scholarly journals An Effective Computational Algorithm for the Global Solution of a Class of Linear Fractional Programming

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
XiaoLi Huang ◽  
YueLin Gao ◽  
Bo Zhang ◽  
Xia Liu
Keyword(s):  
Objective Function ◽  
Global Solution ◽  
Branch And Bound ◽  
Fractional Programming ◽  
Numerical Experiments ◽  
Computational Algorithm ◽  
Decision Variable ◽  
Linear Fractional Programming ◽  
Algorithm Framework ◽  
Output Space

For the minimization of the sum of linear fractions on polyhedra, it is likewise a class of linear fractional programming (LFP). In this paper, we mainly propose a new linear relaxation technique and combine the branch-and-bound algorithm framework to solve the LFP globally. It is worthwhile to mention that the branching operation of the algorithm occurs in the relatively small output space of the dimension rather than the space where the decision variable is located. When the number of linear fractions in the objective function is much lower than the dimension of the decision variable, the performance of the algorithm is better. After that, we also explain the effectiveness, feasibility, and other performances of the algorithm through numerical experiments.

scholarly journals A New Global Optimization Algorithm for a Class of Linear Fractional Programming

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 867 ◽  
Author(s):  
X. Liu ◽  
Y.L. Gao ◽  
B. Zhang ◽  
F.P. Tian
Keyword(s):  
Global Optimization ◽  
Lower Bound ◽  
Objective Function ◽  
Programming Problem ◽  
Optimization Algorithm ◽  
Fractional Programming ◽  
Original Problem ◽  
Global Optimization Algorithm ◽  
Linear Fractional Programming ◽  
Advantages And Disadvantages

In this paper, we propose a new global optimization algorithm, which can better solve a class of linear fractional programming problems on a large scale. First, the original problem is equivalent to a nonlinear programming problem: It introduces p auxiliary variables. At the same time, p new nonlinear equality constraints are added to the original problem. By classifying the coefficient symbols of all linear functions in the objective function of the original problem, four sets are obtained, which are I i + , I i − , J i + and J i − . Combined with the multiplication rule of real number operation, the objective function and constraint conditions of the equivalent problem are linearized into a lower bound linear relaxation programming problem. Our lower bound determination method only needs e i T x + f i ≠ 0 , and there is no need to convert molecules to non-negative forms in advance for some special problems. A output-space branch and bound algorithm based on solving the linear programming problem is proposed and the convergence of the algorithm is proved. Finally, in order to illustrate the feasibility and effectiveness of the algorithm, we have done a series of numerical experiments, and show the advantages and disadvantages of our algorithm by the numerical results.

scholarly journals Crisp-linear-and Models in Fuzzy Multiple Objective Linear Fractional Programming

2020 ◽  
Vol 15 (1) ◽  
Author(s):  
Bogdana Stanojevic ◽  
Simona Dzitac ◽  
Ioan Dzitac
Keyword(s):  
Objective Function ◽  
Fractional Programming ◽  
Linear Models ◽  
Multiple Objective ◽  
Membership Functions ◽  
Tolerance Limits ◽  
Linear Fractional Programming ◽  
Fuzzy Goals ◽  
Fuzzy Goal ◽  
Linear Fractional Programming Problem

The aim of this paper is to introduce two crisp linear models to solve fuzzy multiple objective linear fractional programming problems. In a novel manner we construct two piece-wise linear membership functions to describe the fuzzy goal linked to a linear fractional objective. They are related to the numerator and denominator of the fractional objective function; and we show that using the fuzzy-and operator to aggregate them a convenient description of the original fractional fuzzy goal is obtained. Further on, with the help of the fuzzy-and operator we aggregate all fuzzy goals and constraints, formulate a crisp linear model, and use it to provide a solution to the initial fuzzy multiple objective linear fractional programming problem. The second model embeds in distinct ways the positive and negative information, the desires and restrictions respectively; and aggregates in a bipolar manner the goals and constraints. The main advantage of using the new models lies in the fact that they are linear, and can generate distinct solutions to the multiple objective problem by varying the thresholds and tolerance limits imposed on the fuzzy goals.

scholarly journals An Effective Branch and Bound Algorithm for Minimax Linear Fractional Programming

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hong-Wei Jiao ◽  
Feng-Hui Wang ◽  
Yong-Qiang Chen
Keyword(s):  
Branch And Bound ◽  
Fractional Programming ◽  
Optimal Solution ◽  
Branch And Bound Algorithm ◽  
Linear Relaxation ◽  
Feasible Region ◽  
Test Problems ◽  
Linear Fractional Programming ◽  
Successive Refinement ◽  
Global Optimal

An effective branch and bound algorithm is proposed for globally solving minimax linear fractional programming problem (MLFP). In this algorithm, the lower bounds are computed during the branch and bound search by solving a sequence of linear relaxation programming problems (LRP) of the problem (MLFP), which can be derived by using a new linear relaxation bounding technique, and which can be effectively solved by the simplex method. The proposed branch and bound algorithm is convergent to the global optimal solution of the problem (MLFP) through the successive refinement of the feasible region and solutions of a series of the LRP. Numerical results for several test problems are reported to show the feasibility and effectiveness of the proposed algorithm.

scholarly journals An Efficient Branch-and-Bound Algorithm for Globally Solving Minimax Linear Fractional Programming Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Pujun Jia ◽  
Hongwei Jiao ◽  
Dongwei Shi ◽  
Jingben Yin
Keyword(s):  
Programming Problem ◽  
Branch And Bound ◽  
Fractional Programming ◽  
Outer Space ◽  
Branch And Bound Algorithm ◽  
Linear Relaxation ◽  
Linear Fractional Programming ◽  
Fractional Programming Problem ◽  
Wide Range ◽  
Linear Fractional Programming Problem

This paper presents an efficient outer space branch-and-bound algorithm for globally solving a minimax linear fractional programming problem (MLFP), which has a wide range of applications in data envelopment analysis, engineering optimization, management optimization, and so on. In this algorithm, by introducing auxiliary variables, we first equivalently transform the problem (MLFP) into the problem (EP). By using a new linear relaxation technique, the problem (EP) is reduced to a sequence of linear relaxation problems over the outer space rectangle, which provides the valid lower bound for the optimal value of the problem (EP). Based on the outer space branch-and-bound search and the linear relaxation problem, an outer space branch-and-bound algorithm is constructed for globally solving the problem (MLFP). In addition, the convergence and complexity of the presented algorithm are given. Finally, numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.

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