scholarly journals Solving Linear Programming Problems with Grey Data

2018 ◽  
Vol 3 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Michael Gr. Voskoglou
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Standard Method ◽  
Linear Programming Problem ◽  
Real Life ◽  
Optimal Solution ◽  
Real Numbers ◽  
Decision Variables ◽  
Linear Programming Problems

A Grey Linear Programming problem differs from an ordinary one to the fact that the coefficients of its objective function and / or the technological coefficients and constants of its constraints are grey instead of real numbers. In this work a new method is developed for solving such kind of problems by the whitenization of the grey numbers involved and the solution of the obtained in this way ordinary Linear Programming problem with a standard method. The values of the decision variables in the optimal solution may then be converted to grey numbers to facilitate a vague expression of it, but this must be strictly checked to avoid non creditable such expressions. Examples are also presented to illustrate the applicability of our method in real life applications.

scholarly journals Ranking Function Application for Optimal Solution of Fractional programming problem

2020 ◽  
Vol 25 (1) ◽  
Author(s):  
Rasha Jalal
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Fractional Programming ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimal Solution ◽  
Ranking Function ◽  
Linear Fractional Programming ◽  
Fractional Programming Problem ◽  
Linear Programming Problems

The aim of this paper is to suggest a solution procedure to fractional programming problem based on new ranking function (RF) with triangular fuzzy number (TFN) based on alpha cuts sets of fuzzy numbers. In the present procedure the linear fractional programming (LFP) problems is converted into linear programming problems. We concentrate on linear programming problem problems in which the coefficients of objective function are fuzzy numbers, the right- hand side are fuzzy numbers too, then solving these linear programming problems by using a new ranking function. The obtained linear programming problem can be solved using win QSB program (simplex method) which yields an optimal solution of the linear fractional programming problem. Illustrated examples and comparisons with previous approaches are included to evince the feasibility of the proposed approach.

scholarly journals Modeling of Gauss Elimination Technique and AHA Simplex Algorithm for Multi-objective Linear Programming Problems

2020 ◽  
pp. 1-14
Author(s):  
Sanjay Jain ◽  
Adarsh Mangal
Keyword(s):  
Linear Programming ◽  
Numerical Solution ◽  
Objective Function ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Simplex Algorithm ◽  
Multi Objective ◽  
Linear Objective Function ◽  
Gauss Elimination ◽  
Linear Programming Problems

In this research paper, an effort has been made to solve each linear objective function involved in the Multi-objective Linear Programming Problem (MOLPP) under consideration by AHA simplex algorithm and then the MOLPP is converted into a single LPP by using various techniques and then the solution of LPP thus formed is recovered by Gauss elimination technique. MOLPP is concerned with the linear programming problems of maximizing or minimizing, the linear objective function having more than one objective along with subject to a set of constraints having linear inequalities in nature. Modeling of Gauss elimination technique of inequalities is derived for numerical solution of linear programming problem by using concept of bounds. The method is quite useful because the calculations involved are simple as compared to other existing methods and takes least time. The same has been illustrated by a numerical example for each technique discussed here.

scholarly journals A New Approach to Find an Optimal Solution of a Fuzzy Linear Programming Problem by Fuzzy Dynamic Programming

2018 ◽  
Vol 7 (4.10) ◽  
pp. 360
Author(s):  
T. Nagalakshmi ◽  
G. Uthra
Keyword(s):  
Dynamic Programming ◽  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Optimal Solution ◽  
Fuzzy Linear Programming ◽  
New Approach ◽  
The Cost ◽  
Recursive Equations

This paper mainly focuses on a new approach to find an optimal solution of a fuzzy linear programming problem with the help of Fuzzy Dynamic Programming. Linear programming deals with the optimization of a function of variables called an objective function, subject to a set of linear inequalities called constraints. The objective function may be maximizing the profit or minimizing the cost or any other measure of effectiveness subject to constraints imposed by supply, demand, storage capacity, etc., Moreover, it is known that fuzziness prevails in all fields. Hence, a general linear programming problem with fuzzy parameters is considered where the variables are taken as Triangular Fuzzy Numbers. The solution is obtained by the method of FDP by framing fuzzy forward and fuzzy backward recursive equations. It is observed that the solutions obtained by both the equations are the same. This approach is illustrated with a numerical example. This feature of the proposed approach eliminates the imprecision and fuzziness in LPP models. The application of Fuzzy set theory in the field of dynamic Programming is called Fuzzy Dynamic Programming. 

scholarly journals Two simple ways to find an efficient solution for a multiple objective linear programming problem

2018 ◽  
Vol 23 (1) ◽  
pp. 11-18
Author(s):  
Vasile Căruțașu
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Efficient Solution ◽  
Multiple Objective ◽  
Multiple Objective Linear Programming ◽  
Optimal Solutions ◽  
Nadir Point ◽  
Linear Programming Problems

Abstract A number of methods and techniques for determining “effective” solutions for multiple objective linear programming problems (MPP) have been developed. In this study, we will present two simple methods for determining an efficient solution for a MPP that reducing the given problem to a one-objective linear programming problem. One of these methods falls under the category of methods of weighted metrics, and the other is an approach similar to the ε- constraint method. The solutions determined by the two methods are not only effective and are found on the Pareto frontier, but are also “the best” in terms of distance to the optimal solutions for all objective function from the MPP. Obviously, besides the optimal solutions of linear programming problems in which we take each objective function, we can also consider the ideal point and Nadir point, in order to take into account all the notions that have been introduced to provide a solution to this problem

scholarly journals Generalised Single-Valued Neutrosophic Number and Its Application to Neutrosophic Linear Programming

