Solving a Fully Fuzzy Linear Programming Problem by Ranking

2019

inear programming (LP) is part of an important area of mathematics called ―Optimization technique‖ as it is straightly used to find the most optimized solution to a given problem. It is also one of the simplest ways to perform optimization. A LP may be defined as the problem of maximizing or minimizing a linear function subject to linear constraints. The constraints may be equalities or inequalities. Working with linear programming model requires properly tuning the values of the parameters. Because the real world problems have a high level of complexity and the degree of uncertainty depends on many factors. In order to properly determine the value of these parameters, experts or decision makers needs to deal with this uncertainty and vagueness. Bellman and Zadeh [1] first proposed the concept of decision making in a fuzzy environment as a solution approach for this kind of problems. Zimmermann [2] presented an application of fuzzy optimization technique for multi-objective linear p...

International Journal of Operations Research and Information Systems, 2015

This paper finds solutions to the fuzzy linear program where some parameters are fuzzy numbers. In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either probabilistic programming or multi objective programming methods. Unfortunately all these methods have shortcomings. In this paper, using the concept of comparison of fuzzy numbers, the author introduces a very effective method for solving these problems. This paper extends linear programming based problem in fuzzy environment. With the problem assumptions, the optimal solution can still be theoretically solved using the simplex based method. To handle the fuzzy decision variables can be initially generated and then solved and improved sequentially using the fuzzy decision approach by introducing robust ranking technique. The model is illustrated with an application and a post optimal analysis approach is obtained. The proposed procedure was programmed wi...

International Journal of Advanced Operations Management, 2015

In this paper, I wish to find the solutions to the fuzzy linear program where some parameters are fuzzy numbers. In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either probabilistic programming or multi objective programming methods. Unfortunately, all these methods have shortcomings. In this note, using the concept of comparison of fuzzy numbers, a very effective method is introduced for solving these problems. The model is illustrated with application and a sensitivity analysis is obtained. This paper extends linear programming-based problem in fuzzy environment. With the problem assumptions, the optimal solution can still be theoretically solved using the simplex-based method. To handle the fuzzy decision, variables can be initially generated and then solved and improved sequentially using the fuzzy decision approach by introducing robust ranking technique. The proposed procedure was programmed and through MATLAB (R2009a) version software, the four dimensional slice diagram is represented to the application. The model is illustrated with an application and a post optimal analysis is studied with respect to changes in parameter which incorporates all concepts of a fuzzy arithmetic approach to draw managerial insights.

2015

This paper proposes a new method of Robust ranking technique, which is used for defuzzifying the trapezoidal fuzzy number into a crisp number to represent the fuzzy set. In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either probabilistic programming or multi objective programming methods. Unfortunately all these methods have shortcomings. In this paper, using the concept of comparison of fuzzy numbers, a very effective method is introduced for solving these problems. The model is illustrated with numerical application to generate a good solution and post optimal analyses are obtained. Investigation of the properties of an optimal solution allows developing a simplex algorithm in fuzzy environment. Furthermore, the proposed technique allows the significant ways to help the decision-maker for formulating their decisions and drawing managerial insights efficiently. © 2015 World Academic Press, UK. All rig...

2010

In this paper the shortcomings of an existing method for comparing the generalized fuzzy numbers are pointed out and a new method is proposed for same. Also using the proposed ranking method, a generalized simplex algorithm is proposed for solving a special type of fuzzy linear programming (FLP) problems. To illustrate the proposed algorithm a numerical example is solved and the advantages of the proposed algorithm are discussed. Since the proposed algorithm is a direct extension of classical algorithm so it is very easy to understand and apply the proposed algorithm to find the fuzzy optimal solution of FLP problems occurring in the real life situations.

Hungarian Statistical Review, 2021

In many applications of linear programming, the lack of exact information results in various problems. Nevertheless, these types of problems can be handled using fuzzy linear programming. This study aims to compare different ranking functions for solving fuzzy linear programming problems in which the coefficients of the objective function (the cost vector) are fuzzy numbers. A numerical example is introduced from the field of tourism and then solved using five ranking functions. Computations were carried out using the FuzzyLP package implemented in the statistical software R.

