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Considering the double uncertainty caused by the ambiguity of statistical data and the ambiguity produced by the subjective assessment of decision makers, the crisp values of criteria may be insufficient to model the multi-criteria decision-making (MCDM) problem in the real world. This paper proposes a fuzzy best-worst method (FBWM) based on a dual-interval solution algorithm to extend the best-worst method (the most recent MCDM method) to fuzzy environments. The reference comparisons for the best criteria and for the worst criteria are represented by fuzzy numbers. Then, according to the BWM method, a nonlinear constrained optimization problem with fuzzy parameters is formulated. We decompose the membership function in fuzzy numbers into several interval numbers of special form and solve the aforementioned fuzzy BWM problem by compound interval algorithm to obtain fuzzy weights of different criteria. Meanwhile, an integral type-reduced method is proposed for determining the fuzzy consistency ratio in order to assess the reliability of the FBWM results. The viability of the new algorithm to expand the BWM method into fuzzy environments has been validated through three numerical examples. In contrast to the existing FBWM method, the proposed method avoids the arithmetic operation between fuzzy numbers during the solution process, directly transfers the uncertain information in the membership function corresponding to the fuzzy comparison vector to the result, and generates fuzzy weight value, which indicates that the proposed algorithm is able to obtain accurate BWM results in fuzzy environments. The results of the study provide new solution ideas for multi-criteria optimization problems under uncertainty. [ABSTRACT FROM AUTHOR]

In this study, Interval number and Fuzzy number Linear Programming problems are addressed. We deal with some convenient methods to solve the addressed problems. We focus on preparing an improving approach based on Tang Shaocheng method to solve linear programming problems with interval or fuzzy numbers. Also, some applications of our proposed approach in fractional programming are presented. Particularly, we propose an algorithm using our improved approach to solve a fuzzy multi-objective linear fractional programming problem. Finally, we illustrate the proposed approach using numerical and some real life examples. [ABSTRACT FROM AUTHOR]

When some coefficients of the constraints are uncertain with only their possible ranges being given, a conventional linear programming (LP) problem can be generalised to the one with set-inclusive constraints. We consider the case where the possible ranges are given by fuzzy sets in this paper. The set-inclusive constraints with fuzzy coefficients have been treated by a necessity measure. However, the usual necessity measure cannot express well the decision-maker's requirement about the trade-off between the robustness level and the satisfaction level of the constraints. We extend the necessity measure to incorporate the trade-off between the robustness level and the satisfaction level of the constraints. We apply the extended necessity measure to LP problems with fuzzy coefficients. After the problem is formulated and reduced to the conventional programming problem, we propose a solution algorithm. Numerical examples are given to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

This paper deals with a fully fuzzy linear programming problem (FFLP) in which the coefficients of decision variables, the right-hand coefficients and variables are characterized by fuzzy numbers. A method of obtaining optimal fuzzy solutions is proposed by controlling the left and right sides of the fuzzy variables according to the fuzzy parameters. By using fuzzy controlled solutions, we avoid unexpected answers. Finally, two numerical examples are solved to demonstrate how the proposed model can provide a better optimal solution than that of other methods using several ranking functions.

In this paper we try to define a percentage form of LR fuzzy numbers which is a useful form of fuzzy numbers and its’ arithmetics. So, we show how the maximum variation range of optimal value of fuzzy objective function can be predicted by using this form of fuzzy numbers. Since fuzzy problems are generally solved through a complicated manner, the purpose of this study is releasing a kind of prediction for the final solution in the way that the manager can access to an outlook to optimal solution (Z∗) without solving the problem. Finally, optimal value of fuzzy objective function on fuzzy linear programming is predicted whenmaximum variation range of fuzzy variable have been predetermined.