2020 ◽  
pp. 180-214
Author(s):  
Nirmal Kumar Mahapatra ◽  
Tuhin Bera
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Simplex Method ◽  
Real Life ◽  
Simplex Algorithm ◽  
Real Life Problem ◽  
Crisp Linear Programming ◽  
Neutrosophic Number

In this chapter, the concept of single valued neutrosophic number (SVN-Number) is presented in a generalized way. Using this notion, a crisp linear programming problem (LP-problem) is extended to a neutrosophic linear programming problem (NLP-problem). The coefficients of the objective function of a crisp LP-problem are considered as generalized single valued neutrosophic number (GSVN-Number). This modified form of LP-problem is here called an NLP-problem. An algorithm is developed to solve NLP-problem by simplex method. Finally, this simplex algorithm is applied to a real-life problem. The problem is illustrated and solved numerically.

scholarly journals Solution of Fully Bipolar Fuzzy Linear Programming Models

2021 ◽  
Vol 2021 ◽  
pp. 1-31
Author(s):  
Muhammad Athar Mehmood ◽  
Muhammad Akram ◽  
Majed G. Alharbi ◽  
Shahida Bashir
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimal Solution ◽  
Equality Constraints ◽  
Fuzzy Linear Programming ◽  
Mathematical Tool ◽  
Decision Variables ◽  
Crisp Linear Programming

The Yin-Yang bipolar fuzzy set is a powerful mathematical tool for depicting fuzziness and vagueness. We first extend the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We first define arithmetic operations of unrestricted bipolar fuzzy numbers and multiplication of an unrestricted trapezoidal bipolar fuzzy number (TrBFN) with non-negative TrBFN. We then propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.

scholarly journals A Novel Method for Solving the Fully Fuzzy Bilevel Linear Programming Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Aihong Ren
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Level Sets ◽  
Fuzzy Numbers ◽  
Main Idea ◽  
Bilevel Linear Programming ◽  
Decision Variables ◽  
Upper Level

We address a fully fuzzy bilevel linear programming problem in which all the coefficients and variables of both objective functions and constraints are expressed as fuzzy numbers. This paper is to develop a new method to deal with the fully fuzzy bilevel linear programming problem by applying interval programming method. To this end, we first discretize membership grade of fuzzy coefficients and fuzzy decision variables of the problem into a finite number ofα-level sets. By usingα-level sets of fuzzy numbers, the fully fuzzy bilevel linear programming problem is transformed into an interval bilevel linear programming problem for eachα-level set. The main idea to solve the obtained interval bilevel linear programming problem is to convert the problem into two deterministic subproblems which correspond to the lower and upper bounds of the upper level objective function. Based on theKth-best algorithm, the two subproblems can be solved sequentially. Based on a series ofα-level sets, we introduce a linear piecewise trapezoidal fuzzy number to approximate the optimal value of the upper level objective function of the fully fuzzy bilevel linear programming problem. Finally, a numerical example is provided to demonstrate the feasibility of the proposed approach.

scholarly journals Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 435
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Linear Programming ◽  
Robust Optimization ◽  
Numerical Method ◽  
Programming Problem ◽  
Continuous Time ◽  
Linear Programming Problem ◽  
Optimal Solution ◽  
Robust Counterpart ◽  
Linear Programming Problems ◽  
Time Linear

A robust continuous-time linear programming problem is formulated and solved numerically in this paper. The data occurring in the continuous-time linear programming problem are assumed to be uncertain. In this paper, the uncertainty is treated by following the concept of robust optimization, which has been extensively studied recently. We introduce the robust counterpart of the continuous-time linear programming problem. In order to solve this robust counterpart, a discretization problem is formulated and solved to obtain the ϵ -optimal solution. The important contribution of this paper is to locate the error bound between the optimal solution and ϵ -optimal solution.

scholarly journals The Knowledge of Expert Opinion in Intuitionistic Fuzzy Linear Programming Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
A. Nagoorgani ◽  
J. Kavikumar ◽  
K. Ponnalagu
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Real Life ◽  
Fuzzy Linear Programming ◽  
Limited Resources ◽  
Fuzzy Data ◽  
Intuitionistic Fuzzy ◽  
Competing Demands ◽  
Intuitionistic Fuzzy Data

In real life, information available for certain situations is vague and such uncertainty is unavoidable. One possible solution is to consider the knowledge of experts on the parameters involved as intuitionistic fuzzy data. We examine a linear programming problem in which all the coefficients are intuitionistic in nature. An approach is presented to solve an intuitionistic fuzzy linear programming problem. In this proposed approach, a procedure for allocating limited resources effectively among competing demands is developed. An example is given to highlight the illustrated study.

Construction Method of Constraint Matrices Corresponded by an Optimal Solution

2014 ◽  
Vol 513-517 ◽  
pp. 1617-1620
Author(s):  
Xiao Liu ◽  
Wei Li ◽  
Peng Zhen Liu
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Linear Equations ◽  
Optimal Solution ◽  
Construction Method ◽  
Interval Linear Programming ◽  
Interval Linear

Through deep analysis of the solvability, which is based on interval linear equations and inequalities systems, for a given optimal solution to interval linear programming problem, we propose the construction method of constraint matrices corresponded by the optimal solution in this paper.

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