Approaches to Linear Programming Problems with Interactive Fuzzy Numbers

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

<p>There exist several methods for solving fuzzy linear or nonlinear programming problems under positivity fuzzy variables and restricted fuzzy coefficients. Due to the limitation of these methods, they can’t be applied for solving fully fuzzy linear or non- linear programming problems with unrestricted fuzzy coefficients and fuzzy variables. In this paper an efficient method to find the fuzzy optimal solution for fully fuzzy quadratic programming (FFQP) problem with unrestricted variables and parameters has been proposed. All the coefficients and decision variables of both objective functions and the constraints are triangular fuzzy numbers. The proposed method is based on converted FFQP problem into crisp quadratic programming (CQP) problem. Finally an illustrative numerical example has been given to clarify the proposed solution method.</p>

We present an approach for ordering fuzzy numbers, then we utilize it to suggest two models for solving the fuzzy linear programming (FLP) problem. In this research we present a new approach for ordering two fuzzy numbers and show some practical advantages of it over other methods. We also utilize the suggested approach to introduce two techniques to solve the FLP problem. In the first technique, we convert the fuzzy linear program into a crisp bicriteria linear program. In this new model the fuzzy objective function is replaced by two crisp objectives to be optimized, and each fuzzy constraint is replaced by two or three crisp constraints depending on the level of certainty /spl alpha/-whether /spl alpha/ has or has not been assigned. In the second technique we find the possibility distribution of the objective function over all possible feasible decisions directly.

Linear programming models play an important role in management, economic, data envelopment analysis, operations research and many industrial applications. In many practical situations there is a kind of ambiguity in the parameters of these models which can be expressed by means of fuzzy numbers. In the literature of fuzzy mathematical programming there are many types of the fuzzy linear programming problems. But in this paper, we deal with a kind of linear programming which includes the triangular fuzzy numbers in its parameters. For finding the solution of these problems, we propose a revised simplex algorithm for an extended linear programming problem which is equivalent to the original fuzzy linear programming problem. An illustrative example is presented to clarify the proposed approach. A fuzzy DEA model has been also considered as a practical application to illustrate the effectiveness of the proposed approach. (Received 20 December 2009; Revised 29 September 2010; Accepted 11 November 2010)

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

Fuzzy linear programming problems with fuzzy numbers

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems. Kheirfam and Hasani (Sensitivity analysis for fuzzy linear programming problems with fuzzy variables, Advanced Modeling and Optimization, 12 (2010) 257-272), proposed a method for the sensitivity analysis for fuzzy linear programming (FLP) problems with fuzzy variables. There are some important cases that are not considered by Kheirfam and Hasani. In this paper, those cases are considered and numerical examples are solved.

The aim of this chapter is to study linear programming problem with fuzzy coefficients as one of the convenient model of fuzzy linear programs. In this way, we first introduce a general form of Fuzzy Number Linear programming (FNLP) problems and then give the fundamental concepts which are useful in throughout of the chapter.

Linear programming problems and ranking of fuzzy numbers

This research is about the development of a dynamic programming model for solving fuzzy linear programming problems. Initially, fuzzy dynamic linear programming model FDLP is developed. This research revises the established dynamic programming model for solving linear programming problems in a crisp environment. The mentioned approach is upgraded to address the problem in an uncertain environment. Dynamic programming model can either be passing forward or backward. In the proposed approach backward dynamic programming approach is adopted to address the problem. It is then followed by implementing the proposed method on the education system of Pakistan. The education system of Pakistan comprises of the Primary, Middle, Secondary, and Tertiary education stages. The problem is to maximize the efficiency of the education system while achieving the targets with minimum usage of the constrained resources. Likewise the model tries to maximize the enrollment in the Primary, Middle, Secondary and Tertiary educational categories, subject to the total available resources in a fuzzy uncertain environment. The solution proposes that the enrollment can be increased by an amount 9997130, by increasing the enrollment in the Middle and Tertiary educational categories. Thus the proposed method contributes to increase the objective function value by 30%. Moreover, the proposed solutions violate none of the constraints. In other words, the problem of resources allocation in education system is efficiently managed to increase efficiency while remaining in the available constrained resources. The motivation behind using the dynamic programming methodology is that it always possesses a numerical solution, unlike the other approaches having no solution at certain times. The proposed fuzzy model takes into account uncertainty in the linear programming modeling process and is more robust, flexible and practicable.

The paper presents a microcomputer implementation of a visual interactive method, called ‘FLIP’, for solving MOLP problems with fuzzy coefficients in the objective functions and on the both sides of the constraints. The idea of ‘FLIP’ relies on an observation that for given values of decision variables, the main question to be answered consists in the comparison of objectives and goals, and left-and right-hand-sides of the constraints which are fuzzy numbers. In result of application of a comparison principle, the fuzzy MOLP problem is transformed into a non-fuzzy multiobjective linear fractional programming (MOLFP) problem which is solved using an interactive sampling method. An evaluation of the quality of successive proposals (solutions) is based on the following characteristics: (i) scores of fuzzy objectives in relation to the goals, (ii) dispersion of values of the fuzzy objectives due to uncertainty, (iii) safety of solutions or risk of violation of the constraints. The graphics displayed by ‘FLIP’ provide the most comprehensive synthesis of these characteristics. The aim of this paper is to demonstrate that the kind of visual interaction organized by ‘FLIP’ is the most appropriate for solving MOLP problems under uncertainty and imprecision. In order to support this conclusion, a land-use planning problem is solved using the ‘FLIP’ software on a micro.

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.

In this paper, shortcomings and limitations of the existing methods for solving fuzzy linear programming (FLP) problems are pointed out. To overcome the limitations as well as to resolve the shortcomings, a new method is proposed for solving FLP problems. To show the advantage of the proposed method over existing methods, a FLP problem is solved by the existing methods as well as the proposed method, and the obtained results are compared.

On the equivalence of two optimization methods for fuzzy linear programming problems