We investigate various types of fuzzy linear programming problems based on models and solution methods. First, we review fuzzy linear programming problems with fuzzy decision variables and fuzzy linear programming problems with fuzzy parameters (fuzzy numbers in the definition of the objective function or constraints) along with the associated duality results. Then, we review the fully fuzzy linear programming problems with all variables and parameters being allowed to be fuzzy. Most methods used for solving such problems are based on ranking functions, α -cuts, using duality results or penalty functions. In these methods, authors deal with crisp formulations of the fuzzy problems. Recently, some heuristic algorithms have also been proposed. In these methods, some authors solve the fuzzy problem directly, while others solve the crisp problems approximately. [ABSTRACT FROM AUTHOR]

Fuzzy Linear Programming (FLP) has addressed the increasing complexity of real-world decision-making problems that arise in uncertain and ever-changing environments since its introduction in the 1970s. Built upon the Fuzzy Sets theory and classical Linear Programming (LP) theory, FLP encompasses an extensive area of theoretical research and algorithmic development. Unlike classical LP, there is not a unique model for the FLP problem, since fuzziness can appear in the model components in different ways. Hence, despite fifty years of research, new formulations of FLP problems and solution methods are still being proposed. Among the existing formulations, those using fuzzy numbers (FNs) as parameters and/or decision variables for handling inexactness and vagueness in data have experienced a remarkable development in recent years. Here, a long-standing issue has been how to deal with FN-valued objective functions and with constraints whose left- and right-hand sides are FNs. The main objective of this paper is to present an updated review of advances in this particular area. Consequently, the paper briefly examines well-known models and methods for FLP, and expands on methods for fuzzy single- and multi-objective LP that use lexicographic criteria for ranking FNs. A lexicographic approach to the fuzzy linear assignment (FLA) problem is discussed in detail due to the theoretical and practical relevance. For this case, computer codes are provided that can be used to reproduce results presented in the paper and for practical applications. The paper demonstrates that FLP that is focused on lexicographic methods is an active area with promising research lines and practical implications.

Many ranking methods have been proposed so far. However, there is yet no method that can always give a satisfactory solution to every situation; some are counterintuitive, not discriminating; some use only the local information of fuzzy values; some produce different ranking for the same situation. For overcoming the above problems, we propose a new method for ranking fuzzy quantities based on the distance method. Then, an application of using fuzzy ordering in the fuzzy mathematical programming as well as fuzzy primal simplex algorithm is indicated. In particular, we emphasize that the fuzzy ordering will be useful when a decision maker needs to evaluate the optimality condition in any solving process.

The first researchers to publish an operationalization of the Bellman and Zadeh's original approach to “Decision making in a fuzzy environment” [1] were H. Tanaka and K. Asai in their 1973 article [2] which was published in English in 1974 [3] . These two researchers developed many fundamental theories and methods including novel concepts of fuzzy operations research [4] with their collaborators. They went beyond the original proposal for fuzzy optimization set forth by Bellman/Zadeh. They are important founders of fuzzy optimization. This paper gives an overview of their great contributions to fuzzy optimization. [ABSTRACT FROM AUTHOR]

This paper proposes a novel meta-heuristic based algorithm which provides the optimal solution for several different degrees of feasibility for fuzzy linear and nonlinear programming problems. The proposed method has got the ability for solving those problems in which all coefficients of the objective function and constraints are represented by LR fuzzy numbers with linear and/or non-linear membership function. To solve the fuzzy problems, this paper provides a new hybrid genetic algorithm accompanied by a new proposed method of simulating fuzzy coefficients which 1) eliminates the need of applying defuzzification methods and/or expected interval methods, and 2) allows dealing with different types of fuzzy numbers, properly. In order to show the performance of the proposed method, it is compared with 'M. Jiménez, M. Arenas, A. Bilbao and M.V. Rodríguez, Linear programming with fuzzy parameters: An interactive method resolution, European Journal of Operational Research 177 (2007), 1599-1609.'. Computational results reveal that the proposed method is superior to the Jiménez et al. [13] method from the viewpoint of feasibility and optimality. [ABSTRACT FROM AUTHOR]

