A new method to solve multi-objective linear fractional programming problem in fuzzy stochastic environment

In this paper, a method is proposed for solving multi-objective linear fractional programming (MOLFP) problem. Here, the MOLFP problem is transformed into an equivalent multi-objective linear programming (MOLP) problem. Using the first-order Taylor's series approximation, the MOLFP problem is reduced to single-objective linear programming (LP) problem. Finally, the solution of MOLFP problem is obtained by solving the resultant LP problem. The proposed procedure is verified with the existing methods through the numerical examples.

In the field of operation research, both linear and fractional programming problems have been more encountered in recent years because they are more realistic in expressing real-life problems. Fractional programming problem is used when several rates need to be optimized simultaneously such as resource allocation planning, financial and corporate planning, healthcare, and hospital planning. There are several techniques to solve the multiobjective linear fractional programming problem. However, because of the use of scalarization, these techniques have some limitations. This paper proposed two new mean and median techniques to solve the multiobjective linear fractional programming problem by overcoming the limitations. After utilizing mean and median techniques, the problem is converted into an equivalent linear fractional programming problem; then, the linear fractional programming problem is transformed into linear programming problem and solved by the conventional simplex method or mathematical software. Some numerical examples have been illustrated to show the efficiency of the proposed techniques and algorithm. The performance of these solutions was evaluated by comparing their results with other existing methods. The numerical results have shown that the proposed techniques are better than other techniques. Furthermore, the proposed techniques solve a pure multiobjective maximization problem, which is even impossible with some existing techniques. The present investigation can be improved further, which is left for future research.

ABSTRACTThe computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.

In this paper, we studied a multiobjective linear fractional programming (MOLFP) problem with pentagonal and hexagonal fuzzy numbers, while the decision variables are binary integer numbers. Initially, a multiobjective fuzzy binary integer linear fractional programming (MOFBILFP) problem was transformed into a multiobjective binary linear fractional programming problem by using the geometric average method; second, a multiobjective binary integer linear fractional programming (MOBILFP) problem was converted into a binary integer linear fractional programming (BILFP) problem using the Pearson 2 skewness coefficient technique; and third, a BILFP problem was solved by using LINGO (version 20.0) mathematical software. Finally, some numerical examples and case studies have been illustrated to show the efficiency of the proposed technique and the algorithm. The performance of this technique was evaluated by comparing their results with those of other existing methods. The numerical results have shown that the proposed technique is better than other techniques.

Fuzzy efficient iterative method for multi-objective linear fractional programming problems

In this paper, an attempt has been taken to develop a method for solving fuzzy multi-objective linear fractional programming (FMOLFP) problem. Here, at first the FMOLFP problem is converted into (crisp) multi-objective linear fractional programming (MOLFP) problem using the graded mean integration representation (GMIR) method proposed by Chen and Hsieh. That is, all the fuzzy parameters of FMOLFP problem are converted into crisp values. Then the MOLFP problem is transformed into a single objective linear programming (LP) problem using a proposal given by Nuran Guzel. Finally the single objective LP problem is solved by regular simplex method which yields an efficient solution of the original FMOLFP problem. To show the efficiency of our proposed method, three numerical examples are illustrated and also for each example, a comparison is drawn between our proposed method and the respected existing method.

In this paper, we consider a Multi-Objective Stochastic Interval-Valued Linear Fractional Integer Programming problem (MOSIVLFIP). We especially deal with a multi-objective stochastic fractional problem involving an inequality type of constraints, where all quantities on the right side are log-normal random variables, and the objective functions coefficients are fractional intervals. The proposed solving procedure is divided in three steps. In the first one, the probabilistic constraints are converted into deterministic ones by using the chance constrained programming technique. Then, the second step consists of transforming the studied problem objectives on an optimization problem with an interval-valued objective functions. Finally, by introducing the concept of weighted sum method, the equivalent converted problem obtained from the two first steps is transformed into a single objective deterministic fractional problem. The effectiveness of the proposed procedure is illustrated through a numerical example.

