Multiobjective Linear Fractional Programming Research Articles

A LCA Based Biofuel Supply Chain Analysis Framework

This paper presents a life cycle assessment (LCA) based biofuel supply chain (SC) analysis framework which enables the study of economic, energy and environmental (3E) performances by using multi-objective optimization. The economic objective is measured by the total annual profit, the energy objective is measured by the average fossil energy (FE) inputs per MJ biofuel and the environmental objective is measured by greenhouse gas (GHG) emissions per MJ biofuel. A multi-objective linear fractional programming (MOLFP) model with multi-conversion pathways is formulated based on the framework and is solved by using the ɛ-constraint method. The MOLFP problem is turned into a mixed integer linear programming (MILP) problem by setting up the total annual profit as the optimization objective and the average FE inputs per MJ biofuel and GHG emissions per MJ biofuel as constraints. In the case study, this model is used to design an experimental biofuel supply chain in China. A set of the weekly Pareto optimal solutions is obtained. Each non-inferior solution indicates the optimal locations and the amount of biomass produced, locations and capacities of conversion factories, locations and amount of biofuel being supplied in final markets and the flow of mass through the supply chain network (SCN). As the model reveals trade-offs among 3E criteria, we think the framework can be a good support tool of decision for the design of biofuel SC.

Comment on “Fuzzy mathematical programming for multi objective linear fractional programming problem”

Chakraborty and Gupta, in their paper “Fuzzy mathematical programming for multi objective linear fractional programming problem”, published in Fuzzy Sets and Systems 125 (2002), claimed that their methodology proposed for solving multi objective linear fractional programming problem always yields an efficient solution. This paper indicates that their claim is generally wrong and this is due to the non-equivalence of the original problem and the associated linear problem.

Fuzzy Mathematical Programming approach for Solving Fuzzy Linear Fractional Programming Problem

In this paper, a solution procedure is proposed to solve fuzzy linear fractional programming (FLFP) problem where cost of the objective function, the resources and the technological coefficients are triangular fuzzy numbers. Here, the FLFP problem is transformed into an equivalent deterministic multi-objective linear fractional programming (MOLFP) problem. By using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective linear programming (LP) problem. The proposed procedure illustrated through a numerical example.

Fuzzy efficient and Pareto-optimal solution for multi-objective linear fractional programming problems

Many practical optimisation problems usually have several conflicting objectives. In these multi-objective optimisation problems, solution optimising all the objective functions simultaneously does not exist, in general. Instead, Pareto-optimal solutions, which are efficient in terms of all objective functions, are introduced. Nevertheless, many optimal solutions exist. A final solution among Pareto-optimal solutions is to be selected based on the balance among objective functions. In this paper, we find fuzzy efficient and Pareto-optimal solution to the multi-objective linear fractional programming problem (MOLFP). It has shown that when any fuzzy goal is fully achieved, the fuzzy efficient solution may or may not be Pareto-optimal. Therefore, a procedure is proposed to obtain fuzzy efficient solution which is also Pareto-optimal. The efficiency of proposed method is verified by numerical examples and a practical application in production planning.

A linear modelling to solve multi-objective linear fractional programming problem with fuzzy parameters

The purpose of this paper is to present a linearisation procedure to solve the multi-objective linear fractional programming problem with fuzzy parameters. It is easier to understand and it has less computational complexity in contrast with other existing method to solve the multi-objective linear fractional programming problem with fuzzy parameters. As a result, the problem is firstly transformed into a multi-objective linear fractional programming problem with interval parameters using a-cut method. Then, the multi-objective linear fractional programming problem with interval parameters is reduced to a linear programming problem with interval parameters to provide the optimal interval values for the objective functions of multi-objective linear fractional programming problem with interval parameters for each specified a-cut. The obtained interval parameters can then be aggregated to generate fuzzy numbers for the optimal values of the objective functions of multi-objective linear fractional programming problem with fuzzy parameters. Lastly, a numerical example is given to illustrate the procedure.