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.

International Journal of Computing Science and Mathematics, 2014

Solving fuzzy linear programming (FLP) requires the employment of a consistent ranking of fuzzy numbers. Ineffective fuzzy number ranking would lead to a flawed and erroneous solving approach. This paper presents a comprehensive and extensive review on fuzzy number ranking methods. Ranking techniques are categorised into six classes based on their characteristics. They include centroid methods, distance methods, area methods, lexicographical methods, methods based on decision maker's viewpoint, and methods based on left and right spreads. A survey on solving approaches to FLP is also reported. We then point out errors in several existing methods that are relevant to the ranking of fuzzy numbers and thence suggest an effective method to solve FLP. Consequently, FLP problems are converted into non-fuzzy single (or multiple) objective linear programming based on a consistent centroid-based ranking of fuzzy numbers. Solutions of FLP are then obtained by solving corresponding crisp single (or multiple) objective programming problems by conventional methods.. He has published a number of peer-reviewed papers in the field of operational research, optimisation and soft computing techniques. His current research interest includes expert systems, applications of intelligent systems for optimisation, classification and forecasting.

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.

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.

Applied Mathematical Modelling, 2011

Lotfi et al. [Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. Math. Modell. 33 (2009) 3151-3156] pointed out that there is no method in literature for finding the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints. In this paper, a new method is proposed to find the fuzzy optimal solution of same type of fuzzy linear programming problems. It is easy to apply the proposed method compare to the existing method for solving the FFLP problems with equality constraints occurring in real life situations. To illustrate the proposed method numerical examples are solved and the obtained results are discussed.

2017

In this paper the optimal solution for linear programming is derived where some parameters are fuzzy in numbers. In practice, there are many problems arise if the decision parameters are crisp in nature, and such problems are usually solved by introducing either probabilistic programming or multi objective programming methods. Unfortunately all these methods have shortcomings. In this note, the concept of fuzzy numbers is introduced, which is a very effective method for solving these problems. With the problem assumptions, the optimal solution can still be theoretically solved using the simplex based method. Fuzzy decision variables can be initially generated and then solved and improved sequentially using the fuzzy decision approach by introducing robust ranking technique. The proposed procedure was programmed through MATLAB (R2009a) version software for representing four dimensional slice diagrams to its application. The model is illustrated with an application which incorporates ...

2012

Fuzzy linear programming problems have an essential role in fuzzy modeling, which can formulate uncertainty in actual environment In this paper we present methods to solve (i) the fuzzy linear programming problem in which the coefficients of objective function are trapezoidal fuzzy numbers, the coefficients of the constraints, right hand side of the constraints are triangular fuzzy numbers, and (ii) the fuzzy linear programming problem in which the variables are trapezoidal fuzzy variables, the coefficients of objective function and right hand side of the constraints are trapezoidal fuzzy numbers, (iii) the fuzzy linear programming problem in which the coefficients of objective function, the coefficients of the constraints, right hand side of the constraints are triangular fuzzy numbers. Here we use α –cut and ranking functions for ordering the triangular fuzzy numbers and trapezoidal fuzzy numbers. Finally numerical examples are provided to illustrate the various methods of the fuz...

International Journal of Scientific Research in Science and Technology, 2023

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.

European Journal of Operational Research, 2007

This paper proposes a method for solving linear programming problems where all the coefficients are, in general, fuzzy numbers. We use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. It allows us to work with the concept of feasibility degree. The bigger the feasibility degree is, the worst the objective value will be. We offer the decision-maker (DM) the optimal solution for several different degrees of feasibility. With this information the DM is able to establish a fuzzy goal. We build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal. A reasonable solution is the one that has the biggest membership degree to this fuzzy subset. Finally, to illustrate our method, we solve a numerical example.