The concept of ranking method is an efficient approach to rank fuzzy numbers. In this paper, we have studied stochastic fuzzy multiobjective linear fractional programming problem (SFMOLFPP) where SFMOLFPP is transformed to its equivalent deterministic-crisp multiobjective linear programming problem (MOLPP). To study SFMOLFPP, a SFMOLFPP is presented in which the fuzzy coefficients and scalars in the linear fractional objectives and the fuzzy coefficients are characterised by triangular or trapezoidal fuzzy numbers. The left hand side of the stochastic fuzzy constraints are characterised by triangular or trapezoidal fuzzy numbers, while the right hand sides are assumed to be independent random variable with known distribution function. We have modify Iskander's approach [16] to transform the suggested problem to its equivalence deterministic-crisp MOLPP. We have also used ranking function in SFMOLFPP to find the pareto optimal solution of the reduced multiobjective linear fractional programming problem (MOLFPP). One numerical example is presented to demonstrate two methodologies.

On fuzzy linearization approaches for solving multi-objective linear fractional programming problems

The aims of the article is to the study fuzzy multi-objective linear fractional programming (FMOLFP) problem by using goal programming approach. At first by using Charnes and Cooper [1] transformation, the FMOLFP problem is transformed to fuzzy multi-objective linear programming (FMOLP) problem. The reduced problem is formulated by goal programming approach to find out the solution of FMOLFP problem. By using our proposed GP approach some pareto optimal solution is obtained for the FMOLFP problem. A numerical example is presented to demonstrate our approach.

The multi-objective linear fractional programming is an interesting topic with many applications in different fields. Until now, various algorithms have been proposed in order to solve the multi-objective linear fractional programming (MOLFP) problem. An important point in most of them is the use of non-linear programming with a high computational complexity or the use of linear programming with preferences of the objective functions which are assigned by the decision maker. The current paper, through combining goal programming and data envelopment analysis (DEA), proposes an iterative method to solve MOLFP problems using only linear programming. Moreover, the proposed method provides an efficient solution which fairly optimizes each objective function when the decision maker has no information about the preferences of the objective functions. In fact, along with normalization of the objective functions, their relative preferences are fairly determined using the DEA. The implementation of the proposed method is demonstrated using numerical examples.

ABSTRACTIn this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number and technological coefficients and the quantities on the right side of the constraints are fuzzy random variables with the specific distribution. Here we change an FSLFP problem to an equivalent deterministic Multi-objective Linear Fractional Programming (MOLFP) problem. Then by using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear programming (LP) problem. A numerical example is presented to demonstrate the effectiveness of the proposed method.

This paper presents a method for approximating the nondominated set of a multiobjective linear fractional programming (MOLFP) problem. The resulting approximation set has acceptable quality measures, and is an -approximation to the nondominated set, where is the tolerance vector and can be controlled by the decision-maker. The distance between every two resulting solutions is greater than or equal to a positive value and hence there is no redundant information. Also, for a given upper bound the method controls the cardinality of the resulting approximation set.

This paper presents an interactive dynamic fuzzy goal programming (DFGP) approach for solving bi-level multiobjective linear fractional programming (BL MOLFP) problems with the characteristics of dynamic programming (DP). In the proposed approach, the membership function of the objective goals of a problem with fuzzy aspiration levels are defined first as the membership function 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. 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. Then, under the framework of preemptive priority based GP, a multi stage DP model of the problem is used for achievement of the highest degree (unity) of each of the membership functions. In the decision process, the goal satisficing philosophy of GP is used recursively to arrive at the most satisfactory solution and the suggested algorithm to simplify the solution procedure by DP at each stage is proposed. This paper is considered as an extension work of Mahmoud A. Abo-Sinna and Ibrahim A. Baky (2010) by using dynamic approach. Finally, this approach is illustrated by a given numerical example.

Despite the important role of bi-level multi-objective linear fractional programming (BL-MOLFP) problem for many hierarchical organizations, a very little success has been achieved to deal with this problem. This paper presents a comparative study between two computational approaches, namely fuzzy TOPSIS (technique for order preference by similarity to ideal solution) approach and Jaya (a Sanskrit word meaning victory) approach, for solving BL-MOLFP problem. The fuzzy TOPSIS (FTOPSIS) approach aims to obtain the satisfactory solution of BL-MOLFP problem by using linearization process as well as formulating the membership functions for the distances of positive ideal solution (PIS) and negative ideal solution (NIS) for each level, respectively. In this sense, the deadlock situations among levels are avoided by establishing the membership functions for the upper level decision variables vector with possible tolerances. On the other hand, Jaya algorithm is proposed for solving BL-MOLFP problem based on nested structure scheme to optimize both levels hierarchically. An illustrative example is presented to describe the proposed approaches. In addition, the performances among the proposed approaches are assessed based on ranking strategy of the alternatives to affirm the superior approach. Based on the examined simulation, Jaya algorithm is preferable than the FTOPSIS approach.