Weakly fuzzy efficient conditions for multiobjective fractional programming problem

It is the purpose of this article to establish some weakly fuzzy efficient conditions for multiobjective linear fractional programming problem MOFP under generalized convexity. Osuna et al., [27] derived some results regarding the optimality and duality properties for multiobjective fractional programming in classical sense. In this article, an equivalent fuzzy multiobjective fractional programming problem is formulated by using fuzzy goal and tolerance limits for each objective functions. Some weakly efficient criteria for MOFP are established with fuzzy membership functions parallel to Osuna et al., [27].

Algorithm for Solving Fuzzy Multiobjective Linear Fractional Programming Problem by Additive Weighted Method

In this paper attention has been paid to the study of multiobjective linear fractional programming problem (MOLFPP) by using fuzzy set theoretic approach. In this approach, MOLFPP is transformed into multiobjective linear programming problem (MOLPP) by suitable transformation. In algorithm-I, MOLFPP is transformed into MOLPP by using fuzzy set theory and the pareto optimal solution of the transformed MOLPP is obtained by applying Zimmermann’s min-operator model and simplex method. Further we have used additive weighted method to modify the above approach. Algorithm-II has been presented to find the pareto optimal solution of MOLFPP by applying additive weighted method. To demonstrate the applicability of the proposed approach, one numerical example is solved to find the pareto optimal solution by applying this two algorithms.

On Some Stability Notions of Multiobjective Integer Linear Fractional Programming Problem under Randomness

In this paper a solution algorithm to chance-constrained multiobjective integer linear fractional programming problem is suggested. We assume that there is randomness in the right-hand side of the constraints only and that the random variables are normally distributed. Some stability notions for the problem of concern are defined and characterized. An illustrative numerical example is provided to clarify the developed theory and the suggested solution algorithm.

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.

Enhancing computations of nondominated solutions in MOLFP via reference points

In previous work, Costa and Alves (J Math Sci 161:(6)820–831, 2009; 2011) have presented Branch & Bound and Branch & Cut techniques that allow for the effective computation of nondominated solutions, associated with reference points, of multi-objective linear fractional programming (MOLFP) problems of medium dimensions (ten objective functions, hundreds of variables and constraints). In this paper we present some results that enhance those computations. Firstly, it is proved that the use of a special kind of achievement scalarizing function guarantees that the computation error does not depend on the dimension of the problem. Secondly, a new cut for the Branch & Cut technique is presented. The proof that this new cut is better than the one in Costa and Alves (2011) is presented, guaranteeing that it reduces the region to explore. Some computational tests to assess the impact of the new cut on the performance of the Branch & Cut technique are presented.

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.

On the efficiency test in multi-objective linear fractional programming problems by Lotfi et al. 2010

The solution concept in multi-objective programming is represented by a program which reaches “the best compromise”. Many solving methods find a good compromise feasible solution and then check whether the solution is efficient or not. In this paper, using the efficiency test introduced by Lotfi et al. (2010) [12], we propose two procedures for deriving weakly and strongly efficient solutions in multi-objective linear fractional programming problems (MOLFPP) starting from any feasible solution, and present their possible applications in multiple criteria decision-making process. Then, we discuss a shortcoming of some fuzzy approaches to solving MOLFPP and modify them in order to guarantee the efficiency of the optimal solution.

A Proposal to the Solution of Multiobjective Linear Fractional Programming Problem

We have proposed a new solution to the Multiobjective Linear Fractional Programming Problem (MOLFPP). The proposed solution is based on a theorem that deals with nonlinear fractional programming with single objective function and studied in the work by Dinkelbach, 1967. As a new contribution, we have proposed that is an efficient solution of MOLFPP if is an optimal solution of problem , where is for all . Hence, MOLFPP is simply reduced to linear programming problem (LPP). Some numerical examples are provided in order to illustrate the applications of the proposed method. The optimization software package, namely, WinQSB (Chang, 2001), has been employed in the computations.

