Fuzzy Number Linear Programming: A Probabilistic Approach

Journal of Applied Mathematics and Computing, 2004

In the real world there are many linear programming problems where all decision parameters are fuzzy numbers. Several approaches exist which use different ranking functions for solving these problems. Unfortunately when there exist alternative optimal solutions, usually with different fuzzy value of the objective function for these solutions, these methods can not specify a clear approach for choosing a solution. In this paper we propose a method to remove the above shortcoming in solving fuzzy number linear programming problems using the concept of expectation and variance as ranking functions

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.

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

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 ...

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...

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.

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...

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.

International Journal of Management Science and Engineering Management, 2014

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. In this paper, using the concept of fuzzy numbers comparison, we introduce a very effective method for solving these problems. Then we propose a new method for solving linear programming problems with fuzzy variables. This paper extends linear programming based problem into a fuzzy environment. With the problem assumptions, the optimal solution can still be theoretically solved using the simplex based method. To handle fuzzy decision variables can be generated initially, then solved and improved sequentially using the fuzzy decision approach by introducing a robust ranking technique. The proposed procedure was programmed. The model is illustrated with a numerical example and a sensitivity analysis is of the optimal solution is studied with respect changes in parameter which incorporates all concepts of a fuzzy arithmetic approach to draw managerial insights.

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.