decision variables are assumed to be membership functions. One of thepurposes of this paper is to derive the analytic formula of error estimation regarding the approximateoptimal solution. On the other hand, the existence of optimal solutions is also studied in this paper.Finally we present two numerical examples to demonstrate the usefulness of the numerical method.

An Efficient Algorithm for Solving Fuzzy Linear Programming Problems

In this article, we consider some well-known approaches for solving fuzzy linear programming (FLP) problems. We present some of the difficulties of these approaches. Then, crisp linear programming problems are suggested for solving FLP problems. A new algorithm is also given. The proposed approach has advantages over the other methods. For example, we can achieve higher membership degrees for objective function and constraints. Moreover, we show that the fuzzy optimal solutions obtained by the proposed approach are efficient enough. Also, we see that unlike the previous methods, our method finds efficient solutions by solving only one crisp linear problem instead of solving two or three crisp problems. Finally some numerical examples are presented to show the efficiency of the given approach over the other approaches.

A Note on “A New Method for Solving Fuzzy Linear Programming Problems Based on the Fuzzy Linear Complementary Problem (FLCP)”

Mottaghi et al. (Int J Fuzzy Syst 17:236–245, 2015) proposed some results by considering a pair of fuzzy linear programming problems as a primal—dual pair. In this paper, it is pointed out that the pair of fuzzy linear programming problems, considered by Mottaghi et al. as a primal—dual pair, is not a primal—dual pair. Hence, all the results, proposed by Mottaghi et al. by considering the pair of fuzzy linear programming problems as a primal—dual pair, are not valid.

A New Method For Solving Fuzzy Linear Programming Problems Based on The Fuzzy Linear Complementary Problem (FLCP)

Linear programming (LP) is one of the most widely used methods in the area of optimization. Dealing with the formulation of LP problems, the parameters of objective function and constraints should be assigned by experts. In most cases, precise data have been used, but in most of the real-life situations, these parameters are imprecise and ambiguous. In order to deal with the problem of ambiguity and imprecision, fuzzy numbers can be appropriate. By replacing precise numbers with fuzzy numbers, LP problems change to fuzzy linear programming (FLP) problems. So FLPs can be considered as a broader category in comparison to LPs. Considering the above-mentioned points, FLP problems play an important rule in operational researches hence there is a need to investigate these problems. In this paper, a new method for solving the FLP problems is presented in which the coefficients of the objective function and the values of the right-hand side are represented by fuzzy numbers, while the elements of the coefficient matrix are represented by real numbers. To this end, we develop the Karush–Kuhn–Tucker (KKT) optimality conditions for FLP problems. Then, every FLP problem is converted to a fuzzy linear complementary problem (FLCP) by considering KKT conditions. In order to solve the FLCP problems, ranking functions and Lemke’s algorithm are used. Consequently, the solution to primal and dual problems of FLP is obtained. In addition to simplicity in calculations and feasibility, this method solves the primal and dual problems of FLP simultaneously. In order to illustrate the proposed method, some numerical examples are considered.

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.

Evaluation of Fuzzy Linear Programming Application in Energy Models

Fuzzy Linear Programming (FLP) approach is an effective approach that has not been attended extensively by research and academic centers in energy planning field up to now. It seems that this inattention roots in some unknown and uninvestigated grounds about this approach that should be researched before any sectoral, regional, national or international applications. In this paper the authors tried to investigate different factors influence FLP modeling. Some factors challenge FLP application in energy modeling field. Other factors strengthen it as a serious competitor for other strategies under uncertainty like Stochastic Programming and Minimax Regret Strategy. Unfamiliarity of energy modelers and planners with this method, no favor of academic circles, confusion of modelers and planners stemmed from plurality of fuzzification and defuzzification methods and the lack of effective and comprehensive softwares for solving FLP problems are some of the obstacles against FLP application in energy planning field. In contrast, variant methods for different problems simplicity, flexibility, and possibility of FLP application in current supply energy models are strong points that propound FLP as an effective approach. Some remedies had been proposed to overcome these obstacles. Designing a software is a vital step. GAMS environment help us for this designation. Fuzzy GAMS is an idea to expand GAMS capabilities to fuzzy optimization models.

A New Method to Find the Unique Fuzzy Optimal Value of Fuzzy Linear Programming Problems

In this paper, shortcomings and limitations of the existing methods for solving fuzzy linear programming (FLP) problems are pointed out. To overcome the limitations as well as to resolve the shortcomings, a new method is proposed for solving FLP problems. To show the advantage of the proposed method over existing methods, a FLP problem is solved by the existing methods as well as the proposed method, and the obtained results are compared.

Some new results in linear programs with trapezoidal fuzzy numbers: Finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method

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.

SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS WITH MODIFIED S-CURVE MEMBERSHIP FUNCTION

In this paper, we concentrate on two kinds of fuzzy linear programming problems: linear programming problems with only fuzzy resource variables and linear programming problems in which both the resource variables and the technological coefficients are fuzzy numbers. We consider here only the case of fuzzy numbers with modified s-curve membership functions. We propose here the modified s-curve membership function as a methodology for fuzzy linear programming and use it for solving these problems. We also compare the new proposed method with non-fuzzy linear programming optimization method. Finally, we provide real life application examples in production planning and their numerical solutions.