Fuzzy linear programming is used widedly for the problems of uncertain system's optimization, but the optimization solutions or satisfactory solution are often not "immune" to parameters, that is if the parameters are changed, initial optimal solution is no longer optimal or even infeasible. First of all, for the λ-cut level of fuzzy linear programming, λ-robust optimization solutions are put forward; then article indicates the definition of λ-robust solution by fuzzy structured element, and obtains the solving model. Solution can achieve the degree requirements are different due to the different decision makers, therefore, the measure constraints which can reflect the decision-makers risk preferences are added to the model, the solution of this model is γ-robust solution, and the solution not only has the robustness and optimization, but also reflects the degree of risk preference of the decision makers. By an example, it is found that the optimization robust solutions with measurement are immune to parameters, which are more useful for decision-makers and reflect a better practical value. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Fuzzy linear programming is an application of fuzzy set theory in linear decision making problems and most of these problems are related to linear programming (LP) with fuzzy variables. In this paper, an approximate but convenient method for solving these problems with fuzzy non-negative technical coefficient and without using the ranking functions, is proposed. With the help of numerical examples, the method is illustrated. [ABSTRACT FROM AUTHOR]

Abstract: In the article [G.Q. Zhang, Y.H. Wu, M. Remias, Jie Lu, Formulation of fuzzy linear programming problems as four-objective constrained optimization problems, Applied Mathematics and Computation 139 (2003) 383–399], Guangquan Zhang et al. proposed a number of theorems so as to convert the fuzzy linear programming (FLP) into a multi-objective optimization problems with four-objective functions. However there is a error in the article. [Copyright &y& Elsevier]

In this paper first, we find a canonical symmetrical trapezoidal(triangular) for the solution of the fuzzy linear system A~x = ~b, where the elements in A and ~b are crisp and arbitrary fuzzy numbers, respectively. Then, a model for fuzzy linear programming problem with fuzzy variables (FLPFV), in which, the right hand side of constraints are arbitrary numbers, and coefficients of the objective function and constraint matrix are regarded as crisp numbers, is discussed. A numerical procedure for calculating a canonical symmetrical trapezoidal representation for the solution of fuzzy linear system and the optimal solution of FLPFV, (if there exist) is proposed. Several examples illustrate these ideas. [ABSTRACT FROM AUTHOR]

Abstract: Composite power system reliability evaluation refers to assessment that considers both, generation and transmission facilities. The conventional dc load flow-based crisp linear programming (CLP) model used in composite power system reliability evaluation is modified as a fuzzy linear programming (FLP) model including fuzzy constraints and objective functions. This fuzzy linear programming model can include uncertainties that exist in certain variables and overcome the limitations of minor constraint violations in crisp linear programming model. This fuzzy optimization model is employed to test the system outage contingencies and to determine the degree of difficulty due to these contingencies. Fuzzy-based adequacy indices are obtained after all the selected contingencies have been analyzed using contingency enumeration approach. The effectiveness of the developed model is tested on the IEEE-14 bus system. [Copyright &y& Elsevier]

In the real world there are many linear programming problems where all decision parameters are fuzzy numbers. Several approaches exist which use different ranking functions for solving these problems. Unfortunately when there exist alternative optimal solutions, usually with different fuzzy value of the objective function for these solutions, these methods can not specify a clear approach for choosing a solution. In this paper we propose a method to remove the above shortcoming in solving fuzzy number linear programming problems using the concept of expectation and variance as ranking functions. [ABSTRACT FROM AUTHOR]

In this paper a fuzzy linear programming problem is presented. Then using the concept of comparison of fuzzy numbers, by the aid of the Mellin transform, we introduce a method for solving this problem. [ABSTRACT FROM AUTHOR]

In this paper a fuzzy linear programming problem is presented. Then using the concept of comparison of fuzzy numbers, by the aid of the Mellin transform, we introduce a method for solving this problem. [ABSTRACT FROM AUTHOR]