Fuzzy Goal Programming Procedures for Multi-Level Multi-Objective Linear Fractional Programming Problems

Fuzzy Goal Programming Procedures for Multi-Level Multi-Objective Linear Fractional Programming Problems

GOAL PROGRAMMING APPROACH TO CHANCE CONSTRAINED MULTI-OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEM BASED ON TAYLOR’S SERIES APPROXIMATION

This paper deals with goal programming approach to chance constrained multi-objective linear fractional programming problem based on Taylor’s series approximation. We consider the constraints with right hand parameters as the random variables of known mean and variance. The random variables are transformed into standard normal variables with zero mean and unit variance. We convert the chance constraints with known confidence level into equivalent deterministic constraints. The goals of linear fractional objective functions are determined by optimizing it subject to the equivalent deterministic system constraints. Then the fractional objective functions are transformed into equivalent linear functions at the optimal solution point by using first order Taylor polynomial series. In the solution process, we use three minsum goal programming models and identify the most compromise optimal solution by using Euclidean distance function.

GOAL PROGRAMMING APPROACH TO CHANCE CONSTRAINED MULTI-OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEM BASED ON TAYLOR’S SERIES APPROXIMATION

This paper deals with goal programming approach to chance constrained multi-objective linear fractional programming problem based on Taylor’s series approximation. We consider the constraints with right hand parameters as the random variables of known mean and variance. The random variables are transformed into standard normal variables with zero mean and unit variance. We convert the chance constraints with known confidence level into equivalent deterministic constraints. The goals of linear fractional objective functions are determined by optimizing it subject to the equivalent deterministic system constraints. Then the fractional objective functions are transformed into equivalent linear functions at the optimal solution point by using first order Taylor polynomial series. In the solution process, we use three minsum goal programming models and identify the most compromise optimal solution by using Euclidean distance function.

A Priority based Fuzzy Goal Programming to MultiObjective Linear Fractional Programming Problem

paper deals with priority based fuzzy goal programming approach for solving multi-objective linear fractional programming problem. In the model formulation of the problem, we construct the fractional membership functions by determining the optimal solution of the objective functions subject to the system constraints. The fractional membership functions are then transformed into equivalent linear membership functions at the individual best solution point by first order Taylor series. In the solution process, fuzzy goal programming approach is used to solve problem by minimizing negative deviational variables. Then, sensitivity analysis is performed with the change of priorities of the fuzzy goals. Euclidean distance function is used to identify the appropriate priority structure in the decision-making situation. The efficiency of the proposed approach is illustrated by solving a numerical example.

A linear programming approach to test efficiency in multi-objective linear fractional programming problems

In a multi-objective linear fractional programming problem (MOLFPP), it is often useful to check the efficiency of a given feasible solution, and if the solution is efficient, it is useful to check strong or weak efficiency. In this paper, by applying a geometrical interpretation, a linear programming approach is achieved to test weak efficiency. Also, in order to test strong efficiency for a given weakly efficient point, a linear programming approach is constructed.

Fuzzy Goal Programming Procedure to Bilevel MultiobjectiveLinear Fractional Programming Problems

This paper presents a fuzzy goal programming (FGP) procedure for solving bilevel multiobjective linear fractional programming (BL‐MOLFP) problems. It makes an extension work of Moitra and Pal (2002) and Pal et al. (2003). In the proposed procedure, the membership functions for the defined fuzzy goals of the decision makers (DMs) objective functions at both levels as well as the membership functions for vector of fuzzy goals of the decision variables controlled by first‐level decision maker are developed first in the model formulation of the problem. Then a fuzzy goal programming model to minimize the group regret of degree of satisfactions of both the decision makers is developed to achieve the highest degree (unity) of each of the defined membership function goals to the extent possible by minimizing their deviational variables and thereby obtaining the most satisfactory solution for both decision makers. The method of variable change on the under‐ and over‐deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology. Illustrative numerical example is given to demonstrate the procedure.

A reference point technique to compute nondominated solutions in MOLFP

This paper presents a new technique to compute nondominated solutions in multiobjective linear-fractional programming (MOLFP) by using reference points. The basic idea consists in dividing the nondominated region approximately through the “middle” making two subregions, which are then analyzed in order to try to discard one of them. The process is repeated with the remaining region(s), and it ends when the regions become so small that the differences among their nondominated solutions are lower than a predefined margin of error. The results of several tests carried out to evaluate the performance of the technique are reported.