Multiple Objective Linear Fractional Programming Research Articles

On application of linear fractional programming problem using e-education system

This article focuses on an area of nonlinear programming problems known as linear fractional programming problems with multiple objectives. When tackling real-world linear fractional optimization problems, ambiguity and uncertainty in decision-making are inherent. This research aims to present a simple and computationally quick approach to solving multiple objective linear fractional programming problems with all decision variables and parameters described in terms of crisp. The proposed solution algorithm is based primarily on the fuzzy-based technique, and a membership function strategy. To resolve the multi-objective linear fractional programming problem, first consider the problem as a single objective function and along with the fuzzy programming model obtain the optimal solution using LINGO software. LINGO is a software application primarily used for solving linear, nonlinear, and integer optimization problems Moreover, an e-education setup problem demonstrates the steps of the proposed method.

An Application of Fully Intuitionistic Fuzzy Multi-objective Linear Fractional Programming Problem in E-education System

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

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

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.

A computationally efficient algorithm to approximate the pareto front of multi-objective linear fractional programming problem

The main contribution of this paper is the procedure that constructs a good approximation to the non-dominated set of multiple objective linear fractional programming problem using the solutions to certain linear optimization problems. In our approach we propose a way to generate a discrete set of feasible solutions that are further used as starting points in any procedure for deriving efficient solutions. The efficient solutions are mapped into non-dominated points that form a 0th order approximation of the Pareto front. We report the computational results obtained by solving random generated instances, and show that the approximations obtained by running our procedure are better than those obtained by running other procedures suggested in the recent literature. We evaluated the quality of each approximation using classic metrics.

Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions

This paper integrates fuzzy linearization strategy, goal programming, a membership function and conditional control mechanisms to produce a novel method to deal with the binary behavior of multiple objective fractional programming problems and multiple objective fractional programming problems with a utility function. The major contributions of the proposed method are twofold. (1) The binary behavior of multiple objective fractional programming problems can easily be converted into a linearized program using the proposed fuzzy linearization strategy. The linearized program can easily be solved, using commercial linear programming packages, yielding an approximate global optimal solution, and (2) The utility function is also used to ensure that the qualification requirements for a multiple objective fractional programming problem are met, in contrast to most past mathematical approaches, which only use quantitative approaches to deal with such a problem. In addition, an illustrative example and a practical real case are provided to demonstrate the usefulness of the proposed model. The discussion of the practical problem will help decision makers to realize the usefulness of a utility function and the binary behavior in multiple objective fractional programming problems.

An Interactive Method for Solving Multiple Objective Linear Fractional Programming Problem Based on Linear Optimization Models

An Interactive Method for Solving Multiple Objective Linear Fractional Programming Problem Based on Linear Optimization Models

Solving a special class of multiple objective linear fractional programming problems

In this paper a feasible direction method is presented to find all efficient extreme points for a special class of multiple objective linear fractional programming problems, when all denominators are equal. This method is based on the conjugate gradient projection method, so that we start with a feasible point and then a sequence of feasible directions towards all efficient adjacent extremes of the problem can be generated. Since methods based on vertex information may encounter difficulties as the problem size increases, we expect that this method will be less sensitive to problem size. A simple production example is given to illustrate this method. doi:10.1017/S1446181114000200

SOLVING A SPECIAL CLASS OF MULTIPLE OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEMS

AbstractIn this paper a feasible direction method is presented to find all efficient extreme points for a special class of multiple objective linear fractional programming problems, when all denominators are equal. This method is based on the conjugate gradient projection method, so that we start with a feasible point and then a sequence of feasible directions towards all efficient adjacent extremes of the problem can be generated. Since methods based on vertex information may encounter difficulties as the problem size increases, we expect that this method will be less sensitive to problem size. A simple production example is given to illustrate this method.

Optimization of Multiple-Objective Web Service Selection Using Fractional Programming

The Service Oriented Approach (SOA) is based on the idea that distributed applications can be flexibly composed by integrating independently developed component services. Due to the fact that a large number of available services offer similar functionality, when choosing actual services to be included in the composition their non-functional (Quality of Service – QoS) properties must also be taken into account. On the other hand certain constraints regarding the required performances (i.e. the required QoS levels) may also be given. Therefore, web service selection presents a multiple-objective multiple constraint problem and in this paper we model it as a discrete multiple-objective linear fractional programming (MOLFP) problem. In order to solve this problem we use a previously published technique for generating strongly efficient solutions to a continuous MOLFP problem to the discrete case. We present a complete methodology for solving the multiple criteria web service selection problem and report the experimental results obtained by testing it on 50 random instances generated by varying the input parameters.

A note on ‘Taylor series approach to fuzzy multiple objective linear fractional programming’

Toksari, in his paper ‘Taylor series approach to fuzzy multiobjective linear fractional programming’, published in Information Sciences 178 (2008), proposed a new method to solve multiple objective linear fractional programming problems and declared that it is more effective when compare to previous methods. He constructed fuzzy goals and then used first order Taylor-polynomials to approximate the corresponding linear fractional membership functions by linear functions. Aggregating the linear functions Toksari obtained a crisp linear programming problem and claimed that it is ‘equal’ to the fuzzy fractional one. In this paper, we indicate the fallacy that arises by using Taylor approximation and propose an improved method that guarantees the efficiency of the solution it provides. Both practical applications from Toksari’s paper are recalled to show that the improvements we suggest are effective.

