Fuzzy linear programs with trapezoidal fuzzy numbers

The objective of this paper is to deal with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting results are obtained which in turn lead to a solution of fuzzy linear programming problems without converting them to crisp linear programming problems.     

Some duality results on linear programming problems with symmetric fuzzy numbers

Recently, linear programming problems with symmetric fuzzy numbers (LPSFN) have considered by some authors and have proposed a new method for solving these problems without converting to the classical linear programming problem, where the cost coefficients are symmetric fuzzy numbers (see in [4]). Here we extend their results and first prove the optimality theorem and then define the dual problem of LPSFN problem. Furthermore, we give some duality results as a natural extensions of duality results for linear programming problems with crisp data.     

A revisit of numerical approach for solving linear fractional programming problem in a fuzzy environment

In the article, Veeramani and Sumathi [10] presented an interesting algorithm to solve a fully fuzzy linear fractional programming (FFLFP) problem with all parameters as well as decision variables as triangular fuzzy numbers. They transformed the FFLFP problem under consideration into a bi-objective linear programming (LP) problem, which is then converted into two crisp LP problems. In this paper, we show that they have used an inappropriate property for obtaining non-negative fuzzy optimal solution of the same problem which may lead to the erroneous results. Using a numerical example, we show that the optimal fuzzy solution derived from the existing model may not be non-negative. To overcome this shortcoming, a new constraint is added to the existing fuzzy model that ensures the fuzzy optimal solution of the same problem is a non-negative fuzzy number. Finally, the modified solution approach is extended for solving FFLFP problems with trapezoidal fuzzy parameters and illustrated with the help of a numerical example.     

A new method for solving fully fuzzy linear programming problems

Lotfi et al. [Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. Math. Modell. 33 (2009) 3151–3156] pointed out that there is no method in literature for finding the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints. In this paper, a new method is proposed to find the fuzzy optimal solution of same type of fuzzy linear programming problems. It is easy to apply the proposed method compare to the existing method for solving the FFLP problems with equality constraints occurring in real life situations. To illustrate the proposed method numerical examples are solved and the obtained results are discussed.     

The dual simplex method and sensitivity analysis for fuzzy linear programming with symmetric trapezoidal numbers

In this paper, we first extend the dual simplex method to a type of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. The results obtained lead to a solution for fuzzy linear programming problems that does not require their conversion into crisp linear programming problems. We then study the ranges of values we can achieve so that when changes to the data of the problem are introduced, the fuzzy optimal solution remains invariant. Finally, we obtain the optimal value function with fuzzy coefficients in each case, and the results are described by means of numerical examples.     

Linear programming with triangular fuzzy numbers—A case study in a finance and credit institute

The objective of this paper is to deal with a kind of fuzzy linear programming problem involving triangular fuzzy numbers. Then some interesting and fundamental results are achieved which in turn lead to a solution of fuzzy linear programming models without converting the problems to the crisp linear programming models. Finally, the theoretical results are also supported by a real case study in a banking system. The same idea is emphasized to be also useful when a general LR fuzzy numbers is given.     

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.     

Intuitionistic fuzzy linear regression analysis

Linear regression analysis in an intuitionistic fuzzy environment using intuitionistic fuzzy linear models with symmetric triangular intuitionistic fuzzy number (STriIFN) coefficients is introduced. The goal of this regression is to find the coefficients of a proposed model for all given input–output data sets. The coefficients of an intuitionistic fuzzy regression (IFR) model are found by solving a linear programming problem (LPP). The objective function of the LPP is to minimize the total fuzziness of the IFR model which is related to the width of IF coefficients. An illustrative example is also presented to depict the solution procedure of the IFR problem by using STriIFNs.     

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.     

Optimal compromise solution of multi-objective minimal cost flow problems in fuzzy environment

Ghatee and Hashemi [M. Ghatee, S.M. Hashemi, Ranking function-based solutions of fully fuzzified minimal cost flow problem, Inform. Sci. 177 (2007) 4271–4294] transformed the fuzzy linear programming formulation of fully fuzzy minimal cost flow (FFMCF) problems into crisp linear programming formulation and used it to find the fuzzy optimal solution of balanced FFMCF problems. In this paper, it is pointed out that the method for transforming the fuzzy linear programming formulation into crisp linear programming formulation, used by Ghatee and Hashemi, is not appropriate and a new method is proposed to find the fuzzy optimal solution of multi-objective FFMCF problems. The proposed method can also be used to find the fuzzy optimal solution of single-objective FFMCF problems. To show the application of proposed method in real life problems an existing real life FFMCF problem is solved.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

Editorial

Linear programming problems with fuzzy parameters are formulated by fuzzy functions. The ambiguity considered here is not randomness, but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership. Parameters on constraint and objective functions are given by fuzzy numbers. In this paper, our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters. This fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential.     

Possibility linear programming with trapezoidal fuzzy numbers

Fuzzy linear programming with trapezoidal fuzzy numbers (TrFNs) is considered and a new method is developed to solve it. In this method, TrFNs are used to capture imprecise or uncertain information for the imprecise objective coefficients and/or the imprecise technological coefficients and/or available resources. The auxiliary multi-objective programming is constructed to solve the corresponding possibility linear programming with TrFNs. The auxiliary multi-objective programming involves four objectives: minimizing the left spread, maximizing the right spread, maximizing the left endpoint of the mode and maximizing the middle point of the mode. Three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. An investment example and a transportation problem are presented to demonstrate the implementation process of this method. The comparison analysis shows that the fuzzy linear programming with TrFNs developed in this paper generalizes the possibility linear programming with triangular fuzzy numbers.     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

Some new results in linear programs with trapezoidal fuzzy numbers: Finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method

In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305–315] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems.     

Solving fuzzy multi-objective linear programming problems using deviation degree measures and weighted max–min method

This paper proposes a method for solving fuzzy multi-objective linear programming (FMOLP) problems where all the coefficients are triangular fuzzy numbers and all the constraints are fuzzy equality or inequality. Using the deviation degree measures and weighted max–min method, the FMOLP problem is transformed into crisp linear programming (CLP) problem. If decision makers fix the values of deviation degrees of two side fuzzy numbers in each constraint, then the δ-pareto-optimal solution of the FMOLP problems can be obtained by solving the CLP problem. The bigger the values of the deviation degrees are, the better the objectives function values will be. So we also propose an algorithm to find a balance-pareto-optimal solution between two goals in conflict: to improve the objectives function values and to decrease the values of the deviation degrees. Finally, to illustrate our method, we solve a numerical example.     

Maximum cut in fuzzy nature: Models and algorithms

The maximum cut (Max-Cut) problem has extensive applications in various real-world fields, such as network design and statistical physics. In this paper, a more practical version, the Max-Cut problem with fuzzy coefficients, is discussed. Specifically, based on credibility theory, the Max-Cut problem with fuzzy coefficients is formulated as an expected value model, a chance-constrained programming model and a dependent-chance programming model respectively according to different decision criteria. When these fuzzy coefficients are represented by special fuzzy variables like triangular fuzzy numbers and trapezoidal fuzzy numbers, the crisp equivalents of the fuzzy Max-Cut problem can be obtained. Finally, a genetic algorithm combined with fuzzy simulation techniques is designed for the general fuzzy Max-Cut problem under these models and numerical experiment confirms the effectiveness of the designed genetic algorithm.     

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

On fuzzy random multiobjective quadratic programming

In this paper, a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.