scholarly journals An Equivalent Linear Programming Form of General Linear Fractional Programming: A Duality Approach

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1586
Author(s):  
Mehdi Toloo
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Fractional Programming ◽  
Linear Programming Problem ◽  
Functional Programming ◽  
General Linear ◽  
Data Envelopment ◽  
Linear Fractional Programming ◽  
Duality Approach ◽  
Fractional Functional

Linear fractional programming has been an important planning tool for the past four decades. The main contribution of this study is to show, under some assumptions, for a linear programming problem, that there are two different dual problems (one linear programming and one linear fractional functional programming) that are equivalent. In other words, we formulate a linear programming problem that is equivalent to the general linear fractional functional programming problem. These equivalent models have some interesting properties which help us to prove the related duality theorems in an easy manner. A traditional data envelopment analysis (DEA) model is taken, as an instance, to illustrate the applicability of the proposed approach.

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 THE LINEAR-FRACTIONAL PROGRAMMING PROBLEM UNDER UNCERTAINTY CONDITIONS

2021 ◽  
pp. 20-28
Author(s):  
Oleksandr Pavlov ◽  
Oleksandra Vozniuk ◽  
Olena Zhdanova
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Fractional Programming ◽  
Linear Programming Problem ◽  
Original Problem ◽  
Graphical Form ◽  
Compromise Solution ◽  
Objective Functions ◽  
Linear Fractional Programming ◽  
Optimal Values

This paper addresses the problem of linear-fractional programming under uncertainty. The uncertainty here is understood as the ambiguity of the coefficients’ values in the optimized functional. We give two mathematical formulations of the problem. In the first one, the uncertainty refers to the numerator: there are several sets of objective function coefficients, each coefficient can determine the numerator of the problem’s criterion at the stage of its solution implementation. The uncertainty in the second formulation refers to the denominator of the functional. We propose several compromise criteria for evaluating solutions to the problem we consider. We study the following two criterions in detail: 1) finding a compromise solution in which the deviation of the values of the partial functionals from their optimal values is within the specified limits; 2) finding a compromise solution according to the criterion of minimizing the total weighted excess of the values of partial functionals in relation to the specified feasible deviations from their optimal values (the values of concessions). We formulate an auxiliary linear programming problem to find a compromise solution to the linear-fractional programming problems by these two criteria. The constraints of the auxiliary problem depend on the optimization direction in the original problem. We carried out a series of experiments of four types to study the properties of the problem. The purposes of the experiments were: 1) to study how changes in the values of the specified feasible deviations of partial objective functions impact the values of actual deviations and the values of concessions; 2) to study how changes in the expert weights of partial objective functions impact the values of actual deviations and the values of concessions for the compromise solutions we obtain. We propose in this work the schemes of experiments and present their results in graphical form. We have found that the obtained relations depend on the optimization direction in the original problem. Keywords: optimization, uncertainty, convolution, linear-fractional programming, linear programming problem, compromise solution

The Measurement of Efficiency in Multiunit Public Services

1987 ◽  
Vol 19 (11) ◽  
pp. 1511-1524 ◽  
Author(s):  
W D Macmillan
Keyword(s):  
Linear Programming ◽  
Data Envelopment Analysis ◽  
Fractional Programming ◽  
Policy Formulation ◽  
Operating Conditions ◽  
Data Envelopment ◽  
Linear Fractional Programming ◽  
New Concepts ◽  
Nonlinear Version ◽  
Efficiency And Effectiveness

After a review of the concepts of efficiency and effectiveness in multiunit nonmarket organisations, a recently developed technique known as Data Envelopment Analysis (DEA) is described and illustrated in this paper, and it is shown how this technique may be used in performance measurement. The linear fractional programming and equivalent linear programming formulations of the DEA problem are presented and an illustrative problem is solved. Some new concepts concerned with the application of DEA are introduced. A nonlinear version of the DEA problem is then described and illustrated. A comparison is made between the solutions of the linear and the nonlinear DEA problems. Last, brief consideration is given to the problems of uncontrollable operating conditions, policy formulation, and the acceptability of the technique to unit managers.

scholarly journals A new Mean Deviation and Advanced Mean Deviation Techniques to Solve Multi-Objective Fractional Programming Problem Via Point-Slopes Formula

2021 ◽  
pp. 1051-1064
Author(s):  
Rebaz Mustafa ◽  
Nejmaddin A. Sulaiman
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Fractional Programming ◽  
Linear Programming Problem ◽  
Optimal Solution ◽  
Mean Deviation ◽  
New Techniques ◽  
Multi Objective ◽  
Fractional Programming Problem ◽  
A New Technique

In this paper, we have proposed a new technique to find an efficient solution to fractional programming problems (FPP). The multi-objective fractional programming problem (MOFPP) is converted into multi-objective linear programming (MOLPP) utilizing the point-slopes formula for a plane, which has equivalent weights to the MOFPP. The MOLPP is diminished to a single objective linear programming problem (SOLPP) through using two new techniques for the values of the objective function and suggesting an algorithm for its solution. Finally, we obtained the optimal solution for MOFPP by solving the consequent linear programming problem (LPP). The proposed practicability is confirmed with the existing approaches, with some numerical examples and we indicated comparison with other techniques. 

scholarly journals Additive Algorithm for Solving 0-1 Integer Linear Fractional Programming Problem

2013 ◽  
Vol 61 (2) ◽  
pp. 173-178
Author(s):  
Md Rajib Arefin ◽  
Touhid Hossain ◽  
Md Ainul Islam
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Integer Linear Programming ◽  
Fractional Programming ◽  
Feasible Solution ◽  
Linear Fractional Programming ◽  
Fractional Programming Problem ◽  
Linear Fractional Programming Problem

In this paper, we present additive algorithm for solving a class of 0-1 integer linear fractional programming problems (0-1 ILFP) where all the coefficients at the numerator of the objective function are of same sign. The process is analogous to the process of solving 0-1 integer linear programming (0-1 ILP) problem but the condition of fathoming the partial feasible solution is different from that of 0-1 ILP. The procedure has been illustrated by two examples. DOI: http://dx.doi.org/10.3329/dujs.v61i2.17066 Dhaka Univ. J. Sci. 61(2): 173-178, 2013 (July)

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