An Exact Method for a Discrete Multiobjective Linear Fractional Optimization

Integer linear fractional programming problem with multiple objective (MOILFP) is an important field of research and has not received as much attention as did multiple objective linear fractional programming. In this work, we develop a branch and cut algorithm based on continuous fractional optimization, for generating the whole integer efficient solutions of the MOILFP problem. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions, then generate an integer feasible solution. Using the reduced gradients of the objective functions, an efficient cut is built and a part of the feasible domain not containing efficient solutions is truncated by adding this cut. A sample problem is solved using this algorithm, and the main practical advantages of the algorithm are indicated.

A method of solving fully fuzzified linear fractional programming problems

In this paper, we propose a method of solving the fully fuzzified linear fractional programming problems, where all the parameters and variables are triangular fuzzy numbers. We transform the problem of maximizing a function with triangular fuzzy value into a deterministic multiple objective linear fractional programming problem with quadratic constraints. We apply the extension principle of Zadeh to add fuzzy numbers, an approximate version of the same principle to multiply and divide fuzzy numbers and the Kerre’s method to evaluate a fuzzy constraint. The results obtained by Buckley and Feuring in 2000 applied to fractional programming and disjunctive constraints are taken into consideration here. The method needs to add extra zero-one variables for treating disjunctive constraints. In order to illustrate our method we consider a numerical example.

Computing non-dominated solutions in MOLFP

In this paper we present a technique to compute the maximum of a weighted sum of the objective functions in multiple objective linear fractional programming (MOLFP). The basic idea of the technique is to divide (by the approximate ‘middle’) the non-dominated region in two sub-regions and to analyze each of them in order to discard one if it can be proved that the maximum of the weighted sum is in the other. The process is repeated with the remaining region. The process will end when the remaining regions are so little that the differences among their non-dominated solutions are lower than a pre-defined error. Through the discarded regions it is possible to extract conditions that establish weight indifference regions. These conditions define the variation range of the weights that necessarily leads to the same non-dominated solution. An example, illustrating the concept, is presented. Some computational results indicating the performance of the technique are also presented.

An interactive method for multiple objective linear fractional programming problems

Multiple objective linear fractional programming (MOLFP) is an important field of research. Using some branch and bound techniques, we have developed a new interactive method for MOLFP that drastically reduces the computational effort needed, while providing guidance for the decision maker in the choice of his/her preferred solutions. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions while constraining the others. Several linear programming problems, organized in a tree structure, are generated as the search evolves. The whole idea is simple and it results in a fast and very intuitive approach to exploring the non-dominated set of solutions in MOLFP, and eventually to finding the preferred solution.

A Hybrid Fuzzy-Goal Programming Approach to Multiple Objective Linear Fractional Programming Problems

This paper presents an algorithm for solving a Multiple Objective Fractional Decision Making (MOFDM) problems based on the theory of weakly-efficient points, fuzzy sets, and Goal Programming (GP). Starting from an integer weakly-efficient point, the proposed algorithm combines the attractive features of both fuzzy set theory and GP for MOFDM problem into a lexicographic GP problem by fixing the priorities and aspiration levels using the concept of fuzzy membership function along with the concept of conflict among the objectives. It is illustrated with the help of an example.

On a fuzzy set approach to solving multiple objective linear fractional programming problem

In 1984, Luhandjula used a linguistic variable approach in order to present a procedure for solving multiple objective linear fractional programming problem (MOLFPP). In 1992, Dutta et al. (Fuzzy Sets and Systems 52 (1) (1992) 39–45) modified the linguistic approach of Luhandjula such as to obtain efficient solution to problem MOLFPP. The aim of this paper is to point out certain shortcomings in the work of Dutta et al. and give the correct proof of theorem which validates the obtaining of the efficient solutions. We notice that the method presented there as a general one does only work efficiently if certain hypotheses (restrictive enough and hardly verified) are satisfied. The example considered by Dutta et al. is again used to illustrate the approach.

係数間の関係を考慮したファジィ多目標分数計画法

係数間の関係を考慮したファジィ多目標分数計画法

Multiple objective linear fractional programming — A fuzzy set theoretic approach

A method for optimizing a multiobjective linear fractional programming problem is developed which yields always an efficient solution.

On the use of multiple objective linear programming algorithms to solve problems with fractional objectives

Multiple objective linear fractional programming can be used in a wide range of problem formulations in private and public enterprises. Under certain restrictions, a simpler multiple objective linear programming approach can be used for problem solution. In this note we analyse the use of this simplified approach to MOLFP, and the practical effect of the restrictions implied.

Note—Finding Certain Weakly-Efficient Vertices in Multiple Objective Linear Fractional Programming

Recently Kornbluth and Steuer have developed a simplex-based algorithm for finding all weakly-efficient vertices of an augmented feasible region of a multiple objective linear fractional programming problem. As part of this algorithm, they presented a method for detecting certain weakly-efficient vertices called break points. In this note we show that the procedure used by Kornbluth and Steuer in this method for computing the numbers needed to find these break points may sometimes fail. We also propose a fail-safe method for computing these numbers and give some computational results with this method.